Positivity https://doi.org/10.1007/s11117-018-0564-7

Positivity

Extrapolation results in grand Lebesgue spaces defined on product sets Vakhtang Kokilashvili1 · Alexander Meskhi1,2

Received: 26 March 2017 / Accepted: 3 February 2018 © Springer International Publishing AG, part of Springer Nature 2018

Abstract Extrapolation results in weighted grand Lebesgue spaces defined with respect to product measure μ × ν on X × Y , where (X, d, μ) and (Y, ρ, ν) are spaces of homogeneous type, are obtained. As applications of the derived results we prove new one-weight estimates for multiple integral operators such as strong maximal, Calderón–Zygmund and fractional integral operators with product kernels in these spaces. Keywords Weighted extrapolation · Grand Lebesgue spaces · Strong maximal operators · Multiple integral operators · Calderón–Zygmund operators with product kernels · Fractional integrals with product kernels Mathematics Subject Classification 46E30 · 42B20 · 42B25

1 Introduction Let (X, d, μ) and (Y, ρ, ν) be quasi–metric measure spaces with doubling measures. We establish weighted extrapolation results in grand Lebesgue spaces defined with respect to the product measure μ × ν on X × Y . We show that if the one-weight

B

Alexander Meskhi [email protected] Vakhtang Kokilashvili [email protected]

1

Department of Mathematical Analysis, A. Razmadze Mathematical Institute, I. Javakhishvili Tbilisi State University, 6, Tamarashvili Str., 0177 Tbilisi, Georgia

2

Department of Mathematics, Faculty of Informatics and Control Systems, Georgian Technical University, 77, Kostava St., Tbilisi, Georgia

V. Kokilashvilli, A. Meskhi

inequality holds in the classical weighted Lebesgue space for all weights from the (S) ”strong” Muckenhoupt class Ar defined with respect to products of balls B1 × B2 , then the one-weight estimate is also valid in grand Lebesgue spaces for all weights (S) from the same class Ar . Our results cover both cases: when a weight is written as a measure, or appears in the grand Lebesgue norm. Based on these results we prove one-weight estimates for strong maximal operators, Calderón–Zygmund and potential operators with product kernels. In 1992 Iwaniec and Sbordone [14] introduced grand Lebesgue space L p) (Ω) defined on a bounded domain Ω. Their studies were related with the integrability properties of the Jacobian under minimal hypothesis. A generalized version of the grand Lebesgue space L p),θ (Ω), θ > 0, was appeared in the paper by Greco et al. [10]. During the last decade Harmonic Analysis related to these spaces and their associates spaces (called small Lebesgues spaces) were intensively studied by various authors. Structural properties of grand Lebesgue spaces were investigated in the papers [1,7]. In [8] the authors proved that for the boundedness of the Hardy–Littlewood p) maximal operator defined on [0, 1] in weighted grand Lebesgue spaces L w ([0, 1]) it is necessary and sufficient that the weight w belongs to the Muckenhoupt’s class A p ([0, 1]). The same phenomenon was noticed by the authors of this paper for the Hilbert transform in [20]. We refer to the papers [17,19,30] for one–weight results regarding maximal functions and singular integrals of various type in these spaces. In [28] a variant of the one-weight boundedness for fractional integral operators in grand Lebesgue spaces was established. In that paper the author determined also values of the second parameter for grand Lebesgue spaces governing the boundedness of fractional integral operator in these spaces. We refer also the recent monograph [23] for these and related topics. The paper [21] deals with the boundedness of multiple fractional integral operators p),θ in L w ([0, 1] × [0, 1]) spaces. It should be mentioned that extrapolation theorem (see Theorem 3.2 below) enables us to formulate the boundedness of multiple potential p),θ p),θ operators in new weighted Lw (X ×Y ) spaces which are different from L w (X ×Y ). The results derived in this manuscript can be can be considered as a continuation of the investigation carried out in the recent paper [22], where the extrapolation result p),θ was derived for grand Lebesgue spaces L w (X ).

2 Preliminaries Let X be a set and let d : X ×Y → R+ be a quasi–metric, i.e., d satisfies the following conditions: (i) d(x, y) = 0 if and only if x = y; (ii) There is a finite constant κ > 1 such that for all x, y, z ∈ X , d(x, y) ≤ κ(d(x, z) + d(z, y)); (iii) d(x, y) = d(y, x) for every x, y ∈ X . Let x ∈ X and r > 0. As usual, B(x, r ) := {y ∈ X : d(x, y) < r } denotes a ball with center x and radius r . Suppose that μ is a measure defined on a σ -algebra of

Extrapolation results in grand Lebesgue spaces defined on…

subsets that contains the balls of X . Then the triple X := (X, d, μ) is called a quasi– metric measure space. If μ satisfies the doubling condition μ(B(x, 2r )) ≤ Cμ(B(x, r )),

(1)

with the positive constant C independent of x ∈ X and r > 0, then (X, d, μ) is called a space of homogeneous type (S H T briefly). Throughout the paper we will assume that (X, d, μ) is an S H T , μ is a finite measure (μ(X ) < ∞), μ(B(x, r )) > 0 and μ{x} = 0 for all x ∈ X and r > 0. We will also assume that the class of continuous functions with compact supports is dense in L 1 (X ). For the definition, examples and some properties of an S H T see the paper [27] and the monographs [2,31]. If Cμ is the smallest constant for which (1) holds, then the constant Dμ := log2 Cμ

(2)

is called the doubling order of μ. Suppose that (Y, ρ, ν) be another SHT with finite measure ν having the same restrictions as (X, d, μ) (see above). Suppose that μ × ν be the product measure on X × Y . Let w(x, y) be a weight on X × Y , i.e. w is a.e. positive integrable function on X × Y . Assume that 1 ≤ r < ∞. We denote by L rw (X × Y ) the weighted Lebesgue space defined by the norm: ⎛  f  L rw (X ×Y ) = ⎝

⎞1/r



| f (x, y)|r w(x, y) dμ × ν ⎠

.

X ×Y

Let μ(X ) < ∞, 1 < p < ∞ and let ϕ be a function defined on (0, p − 1) that is non-decreasing on (0, σ ) for some small positive σ < p−1. The generalized weighted p),ϕ grand Lebesgue space L w (X × Y ) is the class of those f : X × Y → R for which the norm ⎛  f  L p),ϕ (X ×Y ) = w

⎝ϕ(ε)

sup 0<ε< p−1



⎞1/( p−ε) | f (x, y)| p−ε w(x, y) dμ × ν ⎠

X ×Y p),ϕ

is finite. For w = const we denote L w (X × Y ) by L p),ϕ (X × Y ). If ϕ(ε) = ε, then L p),ϕ is the Iwaniec–Sbordone [14] space denoted by L p) , while for ϕ(ε) = εθ , θ > 0, this space was introduced by Greco, Iwaniec and Sbordine in [10]. It is easy to check (see also [8]) that, in general, w 1/ p f  L p,ϕ (Ω) =  f  L wp,ϕ (Ω) . p),ϕ

p),ϕ

That is why together with L w (X × Y ) we are interested in the space Lw (X × Y ) which is defined by the norm  f L p),ϕ (X ×Y ) = w f  L p),ϕ (X ×Y ) . w

V. Kokilashvilli, A. Meskhi

If ϕ(x) = x θ , where θ is a positive number, then we denote L p),ϕ (X × Y ) (resp. p),θ × Y )) by L p),θ (X × Y ) (resp. by L w (X × Y )). We do the same when we p),ϕ deal with the spaces L p),ϕ (X × Y ) and Lw (X × Y ) for ϕ(x) = x θ . It is well-known that the space L p),θ (Ω) is a Banach space (see e.g. [7]). The following continuous embeddings hold: p),ϕ L w (X

L wp → L wp),θ1 → L wp),θ2 → L wp−ε , where 0 < ε ≤ p − 1 and θ1 < θ2 . Let 1 < r < ∞. We say that a weight function w defined on X × Y belongs to the (S) Muckenhoupt class Ar if ⎛ ⎜ [w] A(S) := sup ⎝ r B1 ×B2

1 μ(B1 )ν(B2 )

⎛ ⎜ ⎝

1 μ(B1 )ν(B2 )





⎞ ⎟ w dμ × ν ⎠

B1 ×B2

⎞r −1

⎟ w 1−r dμ × ν ⎠

< ∞,

B1 ×B2

where the supremum is taken over all products of balls B1 × B2 ⊂ X × Y . (S) Further, we say that w ∈ A1 (X × Y ) if (M (S) w)(x, y) ≤ Cw(x, y),

f or μ × ν − a.e. (x, y),

(3)

where M (S) is the strong Hardy–Littlewood maximal operator defined on X × Y , i.e., M

(S)

g(x, y) =

sup

B1 x,B2 y

1 (μB1 )(ν B2 )

 |g(t, s)| dμ × ν(t, s). B1 ×B2

We denote by [w] A(S) the best possible constant in (3). 1 Some of the next auxiliary statements can be found in the book [9] for weights defined on products of Euclidean spaces. Since we deal with S H T , we give proofs completely. Lemma 1 Let 1 ≤ s < r < ∞. Then [w] A(S) ≤ [w] A(S) , r

(4)

s

Proof If s = 1, then the result follows from the obvious estimate 1 μ(B1 )ν(B2 )

 w(t, s) dμ × ν ≤ C ess inf w(x, y) B1 ×B2

B1 ×B2

(5)

Extrapolation results in grand Lebesgue spaces defined on…

which is true due to (3). Indeed, ⎛ ⎜ ⎝

1 μ(B1 )ν(B2 )

=



⎞ p−1



⎟ w(t, s) dμ × ν ⎠

B1 ×B2

ess inf w(x, y)

−1

B1 ×B2

≤ ess sup w(x, y)−1 B1 ×B2

⎛ ⎜ ≤C⎝

1 μ(B1 )ν(B2 )

⎞−1



⎟ w(t, s) dμ × ν ⎠

.

B1 ×B2

The proof for s > 1 follows immediately from Hölder’s inequality. The next relation can be checked immediately

[w 1− p ] A(S) = [w] p

p −1 (S) , Ap

1 < p < ∞,

(S)

(6) (S)

(S)

Further, by the definition the class A∞ is defined as follows A∞ := supr ≥1 Ar . Let 1 < p, q < ∞. Suppose that ρ is μ-a.e. positive function such that ρ q is locally (S) integrable. We say that ρ ∈ A p,q if ⎛ ⎜ [ρ]A(S) := sup ⎝ p,q B1 ×B2

1 μ(B1 )ν(B2 )

⎛ ⎝

1 μ(B1 )ν(B2 )





⎞ ⎟ ρ q dμ × ν ⎠

B1 ×B1

⎞q/ p



ρ − p dμ × ν ⎠

< ∞,

B

where the supremum is taken over all products of balls B1 × B2 ∈ X × Y . (S) (S) If p = q, then we denote A p,q by A p . The next relation can be checked immediately [ρ]A(S) = [ρ q ] A(S)

1+q/ p

p,q

, 1 < p ≤ q < ∞.

(7)

Equality (7) for p = q has the form [ρ]A(S) = [ρ p ] A(S) , 1 < p < ∞. p

p

Since the Lebesgue differentiation theorem holds in (X, d, μ), it can be checked that [w] A(S) ≥ 1; [ρ]A(S) ≥ 1. p p,q

V. Kokilashvilli, A. Meskhi

Let us recall that if 1 M f (x) = sup μ(B) Bx

 | f (t)|dμ(t) B

is the Hardy–Littlewood maximal operator defined on an SHT (X, d, μ), then there is a structural constant c such that the following Buckley type estimate holds M L wp (X )→L wp (X ) ≤ c p [w] A p (X ) , 1 < p < ∞, 1/( p−1)

(8)

where [w] A p (X ) is the A p characteristic of a weight w defined on X (see [13]). In fact, in [13] the authors established more general bound for M L wp (X )→L wp (X ) involving A∞ characteristic. It can be observed in [13] that

where

c = 32κ Dμ (2θ ) Dμ (1 + τκ,μ ),

(9)

 D τκ,μ = 6 32κ 4 (4κ + 1) μ ,

(10)

θ = 4κ 2 + κ, Dμ is defined by (2), κ is the triangle inequality constant for the quasi-metric d. Together with the characteristic [w] A(S) we are interested in the following characr teristics: ⎛ ⎞  ⎜ 1 ⎟ [w] A(S1) = ess sup sup ⎝ w(x, y) dμ(x)⎠ r μ(B1 ) y∈B2 B1 B1

⎛ ⎜ ⎝

1 μ(B1 )

 B1

⎟ w 1−r (x, y) dμ(x)⎠

⎛

⎜ [w] A(S2) = ess sup sup ⎝ r

x∈B1

⎛ ⎜ ⎝

1 ν(B2 )

B2



⎞r −1

1 ν(B2 )

 B2

< ∞, ⎞

⎟ w(x, y)dν(y)⎠ ⎞r −1

⎟ w 1−r (x, y) dν(y)⎠

< ∞.

B2

Because of the Lebesgue differentiation theorem and the boundedness of the operator M (S) in L p (X × Y ) (see also [6,16,24]) that there is a positive constant C p depending only on p such that    p −1

. max [w] A(S1) , [w] A(S2) ≤ [w] A(S) ≤ C p [w] A(S1) [w] A(S2) r

r

r

r

r

(11)

Extrapolation results in grand Lebesgue spaces defined on…

Lemma 2 Let 1 < p < ∞. Then M (S)  L wp (X ×Y )→L wp (X ×Y ) ≤ (c p )2 [w]

2/( p−1) , (S) Ap

1 < p < ∞,

where c is defined by (9). Using estimate (8) for each variable separately, the iteration process, estimates (8) and (11) we conclude that Lemma 2 is proved.

Lemma 3 Let 1 < p < ∞ and let w be a weight on X × X . We denote σ = w1− p . If w ∈ A(S) p , Then there is a positive constant ε0 depending on w and p such that (S) w ∈ A p−ε0 . (S1) and w ∈ A(S2) Proof Let w ∈ A(S) p . Then by (11) we have that w ∈ A p p . Hence, by Theorem 1.2 of [13] we have that w(·, y) ∈ A p−σ0 (y) and w(x, ·) ∈ A p−σ0 (x) , for ν- a.e. y and μ- a.e. x respectively, where A p−σ0 (x) and A p−σ0 (y) are Muckenhoupt classes but defined with respect to the first and the second variables respectively,

σ0 (y) =

p−1 p−1 ; σ0 (x) = , 1 + τk,μ [σ (·, y)] A(1) 1 + τk,μ [σ (x, ·)] A(2) ∞

∞

and [σ (·, y)] A(1) , [σ (x, ·)] A(2) are A∞ characteristics of w with respect to the first and ∞ ∞ the second variables respectively. Here τκ,μ is the structural constant defined in (10). Further, since (see [13]), [ρ] A∞ (X ) ≤ C[ρ] A p (X ) for a weight ρ ∈ A p (X ) and with a structural constant C, we have that there are positive structural constants C1 and C2 such that [σ (·, y)] A(1) ≤ C1 [σ (·, y)] A(S1) = C1 [w(·, y)] ∞

p

p −1 (S1) Ap



p −1 ≤ C1 [w] A(S1)

p −1 (S2) Ap



p −1 ≤ C1 [w] A(S2)

p



p −1 ≤ C1 [w] A(S) p

and [σ (x, ·)] A(2) ≤ C2 [σ (x, ·)] A(S2) = C2 [w(x, ·)] ∞

p



p −1 ≤ C2 [w] A(S) . p

Hence, (S)

w ∈ A p−ε0 ,

p

V. Kokilashvilli, A. Meskhi

where ε0 =

p−1 1 + τk,μ [w]

p −1 (S) Ap

.

(S)

For the openness property of the class A p defined with respect to rectangles in Rn see [26]. We now formulate Rubio de Francia’s (see [29]) extrapolation statements with weights in the classical Lebesgue spaces. The following theorems were proved in [4] in the case of Euclidean spaces (see [22] for an SHT) but since we are interested in the Lebesgue spaces defined on product sets/spaces, we give the proofs for completeness. We refer also to [3,11,25] for related topics. Theorem A (Diagonal case) Let for some family F(X × Y ) of pairs of non-negative functions ( f, g), where f and g are defined on X × Y , for p0 ∈ [1, ∞), and for all (S) w ∈ A p0 we have ⎛ ⎝

⎞



1 p0

g p0 (x, y)w(x, y) dμ × ν(x, y)⎠

X ×Y

⎛   ≤ C N [w] A(S) ⎝ p0

⎞

1 p0

f p0 (x, y)w(x, y) dμ × ν(x, y)⎠

,

(12)

X ×Y

where N is a non-decreasing function and the constant C does not depend on w. Then (S) for all 1 < p < ∞, all w ∈ A p and all ( f, g) ∈ F(X × Y ), the inequality ⎛ ⎝



⎞1/ p g p (x, y)w(x, y) dμ × ν(x, y)⎠

X ×Y

⎛

K (w) ⎝ ≤ CC



⎞1/ p f p (x, y)w(x, y) dμ × ν(x, y)⎠

(13)

X ×Y

holds, where C is the same constant as in (12), 

 = max 2, 2( p− p0 )/( pp0 − p0 ) , C

(14)

and ⎧   p −p ⎪ 2M (S)  L wp (X ×Y )→L wp (X ×Y ) 0 , p < p0 ⎨ N [w] A(S) p p0 −1 p− p K (w) = . (15)  0  p−1 p−1 , p > p ⎪ 2M (S)  p ⎩ N [w] (S) p 0 A p (X )

L

w1− p

(X )→L 1− p (X ×Y ) w

Extrapolation results in grand Lebesgue spaces defined on…

Remark 1 Using Lemma 2, we can obtain the following estimate for K (w) in (15): ⎧  ⎨ N 2 p0 − p (c¯ p )2( p0 − p) [w](2((S)p −1)+1)( p0 − p) p < p0 Ap K (w) ≤  ( p− p )/( p−1) , (2 p− p −1)( p−1) 0 2( p− p )/( p−1) 0 0 ⎩ 2 (c¯ p ) [w] (S) p > p0 Ap

where c¯ is the same constant as in Lemma 2, i.e. it does not depend on p and w. Theorem B (Off-diagonal case) Let 0 < q0 < ∞. Assume that for some family F(X × Y ) of pairs of non-negative functions ( f, g), for p0 ∈ [1, ∞), and for all (S) w ∈ A p0 ,q0 the inequality ⎛ ⎝



⎞1

q0

g (x, y)w (x, y) dμ × ν ⎠ q0

q0





X ×Y

⎛

≤ C N [w]A(S) ⎝ p0 ,q0

⎞



f p0 (x, y)w p0 (x, y) dμ × ν ⎠

1 p0

,

(16)

X ×Y

holds, where N is a non-decreasing function and the constant C does not depend on w. Then for all 1 < p < ∞, 0 < q < ∞ such that 1 1 1 1 − = − , q0 q p0 p (S)

all w ∈ A p,q and all ( f, g) ∈ F(X × Y ) we have ⎛ ⎝



⎞1

⎛ ⎞1 p  q q p p ⎠ ⎝ ⎠ g (x, y)w (x, y) dμ × ν ≤ CC K (w) f (x, y)w (x, y) dμ×ν , q

X ×Y

X

(17)

where C is the same constant as in (16),     γq 1− 1 γ p p1 − 1p 0 , C := max 2 q q0 , 2

(18)

and

K (w) =

⎧   γ (q−q0 ) ⎪ N [w]A(S) 2M (S)  L γ qq (X ×Y )→L γ qq (X ×Y ) , q < q0 ⎪ ⎨ p,q w w γ q0 −1   ⎪ γ q−1 ⎪ ⎩ N [w] (S) 2M (S)  L γ p

A p,q

where γ :=

1 q0

+

1 p0

=

1 q

w− p

+

1 p .

γ p

(X ×Y )→L − p (X ×Y ) w

γ (q−q0 ) γ q−1 , q > q0

, (19)

V. Kokilashvilli, A. Meskhi

Remark 2 Lemma 2 implies that for K (w) from (19) we have ⎧  γ q(q −q) p 2 γ (q−q0 ) q 1+2 p0 ⎪ 2 ⎪ 2c N 1 + [w ] , ⎪ (S) ⎪ A q (X ) ⎨ q 1+ p K (w) ≤  

2 γ (q−q0 ) ⎪ γ q−1 (2γ q−γ q0 −1)/(γ q−1) ⎪ q 2 q ⎪ N 2c , 1 + [w ] ⎪ (S) p ⎩ A

q < q0 , (20) q > q0 ,

q 1+ p

where c is the structural constant defined by (9). The proofs of Theorems A and B repeat the arguments of [4] but we give the details for completeness because we deal with products of spaces of homogeneous type, and besides that, we are interested in exact values of constants. The proofs are based on the following Lemmas (see [4] for Euclidean spaces): (S)

(S)

(S)

Lemma 4 (a) Let 1 ≤ p < p0 < ∞. If w ∈ A p and u ∈ A1 , then wu p− p0 ∈ A p0 and [wu p− p0 ] A(S) ≤ [w] A(S) [u] p0

p

(S)

p0 − p (S) . A1

(S)

(b) Let 1 < p0 < p < ∞. If w ∈ A p and u ∈ A1 , then (w p0 −1 u p− p0 )1/( p−1) is in A p(S) and 0

  (w p0 −1 u p− p0 )1/( p−1)

(S)

Ap

≤ [w]

p0 −1 p−1 (S) Ap

[u]

p− p0 p−1 (S) A1

.

The proof of this lemma relies on Hölder’s inequality and the estimates: 1 μ(B1 )ν(B2 )

 u(t, s) dμ × ν ≤ Mu(x, y) ≤ [u] A(S) u(x, y), 1

B

which holds for almost every (x, y) ∈ B1 × B2 . The following lemma is known as Rubio de Francia’s algorithm. p

Lemma 5 Suppose that p > 1. Let f ∈ L w (X × Y ) be a non-negative function and (S) (S) let w ∈ A(S) p . Let Mk be k-th iteration of M with M0 f = f . Define R f (x) =

∞  k=0

Mk(S) f (x)

 k . 2M (S)  L wp (X ×Y )→ L wp (X ×Y )

Then (i) f (x, y) ≤ R f (x, y) a.e., (ii) R f  L wp (X ×Y ) ≤ 2 f  L wp (X ×Y ) ;

Extrapolation results in grand Lebesgue spaces defined on… (S)

(iii) R f is the A1 weight with (S)

[R f ] A(S) ≤ 2Mk  L wp (X ×Y )→ L wp (X ×Y ) .

(21)

1

Proof Everything follows from the sublinearity property of the strong maximal operator, the definition of A1 (X ) condition and the obvious inequality: (S)

Mk  L wp (X ×Y )→ L wp (X ×Y ) ≤ M (S) kL p (X ×Y )→ L p (X ×Y ) . w

w

To prove Theorem A we follow the proof of Theorem 3.1 in [4]. p

Proof of Theorem A Proof of First suppose that p < p0 . Suppose that f ∈ L w (X ×Y ). Using the Hölder inequality, the Factorization Lemma 4 (part (a)) observe that since

(S) (S) (S) w ∈ A p and R f ∈ A1 , then by Lemma 4, w(R f ) p− p0 ∈ A p0 and estimate (21) we have   p( p− p0 ) p( p0 − p) g p w dμ × ν = g p w(R f ) p0 (R f ) p0 dμ × ν X ×Y

X ×Y

⎛

≤⎝

⎞



p p0

⎛ ⎝

g p0 w(R f ) p− p0 dμ×ν ⎠

X ×Y

p  ≤ 2 p C p N w[(R f ) p− p0 ] A(S) ⎛ ⎝

p0



 p− p p ≤ 2 p C p N [w] A(S) [(R f )] (S) 0 p

A1

p p0

(R f ) p w dμ × ν ⎠

X ×Y

⎞

p p0

f p0 w(R f ) p− p0 dμ × ν ⎠

X ×Y

⎞1−





⎛ ⎝



⎞1−

p p0

| f | p w dμ × ν ⎠

X ×Y

| f | p w dμ × ν X ×Y

  p −p p ≤ 2 C N [w] A(S) 2M L wp →L wp 0 | f | p w dμ × ν. p

p

p

X ×Y p/( p− p )

0 Let now p > p0 . Let h ∈ L w (X ) with the norm equal to 1. Define H so that p p 1− p p/( p− p ) 0 =h w. Then H is in L 1− p (X ) with norm equal to 1. Using the H w w pointwise estimate H ≤ RH a.e., the factorization Lemma 4 (part (b)) we have



 g p0 hwdμ × ν ≤

X ×Y

X ×Y

g p0 w( p0 −1)/( p−1) (RH)( p− p0 )/( p−1) dμ × ν

  p ≤ C p0 N w( p0 −1)/( p−1) (RH)( p− p0 )/( p−1) (S) 0 A p0  f p0 w( p0 −1)/( p−1) (RH)( p− p0 )/( p−1) dμ × ν X ×Y

V. Kokilashvilli, A. Meskhi

 ( p0 −1)/( p−1) ≤ C p0 N [w] (S) (2M (S)  ⎛ ×⎝

Ap



f p w dμ × ν ⎠

X ×Y

≤2

p− p0 p−1

⎛ ×⎝

⎞ p0 / p ⎛

p0

C

N



⎝

)( p− p0 )/( p−1)

p p  → L 1− p w1− p w



p

(RH) w

p0

⎞1− p0 / p

1− p

dμ × ν ⎠

X ×Y

p ( p0 −1)/( p−1) [w] (S) (2M (S)  p )( p− p0 )/( p−1) 0 p L 1− p  → L 1− p Ap ⎞ p0 / p ⎛



L

⎝

f p w dμ × ν ⎠

X ×Y

w



p

w

H w

1− p

⎞1− p0 / p

dμ × ν ⎠

.

X ×Y



Taking now the supremum with respect to h we are done. Proof of Theorem B First observe that [w]A(S) On the other hand, [w]A(S)

p0 ,q0

=

[wq0 ]

(S)

A q0 γ

.

p0 ,q0

=

[wq0 ]

(S)

A q0 γ

, where γ :=

1 q0

+

1 . p0

p

Case q < q0 . In this case we have p < p0 . Suppose that f ∈ L w p (X × Y ). Define qγ H so that H qγ wq = f p w p . Then H is in L wq (X × Y ). We construct RH by the Rubio de Francia’s algorithm and apply Hölder’s inequality. Then   (gw)q dμ × ν = (gw)q (RH)qγ (q−q0 )/q0 (RH)qγ (q0 −q)/q0 dμ × ν X ×Y

X ×Y

⎛

≤⎝

⎞q/q0



g q0 wq (RH)γ (q−q0 ) dμ × ν ⎠

X ×Y

⎛ ⎝





(RH)γ w

q

⎞1−q/q0 dμ × ν ⎠

.

(22)

X ×Y (S)

By using Lemma 4 we find that wq (RH)γ (q−q0 ) ∈ Aq0 γ and that   wq (RH)γ (q−q0 )

(S)

A q0 γ

≤ [wq ] A(S) [(RH)] qγ

γ (q0 −q) . (S) A1

Applying now (16) we find that ⎛ ⎝



⎞1/q0 g q0 wq (RH)γ (q−q0 ) dμ × ν ⎠

X ×Y

≤ CN ⎛ ×⎝

  wr (RH)γ (q−q0 ) 

X ×Y

(S) A q0 γ

⎞1/ p0

f p0 wq (RH)γ (q−q0 ) dμ × ν ⎠

.

Extrapolation results in grand Lebesgue spaces defined on…

Taking the latter estimate into account in (22) and using the estimates (RH)−1 ≤ p/qγ RH L γ qq (X ×Y ) ≤ 2H  L γ qq (X ×Y ) = 2 f  L p (X ×Y ) , [RH] A(S) ≤ 2

H −1 ,

w

w

wp

1

M (S)  L γ qq (X )→L γ qq (X ) which follow from the Rubio de Francia’s algorithm, we see w w that ⎛ ⎝



⎞1/q (gw)q dμ × ν ⎠

≤ 2γ (1−q/q0 ) C N [wq ] A(S) (2M (S)  L γ qq (X ×Y ) )γ (q−q0 )

X ×Y

w

qγ

⎛ ⎝

⎞1/ p



( f w) p dμ × ν ⎠

.

X ×Y

Applying equality (8) we get the bound (20). Case q > q0 . It is easy to see that in this case we have p > p0 . Now we use the q/(q−q0 ) (X × Y ) with the duality arguments. Let h be a non-negative function h ∈ L wq  − p  p γ −1 p γ norm equal to 1. Observe that [w]A(S) = w (S) . Define H ∈ L − p by setting w

A p γ

p,q

p γ H − p = h q/(q−q0 ) wq . If we define RH By the Rubio de Francia’s algorithm, then RH w is a A(S) 1 weight and, moreover, H ≤ RH. Consequently,







q−q0 q g q0 H p γ w −( p +q) wq dμ × ν

g q0 hwq dμ × ν = X ×Y

X ×Y



≤

γ (q−q0 ) qγ −1 g q0 wq0 p / p0 RH dμ × ν.

X ×Y q0 γ −1

Observe now that (wq ) qγ −1 (RH)

γ (q−q0 ) qγ −1

 q0 γ −1 γ (q−q0 )  (wq ) qγ −1 (RH) qγ −1

(S)

is the Aq0 γ weight. Moreover,

(S)

A q0 γ

q0 γ −1   (q−q0 )γ qγ −1 ≤ [wr ] qγ(S)−1 RH (S) .

(23)

A1 (X )

Ar γ

Taking (16) into account we get  g q0 hwq dμ × ν ≤ C N X ×Y

 q0 γ −1 (q−q0 )γ  (wr ) qγ −1 (RH) qγ −1

⎛ ×⎝



X ×Y



f p0 w p ( p0 −1) (RH)

q0 A q0 γ

p0 γ (q−q0 ) q0 (qγ −1)

⎞q0 / p0 ⎠

.

V. Kokilashvilli, A. Meskhi

Using Hölder’s inequality with respect to the exponent p/ p0 and taking into account finally we conclude that the theinequality (23) and [RH] A(S) ≤ 2M (S)  p γ L

1

w− p

(X ×Y )



orem is proved. Equality (8) gives the bound also for this case.

3 Extrapolation in weighted grand Lebesgue spaces: main results In this section we derive extrapolation results in weighted grand Lebesgue spaces. As before, (X, d, μ) and (Y, ρ, ν) are spaces of homogeneous type with finite measures. Theorem 1 (Diagonal case) Suppose that p0 ∈ [1, ∞). Let F(X × Y ) be the class of pairs of non-negative functions defined on X × Y . Suppose that for all ( f, g) ∈ (S) F(X × Y ) and for all w ∈ A p0 the inequality ⎛ ⎝

⎞



g p0 w dμ × ν ⎠

1 p0

⎛   ⎝ ≤ C N [w] A(S) p0

X ×Y

⎞

1 p0

f p0 w dμ × ν ⎠

(24)

X ×Y

holds, where N is a non-decreasing function and the constant C does not depend on (S) w. Then for 1 < p < ∞, θ > 0, w ∈ A p we have g L p),θ (X ×Y ) ≤ C f  L p),θ (X ×Y ) , ( f, g) ∈ F(X × Y ), w

(25)

w

where C is the positive constant independent of ( f, g) ∈ F(X × Y ). Theorem 2 (Off-diagonal case) Let 1 < p0 < ∞ and let 1 < q0 < ∞. Assume that F(X × Y ) is the class of pairs of non-negative functions defined on X × Y . Let for all (S) ( f, g) ∈ F(X × Y ), for w ∈ A p0 ,q0 we have ⎛ ⎝



⎞1

q0

(gw)q0 dμ × ν ⎠

X ×Y





⎛

≤ C N [w]A(S) ⎝ p0 ,q0



⎞ ( f w) p0 dμ × ν ⎠

1 p0

,

(26)

X ×Y

where N is a non-decreasing function and the constant C does not depend on w. Then for 1 < p < ∞, 1 < q < ∞ such that 1 1 1 1 − = − , q0 p0 q p for θ > 0 and w ∈ A(S) p,q the inequality gLq),θq/ p (X ×Y ) ≤ C f L p),θ (X ×Y ) , ( f, g) ∈ F(X × Y ) w

w

holds, where the positive constant C is independent of ( f, g).

(27)

Extrapolation results in grand Lebesgue spaces defined on… (S)

Proof of Theorem 1 Let (24) hold for w ∈ A p0 . By Theorem A we have that ⎞1/r ⎛ ⎞1/r ⎛   ⎝ gr w dμ × ν ⎠ ≤ CC(w, r ) ⎝ f r w dμ × ν ⎠ , X ×Y

(28)

X ×Y

(S)

for all w ∈ Ar , where by Theorem A, (14), (15) and Remark 1,

 C(w, r ) ≤ max 2, 2( p− p0 )/( pp0 − p0 ) ⎧  (2/(r −1)+1)( p0 −r ) ⎨ N 2 p0 −r (cr ¯ )2( p0 −r ) [w] (S) r < p0 r  (r − p )/(r −1) 2(r − p A)/(r −1) [w](2r − p0 −1)/(r −1) r > p 0 0 ⎩ 2 (cr ¯ ) 0 (S)

,

Ar

the positive constant c does not depend on r and w, and the positive constant C is the same as in (12), i.e., does not depend on p and w. Let r = p − ε in (28). By Hölder’s inequality we have that ⎛ g L p),θ (X ×Y ) ≤ C sup ⎝εθ w

0<ε<ε0



⎞

1 p−ε

|g(x, y)| p−ε w(x, y) dμ × ν(x, y)⎠

X ×Y

⎛

≤ C sup C(w, p − ε) ⎝εθ 0<ε<ε0



⎞

1 p−ε

| f (x, y)| p−ε w(x, y) dμ × ν(x, y)⎠

X ×Y

≤ C sup C(w, p − ε) f  L p),θ (X ×Y ) , w

0<ε<ε0

where ε0 is sufficiently small and the constant C depends on ε0 , p0 and p. By Lemmas 1 and 3 we have that w ∈ A(S) p−ε0 if ε0 is sufficiently small. Since [w] ArS) ≥ 1 (this is true because of the Lebesgue differentiation theorem) and the inequality [w] A(S) ≤ [w] A(S)

p−ε0

p−ε

holds for ε < ε0 (see (4)), finally we have that sup C(w, p − ε) < ∞. 0<ε<ε0



To prove Theorem 2 we need to introduce some notation. Let 1 < p < q < ∞, ε0 ∈ (0, q − 1) and ε ∈ (0, ε0 ). We denote 

x −q +p Φ(x) = 1 − A(x − q) where number A is defined as follows: A=

1 1 − . p q

1−(x−q)A ,

(29)

V. Kokilashvilli, A. Meskhi

Further,

Ψ (x) := Φ(x θ ),

θ > 0.

(30)

Proof of Theorem 2 First we prove the theorem for p < q. Suppose that (26) holds for all w ∈ A p0 ,q0 (X ). q Let us take w ∈ A(S) . Consep,q . Then by (7) we have that [w]A p,q (X ) = [w ] A(S) quently, by using Lemma 3 we have that wq ∈ A

(S) q−ε 1+ ( p−η 0)

1+q/ p

for some positive ε0 and η0

0

satisfying the condition:

1 1 1 1 − = − := A. p − η0 q − ε0 p q Then wq ∈ A

(S) q−ε , 1+ ( p−η)

for all ε and η with the condition 0 < ε < ε0 , 0 < η < η0 , and 1 1 − = A. p−η q −ε Moreover,  q w A(S) 1+

q−ε ( p−η)

  ≤ wq A(S) 1+

q−ε0 ( p−η0 )

.

Further, Hölder’s inequality yields that [w]A(S)

p−ε,q−η

 1−ε/q ≤ wq (S) A

1+

q−ε ( p−η)

 q 1−ε/q = w q−ε (S)

A p−η,q−ε

.

Suppose that Φ and Ψ are defined as (29) and (30) respectively. Applying now Theorem B and (20) in which p (resp. q ) is replaced by p − η (resp. by q − ε) we have 1

gLq),Φ(ε) (X ×Y ) ≤ C sup Ψ (ε) q−ε wg L q−ε (X ×Y ) w

0<ε<σ0

θ

≤ C sup C1 (w, p − η, q − ε)η p−η w f  L p−η (X ×Y ) 0<η<σ1

≤ C f L p),θ (X ×Y ) , w

where C1 (w, p − η, q − ε) = C K (w), C and K (w) are defined by (18) and (19) respectively, but with p − η and q − ε instead of p and q respectively. In the latter ≥ 1 were used. This inequality estimate (20), Lemma 3 and the fact that [wq ] A(S) 1+q/ p

Extrapolation results in grand Lebesgue spaces defined on…

completes the proof of the case p < q. The proof of the theorem for p = q is similar to that of the previous one. In this case we follow the arguments of the proof of Theorem 1 and use the following easily verifiable chain of inequalities: [w p−ε ] A(S) ≤ [w p ] p−ε

1−ε/ p (S) A p−ε

≤ [w p ]

1−ε/ p (S) A p−ε 0

≤ [w p ] A(S) , p−ε0

(S) p where w p ∈ A(S) p and ε0 is taken so that w ∈ A p−ε0 (see Lemma 3).



4 Some applications in one-weight inequalities Now we give some applications of the extrapolation results in one-weight inequalities for some multiple integral operators. Suppose that k1 and k2 are Calderón–Zygmund kernels defined on X and Y respectively (see, e.g., [5], P. 503 for the definition of Calderón–Zygmund kernel on an SHT). Let us introduce the notation:



E ε,σ (x, y) := X \ B(x, ε) × X \ B(y, σ ) . Denote by K f the multiple Calderón–Zygmund singular integral of f :  K f (x, y) =

lim

(ε,σ )→(0,0) E ε,σ (x,y)

k1 (x, t)k2 (y, τ ) f (t, τ ) dμ × ν(t, τ ), (x, y) ∈ X ×Y.

Together with K we are interested in one-weight estimates for potential operator with product kernels (multiparameter fractional integral operator):  Iα1 ,α2 f (x, y) = X ×Y

f (t, τ ) dμ × ν(t, τ )  1−α1  1−α2 , μ B1 (x, d(x, t) ν B2 (y, ρ(y, τ )

(x, y) ∈ X × Y, 0 < α1 , α2 < 1. As we mentioned above the boundedness results for some multiple integral operap),θ tors operators are known in L w spaces (see [18,21,23]) but Theorem 2 enables us p),θ to formulate the boundedness of T in Lw (X × Y ), where T is M (S) or K or Iα1 ,α2 . To formulate the next results we introduce the following characteristics of weights ⎛ ⎜ [w]A(S1) = ess sup sup ⎝ r,s

y∈B2

⎛ ⎜ ⎝

1 μ(B1 )

B1

 B1

1 μ(B1 )

 B1

⎞ ⎟ wr (x, y)dμ(x)⎠ ⎞r/s

⎟ w −s (x, y) dμ(x)⎠

< ∞,

V. Kokilashvilli, A. Meskhi

⎛ ⎜ [w]A(S2) = ess sup sup ⎝ r,s x∈B1

B2

⎛ ⎜ ⎝

1 ν(B2 )



1 ν(B2 )



⎞ ⎟ wr (x, y)dν(y)⎠

B2

⎞r/s

⎟ w −s (x, y) dν(y)⎠

< ∞.

B2

If r = s, then we denote (S1) = Ar(S1) ; Ar,s

(S2) Ar,s = Ar(S2) . (S)

Theorem 3 Let 1 < p < ∞ and let θ > 0. Suppose that w ∈ A p . Then M (S) is p),θ bounded in Lw (X × Y ). (S)

Proof Let 1 < p0 < ∞ and let w ∈ A p0 . Since the inequality M (S)  L p0p

w 0

p

(X ×Y )→L w0p0 (X ×Y )

≤ (c( p0 ) )2 [w]

2/( p0 −1)

A(S) p0

holds (see Lemma 2) we conclude that Theorem 2 for diagonal case p = q completes the proof. (S)

Theorem 4 Let 1 < p < ∞ and let θ > 0. Let w ∈ A p . Suppose that there is an exponent r0 > 1 such that K f (x, y) exists for a.e. (x, y) ∈ (X × Y ) when f ∈ L r0 (X × Y ) and, besides that, K it is bounded in L r0 (X × Y ). Then the following inequality K f L p),θ (X ×Y ) ≤ C f L p),θ (X ×Y ) w

w

holds for all bounded f defined on X × Y . Proof Recall that the Calderón–Zygmund operator defined on X is bounded in p L w0p0 (X ) for 1 < p0 < ∞ and w p0 ∈ A p0 (X ) (see, e.g., [5,12], P. 503). Using this boundedness in each variable separately together with the iteration (see also [16] p (S) and [6] for Euclidean spaces) we find that K is bounded in L w0p0 (X × Y ) for w ∈ A p0 with desirable bounds for K  L p0p (X ×Y )→L p0p (X ×Y ) . In particular, the following w 0

inequality holds for all bounded f : K f  L p0p

w 0

(X ×Y )

w 0

≤ C N ([w]A(S) ) f  L p0p p0

w 0

(X ×Y )

for some non-decreasing function N . This is enough to apply Theorem 2 for the pairs ( f, K f ). The theorem is proved. Theorem 5 Let 1 < p < ∞ and let 0 < α < 1/ p. Suppose that θ > 0. We set p),θ q),θq/ p (S) p q = 1−αp . Let w ∈ A p,q . Then Iα,α is bounded from Lw (X ×Y ) to Lw (X ×Y ).

Extrapolation results in grand Lebesgue spaces defined on…

Proof It is known (see e.g., [5], P. 412) that for the fractional integral operator on X 

g(y)

1−α dμ(y), 0 < α < 1, μ(B(x, d(x, y))

Iα g(x) = X

p0 the weighted Sobolev inequality holds. More precisely, if q0 = 1−αp , 0 < α < 1/ p0 , 0 p0 q0 then Iα is bounded from L ρ p0 (X ) to L ρ q0 (X ) if and only if ρ ∈ A p0 ,q0 (X ), where A p0 ,q0 (X ) is the Muckenhoupt-Wheeden type class defined on X . Moreover (see [15])

Iα  L p0p

q

ρ 0

(X )→L ρ0q0 (X )

(1−α) max{1,( p0 ) /q0 } . 0 ,q0 (X )

≤ C[ρ]A p

Further, by using weighted Sobolev inequality for fractional integrals in each variable separately, the estimate max{[w]A(S1) , [w]A(S2) } ≤ [w]A(S) p0 ,q0

p0 ,q0

p0 ,q0

(S)

and generalized Minkowki’s inequality we have that the condition w ∈ A p0 ,q0 guarantees the estimate Iα,α  L p0p w

q

(X )→L w0q0 (X ) 0

≤ C[w]

2(1−α) max{1,( p0 ) /q0 }

A(S) p0 ,q0

.

Theorem 2 completes the proof. Let us define strong fractional maximal operator on X × Y : Mα(S) f (x, y) = 1 ,α2

sup

1

1−α1 ν(B )1−α2 2 (B1 ×B2 (x,y) μ(B1 )



B1 ×B2

| f (t, τ )|dtdτ, 0 < α1 , α2 < 1. (S)

(S)

Theorem 6 Let 1 < p < ∞ and let θ > 0. Let w ∈ A p (resp. w ∈ A p ). Then there is a positive constant C such that f  L p),θ (X ×Y ) Iα1 ,α2 f  L p),θ (X ×Y ) ≤ CMα(S) 1 ,α2 w

w

resp.

Iα1 ,α2 f L p),θ (X ×Y ) ≤ CMα(S) f  . p),θ ,α 1 2 Lw (X ×Y ) w Proof The result follows from Theorem 1 (resp. from Theorem 2) and the the inequality (31) Iα f  L rρ0 (X ) ≤ CMα f  L rρ0 (X ) , ρ ∈ A∞ (X ),

V. Kokilashvilli, A. Meskhi

for each variable separately, where A∞ (X ) = ∪r ≥1 Ar (X ) is the Muckenhoupt class of weights defined on X . Indeed, (31) holds for ρ = w with w ∈ A p0 (X ) (resp. ρ = w p0 with w ∈ A p0 (X )). Consequently, there is a positive constant C such that Iα1 ,α2 f  L ρp0 (X ) ≤ CMα(S) f  L ρp0 (X ) 1 ,α2 (S)

(32)

(S)

for ρ = w with w ∈ A p0 (resp. for ρ = w p0 with w ∈ A p0 ). Now Theorem 1 (resp. Theorem 2 completes the proof.

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