J Geom Anal https://doi.org/10.1007/s12220-018-0007-5

Weighted Hardy Spaces Associated with Elliptic p Operators. Part III: Characterisations of HL (w) and the Weighted Hardy Space Associated with the Riesz Transform Cruz Prisuelos-Arribas1

Received: 6 March 2017 © Mathematica Josephina, Inc. 2018

Abstract We consider Muckenhoupt weights w, and define weighted Hardy spaces p HT (w), where T denotes a conical square function or a non-tangential maximal function defined via the heat or the Poisson semigroup generated by a second-order divergence form elliptic operator L. In the range 0 < p < 1, we give a molecular characterisation of these spaces. Additionally, in the range p ∈ Ww ( p− (L), p+ (L)), we see that these spaces are isomorphic to the L p (w) spaces. We also consider the Riesz 1 p transform ∇ L − 2 , associated with L, and show that the Hardy spaces H∇ L −1/2 ,q (w) and HSH ,q (w) are isomorphic, in some range of p  s, and q ∈ Ww (q− (L), q+ (L)). p

Keywords Conical square functions Riesz transform · Muckenhoupt weights · Elliptic operators · Heat and Poisson semigroups · Off-diagonal estimates · Complex interpolation · Tent spaces · Hardy spaces Mathematics Subject Classification 47B06 · 47B38 · 30H10 · 47G10 · 44A15 · 46M35

1 Introduction This work ends a series of three papers, started by [32] and [33], and dedicated to the study of weighted Hardy spaces associated with operators that arise from a secondorder divergence form elliptic operator L. In particular, we consider conical square functions (2.14)–(2.19), non-tangential maximal functions (2.20) and the Riesz trans-

B 1

Cruz Prisuelos-Arribas [email protected] Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, 28049 Madrid, Spain

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C. Prisuelos-Arribas 1

form, ∇ L − 2 . This generalised Hardy space theory has been started by Auscher et al. in an unpublished work [3]. Besides, Auscher and Russ in [8] considered the case on which the heat kernel associated with L is locally Hölder continuous and satisfies pointwise Gaussian bounds, this occurs for instance for real symmetric operators. There, among other things, it was shown that the corresponding Hardy space associated with L agrees with the classical Hardy space. The question of replacing the Laplacian with another second elliptic operator L was also considered in dimension one by Auscher and Tchamitchian in [9]. In the setting of Riemannian manifolds satisfying the doubling volume property, Hardy spaces associated with the Laplace– Beltrami operator are introduced in [7] by Auscher et al. and it is shown that they admit several characterisations. Simultaneously, in the Euclidean setting, the study of Hardy spaces related to the conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by divergence form elliptic operators was taken by Hofmann and Mayboroda in [27], for p = 1. The new point was that only a form of decay weaker than pointwise bounds, and satisfied in many occurrences, was enough to develop a theory. Later on Hofmann et al. in [28] studied that theory for a general p, and simultaneously Jiang and Yang in [29] also considered this case. In the context on weighted Lebesgue measure spaces some progress has been done in [13,14], and [33]. The results obtained in [14] in the particular case ϕ(x, t) := tw(x), where w is a Muckenhoupt weight, give characterisations of the weighted Hardy spaces which, however, only recover part of the results obtained in the unweighted case by simply taking w = 1. In [33], we present a different approach to the theory of weighted Hardy spaces HL1 (w) associated with a secondorder divergence form elliptic operator, which naturally generalises the unweighted setting developed in [27]. We define weighted Hardy spaces associated with the conical square functions considered in (2.14)–(2.19) which are written in terms of the heat and Poisson semigroups generated by the elliptic operator. Also, we use nontangential maximal functions as defined in (2.20). We show that the corresponding spaces are all isomorphic and admit molecular characterisations. This is particularly useful to prove different properties of these spaces as it happens in the classical setting and in the context of second-order divergence form elliptic operators considered in [27]. Some of the ingredients that were crucial in [33] and also in the present work are taken from the first part of this series of papers [32], where we already obtained optimal ranges for the weighted norm inequalities satisfied by the heat and Poisson conical square functions associated with the elliptic operator. In [33] we obtain analogous results for the non-tangential maximal functions associated with the heat and Poisson semigroups. All these weighted norm inequalities for the conical square functions and the non-tangential maximal functions, along with the important fact that our molecules belong naturally to weighted Lebesgue spaces, allow us to impose natural conditions that in particular lead to fully recover the results obtained in [27] and [28] by simply taking the weight identically one. It is relevant to note that in [13,14] their molecules belong to unweighted Lebesgue spaces and also their ranges of boundedness of the conical square functions are smaller. This makes their hypothesis somehow stronger (although sometimes they cannot be compared with ours) and, despite making

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Weighted Hardy Spaces Associated with Elliptic Operators

a very big effort to present a very general theory, the unweighted case does not follow immediately from their work. In this paper we continue with the study of weighted Hardy spaces associated with p conical square functions and non-tangential maximal functions HT (w), where p is an exponent different from one. In Sect. 7, we give a molecular characterisation when 0 < p < 1 (the case p = 1 was consider in [33]). The proofs of these results are analogous to those done in [33]. Therefore, we just sketch them highlighting the main changes. p In Sect. 8, we obtain that the Hardy spaces HT (w) are isomorphic to the L p (w) spaces for p ∈ Ww ( p− (L), p+ (L)). The result is the following. Theorem 1.1 Given w ∈ A∞ , if T is any of the square functions in (2.14)–(2.19) or a non-tangential maximal function in (2.20), then, for all p ∈ Ww ( p− (L), p+ (L)), p the spaces HT (w) and L p (w) are isomorphic with equivalent norms. 1

Finally, in Sect. 9, we consider another operator: the Riesz transform ∇ L − 2 , and study the Hardy spaces associated with it. In particular, we characterise the Hardy space associated with the Riesz transform through the one associated with the square function SH (see below for definitions). The result is the following. Theorem 1.2 Given w ∈ A∞ such that Ww (q− (L), q+ (L)) = ∅, for all   p− (L) < p < q+sw(L) and q ∈ Ww (q− (L), q+ (L)), the spaces max rw , nrnrww+ p− (L) p

p

HSH ,q (w) and H∇ L −1/2 ,q (w) are isomorphic with equivalent norms.

We observe that in view of Theorem 1.1, the dependence on q in the above isomorphism can be omitted when p ∈ Ww (q− (L), q+ (L)) ⊂ Ww ( p− (L), p+ (L)). In order to prove Theorem 1.2 we need to use interpolation between Hardy spaces. p We obtain this from an interpolation result between weighted tent spaces Tq (w), and show that our Hardy spaces are retracts of them (see Sect. 5).

2 Preliminaries First of all we note that along this work, C or c represents general constant independent of the decisive parameters. 2.1 Weights We work with Muckenhoupt weights w, which are locally integrable positive functions. We say that a weight w ∈ A1 if, for every ball B ⊂ Rn , there holds  − w(x) d x ≤ Cw(y), for a.e. y ∈ B, B

or, equivalently, Mu w ≤ C w a.e. where Mu denotes the uncentered Hardy– Littlewood maximal operator over balls in Rn . For each 1 < r < ∞, and r  such that 1/r + 1/r  = 1, we say that w ∈ Ar if

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C. Prisuelos-Arribas

  r −1  1−r  − w(x) − w(x) d x dx ≤ C, ∀B ⊂ Rn . B

B

The reverse Hölder classes are defined as follows: for each 1 < s < ∞, w ∈ R Hs if  1  s s − w(x) d x ≤ C− w(x) d x ∀B ⊂ Rn . B

B

For s = ∞, w ∈ R H∞ provided that there exists a constant C such that for every ball B ⊂ Rn  w(y) ≤ C− w(x) d x, for a.e. y ∈ B. B

Note that we have excluded the case s = 1 since the class R H1 consists of all the weights, and that is the way R H1 is understood in what follows. Besides, if w ∈ Ar , 1 ≤  r < ∞, for every ball B and every measurable set E ⊂ B, if we denote by w(E) = E w(x) dx, then w(E) ≥ [w]−1 Ar w(B)



|E| |B|

r .

(2.1)

This implies in particular that w is a doubling measure: w(λB) ≤ [w] Ar λn r w(B),

∀ B, ∀ λ > 1.

(2.2)

Moreover, if w ∈ R Hs , 1 < s ≤ ∞, w(E) ≤ [w] R Hs w(B)



|E| |B|

 1 s

.

(2.3)

We sum up some of the properties of these classes in the following result, see for instance [20,21], or [23]. Proposition 2.1 (i) A1 ⊂ A p ⊂ Aq , for 1 ≤ p ≤ q < ∞. (ii) R H∞ ⊂ R Hq ⊂ R H p , for 1 < p ≤ q ≤ ∞. (iii) If w ∈ A p , 1 < p < ∞, then there exists 1 < q < p such that w ∈ Aq . (iv) If w ∈ R H s , 1 < s < ∞, then there exists s < r < ∞ such that w ∈ R Hr . Ap = R Hs . (v) A∞ = 1≤ p<∞

1


(vi) If 1 < p < ∞, w ∈ A p if and only if w 1− p ∈ A p . For a weight w ∈ A∞ , define rw := inf{1 ≤ r < ∞ : w ∈ Ar },

123

sw := inf{1 ≤ s < ∞ : w ∈ R Hs  }.

(2.4)

Weighted Hardy Spaces Associated with Elliptic Operators

Note that according to our definition sw is the conjugated exponent of the one defined in [4, Lemma 4.1]. Given 0 ≤ p0 < q0 ≤ ∞, w ∈ A∞ , and according to [4, Lemma 4.1] we have

Ww ( p0 , q0 ) :=

p : p0 < p < q 0 , w ∈ A

p p0

∩

R H

q0 p



 p0 r w ,

=

q0 sw

 . (2.5)

This interval could be empty. Throughout this paper, when we take a point in Ww ( p0 , q0 ), we implicitly understand that we are working with a weight w ∈ A∞ such that Ww ( p0 , q0 ) = ∅, and therefore we can take that point. If p0 = 0 and q0 < ∞ it is understood that the only condition that stays is w ∈ R H q0  . Analogously, if p

0 < p0 and q0 = ∞ the only assumption is w ∈ A p . Finally Ww (0, ∞) = (0, ∞). p0 Besides, by [4, Lemma 4.4], we have that p ∈ Ww ( p0 , q0 ) ⇔ p  ∈ Ww1− p ( p0 , q0 ).

(2.6)

2.2 Weighted Tent Spaces The weighted tent spaces that we consider are defined as follows: given w ∈ A∞ , for 0 < q, p < ∞, p

p Tq (w) := { f measurable in Rn+1 + : Aq ( f ) ∈ L (w)},

(2.7)

endowed with the norm  f Tqp (w) := Aq f  L p (w) , when q = 2 we just write T p (w), = {(y, t) : y ∈ (these spaces were also considered in [16]). We denote by Rn+1 + Rn , 0 < t < ∞}, the upper half space, and, for all α > 0 and 0 < q < ∞, the operator Aqα , Aqα f (x) :=

   α (x)

| f (y, t)|q

dy dt t n+1

1 q

,

where  α (x) := {(y, t) ∈ Rn+1 : |x − y| < αt} is the cone of aperture α > 0 with + vertex at x. We write A f and (x) when q = 2 and α = 1. By [32, Proposition p 3.39] (see also [16,31]), we have that the definition of Tq (w) does not depend on the aperture of the cone  use to define the operator A. This is, for all 0 < α, β < ∞, Aqα f  L p (w) ≈ Aqβ f  L p (w) ,

(2.8)

with constant depending on the weight α and β. Besides, in the same way as in the unweighted case, we can see that these spaces are quasi-Banach spaces for 1 ≤ q, p < ∞. Additionally, note that in our definition of weighted tent spaces the operator A is the same operator as in the unweighted case, i.e. it does not depend on the weight

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C. Prisuelos-Arribas

w. Consequently, we cannot see Rn with a doubling measure given by a weight w and apply the interpolation results obtained for tent spaces defined in metric measure spaces or spaces of homogeneous type (X, μ), since in the definition of those spaces the operator A is modified to depend on the measure μ. See for instance [1], [25, Lemma 4.6, Proposition 4.9], or [34]. That is why, in Sect. 5, we also give an interpolation result for these weighted tent spaces. 2.3 Elliptic Operators Let A be an n × n matrix of complex and L ∞ -valued coefficients defined on Rn . We assume that this matrix satisfies the following ellipticity (or “accretivity”) condition: there exist 0 < λ ≤  < ∞ such that λ |ξ |2 ≤ Re A(x) ξ · ξ¯

|A(x) ξ · ζ¯ | ≤  |ξ | |ζ |,

and

for all ξ, ζ ∈ Cn and almost every x ∈ Rn . We have used the notation ξ · ζ = n ξ1 ζ1 + · · · + ξ n ζn and therefore ξ · ζ¯ is the usual inner product in C . Note that then ¯ A(x) ξ · ζ¯ = j,k a j,k (x) ξk ζ j . Associated with this matrix we define the secondorder divergence form elliptic operator L f = −div(A ∇ f ),

(2.9)

which is understood in the standard weak sense as a maximal-accretive operator on L 2 (Rn , d x) with domain D(L) by means of a sesquilinear form. The operator L has 1 a square root L 2 , defined as the unique maximal-accretive operator such that 1

1

L2L2 = L 1

as unbounded operators (see [2] for a deeper discussion in the operator L 2 , and, for a explicit construction, the two references recommended there: [17, Chap. XIV] and 1 [30, p. 281]). We use the following formula to compute L 2 : 1 2 L2 = √ π



∞ 0

t Le−t

2L

dt . t

(2.10)

Moreover, the operator −L generates a C 0 -semigroup {e−t L }t>0 of contractions on L 2 (Rn ) which is called the heat semigroup. As in [2] and [5], we denote by ( p− (L), p+ (L)) the maximal open interval on which this semigroup is uniformly bounded on L p (Rn ), and by (q− (L), √ q+ (L)) the maximal open interval on which the gradient of the heat semigroup, { t∇ y e−t L }t>0 , is uniformly bounded on L p (Rn ):   2 p− (L) := inf p ∈ (1, ∞) : sup e−t L  L p (Rn )→L p (Rn ) < ∞ , t>0   2 p+ (L) := sup p ∈ (1, ∞) : sup e−t L  L p (Rn )→L p (Rn ) < ∞ , t>0

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Weighted Hardy Spaces Associated with Elliptic Operators

  2 q− (L) := inf p ∈ (1, ∞) : sup t∇ y e−t L  L p (Rn )→L p (Rn ) < ∞ , t>0   2 q+ (L) := sup p ∈ (1, ∞) : sup t∇ y e−t L  L p (Rn )→L p (Rn ) < ∞ . t>0

Note that in place of the semigroup {e−t L }t>0 we are using its rescaling {e−t L }t>0 . We do so since all the “heat” square functions, defined below, are written using the latter, and also because in the context of the off-diagonal estimates it will simplify some computations. Furthermore, for every N ∈ N and 0 < q < ∞, let us set 2

q

N ,∗

:=

qn , n−Nq ∞,

if N q < n,

(2.11)

if N q ≥ n.

Corresponding to the case N = 1, we write q ∗ . Besides, from [2] (see also [5]) we know that p− (L) = 1 and p+ (L) = ∞ if n = 2n 2n and p+ (L) > n−2 . Moreover, q− (L) = p− (L), 1, 2; and if n ≥ 3 then p− (L) < n+2 ∗ q+ (L) ≤ p+ (L), and we always have q+ (L) > 2, with q+ (L) = ∞ if n = 1. w (L) As in [6], given a weight w ∈ A∞ , we also consider the intervals Jw (L) and K 2 −t which are, respectively, (possibly empty) intervals of p ∈ [1, ∞) such that {e L }t>0 2 is a bounded set in L(L p (w)) and {t∇e−t L }t>0 is a bounded set in L(L p (w)), (where L(X ) denotes the space of linear continuous maps on a Banach space X ). 2.4 Off-Diagonal Estimates We briefly recall the notion of off-diagonal estimates. Let {Tt }t>0 be a family of linear operators and let 1 ≤ p ≤ q ≤ ∞. We say that {Tt }t>0 satisfies L p (Rn ) − L q (Rn ) off-diagonal estimates of exponential type, denoted by {Tt }t>0 ∈ F∞ (L p → L q ), if for all closed sets E, F, all f , and all t > 0 we have Tt ( f 1 E ) 1 F  L q (Rn ) ≤ Ct

−n



1 1 p−q



e

−c d(E,F) 2 t

2

 f 1 E  L p (Rn ) .

Analogously, given β > 0, we say that {Tt }t>0 satisfies L p (Rn ) − L q (Rn ) offdiagonal estimates of polynomial type with order β > 0, denoted by {Tt }t>0 ∈ Fβ (L p → L q ) if for all closed sets E, F, all f , and all t > 0 we have

Tt ( f 1 E ) 1 F  L q (Rn ) ≤ Ct

−n



1 1 p−q



d(E, F)2 1+ t2

  β+ n2 1p − q1

−

 f 1 E  L p (Rn ) .

The heat and Poisson semigroups satisfy, respectively, off-diagonal estimates of exponential and polynomial type. The parameters p− (L), p+ (L), q− (L) and q+ (L) besides giving the maximal intervals on which either the heat semigroup or its gradient are uniformly bounded, they characterise the maximal open intervals on which

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C. Prisuelos-Arribas

off-diagonal estimates of exponential type hold (see [2] and [5]). More precisely, for every m ∈ N0 , there hold {(t 2 L)m e−t L }t>0 ∈ F∞ (L p − L q ) 2

for all p− (L) < p ≤ q < p+ (L)

and {t∇ y e−t L }t>0 ∈ F∞ (L p − L q ) 2

for all q− (L) < p ≤ q < q+ (L).

From these off-diagonal estimates we have, for every m ∈ N0 := N ∪ {0}, √ √ {(t L )2m e−t L }t>0 , ∈ Fm+ 1 (L p → L q ), 2

for all p− (L) < p ≤ q < p+ (L), and {t∇ y (t 2 L)m e−t L }t>0 , {t∇ y,t (t 2 L)m e−t L }t>0 ∈ F∞ (L p → L q ), √ √ √ √ {t∇ y (t L )2m e−t L }t>0 ∈ Fm+1 (L p → L q ), {t∇ y,t (t L )2m e−t L }t>0 ∈ Fm+ 1 (L p → L q ), 2

2

2

for all q− (L) < p ≤ q < q+ (L), (see [32, Sect. 2]). Besides, if Ww ( p− (L), p+ (L)) = ∅, there exists a maximal interval of [1, ∞] p− (L),  p+ (L)); and if Ww (q− (L), q+ (L)) = ∅, there exists denoted by Jw (L) = ( q− (L),  q+ (L)), where for all a maximal interval of [1, ∞] denoted by Kw (L) = ( 2 p, q ∈ Jw (L) or p, q ∈ Kw (L) and for all m ∈ N0 , we have that {(t 2 L)m et L }t>0 2 or {t∇(t 2 L)m et L }t>0 satisfy, respectively, off-diagonal estimates on balls and are bounded sets in L(L p (w)), (see [6, Definition 3.2 and Proposition 3.4]). Moreover, in [6] the authors proved the following: Ww ( p− (L), p+ (L)) ⊂ w (L), IntJw (L) = IntJw (L), Jw (L) ⊂ Jw (L), Ww (q− (L), q+ (L)) ⊂ Kw (L) ⊂ K  IntKw (L) = IntKw (L), inf Jw (L) = inf Kw (L) and (supKw (L))∗w = supJw (L), where

qw∗ :=

qnrw nrw −q ,

∞,

nrw > q,

(2.12)

otherwise.

2.5 Operators √

Using the heat semigroup and the corresponding Poisson semigroup {e−t L }t>0 , let us define different conical square functions which all have an expression of the form α

Q f (x) =

123

   α (x)

dy dt |Tt f (y)|2 n+1 t

1 2

,

x ∈ Rn ,

(2.13)

Weighted Hardy Spaces Associated with Elliptic Operators

when α = 1 we just write Q f (x), ( α (x) was defined on page 4). More precisely, we introduce the following conical square functions written in terms of the heat semigroup (hence the subscript H): for every m ∈ N,   Sm,H f (x) =

(x)

|(t 2 L)m e−t

2L

f (y)|2

dy dt t n+1

1 2

,

(2.14)

and, for every m ∈ N0 ,   Gm,H f (x) =

m −t 2 L

|t∇ y (t L) e 2

(x)

  Gm,H f (x) =

m −t 2 L

|t∇ y,t (t L) e 2

(x)

dy dt f (y)|2 n+1 t

1 2

dy dt f (y)|2 n+1 t

,

(2.15)

1 2

.

(2.16)

In the same way, let us consider conical square functions associated with the Poisson semigroup (hence the subscript P): given K ∈ N,   S K ,P f (x) =

√ √ dy dt |(t L )2K e−t L f (y)|2 n+1 t (x)

1 2

,

(2.17)

and for every K ∈ N0 ,   G K ,P f (x) =

√

(x)

|t∇ y (t L )

  G K ,P f (x) =

√ 2K −t L

√

(x)

e

|t∇ y,t (t L )

dy dt f (y)|2 n+1 t

√ 2K −t L

e

1

dy dt f (y)|2 n+1 t

2

,

(2.18)

1 2

.

(2.19)

Corresponding to the cases m = 0 or K = 0 we simply write GH f := G0,H f , GH f := G0,H f , GP f := G0,P f and GP f := G0,P f . Besides, we set SH f := S1,H f and SP f := S1,P f . We also consider the following non-tangential maximal functions.  NH f (x) =

|e−t

sup

(y,t)∈(x)

NP f (x) =

2L

f (z)|2

B(y,t)

 |e

sup

(y,t)∈(x)

B(y,t)

dz tn

1 2

√ −t L

and dz f (z)|2 n t

1 2

,

(2.20)

1

and denote the Riesz transform associated with the operator L by ∇ L − 2 . We have the following representation for it

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C. Prisuelos-Arribas 1 2 ∇ L− 2 f = √ π



∞

t∇ y e−t

2L

f

0

dt . t

(2.21) 1

From [6] we know the following boundedness result for ∇ L − 2 . Theorem 2.2 [6, Theorem 5.2] Let w ∈ A∞ be such that Ww (q− (L), q+ (L)) = ∅. n For all p ∈ Int Kw (L) and f ∈ L ∞ c (R ), 1

∇ L − 2 f  L p (w)   f  L p (w) . 1

Hence ∇ L − 2 has a bounded extension to L p (w). 2.6 Complex Interpolation Let us defined the interpolation space described in [15] by A.P. Calderón. Let A, B be Banach spaces embedded in a complex topological vector space V , and such that  ·  A and  ·  B denote the norm in A and B, respectively. Now, consider the space A + B = {x = y + z : y ∈ A, z ∈ B} endowed with the norm x A+B := inf{y A + z B : x = y + z, y ∈ A, z ∈ B}. Then, the space A + B becomes a Banach space. Now, consider the linear space of functions F := F(A, B) as the space of all functions f (ξ ), ξ = θ + it, defined in the strip 0 ≤ θ ≤ 1 of the ξ − plane, with values in A+ B continuous and bounded with respect to the norm of A+ B in 0 ≤ θ ≤ 1 and analytic in 0 < θ < 1, and such that f (it) ∈ A is A-continuous and tends to zero as |t| tends to infinity and f (1 + it) ∈ B is B-continuous and tends to zero as |t| tends to infinity. The norm that we consider in this space is the following 



 f F := max sup  f (it) A , sup  f (1 + it) B , t

t

under this norm F becomes a Banach space. Finally, for a given θ , 0 ≤ θ ≤ 1, we define the space [A, B]θ := {x ∈ A + B : x = f (θ ), f ∈ F} endowed with the norm x[A,B]θ := inf{ f F : x = f (θ )}. Then [A, B]θ is a Banach space continuously embedded in A + B. 2.7 Extrapolation In some proofs, we shall use the following extrapolation result that appear in [19]. Theorem 2.3 Let F be a given family of pairs ( f, g) of non-negative and not identically zero measurable functions.

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Weighted Hardy Spaces Associated with Elliptic Operators

(a) Suppose that for some fixed exponent q0 , 1 ≤ q0 < ∞, and every weight w ∈ R Hq0 , 



1

Rn

f (x) q0 w(x) dx ≤ Cw

1

g(x) q0 w(x) dx,

Rn

∀ ( f, g) ∈ F.

Then, for all 1 < q < ∞ and for all w ∈ R Hq  , 



1 q

Rn

f (x) w(x) dx ≤ Cw,q

1

Rn

g(x) q w(x) dx,

∀ ( f, g) ∈ F.

(b) Suppose that for some fixed exponent r0 , 0 < r0 < ∞, and every weight w ∈ A∞ 

 Rn

f (x)r0 w(x) dx ≤ Cw

g(x)r0 w(x) dx,

Rn

∀ ( f, g) ∈ F.

Then, for all 0 < r < ∞, and for all w ∈ A∞ , 

 f (x) w(x) dx ≤ Cw,r r

Rn

Rn

g(x)r w(x) d x,

∀ ( f, g) ∈ F.

Part (a) is not written explicitly in [19] but can be easily obtained using [4, Theorem 4.9] and [19, Theorem 3.31] (see also [32, Lemma 3.3, part (b)]); part (b) appears in [19, Corollary 3.15] assuming that the left-hand sides in the inequalities are finite. Here we do not take such assumptions, and in particular, we have that the infiniteness of the left-hand side will imply that of the right-hand one. To prove (b) in the present form just proceed as in the proof of [32, Lemma 3.3, part (a)] but using [19, Corollary 3.15] instead of [19, Theorem 3.9].

3 Weighted Hardy Spaces For w ∈ A∞ , q ∈ Ww ( p− (L), p+ (L)) and 0 < p < ∞. We define the weighted Hardy spaces associated with operators and the molecular weighted Hardy space. 3.1 Weighted Hardy Spaces Associated with Operators Definition 3.1 Given a sublinear operator T acting on functions of L q (w) we define p the weighted Hardy space HT ,q (w) as the completion of the set   p HT ,q (w) := f ∈ L q (w) : T f ∈ L p (w) , with respect to the semi-norm  f H p

T ,q (w)

:= T f  L p (w) .

In our results, T will be any of the square functions in (2.14)–(2.19), or a nontangential maximal function in (2.20).

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C. Prisuelos-Arribas

Remark 3.2 In [28], where the unweighted case was considered, the Hardy spaces are defined taking the completion of a set of functions in L 2 (Rn ). Here we take functions in L q (w), where q ∈ Ww ( p− (L), p+ (L)) because we do not know whether 2 is in Ww ( p− (L), p+ (L)) or not. In any case, we shall show that for 0 < p ≤ 1 or p p ∈ Ww ( p− (L), p+ (L)) this choice of q is irrelevant since all the spaces HT ,q (w) are isomorphic for all q ∈ Ww ( p− (L), p+ (L)). 3.2 Molecular Weighted Hardy Spaces In order to define the molecules and the molecular decompositions, we introduce the following notation: given a cube Q ⊂ Rn we set Q i :=2i+1 Q, for all i ≥ 1, C1 (Q) := 4Q and, for i ≥ 2, Ci (Q) := 2i+1 Q\2i Q.

(3.1)

Besides, (Q) denotes the side length of Q. We next define the notion of molecule and molecular representation. These objects are a weighted version of those defined in [28] in the unweighted case. Definition 3.3 Apart from the conditions stated at the beginning of the section for w,  n rw q and p, let us take ε > 0, and M ∈ N such that M > 2 p − p−1(L) . (a) Molecules We say that a function m ∈ L q (w) (belonging to the range of L M in L q (w)) is a (w, q, p, ε, M) − molecule if there exists a cube Q ⊂ Rn such that m satisfies mmol,w :=

 i≥1

1

2iε w(2i+1 Q) p

− q1

M       ( (Q)2 L)−k m 1Ci (Q)  k=0

L q (w)

< 1.

Henceforth, we refer to the previous expression as the molecular w-norm of m. Besides, any cube Q satisfying that expression, is called a cube associated with m. Note that if m is a (w, q, p, ε, M) − molecule, for all associated cubes Q:      ( (Q)2 L)−k m 1Ci (Q) 

1

L q (w)

≤ 2−iε w(2i+1 Q) q

i = 1, 2, . . . ; k = 0, 1, . . . , M.

− 1p

, (3.2)

 (b) Molecular representations For any function f we say that i∈N λi mi is a (w, q, p, ε, M) − representation of f , if the following conditions are satisfied: (i) {λi }i∈N ∈ p . (ii) For every  i ∈ N, mi is a (w, q, p, ε, M) − molecule. (iii) f = i∈N λi mi in L q (w). We finally define the molecular weighted Hardy spaces.

123

Weighted Hardy Spaces Associated with Elliptic Operators

Definition 3.4 Let w, q, p, ε and M be as in the previous definition, we define the p molecular weighted Hardy space HL ,q,ε,M (w) as the completion of the set p

H L ,q,ε,M (w)

∞ ∞   := f = λi mi : λi mi is a (w, q, p, ε, M) − representation of f , i=1

i=1

with respect to the quasi-norm,  f H p

L ,q,ε,M (w)

:= inf

⎧ ∞ ⎨  ⎩

i=1

1

p

|λi | p

:

∞ 

λi mi is a (w, q, p, ε, M) − representation of f

i=1

⎫ ⎬ ⎭

.

Remark 3.5 Although we shall just show molecular characterisation for weighted Hardy spaces in the range 0 < p ≤ 1, we have given the definition of the molecular weighted Hardy spaces for all 0 < p < ∞. This is because we can always p obtain a molecular decomposition of functions f ∈ HT ,q . This is easily seen by following the proof and noticing that there is no restriction over p. In particular, p p   f H p (w) for all we have that HT ,q (w) ⊂ H L ,q,ε,M (w), with  f H p L ,q,ε,M (w) T ,q 0 < p < ∞. p

Remark 3.6 We shall show below that, for 0 < p ≤ 1, the Hardy space HL ,q,ε,M (w) does not depend on the choice of the allowable parameters q, ε and M. Hence, at this point, it is convenient for us to make a choice of these parameters and define the weighted Hardy space as the one associated with this choice: from now on for every w ∈ A∞ , we fix q0 ∈ Ww ( p− (L), p+ (L)), ε0 > 0 and M0 ∈ N such that p p M0 > n2 rpw − p−1(L) , and set HL (w) := HL ,q0 ,ε0 ,M0 (w), for 0 < p ≤ 1. 3.3 Weighted Hardy Space Associated with the Riesz Transform Definition 3.7 Given w ∈ A∞ , q ∈ Ww (q− (L), q+ (L)) and 0 < p < ∞, we define p the weighted Hardy space associated with the Riesz transform H∇ L −1/2 ,q (w), as the completion of the set 1

H∇ L −1/2 ,q (w) := { f ∈ L q (w) : ∇ L − 2 f  L p (w) < ∞}, p

with respect to the norm  f H p

∇ L −1/2

1

:= ∇ L − 2 f  L p (w) .

Remark 3.8 By [6, Remark 3.5], we know that, for q ∈ Ww ( p− (L), p+ (L)), e−t L has an infinitesimal generator L q,w , on L q (w). In particular e−t L q,w and e−t L agree in L q (w) ∩ L 2 (Rn ). In our definitions of weighted Hardy spaces, we consider

123

C. Prisuelos-Arribas

functions f ∈ L q (w), q ∈ Ww ( p− (L), p+ (L)) (q ∈ Ww (q− (L), q+ (L)) ⊂ Ww ( p− (L), p+ (L)) in the case of the Riesz transform), such that some operator that depends on L, let us write O L , satisfies that O L f ∈ L p (w), 0 < p < ∞. Abusing the notation we write O L , but it should be understood that, when f ∈ L q (w) \ L 2 (Rn ), O L f is in fact O L q,w f . On the other hand, in the case that f ∈ L q (w) ∩ L 2 (Rn ) note that O L q,w f = O L f .

4 Auxiliary Results The proofs of our main results are long and with a lot of technical details. Therefore, in order to facilitate a smooth reading of those proofs, we include some of those technicalities in this section. We also introduce the following notation. Let E ⊂ Rn be a cube or a ball, for any function f and weight w ∈ A∞ , recalling the notation in (3.1), we set   1 − f (x)dw := f (x)w(x)dx and w(E) E E   1 − f (x)dw := f (x)w(x)dx, ∀ j ≥ 2. w(2 j+1 E) C j (E) C j (E) Proposition 4.1 Given w ∈ A∞ such that Ww ( p− (L), p+ (L)) = ∅, for 0 < t, s < ∞, p, q ∈ Jw (L), p ≤ q, M ∈ N and a ball B ⊂ Rn with radius r B , we have M := (e−t 2 L − e−(t 2 +s 2 )L ) M } p q that {Tt,s t>0 satisfies the following L (w) − L (w) offdiagonal estimates on balls: there exist θ1 , θ2 > 0 such that, for all j ≥ 2,  −

C j (B)

1 q

M |Tt,s ( f 1 B )(y)|q dw



2 j rB ,  2 jθ1 max t

θ2   M √ 1 4 j r 2  p s2 s2 + t 2 −c 2 B2 p t +s − e | f (y)| dw , 2 j rB t2 B (4.1)

and

θ2   M  √  1 1 q p rB s2 s2 + t 2 M q p − |Tt,s ( f 1 B )(y)| dw − ,  max | f (y)| dw . 2 t rB t B B (4.2) Proof We have for j ≥ 2,

123

Weighted Hardy Spaces Associated with Elliptic Operators

 −

1 q

C j (B)

M |Tt,s ( f 1 B )(y)|q dw

= w(2 j+1 B)

− q1

⎛  ⎞ M   s2    2 −(r +t )L ⎝ ⎠ 1C j (B)  ∂ e dr f 1 r B   0  

L q (w)

− q1

≤ w(2 B) ⎛  ⎞  M   s 2  s 2   M M   2 −( i=1 ri +Mt 2 )L  ⎝ ⎠ ··· ri + Mt L e ( f 1 B ) 1C j (B)    0 0   i=1 j+1

dr1 · · · dr M

L q (w)

× M ( i=1 ri + Mt 2 ) M ⎛ ⎞θ2  s2  s2 jr 2 B ⎠  2 jθ1 ··· ϒ ⎝ " M 0 0 2 i=1 ri + Mt 2 4 j rB

dr1 · · · dr M −1 w(B) p  f 1 B  L p (w) M ( i=1 ri + Mt 2 ) M

θ2   M √ 1 4 j r 2  p 2 j rB s2 s2 + t 2 −c 2 B2 jθ1 p ,  2 max e t +s − | f (y)| dw , j 2 t 2 rB t B ×e

−c  M

i=1 ri +Mt

2

where θ1 , θ2 > 0, ϒ(u) := max{u, u −1 }, and we have used the fact that (t L) M e−t L satisfies L p (w) − L q (w) off-diagonal estimates on balls (see [5]). The proof of (4.2) follows similarly.   In the unweighted case we have a similar result to the previous one, see [33, (5.12)] (see also [26, Proof of Lemma 2.2]). Proposition 4.2 For 0 < t, s < ∞, p ∈ ( p− (L), p+ (L)), M ∈ N, and for E 1 , E 2 M} closed subsets in Rn , and f ∈ L p (Rn ) such that supp( f ) ⊂ E 1 , we have that {Tt,s t>0 p n p n satisfies the following L (R ) − L (R ) off-diagonal estimates:   M Tt,s

  f

 L p (E 2 )



s2 t2

M e

−c

d(E 1 ,E 2 )2 t 2 +s 2

 f  L p (E 1 ) .

We next give a change of angle result similar to [32, Proposition 3.30], but with the difference that in the one below we keep some control on the support by imposing another condition over the radius of the ball. Proposition 4.3 Let 1 ≤ q ≤ s < ∞, w ∈ R Hs  and 0 ≤ α ≤ 1. Then, for every ball B with radius r B , and 0 < t ≤ r B , there hold, for j ≥ 2,

123

C. Prisuelos-Arribas



 C j (B)

α

1 B(x,αt)

q

q

w(x)d x,

(4.3)

B(x,t)

1 |h(y, t)| dy

4B

1 |h(y, t)| dy

2 j+3 B\2 j−2 B



w(x)d x





n s

and 

q

|h(y, t)| dy

n





1

w(x)d x  α s

|h(y, t)| dy

B(x,αt)

6B

q

w(x)d x.

B(x,t)

(4.4) Proof Note that for every 0 < t ≤ r B , x ∈ C j (B) and 0 < α ≤ 1, we have that B(x, αt) ⊂ 2 j+2 B \ 2 j−1 B, for all j ≥ 2, and B(x, αt) ⊂ 5B, for j = 1. Besides, if y ∈ 2 j+2 B \ 2 j−1 B and 0 < t ≤ r B , then B(y, t) ⊂ 2 j+3 B \ 2 j−2 B, for all j ≥ 2; on the other hand if y ∈ 5B and 0 < t ≤ r B , then B(y, t) ⊂ 6B, for j = 1. Therefore, following the proof of [32, Proposition 3.30] but keeping the above conditions on the support of the integral in x we conclude the proof.   We next establish some results for a general 0 < p < ∞ that were proved in [33] for p = 1. The following lemma related to (w, q, p, ε, M) − molecules is analogous to that of [33, Lemma 4.6]. Lemma 4.4 Given p0 < q, 0 < p < ∞, w ∈ A

q p0

, ε > 0 and M ∈ N. Let m be a

(w, q, p, ε, M) − molecule and let Q be a cube associated with m. For every i ≥ 1 and every 0 ≤ k ≤ M, k ∈ N0 , there holds      ( (Q)2 L)−k m 1Ci (Q)

L p0 (Rn )

 2−iε w(2i+1 Q)

− 1p

1

|2i+1 Q| p0 .

Proof First of all, recall that if m is a (w, q, p, ε, M) − molecule, in particular we have (3.2). That, Hölder’s inequality, and the fact that w ∈ A q imply p0     2 −k ( (Q) L) m 1  Ci (Q)  p n L

 ≤

|( (Q) L) 2

Ci (Q)

 −

2i+1 Q

w(y)

−k

  1− pq

 2−iε w(2i+1 Q)

0

− 1p

0 (R

)

1

m(y)| w(y) dy q

 q1



q p0 −1

dy

q

1

|2i+1 Q| p0

− q1

1

|2i+1 Q| p0 .  

The following propositions are generalisations of [33, Proposition 5.2, Proposition 5.3]. They give us uniform boundedness of the norm on L p (w), for 0 < p ≤ 1, of the square functions applied to (w, q, p, ε, M) − molecules. Proposition 4.5 Let w ∈ A∞ and let {Tt }t>0 be a family of sublinear operators satisfying the following conditions:

123

Weighted Hardy Spaces Associated with Elliptic Operators

(a) {Tt }t>0 ∈ F∞ (L p0 → L 2 ) for all p− (L) < p0 ≤ 2. 1  2 dy dt 2 is bounded on L q (w) for every q ∈ (b)  S f (x) := |T f (y)| t n+1 (x) t Ww ( p− (L), p+ (L)). t2

(c) There exists C > 0 so that for every t > 0 there holds Tt = C T √t ◦ e− 2 L . 2

(d) For every λ > 0, there exists Cλ > 0 such that for every t > 0 it follows that T√1+λ t = Cλ Tt ◦ e−λt L . 2

Then, for every m a (w, q, p, ε, M) − molecule with q ∈ Ww ( p− (L), p+ (L)), 0 < n rw 1 Sm L p (w)  1, with p ≤ 1, ε > 0 and M > 2 p − p− (L) , it follows that  constants independent of m. The proof of this proposition follows line by line that of [33, Proposition 5.2] (i.e. for the case p = 1). It suffices to replace p with q, the L 1 (w) norm with the L p (w) norm, [32, Lemma 4.6] with Lemma 4.4, and [33, (3.3)] with (3.2); apart from other minor and easy changes as consequences of the above replacements. From this result we have: Proposition 4.6 Let S be any of the square functions considered in (2.14)–(2.19). For every w ∈ A∞ and m a (w, q, p, ε, M) − molecule with q ∈ Ww ( p− (L), p+ (L)), 0 < p ≤ 1, ε > 0 and M > n2 rpw − p−1(L) , there hold (a) Sm L p (w) ≤ C. (b) For all f ∈ H1L ,q,ε,M (w), S f  L p (w)   f H1

L ,q,ε,M (w)

.

Proof Note that, in view of [32, Theorems 1.14 and 1.15, and Remark 4.22] and the fact that SH f ≤ 21 GH f , to prove part (a) it suffices to show the desired estimate for GH . To 2 2 2 this end, we observe that |t∇ y,t e−t L f |2 = |t∇ y e−t L f |2 + 4|t 2 Le−t L f |2 . Besides, 2 2 both Tt := t∇ y e−t L and Tt := t 2 Le−t L satisfy the hypotheses of Proposition 4.5: 2 (a) follows from the off-diagonal estimates that the families Tt := t∇ y e−t L and 2 Tt := t 2 Le−t L satisfy (see Sect. 2.4); (b) is contained in [32, Theorem 1.12, part (a)] and finally (c) and (d) follow from easy calculations. Thus, we can apply Proposition 4.5 and obtain the desired estimate. As for part (b), fix w ∈ A∞  and take q ∈ Ww ( p− (L), p+ (L)), 0 < p ≤ 1, ε > 0 p n rw and M ∈ N such that M > 2 p − p−1(L) . Then, for f ∈ H L ,q,ε,M (w), there exists ∞ a (w, q, p, ε, M) − representation of f , f = i=1 λi mi , such that ∞  i=1

1

p

|λi |

p

≤ 2 f H p

L ,q,ε,M (w)

.

∞ λi mi converges in L q (w) and since for any choice of On the other hand, since i=1 S, we have that S is a sublinear operator bounded on L q (w) (see [32, Theorems 1.12 and 1.13]). This, part (a), and the fact that 0 < p ≤ 1 imply

123

C. Prisuelos-Arribas

 ∞       = S λi mi   

S f  L p (w)

i=1

≤C

∞ 

≤

∞ 

L p (w)

1

p

|λi |

p

p Smi  L p (w)

i=1

1

p

|λi | p

  f H p

L ,q,ε,M (w)

i=1

,  

as desired.

As for the non-tangential maximal functions considered in (2.20), we generalise [33, Proposition 7.22]. Proposition 4.7 Let w ∈ A  ∞ , 0 < p ≤ 1, q ∈ Ww ( p− (L), p+ (L)), ε > 0 and n rw M ∈ N such that M > 2 p − p−1(L) , and let m be a (w, q, p, ε, M)-molecule. Then, (a) NH m L p (w) + NP m L p (w) ≤ C p (b) For all f ∈ H L ,q,ε,M (w), NH f  L p (w) + NP f  L p (w) ≤  f H p

L ,q,ε,M (w)

.

The proof of part (a) follows as [33, Proposition 7.22, part (a)], computing the p p norms NH m L p (w) and NP m L p (w) instead of the L 1 (w) norms, replacing p with q, and using Lemma 4.4 instead of [33, Lemma 4.6]. The only major change comes p when, while computing NP m L p (w) , this integral appears 



∞

e

1 4

Rn

−u

√ 4 u SH m(x)du

p w(x)dx,

√ 4 u

(see (2.13) and (2.14) for the definition of SH ). In order to estimate it, note that for all w0 ∈ A∞ , r0 > rw , applying Minkowski’s integral inequality and [32, Proposition 3.2], we have 

 Rn

∞ 1 4



≤  ≤  

123

e

−u

∞ 1 4

e

−u

e

−u

∞ 1 4

Rn

2 √ 4 u SH m(x)du w0 (x)dx  Rn

√

1 √ 2 4 u 2 (SH m(x)) w0 (x)dx

(4 u)

nr0 2

2 du

 Rn

(SH m(x))2 w0 (x)dx.

2

1 (SH m(x)) w0 (x)d x 2

2

du

Weighted Hardy Spaces Associated with Elliptic Operators

Hence, applying Theorem 2.3, part (b), we obtain for all 0 <  r < ∞ and w  ∈ A∞ 



∞ 1 4

Rn

e

−u

r √ 4 u SH m(x)du w (x)dx

In particular, for every 0 < p ≤ 1 and w ∈ A

 

q p− (L)

Rn

(SH m(x))r w (x)dx.

∩ R H p+ (L)  , and by Proposition q

4.6, part (a), we have that 



∞ 1 4

Rn

√ 4 u e−u SH m(x)du

p

 w(x)dx 

Rn

(SH m(x)) p w(x)dx ≤ C.

The proof of part (b) follows as the proof of Proposition 4.6, part (b), but using [33, Proposition 7.1] instead of [32, Theorems 1.12 and 1.13]. Finally, let us see that the Riesz transform applied to (w, q, p, ε, M) − molecules is also uniformly bounded. Proposition 4.8 For every w ∈ A∞ , q ∈ Ww (q− (L), q+ (L)), 0 < p ≤ 1, ε > 0, M ∈ N such that M > n2 rpw − p−1(L) , and m a (w, q, p, ε, M) − molecule, there hold 1

(a) ∇ L − 2 m L p (w) ≤ C. 1 p (b) For all f ∈ H L ,q,ε,M (w), ∇ L − 2 f  L p (w)   f H p

L ,q,ε,M (w)

.

Proof Assuming part (a) the proof of part (b) follows as the proof of Proposition 4.6, part (b), but using Theorem 2.2 instead of [32, Theorems 1.12 and 1.13]. We next prove part (a). Fix w, p, q, ε and M as in the statement of the proposition.  r n w  Note that since w ∈ A q ∩ R H q+ (L)  and M > 2 p − q−1(L) (recall that q− (L)

q

p− (L) = q− (L)), we can take r0 > rw , p0 and q0 , q− (L) < p0 < q < q0 < q+ (L), close enough to rw , q− (L) and q+ (L), respectively, so that w ∈ A q ∩ R H q0  p0

q

and M>

n 2



r0 1 − p p0

 .

(4.5)

Besides, take m a (w, q, p, ε, M)−molecule and Q ⊂ Rn one of its associ M  2 ated cubes, with side length (Q), and consider B Q := I − e− (Q) L and A Q := I − B Q . Additionally, recalling the notation given in (3.1) and con# $ k := k (Q)2 L M e−k (Q)2 L , for every k ∈ {1, . . . , M}, we can sidering A Q write

123

C. Prisuelos-Arribas

1

1

1

1

∇ L − 2 m = ∇ L − 2 BQ m + ∇ L − 2 A Q m = ∇ L − 2 BQ m + =



 1

∇ L − 2 B Q mi +

i≥1

M 

M 

kQ m  Ck,M ∇ L − 2 A 1

k=1

 kQ m i , Ck,M ∇ L − 2 A 1

(4.6)

k=1

 := ( (Q)2 L)−M m and for any function f , we denote f i := f 1Ci (Q) , for all where m i ≥ 1. Thus, we have 1

∇ L − 2 B Q mi  L p (w)  p



1

1C j (Q i ) (∇ L − 2 B Q mi ) L p (w) =:

j≥1

p



Ii j .

(4.7)

j≥1 1

Then, for j = 1, applying Hölder’s inequality, the boundedness of ∇ L − 2 and B Q on L q (w) (see Theorem 2.2 and [5]), and by (3.2) and (2.2), we obtain Ii1  w(2

i+1

Q)

1− qp

 4Q i



p % %q q % − 12 % %∇ L B Q mi (x)% w(x)dx

p 1− p mi  L q (w) w(2i+1 Q) q

≤ 2−i pε .

(4.8)

As for j ≥ 2, denoting Tt := t∇ y e−t L , using (2.21) and splitting the integral in t, we obtain %p %  % (Q) dt %% % Tt B Q mi (x) % w(x)dx Ii j ≤ % t % C j (Q i ) % 0 % % ∞  % dt %% p % T B m (x) w(x)dx =: Ii1j + Ii2j . (4.9) + t Q i % t % C j (Q i ) (Q) 2

The estimate of Ii1j follows applying twice Hölder’s inequality the fact that w ∈ R H q0  , and Minkowski’s integral inequality. Besides, we expand the binomial, apply q

respectively, the L p0 (Rn ) − L q0 (Rn ) and the L p0 (Rn ) − L p0 (Rn ) off-diagonal esti2 2 mates that the families {t∇ y e−t L }t>0 and {e−t L }t>0 satisfy, (see Sect. 2.4), and also [32, Lemma 2.1] (see also [26, Lemma 2.3]), Lemma 4.4, and (2.2). Then, p %q q dt %%  w(2 Qi ) Tt B Q mi (x) % w(x)dx t C j (Q i ) 0 ⎛ ⎞p 1   (Q)  q0 p % % dt − q %Tt B Q mi (x)% 0 dx ⎠  w(2 j+1 Q i ) |2 j+1 Q i | q0 ⎝ t 0 C j (Q i ) ⎛ 1  (Q)  q0 % %q0 p dt 2L − % % j+1 j+1 −t q ⎝  w(2 Q i ) |2 Qi | 0 mi (x)% dx %t∇ y e t C j (Q i ) 0

Ii1j

123

j+1

1− qp



% % % %

(Q)

Weighted Hardy Spaces Associated with Elliptic Operators

+

M 



(Q)

Ck,M

(Q)−1

0

k=1



C j (Q i )

%√ %q0 2 2 % % % k (Q)∇ y e−k (Q) L e−t L mi (x)% dx −

p

1

q0

⎞p dt ⎠

p

 mi  L p0 (Rn ) w(2 j+1 Q i )|2 j+1 Q i | q0   p    i+ j 2 (Q) −c4i+ j n n n n dt te − − − − −c 4 (Q) t2 t p0 q0 + e (Q) p0 q0 (Q) t 0  e−c4

i+ j

.

(4.10)

√ As for Ii2j , we proceed as before but also changing the variable t into M + 1t =: M  2 2 2 C M t and considering B Q,t := e−t L − e−(t + (Q) )L . Next, we apply [32, Lemma 2.1] using that the families {Tt }t>0 and {B Q,t }t>0 , satisfy respectively, L p0 (Rn ) − L q0 (Rn ) and L p0 (Rn )− L p0 (Rn ) off-diagonal estimates (see Sect. 2.4 and Proposition 4.2). Besides, note that (Q) ≤ tC M . Then, by Lemma 4.4 and changing the variable j+i t into 2 t (Q) , we get ⎛ Ii2j

 w(2

j+1

Q i )|2

j+1

Qi |

− qp

0

⎝



∞ (Q) CM

− qp 0

 C j (Q i )

% % %Tt B Q,t mi (x)%q0 dx

1

q0

⎞p dt ⎠ t

p

 w(2 j+1 Q i )|2 j+1 Q i | mi  L p0 (Rn )  p  i+ j 2 ∞  (Q)2  M dt −n p1 − q1 −c 4 (Q) 2 0 0 e t t (Q) t2 t C M

2

 r0 n n −i p(2M+ε) − j p 2M+ p0 − p

2

.

Hence, by this, (4.7), (4.8), (4.9), (4.10) and (4.5), we have 

1

∇ L − 2 B Q mi  L p (w) ≤ C. p

(4.11)

i≥1

k are Now, proceeding as in the estimate of Ii1 , since the Riesz transform and A Q q bounded on L (w) (see Theorem 2.2 and [5, Proposition 5.9]), and by (3.2), we get 1

kQ m i  L p (w)  w(4Q i ) p 14Q i ∇ L − 2 A 1

− q1

 mi  L q (w)  2−iε .

(4.12)

Next, for j ≥ 2, we use (2.21), and proceed as in the estimate of Ii j ,

123

C. Prisuelos-Arribas

kQ m i  L p (w) 1C j (Q i ) ∇ L − 2 A 1

p

 w(2 j+1 Q i )|2 j+1 Q i |

− qp

0

⎛  ⎝

(Q)



0

 +



∞

C j (Q i )

(Q)

%q0 % % % k i (x)% d x %Tt A Q m

=: w(2 j+1 Q i )|2 j+1 Q i |

− qp

0



C j (Q i )

1

q0

I Ii1j + I Ii2j

%q0 % % % k i (x)% d x %Tt A Q m

1

q0

dt t

⎞p dt ⎠ t

p

.

We first estimate I Ii1j , proceeding as in the estimate of Ii1j in (4.10). We apply [32, Lemma 2.1] with the families {t∇ y (t 2 L) M e−t L }t>0 and {e−t L }t>0 that satisfy, respectively, L p0 (Rn ) − L q0 (Rn ) and L p0 (Rn ) − L p0 (Rn ) off-diagonal estimates (see Sect. 2.4). Then, by Lemma 4.4, we obtain 2

I Ii1j

ck,M = (Q)  e−c4



j+i

(Q) 0

(Q)



2

% 1 %q0 % 2 % kQ e−t 2 L m i (x)% dx %k (Q)∇ y A

C j (Q i )  −n p1 − q1 0

0

 mi  L p0 (Rn )  w(Q i )

− 1p

1

n

q0

dt

(Q) q0 e−c4

i+ j

.

(4.13)

√ k Now consider s Q,t := k (Q)2 + t 2 . Then changing the variable t into 2t, and proceeding as in the estimate of Ii2j but applying this time [32, Lemma 2.1] with the families {Tt }t>0 and {(t 2 L) M e−t L }t>0 that satisfy, respectively, L p0 (Rn ) − L q0 (Rn ) and L p0 (Rn ) − L p0 (Rn ) off-diagonal estimates, we have that 2

 I Ii2j



∞



(Q) √ 2

(Q)2 t2 

  mi  L p0 (Rn ) 2

%  %q0  q10 M % % k L dt −s k %Tt s L i (x)%% dx e Q,t m Q,t % t C j (Q i )   M j+i 2 (Q)2 dt −n p1 − q1 −c 4 (Q) 0 0 e t2 t 2 t t

 M 

∞ (Q) √ 2

  − j 2M+n p1 − q1 0

0

n

(Q) q0 w(Q i )

 n − 1p −i 2M− q0 +ε

2

.

Therefore, from this inequality and (4.13), we have that 1C j (Q i ) ∇ L

− 21

i+ j − jp p kQ m i  L p (w)  e−c4 + 2 A



2M+ pn − 0

r0 n p



2−i p(2M+ε) .

 1 p k m This, (4.12) and (4.5) give us i≥1 ∇ L − 2 A Q i  L p (w) ≤ C, which together with (4.11) and in view of (4.6), allows us to conclude the proof.  

123

Weighted Hardy Spaces Associated with Elliptic Operators

5 Interpolation Results The aim of this section is to prove the following interpolation result for Hardy spaces. Theorem 5.1 Given w ∈ A∞ and q ∈ Ww ( p− (L), p+ (L)), suppose 1 ≤ p0 < 2,∗ θ and 1p = 1−θ p1 < p+ (L) sw p0 + p1 , 0 < θ < 1. Then &

p

p

'

HSH0 ,q (w), HSH1 ,q (w)

θ

p

= HSH ,q (w).

We denote by [ , ]θ the complex interpolation method described in [15] (see Sect. 2.6). As we explained in the introduction we obtain this result from the corresponding one for the weighted tent spaces defined in (2.7) , (a real interpolation result involving weighted tent spaces was proved in [16]). Theorem 5.2 Suppose 1 ≤ p0 < p1 < ∞ and

1 p

=

1−θ p0

+

θ p1 ,

0 < θ < 1. Then

[T p0 (w), T p1 (w)]θ = T p (w). Remark 5.3 In the proof of the inclusion T p ⊂ [T p0 , T p1 ]θ , in [18, Lemma 5, Sect. 7] (following the notation there) the authors claim that the support of the function A( f k ) is contained in Ok∗ \ Ok+1 . It is easy to see that supp A( f k ) ⊂ Ok∗ , but it is not clear that the support of A( f k ) is away from Ok+1 . In fact, we can construct 1-dimensional examples which show that this is false in general. This was noticed by A. Amenta in [1, Remark 3.20]. We refer to [12] and [24] for a different proof of that inclusion and hence, of [18, Theorem 4, Sect. 7]. Here, we give a prove of Theorem 5.2 based on the proof of [18, Lemma 4 , Sect. 7] to show that [T p0 (w), T p1 (w)]θ ⊂ T p (w), and on the proof of [18, Lemma 5 , Sect. 7] to show that T p (w) ⊂ [T p0 (w), T p1 (w)]θ . However, in view of Remark 5.3, in order to show this last inclusion, we need to complete and slightly modify the proof given in [18, Lemma 5, Sect. 7]. 5.1 Tent Spaces Interpolation: Proof of Theorem 5.2 As we said a few lines above, following the proof of [18, Lemma 4 , Sect. 7], but using interpolation between L p (w) spaces (note that the proof in [11, Theorem 5.1.1] also works in the weighted case), instead of the usual interpolation in L p (Rn ) spaces, we get [T p0 (w), T p1 (w)]θ ⊂ T p (w), 1 ≤ p0 < p < p1 < ∞, 1 1−θ θ = + , 0 < θ < 1. p p0 p1

123

C. Prisuelos-Arribas

As for the converse inclusion, fix 1 ≤ p0 < p < p1 < ∞ and 0 < θ < 1 such θ p p that 1p = 1−θ p0 + p1 , and take a function f ∈ T (w) such that  f T (w) ≤ 1. Then, we need to find a function F, z → F(z), from the closed strip 0 ≤ Re(z) ≤ 1 to the Banach space T p0 (w)+T p1 (w) (see Sect. 2.6 for definitions). The function F must be continuous and bounded on the full strip, with respect to the norm of T p0 (w)+T p1 (w), and analytic on the open strip, and such that F(i y) ∈ T p0 (w) is continuous in T p0 (w) and tends to zero as |y| → ∞, and F(1 + i y) ∈ T p1 (w) is continuous in T p1 (w) and tends to zero as |y| → ∞. Besides, F must satisfy that F(θ ) = f in T p (w), and F(i y) L p0 (w) + F(1 + i y) L p1 (w) ≤ C, uniformly on f . To this end fix α > 1, to be determined during the proof, and, for each k ∈ Z, consider the sets Ok := {x ∈ Rn: Aα ( f )(x) > 2k }, E k := Rn \ Ok ,and, k ∩B(x,r )| for some fixed γ , 0 < γ < 1, the set E k∗ := x ∈ Rn : ∀ r > 0, |E|B(x,r )| ≥ γ and ∗ ∗ n n its complement Ok := R \ E k = {x ∈ R : M(1 Ok )(x) > 1 − γ }, where M is the centred Hardy–Littlewood maximal operator over balls. Besides, for every k ∈ Z, (∗ := {(y, t) ∈ Rn+1 : d(y, Rn \ we can consider the “tent” over Ok∗ , define by O + k ) (∗ . Additionally, note that Ok∗ ) ≥ t}. Note that R(E k∗ ) := x∈E ∗ (x) = Rn+1 \ O + k k ) ) ∞ dy dt n+1 ∗ (∗ \ O  F, where F ⊂ R and O supp f ⊂ + k∈Z k k+1 0 Rn 1F (y, t) t n+1 = 0. This follows proceeding as in the proof 5.1, part (a)]. Then, we  of [33, Proposition  p (w). =: f in T can write f = k∈Z f 1 O(∗ \ O k ∗  k∈Z k k+1 2 2  k(α(z) p−1) f , where α(z) := Now, consider the function F(z) := e z −θ k k∈Z 2 1−z z + . We shall see that F satisfies all the conditions that we mentioned above. p0 p1 First note that F(θ ) = f in T p (w). Moreover, for all z ∈ C such that 0 < Rez < 1, applying Young’s inequality, we have that % % % % 2 2  k  p 1−Re(z) + p Re(z) −1 % z −θ k(α(z) p−1) % p0 p1 |F(z)| = %e 2 fk % ≤ e 2 | fk | % % k∈Z k∈Z 1−Re(z)   p Re(z)   k  p −1 k −1 2 p0 2 p1 =e | fk | | fk | k∈Z

≤ e(1 − Re(z)) ≤



 k

p p0 −1





2 | f k | + eRe(z) 2 k∈Z k∈Z  k  p −1  k  p −1 p0 e 2 | fk | + e 2 p1 | f k |. k∈Z k∈Z

 k

p p1 −1



| fk | (5.1)

Besides, for all −∞ < y < ∞, |F(i y)| ≤ e

−y 2

 k∈Z



2

k

p p0 −1



| f k | and |F(1 + i y)| ≤ e

1−y 2





2

k

p p1 −1



| f k |.

k∈Z

(5.2)

123

Weighted Hardy Spaces Associated with Elliptic Operators

Then, in order to see that F satisfies the desired conditions, it suffices to show that     p  k p −1   2 0 | f k |    k∈Z

T p0 (w)

    p  k p −1   + 2 1 | f k |   k∈Z

≤ C.

(5.3)

T p1 (w)

Indeed, combining this with (5.2), we obtain that F(i y) ∈ T p0 (w) is continuous in T p0 (w) and tends to zero as |y| → ∞, and F(1 + i y) ∈ T p1 (w) is continuous in T p1 (w) and tends to zero as |y| → ∞. On the other hand, by (5.1), (5.2) and (5.3), we easily obtain that F is a continuous and bounded function with respect to the norm of T p0 (w) + T p1 (w) on the full strip. Finally, to see that F is analytic on the open strip we apply Morera’s theorem for Banach space valued functions. We have that F(z) is continuous, so it just remains to show that for all triangle T in the open set  := {z ∈ C : 0 < Rez < 1}, we have that C T F(z) = 0. To see this, consider for 2 −θ 2 k( pα(z)−1) z each k ∈ Z gk (z) := e 2 f k , we have that these functions are  analytic on  Then, for all T triangle in C  and each k ∈ Z, by Cauchy’s theorem, gk (z) = 0. C. T Hence, it suffices to justify that we can take the sum in k ∈ Z out of the integral. This follows by the dominated convergence theorem for Bochner integrals. Note that  F(z) = T

3  

tj

j=1 0

F(γ j (t))γ j (t)dt,

where γ j is a parametrisation of each side of the triangle T , and that, by (5.1) and (5.3), for j = 1, 2, 3, 

tj

0

F(γ j (t))γ j (t)T p0 (w)+T p1 (w) dt



tj

 0

F(γ j (t))T p0 (w)+T p1 (w) dt < ∞.

Consequently the function t → F(γ j (t))γ j (t), for all t ∈ [0, t j ] is Bochner integrable. Moreover, for all M > 0 and j = 1, 2, 3, again by (5.1) and (5.3),       2 2 γ j (t) −θ k(α(γ j (t)) p−1)   e 2 f k γ j (t)    |k|≤M

 C, T

p0 (w)+T p1 (w)

which implies that we can apply the dominated convergence theorem for Bochner integrals and then conclude that F is analytic. Besides (5.2), (5.3), and the fact that F(θ ) = f in T p (w) with  f T p (w) ≤ 1, imply that f ∈ [T p0 (w), T p1 (w)]θ and that  f [T p0 (w),T p1 (w)]θ   f T p (w) . Thus, let us prove (5.3). Let q be p0 or p1 , then, ∗ ⊂ Ok∗ , we have since supp(A f k ) ⊂ Ok∗ and Ok+1 ⊂ Ok+1     p  k q −1   2 | f k |    k∈Z

≤ T q (w)

ψ

sup

≤1  L q (w)

 k∈Z



2

k

p q −1

 Ok∗

|A f k (x)||ψ(x)|w(x)dx

123

C. Prisuelos-Arribas

≤



sup

ψ

≤1  L q (w)

+

sup

=:

ψ

ψ



2

k

p q −1

k∈Z



≤1  L q (w)

sup

≤1  L q (w)

2



Ok∗ \Ok+1   k qp −1

ψ

|A f k (x)||ψ(x)|w(x)dx

Ok+1

k∈Z

I+

|A f k (x)||ψ(x)|w(x)dx

sup

≤1  L q (w)

I I.

Using that for α > 1, A f k ≤ Aα f k , and that for x ∈ Ok∗ \ Ok+1 , Aα f k (x) ≤ 2k+1 , kp  we obtain I  2 q O ∗ |ψ(x)|w(x)dx. Next we see that I I is controlled by the same k expression in the right-hand side of the previous inequality. For every k ∈ Z, since M : L r (w) → L r,∞ (w), for all r > rw , we have that w(Ok∗ )  w(Ok ) < ∞. Then, ) j ∗ : O∗ we can take a Whitney decomposition of Ok+1 j∈N Q k+1 , which satisfies k+1 = √

√ j j j ∗ n (Q k+1 ) ≤ d(Q k+1 , Rn \ Ok+1 ) ≤ 4 n (Q k+1 ),

j

j

and the Q k+1 have disjoint interiors. Now, for every x ∈ Q k+1 , we split A f k (x) as follows 

(Q k+1 )/2  j

A f k (x) ≤ 0

 +

dy dt | f k (y, t)|2 n+1 t B(x,t)



∞ j

(Q k+1 )/2

B(x,t)

 21

dy dt | f k (y, t)|2 n+1 t

1 2

=: G 1 (x) + G 2 (x).

On one hand, note that E

*

*  j j ∗ ∗ (∗ \ O (∗ \ O   O O k k+1 := (y, t) ∈ (x) : x ∈ Q k+1 , 0 < t < (Q k+1 )/2 k k+1 = ∅. j

Indeed for (y, t) ∈ E, since d(y, Q k+1 ) ≤ t, we have ∗ ∗ d(y, Rn \ Ok+1 ) ≥ d(Q k+1 , Rn \ Ok+1 ) − d(y, Q k+1 ) √ √ j ≥ n (Q k+1 ) − t > (2 n − 1)t ≥ t, j

j

j ∗  which implies that (y, t) ∈ O k+1 . Hence G 1 (x) = 0, for all x ∈ Q k+1 .

√ j j ∗ such that d(x j , Q k+1 ) ≤ 4 n (Q k+1 ), On the other hand, take x j ∈ Rn \ Ok+1 j

j

and note that if (Q k+1 )/2 ≤ t < ∞ and x ∈ Q k+1 , then B(x, t) ⊂ B(x j , αt), for √ α ≥ 11 n. Indeed, for x0 ∈ B(x, t), we have

123

Weighted Hardy Spaces Associated with Elliptic Operators

√ √ j j |x0 − x j | ≤ |x0 − x| + |x − x j | < t + n (Q k+1 ) + 4 n (Q k+1 ) √ √ √ ≤ t (1 + 2 n + 8 n) ≤ 11 nt. ∗ Hence, G 2 (x) ≤ Aα f k (x j ) ≤ 2k+1 , ∀ x ∈ Q k+1 . Therefore, since Ok+1 ⊂ Ok+1 ⊂ ∗ Ok j



II ≤ 2

k

p q −1





|A f k (x)||ψ(x)|w(x)dx

j

j∈N Q k+1



≤2

k

p q −1



  

j

2

k qp



j

|G 1 (x)||ψ(x)|w(x)dx +

Q k+1

j∈N

|ψ(x)|w(x)dx = 2

j∈N Q k+1

≤2

k qp



Ok∗





k qp

j

|G 2 (x)||ψ(x)|w(x)dx

Q k+1

 ∗ Ok+1

|ψ(x)|w(x)dx

|ψ(x)|w(x)dx.

Now consider, respectively, Md and Mc the dyadic maximal function and the centred maximal function over cubes. For some dimensional constant cn , we have that 1−γ ,k := M(1 Ok )(x) ≤ cn Mc (1 Ok )(x).Next, for each k ∈ Z, we define the set O  x ∈ Rn : Md (1 Ok )(x) >

1−γ 4 n cn

1−γ  4n cn : O1−γ ,k

of this set at height dyadic cubes such that

; and we take a Calderón–Zygmund decomposition ) l }l∈N is a collection of disjoints l , where { Q = l∈N Q k k

 − 1 Ok (x)d x ≈ 1 − γ . l Q k

Then, since   1−γ lk , Ok∗ ⊂ x ∈ Rn : Mc (1 Ok )(x) > ⊂ 2Q cn l∈N

(see [20, proof of Lemma 2.12]), for r > rw , we have that    1 |ψ(x)|w(x)d x ≤ (1 − γ )r |ψ(x)|w(x)dx r l ∗ (1 − γ )  Ok 2 Qk l∈N r    1 − 1 Ok (y)dy |ψ(x)|w(x)dx ≈ (1 − γ )r l l Q 2Q k k l∈N

1 ≤ (1 − γ )r  l∈N

l 2Q k

  r −1  1−r  − w − 1 Ok (y)w(y)dy (y)dy |ψ(x)|w(x)dx l Q k

l Q k

123

C. Prisuelos-Arribas

   1 lk )−1 |ψ(x)|w(x)dx  1 Ok (y)w(y)dy w( Q (1 − γ )r l l Q 2 Q k k l∈N  1  1 Ok (y)Mw (ψ)(y)w(y)dy (1 − γ )r l Q k l∈N  1 ≤ Mw (ψ)(y)w(y)dy, (1 − γ )r Ok

(5.4)

where Mw f (x) := sup Qx

1 w(Q)

 | f (y)|w(y)dy.

(5.5)

Q 



Moreover, by (2.2) and since 1 < q  ≤ ∞, Mw : L q (w) → L q (w). Therefore, by the estimates obtained for I and I I , by (5.4), and by (2.8), we conclude     p  k q −1   2 | f k |    k∈Z T q (w)  kp   sup 2 q Mw (ψ)(x)w(x)dx ψ

 = ≈

ψ

ψ

ψ

≤1  L q (w)

sup

≤1  L q (w)



sup

≤1  L q (w)

2k

p

λq

k−1 k∈Z 2 ∞ p 



λq

sup

≤1  L q (w)

Ok

k∈Z

Rn

{x∈Rn :Aα f (x)>λ}

{x∈Rn :Aα

0





f (x)>λ}

Mw (ψ)(x)w(x)dx

ψ

sup

≤1  L q (w)

dλ λ

dλ λ

p q

|Aα f (x)| Mw (ψ)(x)w(x)dx p



Mw (ψ)(x)w(x)dx

p

p

Aα f  Lq p (w) ψ L q  (w)  A f  Lq p (w) =  f Tq p (w) ≤ 1.  

5.2 Hardy Spaces Interpolation: Proof of Theorem 5.1 As we explained above, in view of Theorem 5.2, it is enough to show that the Hardy spaces are retracts of the weighted tent spaces, i.e. that there exists an operator from any tent space to the corresponding Hardy space having a right inverse. Fix q ∈ Ww ( p− (L), p+ (L)). Note that for a function f ∈ L q (w) and m ∈ N, we have the following Calderón reproducing formula of f ,  f = Cm 0

123

∞

(t 2 L)m e−t

2L

2

f

dt in L q (w), t

(5.6)

Weighted Hardy Spaces Associated with Elliptic Operators

where Cm is a positive constant and the equality is in L q (w). Remark 5.4 A priori, by L 2 (Rn ) functional calculus we have the above equality for functions in L 2 (Rn ). But, as in Remark 3.8, for q ∈ Ww ( p− (L), p+ (L)), we consider the infinitesimal generator L q,w of the e−t L on L q (w) (see [6, Remark 3.5]). Hence, by abuse of notation, we have the above Calderón reproducing formula for functions in L q (w), understanding that in L q (w) \ L 2 (Rn ), L denotes L q,w . 2 m −t L f (y) and the Besides, if we define for each (y, t) ∈ Rn+1 + , F(y, t) := (t L) e operator Q L ,m f := F acting over functions in L q (w), by [32, Theorem 1.12, part (b)], 2

Q L ,m f T p (w) = Sm,H f  L p (w) ≤ SH f  L p (w) , p

then Q L ,m is bounded from HSH ,q (w) to T p (w), for all q ∈ Ww ( p− (L), p+ (L)) and p 1 ≤ p < ∞. Thus, by the definition of HSH ,q (w), it can be extended to a bounded p operator, denoted by Q L ,m from HSH ,q (w) to T p (w). Similarly if we consider Q L ∗ ,m , 



2 ∗

defined for all functions f ∈ L q (w 1−q ) by Q L ∗ ,m f (y, t) := (t 2 L ∗ )m e−t L f (y), for all (y, t) ∈ Rn+1 + . Again by [32, Theorem 1.12, part (b)], by [4, Lemma 4.4] and     since p± (L ∗ ) = p∓ (L) , see [2], we have that Q L ∗ ,m : L q (w 1−q ) → T q (w 1−q )  ∗ ∗ 2 n for all q ∈ Ww1−q  ( p− (L ), p+ (L )). Moreover, for all F ∈ T (R ), its adjoint operator (Q L ∗ ,m )∗ , has the following representation  ∞ dt 2 (Q L ∗ ,m )∗ F(y) = (5.7) (t 2 L)m e−t L F(y, t) . t 0 Then since for all F ∈ T 2 (Rn ) and g ∈ L 2 (Rn ), %  ∞ % % ∞ %  % % % dt dy %% 2 L)m e−t 2 L F(y, t) dt % = % 2 L ∗ )m e−t 2 L ∗ g(y) % (t F(y, t)(t dx % n % t % t n+1 % R 0 B(y,t) 0  2 ∗ ≤ |||F|||(x) |||(t 2 L ∗ )m e−t L g|||(x) dx, Rn

where |||F|||(x) =



2 dy dt (x) |F(y, t)| t n+1

1 2

. By a density argument, we conclude L ,m , from T q (w) to L q (w), that (Q L ∗ ,m )∗ has a bounded extension, denoted by Q for all q ∈ Ww ( p− (L), p+ (L)). If we replace L by L q,w , this extension satisfies the equality (5.7) for functions on T q (w). But, abusing the notation (see Remarks 3.8 and 5.4), we just write L. Besides we shall show that for all functions F ∈ T p (w) ∩ T q (w) and m ∈ N big enough, 2,∗ L ,m F L p (w)  FT p (w) , for all 1 ≤ p < p+ (L) , SH Q sw

(5.8)

where p+ (L)2,∗ is defined in (2.11). Assuming this, since T p (w) ∩ T q (w) is dense  L ,m , L ,m| p can be extended to a bounded operator, denoted by Q in T p (w), Q T (w)

123

C. Prisuelos-Arribas p  L ,m ◦ Q L ,m = I in from T p (w) to HSH ,q (w). Then, by (5.6), we have that Cm Q

HSH ,q (w), and by density in HSH ,q (w). Hence, for w ∈ A∞ , 1 ≤ p < p+ (L) and sw p q ∈ Ww ( p− (L), p+ (L)), the Hardy spaces HSH ,q (w) are retracts of the tent spaces T p (w). Therefore, to finish the proof it just remains to show (5.8). Applying Minkowski’s integral inequality we obtain p

2,∗

p

L ,m F L p (w) S H Q ⎛ ⎛  ∞  t   ⎜ ⎝ ≤⎝ Rn

⎛ ⎜ +⎝

0

⎛



⎝

0



Rn



∞ 0

∞

|t 2 Le−t

2L

(s 2 L)m e−s

2L

1 2

F(y, s)|2 dy

B(x,t)



t

|t 2 Le−t

2L

(s 2 L)m e−s

2L

ds s 1

F(y, s)|2 dy

2

B(x,t)

2

ds s

⎞ 1p

⎞ 2p

dt ⎠ ⎟ w(x)dx ⎠ t n+1 2

⎞ 1p

⎞ 2p

dt ⎠ ⎟ w(x)dx ⎠ t n+1

=: I + I I.

We first show by extrapolation that I I  FT p (w) , for every p as in (5.8) and every m ∈ N. To this end, in view of Theorem 2.3, part (a), (or part (b) if p+ (L)2,∗ = ∞) it is enough to consider the case p = 2 and w ∈ R H p (L)2,∗  . That is, to prove that, +

for every w ∈ R H p

2,∗ + (L) 2

⎛  II := ⎝

 Rn

dt

t

q0 2

0

∞





∞ 

|t Le 2

−t 2 L

m −s 2 L

(s L) e 2

1 2

F(y, s)| dy

B(x,t)

t

w(x)dx n+1

2

and m ∈ N,

2

ds s

2

1 2

 FT 2 (w) .

Under this assumption, note that we can find q0 and r so that 2 < q0 < p+ (L), ≤ r < ∞, w ∈ R Hr  , and 2+

n n > 0. − 2r q0

Indeed, if n > 2 p+ (L), since w ∈ R H p

+ (L) 2

2,∗

 ,

(5.9) we have that sw <

np+ (L) 2(n−2 p+ (L)) .

Therefore, there exist ε1 > 0 small enough and 2 < q0 < p+ (L) close enough to p+ (L) so that sw <

123

nq0 . 2(1 + ε1 )(n − 2q0 )

Weighted Hardy Spaces Associated with Elliptic Operators

Besides, there exists ε2 > 0 so that q0 <

nq0 . (1 + ε2 )(n − 2q0 )

0 Hence, taking ε0 := min{ε1 , ε2 } and r := 2(1+ε0nq )(n−2q0 ) , we have that 2 < q0 < p+ (L), q0 /2 ≤ r < ∞, and w ∈ R Hr  . Moreover

n n − = ε0 2+ 2r q0



   n n − 2 > 0. − 2 > ε0 q0 p+ (L)

If now we consider n ≤ 2 p+ (L), we have that p+ (L)2,∗ = ∞. Then, note that the assumption w ∈ R H p (L)2,∗  becomes w ∈ A∞ . Hence, we fix r > sw , and q0 +



2r p+ (L) p+ (L)+2r

2



< q0 < min { p+ (L), 2r } if p+ (L) < ∞, and q0 = 2r satisfying max 2, if p+ (L) = ∞. Therefore, we have that 2 < q0 < p+ (L), q0 /2 ≤ r < ∞, and w ∈ R Hr  . Besides, 2+

n n n − ≥ 0. >2− 2r q0 p+ (L)

Keeping these choices in mind, we apply the L 2 (Rn ) − L 2 (Rn ) off-diagonal 2 estimates satisfied by {e−t L }t>0 , change the variable s into st, and apply Jensen’s inequality and Minkowski’s integral inequality. Then, we have II 



e−c4

j

j≥1

⎛  ⎝ 





∞

Rn

0

e−c4

j



∞

t2 s2

t



m+1 −s 2 L

|(s L) 2

B(x,2 j+1 t)

e

1 2

2

F(y, s)| dy

ds s

2

dt t n+1

⎞ 21 w(x)d x ⎠

j≥1



∞

s

−2

 Rn

1

=:

 j≥1

e−c4

j



∞ 1

∞ 



B(x,2 j+1 t)

0

s −2

e

1

 Rn

m+1 −(st)2 L

|((st) L) 2

J (x, s)2 w(x)dx

2

dy F(y, st)| n t q0



2 q0

dt w(x)dx t

 21

ds s

ds . s

Note now that, applying Fubini’s theorem, [32, Proposition 3.30], and changing the variable t into t/s; and next, applying the L 2 (Rn ) − L q0 (Rn ) off-diagonal estimates 2 satisfied by the family {(t 2 L)m+1 e−t L }t>0 and [32, Proposition 3.2], taking r0 > rw , and also recalling our choices of q0 and r , we obtain that, for every s > 1,

123

C. Prisuelos-Arribas

 Rn

J (x, s)2 w(x)dx

2 2 dy q0 dt |((st)2 L)m+1 e−(st) L F(y, st)|q0 n w(x)d x n j+1 t t R B(x,2 st) 0 2   ∞  n 2n 2 dy q0 dt − + w(x)dx |(t 2 L)m+1 e−t L F(y, t)|q0 n ≈ s r q0 n j+1 t t R 0 B(x,2 t)    ∞ dy dt − n + 2n  −c4l  s r q0 e |F(y, t)|2 n+1 w(x)dx t Rn 0 B(x,2 j+l+2 t) n

 s− r



 ∞

l≥1

 2 jnr0



e−c4 s l

− nr + q2n 0

l≥1

F2T 2 (w) .

Hence, by (5.9), we have II 



e−c4

j



∞

s

ds FT 2 (w)  FT 2 (w) s

n −2− 2r + qn

0

1

j≥1

which, as we observed above, implies that I I  FT p (w) , for all 1 ≤ p < p+ (L) sw and all m ∈ N. Next, in order to estimate I we apply the L 2 (Rn ) − L 2 (Rn ) off-diagonal estimates 2 2 satisfied by {(t 2 L)m+1 e−t L }t>0 and {e−s L }s>0 , and [26, Lemma 2.3] recalling that, in this case, s < t. Then, 2,∗

 |t Le 2

B(x,t)



=  

s2 t2

−t 2 L

m −s 2 L

(s L) e 2

m  |(t L) 2

B(x,t)

2 m 

s t2

e−c4

j

 21

2

F(y, s)| dy

m+1 −t 2 L −s 2 L

e

e

1 2



j≥1

2

F(y, s)| dy 1

B(x,2 j+1 t)

|F(y, s)|2 dy

2

.

Using this, changing the variable s into st, applying Minkowski’s integral inequality twice, changing the variable t into t/s, applying [32, Proposition 3.2] and taking 2m > nrp0 − n2 , where r0 > max{ p/2, rw }, we obtain that I 



e−c4

j

j≥1

⎛  ⎜ ⎝

⎛ ⎝



Rn

123

0

∞



1 0



1

s 2m B(x,2 j+1 t)

|F(y, st)|2 dy

2

ds s

2

⎞ 2p

⎞ 1p

dt ⎠ ⎟ w(x)d x ⎠

t n+1

Weighted Hardy Spaces Associated with Elliptic Operators





e

−c4 j

=

e

−c4 j



s 

1

s



2

j

nr0 p

e−c4

j

2m+ n2

Rn

0



 Rn



1

s

B(x,2 j+1 t)

nr0 p

dy dt |F(y, st)|2 n+1 t

∞

0

2m+ n2 −

0

j≥1

∞

2m

0

j≥1





1 0

j≥1





B(x,2 j+1 t/s)

 1p

p 2

dy dt |F(y, t)|2 n+1 t

w(x)dx

 1p

p 2

ds s

w(x)dx

ds s

ds FT p (w)  FT p (w) , s  

which finishes the proof.

6 Boundedness Improvement for Conical Square Functions In [6] the authors observed that if L is a real operator and w ∈ A∞ such that p− (L),  p+ (L)) Ww ( p− (L), p+ (L)) = ∅, then Ww ( p− (L), p+ (L)) = IntJw (L) =: ( (see page 5 for definitions). However, in the case of complex operators we do not know whether Jw (L) and Ww ( p− (L), p+ (L)) have different end-points. Therefore, motivated by getting the expected lower exponent in Proposition 9.1 in the case of complex operators (see for instance [6, Theorem 6.2]), we improve the range of boundedness, obtained in [32, Theorem 1.12], of SH in the case that w ∈ A∞ with Ww ( p− (L), p+ (L)) = ∅. Theorem 6.1 Given w ∈ A∞ such that Ww ( p− (L), p+ (L)) = ∅, for all p ∈ ( p− (L), ∞), there hold: (a) SH is bounded on L p (w). (b) Given m ∈ N, Sm,H , Gm,H and Gm,H are bounded on L p (w). If we further assume that Ww (q− (L), q+ (L)) = ∅ we have that, for p ∈ ( p− (L), ∞), GH and GH are bounded on L p (w). Proof Note that GH ≤ 2SH + GH , then by [32, Theorem 1.14], we just need to prove the theorem for SH and GH . Let Q be SH or GH . By [6, Theorem 2.4], to conclude our result, it is enough to prove that for every ball B = B(x B , r B ) ⊂ Rn 1

 −

p

|QBr B f (x)| dw p

C j (B)

 1 p p  g( j) − | f (x)| dw , for all j ≥ 2,

(6.1)

B

and  −

C j (B)

1 q

|Ar B f (x)| dw q

 1 p  g( j) − | f (x)| p dw , for all j ≥ 1,

(6.2)

B

123

C. Prisuelos-Arribas

where Ar B := I − (I − e−r B L ) M and Br B := I − Ar B , for some M ∈ N arbitrarily n large, q is such that on L q (w), f ∈ L ∞ c (R ) such that supp f ⊂ B and  Q is bounded nr g( j) is such that j≥1 g( j)2 , for some r > rw . We start by taking q ∈ Ww ( p− (L), p+ (L)) when Q = SH and q ∈ Ww (q− (L), q+ (L)) when Q = GH , by [32, Theorem 1.12], we know that, in any case, Q is bounded on L q (w). Besides, also by that result, we only need to consider the case  p− (L) < p ≤ rw p− (L) (we recall that p− (L) = q− (L)). Next, we fix p0 so that p− (L) < p0 < min{2, q} and w ∈ Aq/ p0 . The proof of (6.2) follows by expanding the binomial and using that, for 1 ≤ k ≤ M, √ − e kr B L satisfies L p (w) − L q (w) off-diagonal estimates on balls, (see [5,6]). As for (6.1), first note that it is enough to prove 2

 I := −



C j (B)

∞ B(x,t)

0

for Tt being t 2 Le−t  I ≤ −

2L

2

dw

 1 p p  g( j) − | f (x)| dw , B

2

rB



0

 + −

 1p

p

or t∇ y e−t L . Splitting the integral in t we have that



C j (B)

|Tt Br B

dy dt f (y)|2 n+1 t

B(x,t)



C j (B)

|Tt Br B

∞

rB

B(x,t)

dy dt f (y)|2 n+1 t

|Tt Br B

 1p

p 2

dw

dy dt f (y)|2 n+1 t

 1p

p 2

=: I1 + I2 . (6.3)

dw

In order to estimate I2 , consider Br B ,t := (e−t L −e−(t +r B )L ) M . Then, changing the √ variable t into t M + 1 =: tC M and applying that {Tt }t>0 satisfies L p0 (Rn )−L 2 (Rn ) off-diagonal estimates (see Sect. 2.4), we have 2

2

⎛  ⎝ I2  −



C j (B)





e

−c4i

∞ rB CM

 B(x,tC M )

⎛  ⎝−



rB CM

C j (B)

i≥1

∞

|Tt Br B ,t

dy dt f (y)|2 n+1 t

 B(x,2i+1 tC M )

|Br B ,t

p 2

2

⎞ 1p dw ⎠

dy f (y)| p0 n t



2 p0

dt t

 2p

⎞ 1p dw ⎠ .

Besides, since w ∈ A q , note that we have the following estimate for the integral p0 in y:  B(x,2i+1 C M t) i2n

 2 p0

123

dy |Br B ,t f (y)| p0 n t



2 p0



B(x,2i+1 C M t)

2 |Br B ,t f (y)|q w(y)dy

q

Weighted Hardy Spaces Associated with Elliptic Operators

 −

w(y)

B(x,2i+1 tC M )

2

  1− pq

 −

i2n p0

0

 q2



q p0 −1



(2i t)

dy

− 2n q

2 |Br B ,t f (y)| dw q

B(x,2i+1 C M t)

q

.

(6.4)

By (6.4), we can split I2 as follows:

I2 

j−2 

e

−c4i

2

⎛  ⎝−

in p0

+e



e

2

j−2 

e−c4 I2i1 + e−c4 i

j



i=1

B(x,2i+1 tC M )



∞ rB CM

C j (B)

i≥ j−1

=:

2

 −

⎛  ⎝−

in p0

−c4i

∞ rB CM

C j (B)

i=1

−c4 j



|Br B ,t f (y)|q dw

q

dt t

⎞ 1p

 2p

dw ⎠ 2

 −

B(x,2i+1 tC M )

|Br B ,t f (y)|q dw

q

dt t

⎞ 1p

 2p

dw ⎠

e−c4 I2i2 . i

i≥ j−1

 j−2 i The sum i=1 e−c4 I2i1 only appears when j ≥ 3. In this case, we split the integral −1 −1 j−i−2 in t and observe that for x ∈ C j (B), and r B C M < t < rBCM 2 , we have that −1 j−i−2 2 , B(x, 2i+1 C M t) ⊂ 2 j+2 B \ 2 j−1 B; besides, for 1 ≤ i ≤ j − 2 and t ≥ r B C M B, B(x, 2i+1 C M t) ⊂ B(x B , 2 j+2 C M t). Then, applying (2.2), Proposition 4.1 and the fact that tC M ≥ r B , we obtain

I2i1  2

jn p0

⎞1 ⎛ ⎛ ⎞ 2 1 2  2 j−i−2 r B  2n j+1   q CM r B p0 ⎝ dt ⎟ ⎜ q − |Br B ,t f (y)| dw ⎠ ⎠ ⎝ r B t t Cl (B) C l= j−1

M

+2 



2

j

jn p0



n p0 +θ2



∞ 2 j−i−2 r B CM

 −

2

B(x B ,2 j+2 C M t)

⎛

⎛

 f  L p (w) ⎜ jθ1 ⎝ ⎝2 1 w(B) p  +

∞ 2 j−i−2 r B CM

2

2 j−i−2 r B CM rB CM

 r 4M dt B t t

  n i2M − j 2M− p0 −θ1 −θ2

2



|Br B ,t ( f 1 B(x B ,2 j+2 C M t) )(y)|q dw  r 2n +4M+2θ2 B

t

p0

e

4 j r2 −c 2B t

q

dt t

 21

⎞ 21 dt ⎠ t

1 ⎞ 2 ⎠ 1

− | f (y)| dw p

p

.

B

The estimate of I2i2 follows applying Proposition 4.1 and the fact that for x ∈ C j (B), j ≥ 2, i ≥ j − 1 and tC M ≥ r B , we have that B, B(x, 2i+1 C M t) ⊂ B(x B , 2i+3 C M t)

123

C. Prisuelos-Arribas

I2i2

2



in p0

∞ rB CM

 −

2

B(x B ,2i+3 C M t)



q

|Br B ,t ( f 1 B(x B ,2i+3 C M t) )(y)| dw q

dt t

 21

1 1  r 4M dt 2  p B p − | f (y)| dw 2 rB t t B CM 1    p i n +θ  2 p0 2 − | f (y)| p dw . 

i

n p0 +θ2

∞

B

Therefore, for all j ≥ 2, we have     1 p − j 2M− pn −θ1 −θ2 −c4 j p 0 − | f (y)| dw I2  2 +e .

(6.5)

B

In order to estimate I1 , we expand the binomial. Then,  I1 ≤ −



C j (B)

+



rB 0

dy dt |Tt f (y)|2 n+1 t B(x,t)

 Ck,M −

M 



C j (B)

k=1

=: I11 +

M 

rB 0

 |Tt e

 1p

p 2

dw

−kr B2 L

B(x,t)

 1p

p

dy dt f (y)|2 n+1 t

2

dw

Ck,M Ik .

(6.6)

k=1 t2

We first estimate I11 , noticing that Tt = cTt/√2 e− 2 L , and applying the L p0 (Rn ) − 2 L (Rn ) off-diagonal estimates satisfied by Tt/√2 , we have

I11 



⎛  i ⎜ e−c4 ⎝−

 r B 

⎛  i ⎜ e−c4 ⎝−

 r B 

C j (B)

i≥1



j−2 

C j (B)

i=1

j + e−c4



j−2 

⎛  i ⎜ e−c4 ⎝−

e−c4 I Ii + e−c4

i=1

123

i

B(x,2i+1 t)

0

C j (B)

i≥ j−1

=:

B(x,2i+1 t)

0

j

 i≥ j−1



t2 |e− 2

t2 |e− 2

L f (y)| p0 dy n

L f (y)| p0 dy n

B(x,2i+1 t)

e−c4 I I Ii , i

2 p0

t

r B 

0



t

t2 |e− 2



2 p0

dt t

dt t

 2p

 2p

L f (y)| p0 dy n

t

⎞1

p

⎟ dw⎠ ⎞1

p

⎟ dw⎠



2 p0

dt t

 2p

⎞1

p

⎟ dw⎠

Weighted Hardy Spaces Associated with Elliptic Operators

 j−2 i where the sum i=1 e−c4 I Ii only appears if j ≥ 3. Then, proceeding as in (6.4), and noticing that for x ∈ C j (B), j ≥ 3, 1 ≤ i ≤ j − 2 and 0 < t < r B , we have that B(x, 2i+1 t) ⊂ 2 j+2 B \ 2 j−1 B, applying the L p (w) − L q (w) off-diagonal estimates t2

on balls satisfied by e− 2 ⎛ ⎜ I Ii  ⎝



rB



(see [5,6]), we obtain that, for some constants θ1 , θ2 > 0,

L

2jr

B

 2n p

0

t

0

 2 jθ1 e

−c4 j

⎛ ⎝

j+1 

 −

l= j−1

Cl (B)

⎛  1  p ⎝ − | f (y)| p dw B

 e−c4

j

|e

2

− t2 L



rB 0

1 q

f (y)| dw

2 j rB t

q

 2n p +2θ2 0

e

⎞1

⎞2

2

⎠ dt ⎟ ⎠ t

−c

2 4 j rB t2

⎞1 2 dt ⎠ t

1  p − | f (y)| p dw . B

Now we split I I Ii as follows ⎛  ⎝ I I Ii  −



C j (B)

⎛  ⎝ + −

rB 2i+1

 B(x,2i+1 t)

0



C j (B)

rB rB 2i+1

|e

2 − t2



L

f (y)| p0

t2

B(x,2i+1 t)

|e− 2

L

dy tn

f (y)| p0



dy tn

2 p0



dt t 2 p0

⎞ 1p

 2p

dw ⎠

dt t

 2p

⎞ 1p dw ⎠

=: I I Ii1 + I I Ii2 . Note that for x ∈ C j (B) and 0 < t < r B /2i+1 we have that B(x, 2i+1 t) ⊂ 2 j+2 B \ 2 j−1 B. Then, if j ≥ 3, the estimate of I I Ii1 follows as the estimate of I Ii . If j = 2, 3 we write B(x, 2i+1 t) ⊂ l=2 Cl (B) ∪ (4B \ 2B) and proceed as in the estimate of $1 j # I Ii , applying [5, Lemma 6.5]. Hence, we obtain I I Ii1  e−c4 −B | f (y)| p dw p . In order to estimate I I Ii2 , we observe that for x ∈ C j (B), j ≥ 2, i ≥ j − 1 and r B /2i+1 ≤ t < r B , we have that B, B(x, 2i+1 t) ⊂ 2i+3 B. Thus, proceeding as before, 

 2  21  r 2n  2 q dt t B p0 − I I Ii2  2 |e− 2 L ( f 12i+3 B )(y)|q dw rB t t 2i+3 B 2i+1 1 ⎛ ⎞ 2    1  r B  i 2θ2 + 2n 1 p0 p p 2 rB dt ⎠ ic p ⎝ −  − | f (y)| p dw  2 | f (y)| dw . rB t t B B i+1 in p0

rB

2

Consequently, we conclude that

123

C. Prisuelos-Arribas

I11  e−c4

j

 1 p − | f (y)| p dw .

(6.7)

B

Let us now estimate Ik . We shall use extrapolation to show that Ik  e−c4 $1 − | f (y)| p dw p for all k ∈ N. To this end, we first show that for every w0 ∈ B 2 2 R H p+ (L)  if Tt = t 2 Le−t L (or w0 ∈ R H q+ (L)  if Tt = t∇ y e−t L ), and k ∈ N, j

#

2

2





rB

C j (B) 0



|Tt e−kr B L f (y)|2 2

B(x,t)

⎛   i ⎝ e−c4

 

2 j+3 B\2 j−2 B

i≥1

dy dt w0 (x)dx t n+1

B(x,2i+1 r B )

|e

kr 2 − 2B

L

f (y)| p0

dy r Bn

 p1

0

⎞2 ⎠ w0 (x)dx. (6.8)

Then, note that since p ≤ rw p− (L) < q < p+sw(L) (or p ≤ rw p− (L) < q < q+sw(L) ), we have that w ∈ R H p+ (L)  (or w ∈ R H q+ (L)  ). Hence, (6.8) and Theorem 2.3, p

p

part (a), (or part (b) if q+ (L), p+ (L) = ∞), imply that for all k ∈ N, p

w(2 j+1 B)Ik  

⎛   −c4i ⎝ e

2 j+3 B\2 j−2 B

i≥1

⎞p % kr B2 % p0 dy  p10 % − 2 L % ⎠ w(x)d x =: I I. f (y)% n %e rB B(x,2i+1 r B )

Thus, once proved (6.8), to estimate Ik we just need to consider I I . Let us postpone the proof of (6.8) until later and continue with the estimate of Ik . Since w ∈ A q , p0 proceeding as in (6.4), we have ⎛ ⎜ −c4i pin0 e 2 II  ⎝

 2 j+3 B\2 j−2 B

i≥1

 −

B(x,2i+1 r

B)

%q  qp % kr B2 % % − 2 L f (y)% dw dw %e

 1p

⎞p ⎟ ⎠ .

For 2 ≤ j ≤ 4, note that if x ∈ 2 j+3 B \2 j−2 B then B, B(x, 2i+1 r B ) ⊂ 2i+7 B. Hence, using (2.2), (2.1) and the L p (w) − L q (w) off-diagonal estimates on balls satisfied by e−

2 kr B 2

L

, we get

II

1 p





e

−c4i

2i+7 B

i≥1



 i≥1

123

 − w(B) 1 p

e

−c4i

%q  q1 % kr B2 % % − 2 L ( f 12i+7 B )(y)% dw %e



1 | f (y)| dw p

B

p



1



| f (y)| dw p

B

p

.

Weighted Hardy Spaces Associated with Elliptic Operators

And, for j ≥ 5, we proceed as before, but noticing this time that for x ∈ 2 j+3 B \ 2 j−2 B, if 1 ≤ i ≤ j − 4 then B(x, 2i+1 r B ) ⊂ 2 j+4 B \ 2 j−3 B; and if i ≥ j − 3 B, B(x, 2i+1 r B ) ⊂ 2i+7 B. Hence,

II

1 p

2

jn p0

w(2

j+1

B)

1 p

j−4 

e

i=1

+e

−c4 j

w(2

j+1

B)

1 p

j+3   −

−c4i

Cl (B)

l= j−3



e

−c4i

 −

2i+7 B

i≥ j−3

e

−c4 j

w(2

j+1

%q  q1 % kr B2 % % − 2 L f (y)% dw %e

% kr B2 %q  q1 % − 2 L % ( f 12i+7 B )(y)% dw %e

 1 p p − | f (y)| dw B) . 1 p

B

Let us next prove (6.8). When Tt = t 2 Le−t L , for w0 ∈ R H p+ (L)  , if p+ (L) < ∞, 2

2

we chose 2 <  q < p+ (L) so that w0 ∈ R H q  ; if p+ (L) = ∞, the condition 2

w0 ∈ R H p+ (L)  becomes w0 ∈ A∞ . In this case, we take  q /2 > sw , consequently 2

w ∈ R H q  and  q /2 > 1. When Tt = t∇ y e−t L , we do the same but replacing 2

2

q (Rn ) offp+ (L) with q+ (L). Hence, by Proposition 4.3 and applying the L p0 (Rn )−L  2 diagonal estimates satisfied by T" 2 2 , we have that, for η = 2 if Tt = t∇ y e−t L t +kr B /2

or η = 4 if Tt = 

2 t 2 Le−t L ,

 rB 

%2 dy dt % % % −kr B2 L f (y)% n+1 w0 (x)dx %Tt e t C j (B) 0 B(x,t) ⎛ ⎞2 % %  r B  η   2 % %q dy q kr B t dt % " % ⎝ e− 2 L f (y)% n ⎠ w0 (x)dx  %T 2 2 % % t +kr /2 rB t t 0 C j (B) B(x,r B t/r B ) B  2  r B  η  2 % % q kr B q dy  t dt % " % − L  e 2 f (y)% n w0 (x)dx %T 2 2 t +kr B /2 rB rB t 0 2 j+3 B\2 j−2 B B(x,r B )  r B  η t dt  r t 0 B ⎛   1 ⎞2  % p0 dy p0 % kr B2  i % % − L −c4 ⎝ ⎠ w0 (x)dx e %e 2 f (y)% r Bn 2 j+3 B\2 j−2 B B(x,2i+1 r B ) i≥1

⎛

 



⎝ 2 j+3 B\2 j−2 B

i≥1

i e−c4



% kr B2 % p0 dy % − 2 L % f (y)% %e r Bn B(x,2i+1 r B )



1 p0

⎞2 ⎠ w0 (x)dx.

123

C. Prisuelos-Arribas

$1 j # Therefore, we conclude that Ik  e−c4 −B | f (y)| p dw p , for all k ∈ N. This, (6.3), nq (6.5) with 2M > n/ p0 + p0 + θ1 + θ2 , (6.6) and (6.7) allow us to conclude the proof.   Remark 6.2 Note that, from the previous result and [32, Theorem 1.15], in the case that w ∈ A∞ satisfying Ww ( p− (L), p+ (L)) = ∅, we also improve the lower exponent of the range of p  s where the conical square function associated with the Poisson semigroup (2.17)–(2.19) are bounded on L p (w). With the exception that in the case of GP and GP , we need to assume further that Ww (q− (L), q+ (L)) = ∅. p

7 Characterisation of HL (w), 0 < p < 1 In this section we include the case p = 1 in the statement of our results, because they are also true on that case, but it should be noticed that, as we said above, the case p = 1 was already obtained in [33]. p

Theorem 7.1 Given w ∈ A∞ and 0 < p ≤ 1, let HL (w) be the fixed molecular Hardy space as inRemark 3.6. For every q ∈ Ww ( p− (L), p+ (L)), ε > 0 and M ∈ N p such that M > n2 rpw − p−1(L) , the following spaces are isomorphic to HL (w) (and therefore one another) with equivalent norms p

p

HL ,q,ε,M (w); and

HSm,H ,q (w), m ∈ N;

p

HGm,H ,q (w), m ∈ N0

p

HGm,H ,q (w), m ∈ N0 .

In particular, each of the previous spaces does not depend (modulo isomorphisms) on the choice of the allowable parameters q, ε, M and m. p

Theorem 7.2 Given w ∈ A∞ and 0 < p ≤ 1, let HL (w) be the fixed molecular Hardy space as in Remark 3.6. For every q ∈ Ww ( p− (L), p+ (L)), the following p spaces are isomorphic to HL (w) (and therefore one another) with equivalent norms p

HS K ,P ,q (w), K ∈ N;

p

HG K ,P ,q (w), K ∈ N0

p

HG K ,P ,q (w), K ∈ N0 .

and

In particular, each of the previous spaces does not depend (modulo isomorphisms) on the choice of q, and K . p

Theorem 7.3 Given w ∈ A∞ and 0 < p ≤ 1, let HL (w) be the fixed molecular Hardy space as in Remark 3.6. For every q ∈ Ww ( p− (L), p+ (L)), the following p spaces are isomorphic to HL (w) (and therefore one another) with equivalent norms p

HNH ,q (w)

and

p

HNP ,q (w).

In particular, each of the previous spaces does not depend (modulo isomorphisms) on the choice of q.

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Weighted Hardy Spaces Associated with Elliptic Operators

The proofs of these theorems are analogous to those of the case p = 1 (obtained in [33]). Hence, we need similar results to [33, Propositions 5.1, 6.1, and 7.31]. These are the following. Proposition 7.4 Let w ∈ A∞  , q1 , q2 ∈ W w ( p− (L), p+ (L)), 0 < p ≤ 1, ε > 0 and M ∈ N be such that M > n2 rpw − p−1(L) , there hold p

p

(a) H L ,q1 ,ε,M (w) = HSm,H ,q1 (w) with equivalent norms, for all m ∈ N; p

p

(b) the spaces HSm,H ,q1 (w) and HSm,H ,q2 (w) are isomorphic, for all m ∈ N; p

p

p

(c) H L ,q1 ,ε,M (w) = HGm,H ,q1 (w) = HGm,H ,q1 (w), with equivalent norms, for all m ∈ N0 .

Proposition 7.5 Given  w ∈ A∞ , q1 , q2 ∈ Ww ( p− (L), p+ (L)), 0 < p ≤ 1 K , M ∈ n rw N such that M > 2 p − 21 , and ε0 = 2M + 2K + n2 − nrpw , there hold p

p

(a) H L ,q1 ,ε0 ,M (w) = HS K ,P ,q1 (w), with equivalent norms; p

p

p H L ,q1 ,ε0 ,M (w)

p HG K −1,P ,q1 (w)

(b) the spaces HS K ,P ,q1 (w) and HS K ,P ,q2 (w) are isomorphic; (c)

=

p

= HG K −1,P ,q1 (w), with equivalent norms.

The proofs of these propositions follow as [33, Proposition 5.1 and Proposition 6.1], respectively, just doing the obvious changes. In particular, in order to prove Proposition 7.4 we need to change, in [33, Proposition 5.1], 1 to p, p and q to q1 and q2 , respectively, j

j

j

1

and replace the definition of λl in [33, (5.21)] with λl := 2l w(Q l ) p . Proposition 7.5 follows as [33, Proposition 6.1] doing the same changes as the ones indicated for the proof of Proposition 7.4, and also considering the ε0 defined in the statement of Proposition 7.5 instead of the one in the statement of [33, Proposition 6.1]. From these proposition Theorems 7.1 and 7.2 follow at once. Finally, we characterise the Hardy spaces associated with non-tangential maximal functions. We need the following result. Proposition7.6 Let w ∈ A∞ , q ∈ Ww ( p− (L), p+ (L)), 0 < p ≤ 1, M ∈ N such that M > n2 rpw − 21 , and ε0 = 2M + 2 + n2 − rwpn , there hold p

p

p

(a) HNH ,q (w) = HSH ,q (w) = H L ,q,ε0 ,M (w), with equivalent norms. p p p (b) HNP ,q (w) = HGP ,q (w) = H L ,q,ε0 ,M (w), with equivalent norms. This last proposition follows as Proposition [33, Proposition 7.31], just replacing 1 with p, and p with q, and obviously using the ε0 defined here. Theorem 7.3 follows at once from Proposition 7.6. p

8 Characterisation of HT (w), p ∈ Ww ( p− (L), p+ (L)) In this section we prove Theorem 1.1. That is, for T being any square function in (2.14)–(2.19) or a non-tangential maximal function in (2.20), we show that the Hardy p spaces HT (w) are isomorphic to the L p (w) spaces, for an appropriate range of p.

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C. Prisuelos-Arribas

8.1 Proof of Theorem 1.1 For w ∈ A∞ and p, q ∈ Ww ( p− (L), p+ (L)), we claim that L q (w) ∩ L p (w) = p HT ,q (w) with  f H p

T ,q (w)

≈  f  L p (w) ,

(8.1)

where T is any function defined in (2.14)–(2.19) or (2.20). Then, taking the closure we would conclude the desired isomorphism p

HT ,q (w) ≈ L p (w), for all p, q ∈ Ww ( p− (L), p+ (L)), with constants independent of q, so we can drop the dependence on q and just write p

HT (w) ≈ L p (w), for all p ∈ Ww ( p− (L), p+ (L)). Let us prove our claim. If f ∈ L p (w) ∩ L q (w), since T f  L p (w)   f  L p (w) < ∞, p (see [32, Theorems 1.12 and 1.13] and [33, Proposition 7.1]), then f ∈ HT ,q (w). In order to show the converse inclusion, let us first consider the particular case of p T ≡ Sm,P , for m ∈ N. Then, take f ∈ HSm,P ,q (w), and consider the operator Q L defined by Q L h(x, t) := Tt ∗ h(x), for all (x, t) ∈ Rn+1 + , √



∗



where Tt ∗ := (t 2 L ∗ )m e−t L . This operator is bounded from L p (w 1− p ) to   T p (w 1− p ), for all p  ∈ Ww1− p ( p− (L ∗ ), p+ (L ∗ )). Indeed, by [32, Theorem 1.13],   we have that, for every h ∈ L p (w 1− p ), ⎛ Q L hT p (w1− p ) = ⎝

 Rn

 

⎞ 1  p % %2 dy dt  p2 √ % 2 ∗ m −t L ∗ % 1− p  ⎠ h(y)% n+1 w (x)dx %(t L ) e t (x)

 h L p (w1− p ) . Then, if Q∗L denotes its adjoint operator with respect to d x, we have that for every H ∈ T 2 (Rn ) Q∗L H (x) =

 0

∞

(t 2 L)m e−t

√

L

H (x, t)

dt . t

(8.2)

Similarly as in the proof of Theorem 5.1, we conclude that Q∗L has a bounded extension from T p (w) to L p (w), for all p ∈ W ( p (L), p+ (L)). Next, since the √w − 2 m −t L in bounded on L q (w) (see [33, vertical square function defined by (t L) e

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Weighted Hardy Spaces Associated with Elliptic Operators

(6.3)] and [6]), we can consider the following Calderón reproducing formula of f (see [33, Remark 5.21]), 

∞

(t 2 L)m e−t

f (x) = Cm

√

L

2

f (x)

0

dt , t

(8.3)

where the equality is in L q (w). Note that, as we explain in Remarks 3.8 and 5.4, and in the proof of Theorem 5.1, for q ∈ Ww ( p− (L), p+ (L)), we have (8.2) and also (8.3), for functions in T q (w) ∩ T p (w), and in L q (w), respectively, understanding that, by abuse of notation, L denotes L q,w . p f (x, t) := Now, since for f ∈ HSm,P ,q (w), we have that f ∈ L q (w) and that  √

(t 2 L)m e−t L f (x) ∈ T p (w) ∩ T q (w) (  f T p (w) = Sm,P f  L p (w) and   f T q (w) =   p q q q Sm,P f  L (w)   f  L (w) ), we get, for every g ∈ L (w) ∩ L (w), % % % %

Rn

% % % % f (y)g(y)w(y) ¯ dy %% = Cm %%

Rn



∞

 f (y, t)(t 2 L ∗ )m e−t

√

L ∗ (gw)(y)

0

% dt %% dy % t

p      Sm,P f  L p (w) Q L (gw) ¯ T p (w1− p )  Sm,P f  L (w) gw L p (w1− p )

= Sm,P f  L p (w) g L p (w) .





Then, taking the supremum over all g ∈ L p (w) ∩ L q (w) such that g L p (w) = 1 q

p

p

(note that L (w) ∩ L (w) is dense in L (w)), we obtain that  f  L p (w)  Sm,P f  L p (w) .

(8.4)

p

Therefore, we have that, for all m ∈ N, HSm,P ,q (w) = L p (w)∩ L q (w), with equivalent norms. Now noticing that, in the proof of [32, Theorem 1.15, part (b)], the authors showed that, for every m ∈ N, Sm,P f  L p (w) ≤ Sm,H f  L p (w) . From this, (8.4), [32, Theorems 1.14 and 1.15, Remark 4.22] and [33, Lemma 4.4 and Proposition p 7.1], we obtain that HT ,q (w) = L p (w) ∩ L q (w) with equivalent norms, for all p, q ∈ Ww ( p− (L), p+ (L)) and T being any function in (2.14)–(2.19), or a nontangential maximal function in (2.20). From the observations made at the beginning of the proof, this allows us to conclude the desired isomorphism.   p

Remark 8.1 As we explain in the proof we have obtained the isomorphism HT ,q (w) ≈ L p (w) for all p, q ∈ Ww ( p− (L), p+ (L)). In particular, this implies that p

p

HT ,q1 (w) ≈ HT ,q2 (w), for all p, q1 , q2 ∈ Ww ( p− (L), p+ (L)). for T being any function in (2.14)–(2.19), or a non-tangential maximal function in (2.20).

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C. Prisuelos-Arribas

9 Characterisation of the Weighted Hardy Space Associated with the Riesz Transform In order to characterise the weighted Hardy space associated with the Riesz transform, we proceed as in [28], where the unweighted case was consider. First of all, we need to prove the following weighted versions of [28, Propositions 5.32 and 5.34] from which we obtain at once Theorem 1.2. Proposition 9.1 Given w ∈ A∞and q ∈ Ww (q− (L), q+ (L)), we have that, for all p  p− (L) p+ (L) p satisfying max rw , nrnrww+ p− (L) < p < sw and f ∈ H∇ L −1/2 ,q (w), 1

SH f  L p (w)  ∇ L − 2 f  L p (w) .

(9.1)

   p− (L) In particular, we conclude that, for all max rw , nrnrww+ < p < p− (L) p

p

H∇ L −1/2 ,q (w) ⊂ HSH ,q (w).

Proposition 9.2 Given w ∈ A∞ and q ∈ Ww (q− (L), q+ (L)), for all 0 < p < p and f ∈ HSH ,q (w), we have that

p+ (L) sw ,

q+ (L) sw

1

∇ L − 2 f  L p (w)  SH f  L p (w) . In particular, we conclude that, for all 0 < p <

q+ (L) sw ,

p

p

HSH ,q (w) ⊂ H∇ L −1/2 ,q (w).

We start by proving Proposition 9.1. To this end, consider the following conical square function: S f (x) :=

 

√ dy dt 2 |t Le−t L f (y)|2 n+1 t (x)

1 2

.

We show that, in some range of p, its norm is comparable with the norm of SH in L p (w). Proposition 9.3 Given w ∈ A∞ , for all q ∈ Ww ( p− (L), p+ (L)) and f ∈ L q (w), there hold (a) SH f  L p (w)  S f  L p (w) , for all p ∈ Ww (0, p+ (L)2,∗ ); (b) S f  L p (w)  SH f  L p (w) , for all p ∈ Ww (0, p+ (L)∗ ). In particular S f  L p (w) ≈ SH f  L p (w) , for all p ∈ Ww (0, p+ (L)∗ ). Proof We first prove part (a). Note that since 2 < p+ (L)2,∗ , in view of Theorem 2.3, part (a), (or part (b) if p+ (L)2,∗ = ∞), it is enough to prove it for p = 2 and all w ∈ R H p (L)2,∗  . +

123

2

Weighted Hardy Spaces Associated with Elliptic Operators

Assuming this, note that, proceeding as in the estimate of term I I when proving (5.8), given w ∈ R H p (L)2,∗  , we can find q0 and r , so that 2 < q0 < p+ (L), +

2

q0 /2 ≤ r < ∞, w ∈ R Hr  and n n > 0. − 2r q0

2+

(9.2)

After this observation we show the desired estimate. By (2.10) and Minkowski’s integral inequality, we obtain that ⎛

 ∞  t 

SH f (x)  ⎝

0

B(x,t)

0

|s Le

⎛

 ∞  ∞ 

+⎝

0

B(x,t)

t

−s 2 L

t

√ 2

Le

1

−t 2 L

f (y)|2 dy

2

√ 2 2 |s Le−s L t 2 Le−t L f (y)|2 dy

ds s 1 2

⎞ 21 dt ⎠

2

t n+1 ds s

2

⎞ 21

dt ⎠ t n+1

=: I + I I.

In the case that s < t, use the L 2 (Rn ) − L 2 (Rn ) off-diagonal estimates satisfied by 2 2 the families {t 2 Le−t L }t>0 and {e−s L }s>0 , and apply [26, Lemma 2.3] to get ⎛  I ≤⎝

∞



0





e

0

−c4 j

t



s t



|e ∞  t

I  L 2 (w) 

0

e

−c4 j





√

t Le √

B(x,2 j+1 t)

 Rn

j≥1



L

2 

s ds t s

Then, changing the variable t into 3.2], we conclude that 

t 2 Le

2 − t2

2 − t2

1 L

f (y)|2 dy

B(x,t)

0

j≥1

−s 2 L

0

|t Le

2

− t2 L

2

ds s

dy dt f (y)|2 n+1 t

2

⎞ 21 dt ⎠

t n+1

1 2

.

√ 2t and applying change of angles [32, Proposition ∞

√

B(x,2 j+2 t)

|t Le

−t 2 L

dy dt f (y)|2 n+1 w(x)dx t

1 2

e−c4 S f  L 2 (w)  S f  L 2 (w) . j

j≥1

√ s2 As for the estimate of I I , consider  f (y, s) := s Le− 2 L f (y), apply the L 2 (Rn ) − 2 L 2 (Rn ) off-diagonal estimates satisfied by the family {e−t L }t>0 and Jensen’s inequality. Besides, change the variable s into st, apply Minkowski’s integral inequality and then change the variable t into t/s. Hence, we have

123

C. Prisuelos-Arribas

II 



⎛ e

−c4 j

e

−c4 j

e

−c4 j

j≥1





⎛

j≥1







j e−c4

j≥1

=:

2 ⎞ 21 1  ∞  ∞ 2  q0 ds s2 t dy dt ⎠ ⎝ |s 2 Le− 2 L  f (y, s)|q0 n 2 j+1 t s t s B(x,2 t) 0 t  ∞ 1

j≥1



⎞ 21 2 1  ∞  ∞ 2  2 ds s2 t dt ⎝ ⎠ |s 2 Le− 2 L  f (y, s)|2 dy s s2 t n+1 B(x,2 j+1 t) 0 t

 j≥1

⎛

 ∞ 

s −2 ⎝

B(x,2 j+1 t)

0

  ∞ ∞  −2+ qn 0 s 1

0

(st)2 |(st)2 Le− 2

dy L f (y, st)|q0 n t

2

B(x,2 j+1 t/s)

t dy |t 2 Le− 2 L  f (y, t)|q0 n

t





2 q0

2 q0

⎞1 2 dt ⎠ ds t s dt t

 21

ds s

 ∞ j ds −2+ qn 0 J (x, s) . e−c4 s s 1

In order to estimate the norm in L 2 (w) of the above integral, we first apply √ Minkowski’s inequality, [32, Proposition 3.30], and change the variable t into 2t. Next, we apply the L 2 (Rn ) − L q0 (Rn ) off-diagonal estimates satisfied by the family 2 {t 2 Le−t L }t>0 , and recall that q0 and r satisfy 2 < q0 < p+ (L), q20 ≤ r , w ∈ R Hr  and (9.2). Finally, we apply [32, Proposition 3.2]. Thus, we have, for r0 > rw , 1   ∞ 2 ds 2 −2+ qn 0 J (x, s) s w(x)dx s Rn 1   ∞ n n ds −2− 2r + q 0  s s 1   21 2 ∞   √ q0 2 2 dy dt |t 2 Le−t L t Le−t L f (y)|q0 n w(x)dx t t Rn B(x,2 j+2 t) 0 1   ∞    √ 2 l 2 dy dt  e−c4 |t Le−t L f (y)|2 n+1 w(x)dx n j+l+3 t R 0 B(x,2 t) l≥1 nr0  nr l j 20 S f  2 f  2 .  2j 2 e−c4 S L (w)  2 L (w)



l≥1

Consequently, I I  L 2 (w)  S f  L 2 (w) , which, together with the estimate obtained for I  L 2 (w) , gives us the desired inequality. As for proving part (b), note that again it is enough to consider the case p = 2 and w ∈ R H p+ (L)∗  . In this case we proceed as in the proof of part (a), so we skip some 2

details. For n > p+ (L), note that (as in the proof of (5.8)) we can take ε0 > 0 small n , enough and 2 < q0 < p+ (L), close enough to p+ (L) so that for r := 2(1+εq00)(n−q 0) we have that 2 < q0 < p+ (L), q0 /2 ≤ r < ∞, w ∈ R Hr  and

123

Weighted Hardy Spaces Associated with Elliptic Operators

n n > 0. − 2r q0

1+

If now n ≤ p+ (L), our condition w becomes w ∈ A∞ . Then, we take   over the weight 2r p+ (L) r > sw , and q0 satisfying max 2, p+ (L)+2r < q0 < min { p+ (L), 2r } if p+ (L) < ∞ and q0 = 2r if p+ (L) = ∞. Therefore, we have that 2 < q0 < p+ (L), q0 /2 ≤ r < ∞ and w ∈ R Hr  . Besides, 1+

n n n >1− − ≥ 0. 2r q0 p+ (L)

Hence, we have found q0 and r so that 2 < q0 < p+ (L), q0 /2 ≤ r < ∞, w ∈ R Hr  and n n − > 0. 2r q0

1+

(9.3)

Keeping these choices of q0 and r we prove part (b). Using again (2.10) and Minkowski’s integral inequality, we obtain ⎛ S f (x)  ⎝



∞

  t

0

⎛ +⎝

0



∞

|ts Le

−s 2 L

e

−t 2 L

1

ds s

2

f (y)|2 dy

B(x,t)



0

∞  t

|ts Le

−s 2 L

e

−t 2 L

2

t n+1

1

ds s

2

f (y)|2 dy

⎞ 21 dt ⎠

B(x,t)

2

⎞ 21 dt ⎠

t n+1

=: I + I I. We first estimate I . Using that s < t and applying the L 2 (Rn ) − L 2 (Rn ) off-diagonal 2 estimates satisfied by the family {e−s L }s>0 , we have ⎛  ⎝ I ≤

∞



0





t 0

e

−c4 j

s t



 j≥1

|e

e−c4

j

∞  t



−s 2 L

t 2 Le

−t 2 L

1 f (y)|2 dy

2

B(x,t)

0

j≥1





0

s ds t s

∞ 0

B(x,2 j+1 t)

2  |t Le 2

B(x,2 j+1 t)

|t 2 Le−t

2L

f (y)|2

−t 2 L

dy dt t n+1

ds s

2

⎞ 21 dt ⎠

t n+1

dy dt f (y)|2 n+1 t

1 2

1 2

.

Therefore, applying change of angles [32, Proposition 3.2], we get I  L 2 (w) 



e−c4 SH f  L 2 (w)  SH f  L 2 (w) . j

j≥1

123

C. Prisuelos-Arribas

As for the second term, we first apply the L 2 (Rn ) − L 2 (Rn ) off-diagonal estimates 2 satisfied by the family {e−t L }t>0 , change the variable s into st and apply Jensen’s inequality. Next, we apply Minkowski’s integral inequality and change the variable t into t/s. Hence, we have II 



⎛ e

 ∞  ∞

−c4 j

⎝

0

j≥1





s −1



1

B(x,2 j+1 t)

  ∞ ∞  −1+ qn 0 s

j e−c4

1

j≥1

0

|(st)2 Le

B(x,2 j+1 t/s)

−(st)2 L

1 2

f (y)|2 dy

2 dy |t 2 Le−t L f (y)|q0 n



ds s

2 q0

t

2

dt t

⎞ 21

dt ⎠ t n+1

 21

ds s

 ∞  j ds −1+ qn 0 J (x, s) . =: e−c4 s s 1 j≥1

Thus, applying first Minkowski’s integral inequality, [32, Proposition 3.30], and √ changing the variable t into 2t, next applying the L 2 (Rn ) − L q0 (Rn ) off-diagonal 2 estimates satisfied by the family {e−t L }t>0 , recalling our choices of q0 and r and (9.3) and applying [32, Proposition 3.2], we obtain, for r0 > rw , 



∞

ds 0 J (x, s) s n s R 1   ∞ n −1− 2r + qn ds 0  s s 1    −1+ qn

∞

Rn

 2j

nr0 2

B(x,2 j+2 t)

0



|e

1

2

2

w(x)dx

−t 2 L 2

t Le

e−c4 SH f  L 2 (w)  2 j l

nr0 2

−t 2 L

dy f (y)|q0 n t

2

q0

dt w(x)d x t

 21

SH f  L 2 (w) .

l≥1

Using this, we obtain I I  L 2 (w)  SH f  L 2 (w) . Gathering this and the estimate obtained for I  L 2 (w) gives us that, for all w ∈ R H p+ (L)∗  , 2

S f  L 2 (w)  SH f  L 2 (w) ,  

which, from the observations made at the beginning, finishes the proof. 9.1 Proof of Proposition 9.1 1

1

First of all note that if f is such that ∇ L − 2 f  L p (w) < ∞, then for h := L − 2 f , we have that h ∈ W˙ 1, p (w) (the space W˙ 1, p (w) is defined as the completion of {h ∈ C0∞ (Rn ) : ∇h ∈ L p (w)} under the semi-norm hW˙ 1, p (w) := ∇h L p (w) ). Additionally, note that applying Proposition 9.3, Theorem 6.1 and [6, Theorem 6.2], for

123

Weighted Hardy Spaces Associated with Elliptic Operators

all w ∈ A∞ such that Ww ( p− (L), p+ (L)) = ∅ and max{rw ,  p− (L)} < p < we have that

p+ (L) sw ,

√ √ √ S Lh L p (w) ≈ SH Lh L p (w)   Lh L p (w)  ∇h L p (w) . This gives us that √ p+ (L) S L : W˙ 1, p (w) → L p (w), ∀ max{rw ,  p− (L)} < p < . sw

(9.4)

Therefore, if we show that, for every  p− (L) <  p < q+sw(L) , r0 > rw , so that rw q− (L) <    p r0 q− (L) < q+sw(L) , and for p0 := max r0 , nrnr00+ p , √ S L : W˙ 1, p0 (w) → L p0 ,∞ (w),

(9.5)

then, by interpolation (see [10]), applying Proposition 9.3, and by the observation made at the beginning of the proof, we will conclude (9.1). Besides, note that Ww (q− (L), q+ (L)) = ∅ implies Ww ( p− (L), p+ (L)) = ∅ (recall that Ww (q− (L), q+ (L)) ⊂ Ww ( p− (L), p+ (L))). We fix  p and r0 satisfying the above restrictions. Additionally, we take r , q− (L) < r < 2, close enough to q− (L) so that rr0 < q+sw(L) . Then, if we consider p1 so that p } < p1 < max{rr0 , 

q+ (L) sw ,

we have that w ∈ A p1 ∩ R H q+ (L)  , and p1 > p0 . r

p1

Recalling these choices of  p , r0 , r , p1 and p0 , note that in order to prove (9.5) it suffices to show that, for every α > 0 and h ∈ W˙ 1, p0 (w), w



  √ 1 x ∈ Rn : S Lh(x) > α  p |∇h(x)| p0 w(x)dx. α 0 Rn

To this end, consider the following Calderón–Zygmund decomposition of h (see [6, Lemma 6.6]). Lemma 9.4 Let n ≥ 1, w ∈ A∞ , μ := wd x and rw < p0 < ∞ (with the possibility of taking p0 = 1 if rw = 1). Assume that h ∈ W˙ 1, p0 (w), and let α > 0. Then, one can find a collection of balls {Bi }i∈N (with radii r Bi ), smooth functions bi and a function 1 (w) such that g ∈ L loc h=g+



bi

i∈N

and the following properties hold |∇g(x)| ≤ Cα, for μ − a.e. x,  supp bi ⊂ Bi and |∇bi (x)| p0 w(x)dx ≤ Cα p0 w(Bi ),

(9.6) (9.7)

Bi

123

C. Prisuelos-Arribas



w(Bi ) ≤

i∈N



C α p0

 Rn

|∇h(x)| p0 w(x)dx,

(9.8)

1 Bi ≤ N ,

(9.9)

i∈N

where C and N depend only on the dimension, the doubling constant of μ and p0 . In addition, for 1 ≤ q < ( p0 )∗w , where ( p0 )∗w is defined in (2.12), we have 1  q − |bi (x)|q dw  αr Bi .

(9.10)

Bi

Applying this lemma to our function h and to our choice of p0 , and considering for M ∈ N, arbitrarily large, and for every i ∈ N, Br Bi := (I − e Ar Bi := I − Br Bi , we can write bi = Br Bi bi + Ar Bi bi . Hence, h=g+



Br Bi bi +

i∈N



−r B2 L M i )

and

Ar Bi bi .

i∈N

Then, w



  √ √ α  x ∈ Rn : S Lh(x) > α ≤ w x ∈ Rn : S Lg(x) > 3 

 

  √ α +w x ∈ Rn : S L Ar Bi bi (x) > 3 i∈N   

  √ α n  Br Bi bi (x) > +w x ∈R :S L =: I + I I + I I I. (9.11) 3 i∈N

By our choice of p1 , we have that p1 ∈ Ww (q− (L), q+ (L)) ⊂ Ww ( p− (L), p+ (L)). Then, applying Chebyshev’s inequality, (9.4), (9.6), (9.7) and (9.8), we obtain  √ 1 p1  |S Lg(x)| w(x)dx  p |∇g(x)| p1 w(x)dx α 1 Rn Rn     1 1 p0 p0 |∇h(x)| w(x)dx + α w(Bi )  p |∇h(x)| p0 w(x)dx.  p 0 n n α 0 α R R

1 I  p α 1



i∈N

(9.12) 

In order to estimate the remaining terms, we take 1 < p < ∞ and u ∈ L p (w) such that u L p (w) = 1. Besides, we denote by Mw the weighted maximal operator defined as in (5.5) but taking the supremum over balls instead of over cubes. Then, using a Kolmogorov type inequality (see [22, Exercise 2.1.5], and follow the proof suggested there replacing the Lebesgue measure with the measure given by the weight w) and (9.8), we have that

123

Weighted Hardy Spaces Associated with Elliptic Operators

  i∈N

 1  p Mw (|u| p )(x) w(x)d x Bi

 w(∪i∈N Bi )u

p  L p (w)



1 α p0

p

 

 Rn

∪i∈N Bi

p  1  p Mw (|u| p )(x) w(x)dx

|∇h(x)| p0 w(x)dx.

(9.13)

Moreover, note that 1 < max{r0 ,  p } < ( p0 )∗w and hence by (9.10),  1  1  p p2 p − |bi (x)|  dw  − |bi (x)| p2 dw  αr Bi , Bi

(9.14)

Bi

where p2 := max{r0 ,  p }, and the first inequality follows applying Jensen’s inequality, p . Trivially 1 < max{r0 ,  p }. In order to prove p } < ( p0 )∗w , since p2 ≥   0,   that max{r

 p = r0 , and thus p } = r0 . Then, p0 = max r0 , nrnr00+ first assume that max{r0 ,  p ∗ ∗ p } = r0 < (r0 )w = ( p0 )w . If max{r0 ,  p} =  p , we assume that nrw > p0 , max{r0 ,   p otherwise ( p0 )∗w = ∞ >  p = max{r0 ,  p }. Besides, since p0 ≥ nrnr00+ p , note that

nrw > p0 implies nrw r0 >  p (r0 − rw ). Consequently, using again that p0 ≥ we obtain that ( p0 )∗w =

nrw p0 nrw − p0

≥

nrw r0  p nrw r0 − p (r0 −rw )

nr0  p nr0 + p,

> p = max{r0 ,  p }.

Therefore, in order to estimate I I , we first √ apply Chebyshev’s inequality. Next, by (9.4), expanding the binomial, using that { t∇ y e−t L }t>0 satisfies L p2 (w) − L p2 (w) off-diagonal estimates on balls (note that  p− (L) < p2 < p1 < q+ (L)/sw and see [5,6]), by (2.2), (9.14) and (9.13) with p = p2 , we have % ⎛ ⎞ % p2 %  % √  % % 1 % w(x)dx %S  L⎝ ⎠ II  p A b (x) r Bi i % % 2 α Rn % % i∈N % ⎛ ⎞ % p2 %  % M  2 L % % 1 −kr Bi b ⎠ (x)% %∇ ⎝  p C e k,M i % w(x)dx % α 2 Rn % % i∈N k=1 ⎞ p2 ⎛ %   % M    %√ 2 L % b 1 −kr i % kr B ∇ y e Bi ⎝ (x)%% |u(x)|w(x)dx ⎠  p sup Ck,M i % α 2 u  =1 r Bi Rn p L 2 (w)

k=1

i∈N

⎛ M    jnp1 ⎝ Ck,M 2 r w(Bi )

1  p sup α 2 u  =1 p k=1 L 2 (w)

i∈N j≥1

⎞p2 %   % p2 p12 u1C (B )  p 2 √ j i % % b L 2 (w)⎟ i % kr B ∇ y e−kr Bi L (x)%% dw i % 1 ⎠ r Bi C j (Bi ) p2 j+1 w(2 Bi ) ⎛ 1 % % p2     p % bi (x) % j 1 % dw 2 ⎝  p sup e−c4 w(Bi ) − %% % α 2 u  =1 r Bi Bi p i∈N j≥1 L 2 (w)   1 p2  inf Mw (|u| p2 )(x) p2  −

x∈Bi

123

C. Prisuelos-Arribas

⎛  

u

⎝

sup

=1 p L 2 (w)





 Mw (|u| p2 )(x)

i∈N Bi



1 p2

⎞ p2 w(x)dx ⎠

 1 |∇h(x)| p0 w(x)dx. α p0 R n

(9.15)

Next, we estimate I I I . Note that,  III  w







+w

16Bi

x ∈R

n

-

i∈N

i∈N

 

  √ α 16Bi : S L Br Bi bi (x) > 3 i∈N

= I I I1 + I I I2 .

(9.16)

Applying (9.8) we have that I I I1 

1 α p0

 Rn

|∇h(x)| p0 w(x)dx.

(9.17)

Hence it just remains to control I I I2 . Applying Chebyshev’s inequality, we obtain I I I2

⎛



  ⎜ ∞ ⎝

⎞ p1 % ⎛ ⎞ %2 2 % %  % dy dt ⎟ % % %t Le−t 2 L ⎝ ⎠ Br Bi bi (y)% n+1 ⎠ w(x)dx % B(x,t) % % t i∈N

1 α p1 Rn \)i∈N 16Bi 0 ⎛ p1   ∞ %  %2 dydt p21 1 2L 1 % % ⎜  −t  p Br Bi bi (y)% n+1 w(x)dx sup %t Le ⎝ 1 α u  =1 t C j (Bi ) 0 B(x,t) 

i∈N j≥4

p L 1 (w)

u1C j (Bi ) 

p1 

L p1 (w)

⎛ ⎞ p1  1 ⎝ ⎠ . =: p sup III i j u1C j (Bi )  p L 1 (w) α 1 u  =1 p1 i∈N j≥4 L (w)

(9.18)

Splitting the integral in t (recall that j ≥ 4), we have ⎛





III i j  ⎝

C j (Bi )

⎛



+⎝

C j (Bi )



0 ∞

2 j−2 r Bi

= III i1j + III i2j .

123

2 j−2 r Bi 



%  %2 dy dt 2 % % %t Le−t L Br Bi bi (y)% n+1 t B(x,t)

 p1 2

⎞ p1

1

w(x)dx ⎠

⎞ p1  p1 %    %2 1 2 % % 2 −t 2 L b dy dt i % %t Le ⎠ B (y) w(x)dx r B % t n+1 % i r Bi B(x,t)

Weighted Hardy Spaces Associated with Elliptic Operators

We first estimate III i1j . Recall that w ∈ A p1 ∩ R H q+ (L)  . Hence, we can take r

p1

q0 , max{2, p1 } < q0 < q+ (L), close enough to q+ (L) so that w ∈ R H q0  . Then, p1

applying Jensen’s inequality, Fubini’s theorem and noticing that for x ∈ C j (Bi ) and 0 < t ≤ 2 j−2 r Bi we have that B(x, t) ⊂ 2 j+2 Bi \ 2 j−1 Bi , we get −

1

1

III i1j  |2 j+1 Bi | q0 w(2 j+1 Bi ) p1 ⎛  j−2   q0 ⎞ q10  2 % %  2 r Bi 2 dy dt 2 % % ⎝ dx ⎠ %t Le−t L Br Bi bi (y)% n+1 t C j (Bi ) 0 B(x,t) −

1

1

−

1

1

 |2 j+1 Bi | q0 w(2 j+1 Bi ) p1 ⎞1 ⎛ q0   2 j−2 r B  j  q20 −1 % %  q i 0 2 r Bi 2 % dy dt ⎠ % ⎝ d x %t Le−t L Br Bi bi (y)% t t n+1 C j (Bi ) 0 B(x,t)  |2 j+1 Bi | q0 w(2 j+1 Bi ) p1 ⎛ ⎞1 q0  2 j−2 r B  j  q20 −1  % %  q i 0 2 r Bi % dy dt ⎠ % 2 −t 2 L ⎝ Br Bi bi (y)% t −q0 . %t Le t t 0 2 j+2 Bi \2 j−1 Bi

We estimate the integral in y by using functional calculus. We use the notation in [2] and [6, Sect. 7]. We write ϑ ∈ [0, π/2) for the supremum of |arg(L f, f  L 2 (Rn ) )| over all f in the domain of L. Let 0 < ϑ < θ < ν < μ < π/2 and note that, −r 2 z

for a fixed t > 0, φ(z, t) := e−t z (1 − e Bi ) M is holomorphic in the open sector μ = {z ∈ C \ {0} : |arg(z)| < μ} and satisfies |φ(z, t)|  |z| M (1 + |z|)−2M (with implicit constant depending on μ, t > 0, r Bi and M) for every z ∈ μ . Hence, we can write 2

 φ(L , t) =



e−z L η(z, t)dz,

 where η(z, t) =

γ

eζ z φ(ζ, t)dζ.

Here  = ∂ π2 −θ with positive orientation (although orientation is irrelevant for our computations) and γ = R+ ei sign(Im(z)) ν . It is not difficult to see that for every z ∈ ,

|η(z, t)| 

r B2M i (|z| + t 2 ) M+1

.

Consequently, we can write

123

C. Prisuelos-Arribas

 2 j+2 Bi \2 j−1 Bi

% %q0 1 q % 2 −t 2 L % Br Bi (bi ) (y)% dy 0 %t Le

 

%z  z %q0  q10 t 2 r B2M % % i − 2z L −2 L e  bi (y)% dy |dz| % Le |z| (|z| + t 2 ) M+1  2 j+2 Bi \2 j−1 Bi 2 j−3   %z  %q0  q10  z z % %  % Le− 2 L 1Cl (Bi ) e− 2 L bi (y)% dy  2 j+2 Bi \2 j−1 Bi 2 l=1

r B2M i

t2

|dz|

|z| (|z| + t 2 ) M+1    + l≥ j−2 

%z  %q0  q10 % % − 2z L − 2z L 1Cl (Bi ) e bi (y)% dy % Le 2

2 j+2 Bi \2 j−1 Bi

r B2M t2 i |dz|. |z| (|z| + t 2 ) M+1 4, for 1 ≤ l ≤ j − 3 we have that d(2 j+2 Bi \ ≥ 2l+1r Bi . Then, in that case, applying the fact that L q0 (Rn ) off-diagonal estimates (see [2]), splitting the w ∈ A p1 , changing the variable s into 4 j r B2 i /s 2 and

Note now that since j ≥ 2 j−1 Bi , Cl (Bi )) ≥ 2 j−2 r Bi z − 2z L satisfies L r (Rn ) − 2 Le exponential term, using that

r

z

p (w) − L p1 (w) off-diagonal estimates on balls (see applying that e− 2 L satisfies L  [5,6]), and by (9.14), we obtain

  

%z  %q0  q10 t 2 r B2M % % i − 2z L − 2z L 1 (y) e b dy |dz| Le % % i Cl (Bi ) 2 ) M+1 j+2 j−1 2 |z| (|z| + t 2 Bi \2 Bi  2   4 j rB % z %r  r1 2 r B2M − n2 r1 − q1 % −2 L % −c |z| i t i 0 bi (y)% dy |z| e |dz|  %e |z| (|z| + t 2 ) M+1  Cl (Bi )  2 2 1   4 j rB 4l r B % z % p1 p1 n − n2 r1 − q1 % −2 L % −c |z| i −c |z| i l 0 r −  (2 r Bi ) bi (y)% dw |z| e e %e  Cl (Bi ) 2M r Bi |dz| (|z| + t 2 ) M+1

t2 |z|

 1  p  p  2 (2 r Bi ) − |bi (y)| dw lθ1



n r

l



∞

ϒ

s

0 l(

 αr Bi 2  ∞ 0

123

Bi

2l r

θ1 + nr

 ϒ

Bi

θ2 s

1 2

− n2

n

)r q0 2− jn Bi

θ 2l s 2 2j



s

n





1 1 r − q0

1 1 r − q0

1 1 r − q0



e

−c

2 4 j rB i s

e

−c





e−cs e 2

l s2 4j

−c 4

t2

2 4l r B i s

t2

r B2M i

ds (s + t 2 ) M+1 s

r B2M i

ds , (4 j r B2 i /s 2 + t 2 ) M+1 s

Weighted Hardy Spaces Associated with Elliptic Operators

recall that ϒ(u) = max{u, u −1 }. If we now consider l ≥ j − 2, in this case, we do not have distance between 2 j+2 Bi \ 2 j−1 Bi and Cl (Bi ), but we do have between Cl (Bi ) and Bi . Indeed, since l ≥ j − 2 ≥ 2, we have that d(Cl (Bi ), Bi ) > 2l−1r Bi ≥ 2 j−3r Bi . Hence, proceeding as in the above computation, we obtain  

%z  %q0  q10 t 2 r B2M % % i − 2z L − 2z L 1Cl (Bi ) e bi (y)% dy |dz| % Le |z| (|z| + t 2 ) M+1 2 j+2 Bi \2 j−1 Bi 2    % z %r  r1 2 r B2M − n2 r1 − q1 t % −2 L % i 0 bi (y)% dy |z| |dz|  %e |z| (|z| + t 2 ) M+1  Cl (Bi )  1   % z % p1 2 p1 r B2M n − n2 r1 − q1 t % % i l − L 0 −  (2 r Bi ) r |z| |dz| %e 2 bi (y)% dw |z| (|z| + t 2 ) M+1  Cl (Bi ) 1   p n p  2lθ1 (2l r Bi ) r − |bi (y)|  dw





∞

 ϒ

Bi

2l r

θ2

Bi

s

1 2

− n2



1 1 r − q0



e−c

2 4l r B i s

t2

r B2M i

ds (s + t 2 ) M+1 s

0 s n  αr Bi 2lθ1 (2l r Bi ) r  ∞  l θ2 n  1 1 2 2 4 j rB 4l r B r B2M ds 2 r Bi −2 r −q i −c s i −c s i 2 0 ϒ s e e t 1 2 ) M+1 s (s + t 0 s2  n q

− jn

1

−

1

r q0  αr Bi 2l (θ1 + r )r B0i 2  ∞  l θ2  1 1 l 2 r B2M ds 2s 2 −c 4 s n r −q i 0 e −cs e 4j t2 . ϒ s j j r 2 /s 2 + t 2 ) M+1 s 2 (4 0 Bi n

 > 0 large enough to be Next, changing the variable t into 2 j r Bi t, we have for M chosen later, ⎛ ⎝

2 j−2 r Bi 2 j r



 q20 −1 t −q0

Bi

t

0



∞

 ϒ

0

2l s 2j

θ2



s

n

1 1 r − q0



e

−cs 2

e

l s2 4j

−c 4

r B2M i

ds t 2 j 2 2 2 M+1 s (4 r Bi /s +t )

q0

2− j(2M+1)r B−1 i

 

1

t 0



q 1+ 20



∞

ϒ

0



2l s 2j

θ2



s

n

1 1 r − q0



e

−cs 2

e

l s2 4j

−c 4

1 ds (1/s 2 + t 2 ) M+1 s

dt t

 q1

q0

0

dt t

 q1

0

  2−l (2 M−θ2 ) 2− j (2M+1−θ2 −2 M )r B−1 i

123

C. Prisuelos-Arribas

⎛   1 q 1+ 20 ⎝ t 0

 +



1

s

n

1 1 r − q0



 −2 M+2M+2−θ 2 ds

t

q 1+ 20





∞

s

0

dt t

s

0 1

q0

n

1 1 r − q0



 −2 M+2M+2+θ 2 −cs 2 ds

e

s

1

 = θ1 + θ2 + Therefore, taking 2 M 1

n r

1

q0

q0

dt t

 q1

0

⎞ ⎠.

+ 1, and 2M > 2θ2 + θ1 + nr , we have

#

n  III i1j  αw(2 j+1 Bi ) p1 2− j 2M+ r +1−θ2 −2 M

$

1

2−l  αw(2 j+1 Bi ) p1 2− j(2M−θ1 −2θ2 ) .

l≥1

(9.19) In order to estimate III i2j , we consider θ M := −(t 2 +r B2 )L i

√

M + 2 and Br Bi ,t := (e−t

2L

−

−t 2 L

) M . Hence, applying the fact that {t 2 Le }t>0 ∈ F(L r −L 2 ), Proposition e 4.2 with s = r Bi and p = r , [32, Lemma 2.1], and next using that w ∈ A p1 , applying r

p (w)− L p1 (w) off-diagonal estimates on balls, and by (9.14), that {e−t L }t>0 satisfies L  we obtain

 B(x,θ M t)

1 %    %2 % dy 2 % 2 −t 2 L b 2L i −t %t Le 1 B(x,9θ M t) (y)%% n Br Bi ,t e % r Bi t  2 M %   %r 1   r Bi % dy r % −t 2 L bi % %e 1 (y)  B(x,9θ M t) % tn % t2 r Bi Cl (B(x,9θ M t)) l≥1  2 M %   % p1  p1  ln  r Bi % % −t 2 L 1 bi % % e 1 B(x,9θ M t) (y)% dw  2r − % 2 t r Bi Cl (B(x,9θ M t))  

α

r B2 i

M

l≥1



t2 l≥1  2 M  r Bi t2

e

−c4l

1 % %p  p % bi (y) %  % % dw % % B(x,9θ M t) r Bi

 −

w(Bi ) w(B(x, 9θ M t))

1

 p

.

Therefore, changing the variable t into tθ M and noticing that, for x ∈ C j (Bi ) and t>

2 j−2 r Bi θM

123

, we have that Bi ⊂ B(x, 9θ M t), using the estimate above, we get

Weighted Hardy Spaces Associated with Elliptic Operators

 III i2j





2 j−2 r

C j (Bi )

θM

dy dt t n+1 ⎛  αw(2 j+1 Bi )

1 p1



∞

⎝

Bi

B(x,θ M t)



Bi

 p1

 p1 2

%    %2 % % 2 −t 2 L bi −t 2 L %t Le e (y)%% B r Bi ,t % r

1

w(x)dx 

∞ 2 j−2 r B i θM

r B2 i t2

2M

⎞1 2

1 dt ⎠  αw(2 j+1 Bi ) p1 2− j2M . t

1

This and (9.19) imply that III i j  αw(2 j+1 Bi ) p1 2− j (2M−θ1 −2θ2 ) . Therefore, in view of (9.18), and by (2.2) and (9.13) with p = p1 , taking 2M > npr 1 + θ1 + 2θ2 , we obtain that ⎛ ⎞ p1 1   $ #  np   1 ⎝ sup w(Bi ) inf Mw (|u| p1 )(x) p1 2− j 2M− r −θ1 −2θ2 ⎠ I I I2  u

 

u

=1 p L 1 (w)

sup

 

=1 p L 1 (w)

1 α p0

i∈N



Rn

x∈Bi

i∈N

j≥4

 1  Mw (|u| p1 )(x) p1 w(x)dx

 p1

Bi

|∇h(x)| p0 w(x)dx.

 Plugging this and (9.17) into (9.16) gives us I I I  α − p0 Rn |∇h(x)| p0 w(x)dx. Hence, by this, (9.15), (9.12) and (9.11), weconclude (9.5).  p− (L) p+ (L) To complete the proof note that for max rw , nrnrww+ p− (L) < p < sw and q ∈ p

Ww (q− (L), q+ (L)), if we take f ∈ H∇ L −1/2 ,q (w), by (9.1), we have that 1

SH f  L p (w)  ∇ L − 2 f  L p (w) , p

consequently f ∈ HSH ,q (w).

 

9.2 Proof of Proposition 9.2 Given w ∈ A∞ satisfying that Ww (q− (L), q+ (L)) = ∅, and q ∈ Ww (q− (L), q+ (L)) p ⊂ Ww ( p− (L), p+ (L)), if we take p ∈ Ww (q− (L), q+ (L)) and f ∈ HSH ,q (w), applying Theorems 2.2 and 1.1, we obtain 1

∇ L − 2 f  L p (w)   f  L p (w) ≈ SH f  L p (w) =  f H p

SH ,q (w)

.

(9.20)

On the other hand, for 0 < p ≤ 1 by Propositions 4.8, part (b) and 7.4, part (a) (see also [33, Proposition 5.1, part (a)], where the case p = 1 was considered), we have

123

C. Prisuelos-Arribas 1

∇ L − 2 f  L p (w)   f H p

SH ,q (w)

.

Therefore, applying Theorem 5.1 with p0 = 1 and p1 ∈ Ww (q− (L), q+ (L)), we p conclude, for all 0 < p < q+sw(L) , q ∈ Ww (q− (L), q+ (L)) and f ∈ HSH ,q (w), 1

∇ L − 2 f  L p (w)  SH f  L p (w) , p

consequently f ∈ H∇ L −1/2 ,q (w).

 

Acknowledgements I want to thank my advisor José María Martell for his useful comments and corrections, Li Chen for some conversations and help with references, and Pascal Auscher for some valuable comments. Funding The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Agreement No. 615112 HAPDEGMT. The author acknowledges receiving financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554).

References 1. Amenta, A.: Tent spaces over metric measure spaces under doubling and related assumptions. Operator Theory: Advances and Applications, “Operator Theory in Harmonic and Non-commutative Analysis”, vol. 240, pp. 1–29 (2014) 2. Auscher, P.: On necessary and sufficient conditions for L p estimates of Riesz transform associated to elliptic operators on Rn and related estimates. Mem. Am. Math. Soc. 186(871) (2007) 3. Auscher, P., Duong, X.T., McIntosh, A.: Boundedness of Banach space valued singular integral operators and applications to Hardy spaces (unpublished manuscript) 4. Auscher, P., Martell, J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: general operator theory and weights. Adv. Math. 212, 225–276 (2007) 5. Auscher, P., Martell, J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part II: off-diagonal estimates on spaces of homogeneous type. J. Evol. Equ. 7(2), 265–316 (2007) 6. Auscher, P., Martell, J.M.: Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: harmonic analysis of elliptic operators. J. Funct. Anal. 241, 703–746 (2006) 7. Auscher, P., McIntosh, A., Russ, E.: Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 18(1), 192–248 (2008) 8. Auscher, P., Russ, E.: Hardy spaces and divergence operators on strongly Lipschitz domain of Rn . J. Funct. Anal. 201(1), 148–184 (2003) 9. Auscher, P., Tchamitchian, P.: Calcul fonctionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux). Ann. Inst. Fourier 45(3), 721–778 (1995) 10. Badr, N.: Real interpolation of Sobolev spaces. Math. Scand. 31(4), 235–264 (2009) 11. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976) p 12. Bernal, A.: Some results on complex interpolation of Tq spaces, interpolation spaces and related topics (Ramat-Gan). Isl. Math. Conf. Proc. 5, 1–10 (1992) 13. Bui, T.A., Cao, J., Ky, L.D., Yang, D., Yang, S.: Weighted Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Taiwan. J. Math. 17(4), 1127–1166 (2013) 14. Bui, T.A., Cao, J., Ky, L.D., Yang, D., Yang, S.: Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Anal. Geom. Metr. Spaces 1, 69–129 (2013) 15. Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)

123

Weighted Hardy Spaces Associated with Elliptic Operators 16. Cao, J., Chang, D.-C., Fu, Z., Yang, D.: Real interpolation of weighted tent spaces. Appl. Anal. 95(11), 2415–2443 (2015) 17. Coifman, R.R., Meyer, Y.: Ondelettes et opérateurs, vol. 3. Hermann, Paris (1990) 18. Coifman, R.R., Meyer, Y., Stein, E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62(2), 304–335 (1985) 19. Cruz-Uribe, D.V., Martell, J.M., Pérez, C.: Weights Extrapolation and the Theory of Rubio de Francia. Operator Theory: Advances and Applications. Birkhäuser/Springer Basel AG, Basel (2011) 20. Duoandikoetxea, J.: Fourier Analysis. Graduate Students Mathematics, vol. 29. American Mathematical Society, Providence (2000) 21. García-Cuerva, J., Rubio de Francia, J.: Weighted Norm Inequalities and Related Topics. North Holland, Amsterdam (1985) 22. Grafakos, L.: Classical Fourier analysis, 2nd edn. Graduate Texts in Mathematics, vol. 249. Springer, New York (2008) 23. Grafakos, L.: Modern Fourier Analysis, 2nd edn. Graduate Texts in Mathematics, vol. 250, Springer, New York (2009) 24. Harboure, E., Torrea, J., Viviani, B.: A vector-valued approach to tent spaces. J. Anal. Math. 56, 125–140 (1991) 25. Hofmann, S., Lu, G., Mitrea, D., Mitrea, M., Yan, L.: Hardy spaces associated to non-negative selfadjoint operators satisfying Davies-Gaffney estimates. Mem. Am. Math. Soc., vol. 214(1007) (third of 5 numbers) (2011) 26. Hofmann, S., Martell, J.M.: L p bounds for Riesz transforms and square roots associated to second order elliptic operators. Publ. Mat. 47(2), 497–515 (2003) 27. Hofmann, S., Mayboroda, S.: Hardy and B M O spaces to divergence form elliptic operators. Math. Ann. 344(1), 37–116 (2009) 28. Hofmann, S., Mayboroda, S., McIntosh, A.: Second order elliptic operators with complex bounded measurable coefficients in L p , Sobolev and Hardy spaces. Ann. Sci. École. Norm. Sup. 44(5), 723–800 (2011) 29. Jiang, R., Yang, D.: New Orlicz-Hardy spaces associated with divergence form elliptic operators. J. Funct. Anal. 258(4), 1167–1224 (2010) 30. Kato, T.: Perturbation theory for linear operators. Springer, New York (1966) 31. Latter, R.H.: A characterization of H p (Rn ) in terms of atoms. Stud. Math. 62, 93–101 (1978) 32. Martell, J.M., Prisuelos-Arribas, C.: Weighted Hardy spaces associated with elliptic operators. Part I: weighted norm inequalities for conical square functions. To appear in Trans. Am. Soc. arXiv:1406.6285 33. Martell, J.M., Prisuelos-Arribas, C.: Weighted hardy spaces associated with elliptic operators. Part II: characterizations of HL1 (w). To appear in Publ. Mat. arXiv:1701.00920 34. Russ, E.: The atomic decomposition for tent spaces on spaces of homogeneous type. In: Proceedings of the Centre for Mathematical Analysis. Australian National University, 42. Australian National University, Canberra (2007)

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