Math. Z. (2007) 255:133–159 DOI 10.1007/s00209-006-0017-z

Mathematische Zeitschrift

New Sobolev spaces via generalized Poincaré inequalities on metric measure spaces Lixin Yan · Dachun Yang

Received: 15 August 2005 / Accepted: 9 January 2006 / Published online: 11 July 2006 © Springer-Verlag 2006

Abstract In this paper, we introduce some new function spaces of Sobolev type on metric measure spaces. These new function spaces are defined by variants of Poincaré inequalities associated with generalized approximations of the identity, and they generalize the classical Sobolev spaces on Euclidean spaces. We then obtain two characterizations of these new Sobolev spaces including the characterization in terms of a variant of local sharp maximal functions associated with generalized approximations of the identity. For the well-known Hajłasz–Sobolev spaces on metric measure spaces, we also establish some new characterizations related to generalized approximations of the identity. Finally, we clarify the relations between the Sobolev-type spaces introduced in this paper and the Hajłasz–Sobolev spaces on metric measure spaces. Mathematics Subject Classification (2000) Primary 42B35; Secondary 46E35 · 43A99

1 Introduction Sobolev spaces on Rn play a fundamental role in many fields of mathematics such as harmonic analysis and partial differential equations ([12,26,33]). There are various generalizations of Sobolev spaces on Rn to the setting of metric spaces including some fractals and manifolds equipped with a measure; see, for examples, [3,11,13,14,17,18, 19,22,23,24,28,31,34,35].

L. Yan Department of Mathematics, Zhongshan University, Guangzhou 510275, People’s Republic of China e-mail: [email protected] D. Yang (B) School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China e-mail: [email protected]

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Let p ∈ (1, ∞) and (X, d) be a metric space equipped with a doubling measure μ. Following [17], an Lp (X) function f belongs to the Sobolev space P1,p (X) if there exist constants C > 0 and λ ≥ 1, 0 ≤ g ∈ Lp (X), and q ∈ (0, p) such that the Poincaré inequality ⎧ ⎫1/q  ⎨ 1  ⎬ 1 |f − fB |dμ ≤ CrB gq dμ (1) ⎩ μ(λB) ⎭ μ(B) λB

B

holds on every ball B of radius rB , where μ(B) is the measure of the ball B and 1 fB stands for the mean of f over B with respect to μ, i. e., fB = μ(B) B f (y) dμ(y).

It is well-known that Sobolev spaces P1,p (X) coincide with Hajłasz–Sobolev spaces M1,p (X) ([18], Theorem 3.1). See also Lemma 3.1 below. Moreover, some relevant results from the classical theory of Sobolev spaces on Rn have their counterparts on Sobolev spaces P1,p (X) including Sobolev-Poincaré type embedding theorem, Rellich–Kondrachov compact embedding theorem and so on. See, for examples, [18] and its references. The main purpose of this paper is to introduce some new function spaces of Sobolev type on metric measure spaces, which generalize the classical Sobolev spaces on Euclidean spaces. Roughly speaking, given an appropriate family of operators At with kernels at (which decay fast enough), we can view At f (x) for x ∈ X as an average version of f and use AtB f (x) = X atB (x, y)f (y)dμ(y) to replace the mean value fB in (1) of the definition of the Sobolev space P1,p (X), where tB is scaled to the radius rB 1,p of the ball B. We say that a function f ∈ Lp (X), p ∈ (1, ∞], is in the space PA (X) if p there exist some q ∈ [1, p), λ ≥ 1 and a non-negative function g ∈ L (X) such that for every ball B ⊂ X, ⎫1/q ⎧  ⎬ ⎨ 1 



1

f − At f dμ ≤ rB gq dμ ; (2) B ⎭ ⎩ μ(λB) μ(B) λB

B

see Definition 2.2 below. (Throughout this paper, n, N,  and m are always taken to be the same as in (4), (5) and (6), respectively.) We then proceed to prove the following: (i)

1,p

In Sect. 2 of this paper, we obtain two characterizations of PA (X). More spe1,p

(α) (β)

cifically, f ∈ PA (X) with p ∈ (n/, ∞] if and only if either of the following (α) and (β) holds: f ∈ Lp (X) and there exists a function g ∈ Lp (X)

such that for all balls B ⊂ X and almost everywhere x ∈ B, f (x) − AtB f (x) ≤ rB g(x) (see Theorem 2.2);  f ∈ Lp (X) and fγ ,A ∈ Lp (X), where 1 ≤ γ < p, γ > n/, ⎫1/γ ⎧ ⎪ ⎪  ⎬ ⎨ 1  |f (y) − Atm f (y)|γ dμ(y) fγ ,A (x) = sup t−1 , ⎪ ⎪ 0
(ii)

and C > 0 is a constant (see Theorem 2.3). Section 3 of this paper is devoted to establishing the corresponding characterizations of Hajłasz–Sobolev spaces via Poincaré-type inequalities and local sharp maximal functions associated with generalized approximations of the identity on metric measure spaces.

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(iii)

135

In Sect. 4 of this paper, we discuss the relations between the Hajłasz–Sobolev 1,p spaces P1,p (X) and the Sobolev-type spaces PA (X) on metric measure spaces. When the kernel at of At satisfies X at (x, y)dμ(y) = 1 for almost all x ∈ X and 1,p

2N +  > 1, it can be proved that P1,p (X) ⊂ PA (X) for max{1, n/} < p ≤ ∞; see Proposition 4.1 below. With the choices of X = Rn and the convolution At f = ϕt ∗ f in which ϕ is some 1,p suitable function on Rn , we then prove that the space PA (Rn ) coincides with the classical Sobolev space W 1,p (Rn ), p ∈ (1, ∞). We also show that if At = e−tN f in which N is the Neumann Laplacian on Rn , P1,p (Rn ) (= W 1,p (Rn )) is a proper subspace of 1,p PA (Rn ), where n < p < ∞. In [20], the fractional versions of Sobolev spaces on subsets of Rn are introduced and are generalized in [36] and [37] to spaces of homogeneous type in the sense of Coifman and Weiss [5]. Thus, in this paper, we also work on fractional versions of Sobolev spaces on metric measure spaces. We remark that in order to give a sufficient condition without involving the regularity of the kernel for weak (1, 1) estimates on certain singular integrals defined on irregular domains, Duong and McIntosh in [8] first used the estimate (f −At f ) in (2) on the kernel to replace the classical Hörmander condition. Later, replacing the average of functions on balls by At f , Martell in [25] introduce new sharp maximal functions on spaces of homogeneous type, which play a key role in establishing weighted norm estimates for singular integrals of Duong and McIntosh in [8]; see also [1,21]. Recently, this idea was further used in [9] (see also [7]) to introduce some new function spaces of BMO type related to a given generalized approximation of the identity on spaces of homogeneous type, which generalize the classical BMO space and characterize the boundedness of singular integral operators of Duong and McIntosh in [8] when p = ∞. Moreover, it was proved in [10] that these new BMO-type spaces are the dual spaces of associated Hardy spaces if X = Rn . In comparison with the corresponding classical spaces, an important feature of these new function spaces is that they tightly connect the operators considered. To be precise, for a given operator, by skillfully choosing an approximation of the identity corresponding to the given operator, one can then have a function space which suitably characterize the boundedness of the given operator; see [9] for more details. We finally make some conventions. Throughout the paper, A ∼ B means that the ratio A/B is bounded and bounded away from zero by constants that do not depend on the relevant variables in A and B. Similarly is A  B. We also denote by C a positive constant which is independent of main parameters, but it may vary from line to line. Constants with subscripts, such as C1 , do not change in different occurrences. For a set A, we define diam A = supx, y∈A d(x, y). If X 1 and X2 are two norm spaces, X1 ⊂ X2 means that for all f ∈ X1 , f X2  f X1 , where f Xj is the norm of f in Xj , j = 1, 2. 2 New Sobolev-type spaces In this section, we introduce some new versions of Hajłasz–Sobolev spaces on metric measure spaces via Poincaré-type inequalities related to generalized approximations of the identity. The characterizations of these new Sobolev-type spaces including the local sharp maximal function related to generalized approximations of the identity are also presented in this section.

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Let us first recall some definitions and notation on metric measure spaces. A metric measure space (X, d, μ) is a set X endowed with a distance d and a nonnegative Borel regular measure μ on X with that there exists a constant C1 > 0 such that for all r > 0 and all x ∈ X, μ(B(x, 2r)) ≤ C1 μ(B(x, r)),

(3)

where B(x, r) = {y ∈ X : d(y, x) < r}. A more general definition and further studies of these spaces can be found in [4,5]. In what follows, we denote by CB(x, r) the ball, {y ∈ X : d(x, y) < Cr}, and by – B(x,t) f dμ the average on B(x, t) of f , that is,   1 f dμ. – f dμ = μ(B(x, t)) B(x,t)

B(x,t)

Note that the doubling property (3) implies the following strong homogeneity property that there exist C2 > 0 and n > 0 such that for all λ ≥ 1 and x ∈ X, μ(B(x, λr)) ≤ C2 λn μ(B(x, r)).

(4)

The parameter n is a measure of the dimension of the space X. There also exist constants C3 > 0 and 0 ≤ N ≤ n such that

d(x, y) N μ(B(x, r)) (5) μ(B(y, r)) ≤ C3 1 + r uniformly for all x, y ∈ X and r > 0. Indeed, (5) with N = n is a simple consequence of (4) and the triangle inequality of the quasi-metric d. In the cases of Euclidean spaces and Lie groups of polynomial growth, N can be chosen to be 0; see [8,27]. We also need the following generalized approximations of the identity which previously appeared in [8]. Definition 2.1 A family of operators {At }t>0 is said to be a generalized approximation of the identity if, for every t > 0, At is represented by the kernel at , which is a measurable function defined on X × X, in the following sense: for any f ∈ Lp (X) with p ∈ [1, ∞] and all x ∈ X,  At f (x) = at (x, y)f (y) dμ(y), X

and the following condition holds: |at (x, y)| ≤ ht (x, y) =

1 ϕ μ(B(x, t1/m ))



d(x, y)m t

,

where m is a positive fixed constant and ϕ is a positive, bounded, decreasing function satisfying that for some  > 0, lim rn+2N+ ϕ(rm ) = 0,

r→∞

(6)

where n and N are the powers appearing in (4) and (5), respectively. We recall our convention in the introduction again that throughout this paper, n, N,  and m are always taken to be the same as in (4), (5) and (6).

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exists a constant C > 0 such that C−1 ≤ Remark 2.1 It is easy to check that there −1 ≤ X ht (x, y) dμ(y) ≤ C for all x ∈ X; X ht (x, y) dμ(x) ≤ C for all y ∈ X, and C see [8]. Moreover, from the Hölder inequality and (6), it is easy to check that for any f ∈ Lp (X) with p ∈ [1, ∞] and any x ∈ X, At f (x) is well-defined. In the sequel, let {At }t>0 be a generalized approximation as in Definition 2.1. We assume that A0 is the identity operator and the operators {At }t>0 satisfy the commutative law (for short, CLP), that is, for any t, s > 0 and any f ∈ L1loc (X), At As f (x) = As At f (x) for almost everywhere x ∈ X. Note that we do not assume the semigroup property At As f = At+s f for any t, s > 0. See [7,8,9,10]. This allows flexibility on the choice of At in applications. We now introduce some Sobolev-type spaces via the Poincaré inequality related to the above generalized approximation of the identity. Definition 2.2 Let 1 < p ≤ ∞, s > 0 and A ≡ {At }t>0 be the same as in Definition 2.1. s,p The Sobolev-type space PA (X) is the set of functions f ∈ Lp (X) satisfying that there exist some q ∈ [1, p), λ ≥ 1 and a non-negative function g ∈ Lp (X) such that for every ball B ⊂ X, ⎫1/q ⎧  ⎬ ⎨



s , (7) – f (x) − AtB f (x) dμ(x) ≤ rB – g(x)q dμ(x) ⎭ ⎩ λB

B

m . Moreover, if f ∈ P s,p (X), its where rB is the radius of the ball B and tB = rB A defined by f P s,p (X) = f Lp (X) + inf g g Lp (X) , where the infimum is taken A

norm is over all

functions g satisfying (7).

Next, let us introduce another kind of Sobolev spaces, which are finally proved to s,p be the same as PA (X) when the generalized approximation of the identity A has CLP. Definition 2.3 Let 1 ≤ p ≤ ∞, s > 0 and A ≡ {At }t>0 be the same as in Definition 2.1. s,p The Sobolev-type space HSA (X) is the set of functions f ∈ Lp (X) satisfying that there exists a nonnegative function g ∈ Lp (X) such that for all balls B ⊂ X and a. e. x ∈ B,



f (x) − At f (x) ≤ rs g(x). (8) B B s,p

Moreover, if f ∈ HSA (X), its norm is defined by f HSs,p (X) = f Lp (X) +inf g g Lp (X) , A where the infimum is taken over all functions g satisfying (8). s,p

s,p

If we replace AtB f (x) in the definitions of PA (X) and HSA (X) by the average of s,p s,p f on the ball B, then the spaces PA (X) and HSA (X) are just the spaces Asp (X) and s Bp (X) in [36], respectively; see also [24] and [37] for the case s = 1. s,p s,p To verify that the spaces PA (X) and HSA (X) coincide, we first establish the following theorem. In the sequel, let M be the Hardy–Littlewood maximal function on X. Theorem 2.1 Let 1 ≤ q < ∞ and n/q < , s > 0, A ≡ {At }t>0 be the same as in Definition 2.1 with CLP. Assume that f is a locally integral function on X for which there are a non-negative function g ∈ Lq (X) and some λ ≥ 1 such that the Poincaré inequality ⎫1/q ⎧  ⎬ ⎨



s – f (x) − AtB f (x) dμ(x) ≤ CrB (9) – g(x)q dμ(x) ⎭ ⎩ B

λB

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holds for every ball B ⊂ X. Then for a. e. x ∈ B,



 

f (x) − At f (x) ≤ Crs M gq (x)1/q , B B where C is independent of x, B and f . To prove Theorem 2.1, we need the following construction given by Christ in [4], which provides an analogue of the grid of Euclidean dyadic cubes on spaces of homogeneous type. In fact, we can think of Qkβ as being a dyadic cube with diameter rough δ k and centered at zkβ . Lemma 2.1 Let X be a metric measure space as above. Then there exists a collection {Qkβ ⊂ X : k ∈ Z, β ∈ Ik } of open subsets, where Ik is some index set, and constants δ ∈ (0, 1), C4 > 0 and a0 ∈ (0, 1) such that (i) (ii) (iii) (iv) (v)

μ(X \ ∪β Qkβ ) = 0 for each fixed k and Qkβ ∩ Qkτ = ∅ if β  = τ ; for any β, τ , k, l with l ≥ k, either Qlτ ⊂ Qkβ or Qlτ ∩ Qkβ = ∅; for each (k, β) and each l < k there is a unique τ such that Qkβ ⊂ Qlτ ; diam (Qkβ ) ≤ C4 δ k ; each Qkβ contains some ball B(zkβ , a0 δ k ), where zkβ ∈ X.

The following technical lemma was established in [9] by using Lemma 2.1. l

l

Lemma 2.2 Let y ∈ Qα00 for some l0 ∈ Z and some α0 ∈ Il0 , where Qα00 is a dyadic   l cube as in Lemma 2.1. For any k ∈ N, define Mk = β : Qβ0 ∩ B(y, C4 δ l0 −k )  = ∅ . There exists a constant C > 0, independent of l0 , α0 and k, such that the number of open l l subsets {Qβ0 }β∈Mk is less than Cδ −k(n+N) , i. e., mk = #{Qβ0 : β ∈ Mk } ≤ Cδ −k(n+N) . We now return to the proof of Theorem 2.1. Proof of Theorem 2.1 Let x ∈ B be a Lebesgue point of functions f (y) − AtB f (y) and g(y)q . Let B = B0 and Bj = B(x, 2−j/m rB ) for j ∈ N. Then 





f (x) − At f (x) = lim – f (y) − At f (y) dμ(y) B B j→∞







≤ lim sup – f (y) − AtBj f (y) dμ(y) Bj

j→∞

Bj 





+ lim sup – AtBj f (y) − AtB f (y) dμ(y) j→∞

Bj

= I11 + I12 . For I11 , by (9) and the definition of the Lebesgue point, we have 1/q  s q g(y) dμ(y) = 0g(x) = 0. I11  lim sup rB – j j→∞

λBj

To estimate I12 , we first have 

j 









– AtBl f (y) − AtBl−1 f (y) dμ(y). – AtBj f (y) − AtB f (y) dμ(y) ≤ Bj

l=1 B j

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139

For y ∈ Bj and l = 1, 2, . . . , j, from CLP of At , it follows that





AtBl f (y) − AtBl−1 f (y)

  







≤ AtBl f − AtBl−1 f (y) + AtBl−1 f − AtBl f (y)

= I21 + I22 .

The estimates for I2i with i = 1, 2 are similar and we only estimate I21 . To this end, by Definition 2.1, we have that for y ∈ Bj and l = 1, 2, . . . , j,





 



I21 =

atBl (y, z) f (z) − AtBl−1 f (z) dμ(z)



X 





 –

f (z) − AtBl−1 f (z) dμ(z) B(y,(2−l tB )1/m )

1 + μ(B(y, (2−l tB )1/m ))



 ϕ B(y,(2−l tB )1/m )

d(y, z)m 2−l tB







f (z) − AtBl−1 f (z) dμ(z)

= I31 (y) + I32 (y), where B(y, (2−l tB )1/m ) = X \ B(y, (2−l tB )1/m ). Applying (9) to I31 (y) gives j   – I31 (y) dμ(y) l=1 B j



  j  (2−l tB )s/m – l=1

Bj

1/q g(z)q dμ(z) dμ(y).

 – B(y,λ(2−l tB )1/m )

The facts that λ ≥ 1, y ∈ Bj and l ∈ {1, . . . , j} imply x ∈ B(y, λ(2−l tB )1/m ). From this together with the definition of the Hardy–Littlewood maximal function and the assumption that s > 0, it follows that j  j   s – I31 (y) dμ(y)  (2−l tB )s/m M(gq )(x)1/q  rB M(gq )(x)1/q , l=1 B j

l=1

which is a desired estimate. We now estimate I32 (y). With the notation same as in Lemma 2.1, we fix some l0 ∈ Z such that C4 δ l0 ≤ (2−l tB )1/m < C4 δ l0 −1 . By Properties (i) and (iv) of Lemma 2.1, for l a. e. y ∈ Bj , there exist some α0 ∈ Il0 and a dyadic cube Qα00 same as in Lemma 2.1 l0 l0 such that y ∈ Qα0 and Qα0 ⊂ B(y, C4 δ l0 ). For any k ∈ N, we let Mk to be the same as in Lemma 2.2. We first claim that for β ∈ Mk+1 , we have Qβ0 ⊂ B(zβ0 , C4 δ l0 ) ⊂ B(zβ0 , (2−l tB )1/m ),

(10)

−1    −1    l μ B y, (2−l tB )1/m  δ −kN μ B zβ0 , (2−l tB )1/m ,

(11)

l

l

l

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and 

1/q g(z) dμ(z)  δ −kn/q M(gq )(x)1/q .

 –

q

(12)

l B(zβ0 ,λ(2−l tB )1/m )

In fact, β ∈ Mk+1 tells us that Qβ0 ∩ B(y, C4 δ l0 −k−1 )  = ∅. Let z ∈ Qβ0 ∩ B(y, C4 δ l0 −k−1 ). Then for y ∈ Bj and j = 1, . . . , l, l

l

d(x, zβ0 ) ≤ d(x, y) + d(y, zβ0 ) ≤ d(x, y) + d(y, z) + d(z, zβ0 ) ≤ 3(2−l/m )rB δ −k−1 . l

l

l

That is, x ∈ B(zβ0 , 3(2−l/m )rB δ −k−1 ). From this, (5) and the definition of M together with (4), it follows that (11) and (12) hold. The inclusion relation (10) is obvious. Noting that for any z ∈ B(y, C4 δ l0 −k−1 ) \ B(y, C4 δ l0 −k ), one has d(y, z) ≥ C4 δ l0 −k . This fact together with the decreasing property of ϕ yields that



d(y, z)m



(z) − A ϕ f (z)

f

dμ(z) t Bl−1 2−l tB l

B(y,C4 δ l0 )

≤





∞ 

ϕ

k=0 B(y,C4 δ l0 −k−1 )\B(y,C4 δ l0 −k )

≤

k=0

≤



∞    ϕ δ −(k+1)m

d(y, z)m 2−l tB







f (z) − AtBl−1 f (z) dμ(z)







f (z) − AtBl−1 f (z) dμ(z)

B(y,C4 δ l0 −k−1 )

∞   k=0 β∈Mk+1







−(k+1)m ϕ δ

f (z) − AtBl−1 f (z) dμ(z), l

Qβ0

which together with (10), (11), (9), (12), Lemma 2.2, (6) and the assumption that  > n/q further indicates that for a. e. y ∈ Bj and l = 1, 2, . . . , j, I32 (y) ≤

∞  

  ϕ δ −(k+1)m

k=0 β∈Mk+1

≤

∞  

1 μ(B(y, (2−l tB )1/m ))  –

  ϕ δ −(k+1)m δ −kN

k=0 β∈Mk+1

 (2−l tB )s/m







f (z) − AtBl−1 f (z) dμ(z) l

Qβ0







f (z) − AtBl−1 f (z) dμ(z)

l

B(zβ0 ,(2−l tB )1/m )

∞  

   ϕ δ −(k+1)m δ −kN

k=0 β∈Mk+1

 (2−l tB )s/m M(gq )(x)1/q

 – l

B(zβ0 ,λ(2−l tB )1/m ) ∞  k=0

 (2−l tB )s/m M(gq )(x)1/q .

  mk+1 ϕ δ −(k+1)m δ −k(N+n/q)

1/q g(z)q dμ(z)

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141

From this and the assumption that s > 0, it follows that j  j   s – I32 (y) dμ(y)  (2−l tB )s/m M(gq )(x)1/q  rB M(gq )(x)1/q , l=1 B j

l=1

 

which completes the proof of Theorem 2.1. s,p

s,p

Using Theorem 2.1, we can establish the relation between PA (X) and HSA (X). Theorem 2.2 Let max{1, n/} < p ≤ ∞, s > 0 and A ≡ {At }t>0 be the same as in s,p s,p Definition 2.1 with CLP. Then PA (X) = HSA (X) with equivalent norms. s,p

Proof Let f ∈ PA (X). For any given δ > 0, Definition 2.2 then tells us that there exist some q ∈ [1, p), λ ≥ 1 and a non-negative function g ∈ Lp (X) such that (7) holds for any ball B ⊂ X, and f Lp (X) + g Lp (X) < f Ps,p (X) +δ. By the Hölder inequality and A the assumption that p > n/, we may also assume that q > n/. Therefore,

  Theorem s M gq (x)1/q . Let 2.1 yields that for all B ⊂ X and a. e. x ∈ B, f (x) − AtB f (x) ≤  CrB   h= CM gq (x)1/q . Then (8) holds with g replaced by h; and by the Lp/q (X)-boundedness of M,     1/q  

f Lp (X) + h Lp (X)  f Lp (X) + M gq  f Ps,p (X) + δ.  p L (X)

s,p

A

s,p

Let δ → 0. We therefore obtain that PA (X) ⊂ HSA (X) and f HSs,p (X)  f Ps,p (X) . A

A

s,p ∈ HSA (X). Then, by Definition 2.3, there is a function g ∈ Lp (X) and f Lp (X) + g Lp (X) < f HSs,p (X) + δ. Then the estimate (8)

Conversely, let f such that (8) holds A and the Hölder inequality tell us that for any ball B ⊂ X and any q ∈ [1, p), ⎫1/q ⎧   ⎬ ⎨



s s – f (x) − AtB f (x) dμ(x) ≤ rB – g(x) dμ(x) ≤ rB . – g(x)q dμ(x) ⎭ ⎩ B

Thus, f ∈

B s,p PA (X)

B

and f Ps,p (X) ≤ f Lp (X) + g Lp (X) < f HSs,p (X) + δ. Letting A

s,p

A

s,p

δ → 0, we finally obtain that HSA (X) ⊂ PA (X) and f Ps,p (X) ≤ f HSs,p (X) . This A A finishes the proof of Theorem 2.2.   s,p

To give another new characterization of PA (X), we first introduce a local version of the sharp maximal function associated with a generalized approximation to the identity {At }t>0 ; see [9] for the definition of the related sharp maximal function. For s ∈ R, C5 > 0, γ ∈ (0, ∞] and x ∈ X, we introduce the local sharp maximal function 1/γ   s, . (13) fγ ,A (x) = sup t−s – |f (y) − Atm f (y)|γ dμ(y) 0
B(x,t)

Theorem 2.3 Let 1 ≤ γ < p ≤ ∞ and γ > n/, s > 0, C5 > 0 and A ≡ {At }t>0 be s,p the same as in Definition 2.1 with CLP. Then f ∈ PA (X) if and only if f ∈ Lp (X) and s, fγ ,A ∈ Lp (X). Moreover,    s, 

f Ps,p (X) ∼ f Lp (X) + fγ ,A  p . A

L (X)

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Proof Note that the condition of the theorem implies that max{1, n/} < p ≤ ∞. By s,p s,p Theorem 2.2, PA (X) = HSA (X). Thus, to prove the theorem, we only need to verify s,p s,p Theorem 2.3 is still true if the space PA (X) is replaced by the space HSA (X). To this s,p p end, let f ∈ HSA (X). By Definition 2.3, f ∈ L (X) and for any given δ > 0, there is a function g ∈ Lp (X) such that (8) holds for all balls B ⊂ X and a. e. x ∈ B, and

f Lp (X) + g Lp (X) < f HSs,p (X) + δ. Therefore, A

⎤1/γ  ⎥ ⎢ s, fγ ,A (x) = sup t−s ⎣ – |f (y) − Atm f (y)|γ dμ(y)⎦ ⎡

0
B(x,t)

⎤1/γ  ⎥ ⎢ ≤ sup ⎣ – g(y)γ dμ(y)⎦ ⎡

0
  B(x,t) ≤ M gγ (x)1/γ . s,

Thus, the Lp/γ (X)-boundedness of M implies that fγ ,A ∈ Lp (X) and    s, 

f Lp (X) + fγ ,A 

Lp (X)

    1/γ   ≤ f Lp (X) + M gγ 

Lp (X)

 f HSs,p (X) + δ. A

 s,  Letting δ → 0, we obtain f Lp (X) + fγ ,A Lp (X)  f HSs,p (X) . A

s,

Let now f ∈ Lp (X) and fγ ,A ∈ Lp (X). For any fixed 0 < t < C5 , let y be a Lebesgue point of the function f (v) − Atm f (v). Moreover, let B0 = B(x, t) for some x ∈ X and Bj = B(y, 2−j/m t) for y ∈ B0 and j ∈ N. Then for a. e. y ∈ B(x, t),  |f (y) − Atm f (y)| = lim – |f (v) − Atm f (v)| dμ(v) j→∞





≤ lim sup – f (v) − A2−j tm f (v) dμ(v) Bj

j→∞

Bj 



+ lim sup – A2−j tm f (v) − Atm f (v) dμ(v) j→∞

Bj

= I41 + I42 . The Hölder inequality tells us that I41 ≤ lim sup j→∞

1/γ 

γ

s, ≤ lim sup 2−js/m ts fγ ,A (y) = 0. – f (v) − A2−j tm f (v) dμ(v) j→∞

Bj

By a literal repetition of the proof of Theorem 2.1, we obtain that for v ∈ Bj and 1 ≤ l ≤ j,





AtBl f (v) − AtBl−1 f (v)

 







≤ AtBl f − AtBl−1 f (v) + AtBl−1 f − AtBl f (v)

 2−ls/m ts fγ ,A (y). s,

New Sobolev spaces

143

From this and the assumption that s > 0, it follows that

I42

j 

∞

 



s, s, ≤ lim sup – AtBl f (v) − AtBl−1 f (v) dμ(v)  2−ls/m ts fγ ,A (y)  ts fγ ,A (y). j→∞

l=1 B j

l=1



s, That is, for a. e. y ∈ B(x, t), f (y) − Atm f (y) ≤  Cts fγ ,A (y), where  C > 0 is a constant s, p  independent of t, f and y. Letting g(y) = Cfγ ,A (y), by g ∈ L (X) and Definition 2.3, we  s,  s,p obtain that f ∈ HSA (X) and f HSs,p (X)  f Lp (X) + fγ ,A Lp (X) , which completes A the proof of Theorem 2.3.   The following conclusion is convenient in applications. s,p

Corollary 2.1 Let all the assumptions be the same as in Theorem 2.3. Then f ∈ PA (X) s, if and only if f ∈ Lp (Rn ) and fγ ,A,k ∈ Lp (Rn ), where for any k > 0,

s,

fγ ,A,k (x) =

sup

t−s

1/γ   . – |f (y) − Atm f (y)|γ dμ(y)

0
(14)

B(x,t)

 s,  Moreover, f Ps,p (X) ∼ f Lp (X) + fγ ,A,k Lp (X) . A

s,

s,

Proof Obviously, when k = 1, then fγ ,A,k is just fγ ,A in (13). We only prove Corollary 2.1 when 0 < k < 1 by similarity. If 0 < k < 1, then obviously s,

s,

fγ ,A,k (x) ≤ fγ ,A (x).

(15)

On the other hand, by k < 1 and the Minkowski inequality, we have

s,

s,

fγ ,A (x) ≤ fγ ,A,k (x) + 

sup

kC5 ≤t
t−s

  1/γ – |f (y) − Atm f (y)|γ dμ(y) B(x,t)

s, fγ ,A,k (x) + M(|f |γ )(x)1/γ

+

sup

kC5 ≤t
  1/γ γ – |Atm f (y)| dμ(y) . B(x,t)

Since ϕ satisfies (6), without loss of generality, we may assume that there exists some r0 ≥ 1 such that for all r ≥ r0 , ϕ(rm ) ≤ r−n−2N− .

(16)

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L. Yan, D. Yang

From this, it follows that for all y ∈ B(x, t),







|Atm f (y)| =

atm (y, z)f (z) dμ(z)



X  1  |f (z)| dμ(z) μ(B(y, t)) d(y,z)≤r0 t

+

∞  k=1

1 μ(B(y, t))

 –





2k−1 r0 t
|f (z)| dμ(z) +

∞  k=1

B(x,2r0 t)

tn+2N+ |f (z)| dμ(z) d(y, z)n+2N+ 1 2k(2N+)

 – B(x,2k+1 r

|f (z)| dμ(z) 0 t)

 Mf (x).

Thus, for all x ∈ X, fγ ,A (x)  fγ ,A,k (x) + M(|f |γ )(x)1/γ + Mf (x). From this, (15), the Lp (X)-boundedness of M and Theorem 2.3, it is easy to deduce the conclusion of this corollary.   s,

s,

3 New characterizations of Hajłasz–Sobolev Spaces In this section, we establish some new characterizations of Hajłasz–Sobolev spaces on spaces of homogeneous type in terms of the Poincaré inequality and the local sharp maximal function related to generalized approximations of the identity. We first recall the definition of Hajłasz-Sobolev spaces on metric measure spaces in [36]. Definition 3.1 Let 1 ≤ p ≤ ∞ and s > 0. The Sobolev space Ms,p (X) is the set of functions f ∈ Lp (X) satisfying that there exists a non-negative function g ∈ Lp (X) such that for a. e. x, y ∈ X, # $ |f (x) − f (y)| ≤ d(x, y)s g(x) + g(y) . (17) Moreover, if f ∈ Ms,p (X), its norm is defined by f Ms,p (X) = f Lp (X) + inf g g Lp (X) , where the infimum is taken over all functions g satisfying (17). The Sobolev space M1,p (X) was first introduced by Hajłasz in [14]; see also [11,16, 18,19]. It was proved by Hajłasz in [14] that if X = Rn and 1 < p < ∞, M1,p (Rn ) is just the classical Sobolev space W 1,p (Rn ); while the space M1,1 (Rn ) is different from W 1,1 (Rn ); see [15]. Obviously, M1,∞ (X) is just the space Lip 1 (X); see also Corollary 1 in [14]. When s > 0 and X is an open set of Euclidean spaces, the space Ms,p (X) was introduced by Hu in [20] with a different notation there. It was proved in [36] that s (Rn ) when 0 < s < 1 and 1 < p < ∞, where F s (Rn ) is the classical Ms,p (Rn ) = Fp,∞ p,∞ Triebel–Lizorkin space; see [33]. Obviously, if X = Rn or X is a connected open set of Rn , and s > 1, then f ∈ Ms,p (X) must be a constant. However, if X is a totally disconnected subset of Rn , for example, X is a fractal of Rn , then even for s > 1, f ∈ Ms,p (X) is not necessary to be a constant; see [38] for an example.

New Sobolev spaces

145

The following spaces Ps,p (X) and HSs,p (X) were introduced in [36] with other notations there and proved that they are the same spaces as Hajłasz–Sobolev spaces Ms,p (X). Definition 3.2 Let 1 < p ≤ ∞ and s > 0. The space Ps,p (X) is the set of functions f ∈ Lp (X) satisfying that there exist some q ∈ [1, p), λ ≥ 1 and a non-negative function g ∈ Lp (X) such that for every ball B ⊂ X,  – B



 1/q 



f (x) − – f (y) dμ(y) dμ(x) ≤ rs – g(x)q dμ(x) , B



(18)

λB

B

where rB is the radius of the ball B. Moreover, if f ∈ Ps,p (X), its norm is defined by

f Ps,p (X) = f Lp (X) + inf g g Lp (X) , where the infimum is taken over all functions g satisfying (18). Remark 3.1 It is easy to see that we can replace the left-hand side of (18) by  inf – |f (x) − A| dμ(x), A∈C

B

we then get the same space; see Remark 1.2 in [36]. Definition 3.3 Let 1 ≤ p ≤ ∞ and s > 0. The space HSs,p (X) is the set of functions f ∈ Lp (X) satisfying that there exists a function g ∈ Lp (X) such that for all balls B ⊂ X and a. e. x ∈ B,







f (x) − – f (z) dμ(z) ≤ rs g(x). (19) B



B

HSs,p (X), its norm is defined by f

Moreover, if f ∈ HSs,p (X) = f Lp (X) +inf g g Lp (X) , where the infimum is taken over all functions g satisfying (19). The following result was proved in [36]. Lemma 3.1 Let 1 < p ≤ ∞ and s > 0. Then Ms,p (X) = Ps,p (X) = HSs,p (X) with equivalent norms. A new characterization of the Hajłasz–Sobolev spaces Ms,p (X) can be stated as follows. Theorem 3.1 Let 1 < p ≤ ∞, 0 < s < 2N + , and A ≡ {At }t>0 be the same as in Definition 2.1 with At (1) = 1 for all t > 0. Then f ∈ Ms,p (X) if and only if f ∈ Lp (X) and there exists a function g ∈ Lp (X) such that for all balls B ⊂ X and a. e. x ∈ B,



f (x) − At f (xB ) ≤ rs g(x). (20) B B Moreover, if we define f

s,p  HS A (X)

= f Lp (X) + inf g g Lp (X) , where the infimum is

taken over all functions g satisfying (20), then f Ms,p (X) ∼ f

. s,p  HS A (X)

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L. Yan, D. Yang

Proof Let f ∈ Ms,p (X) and g be any function follows that for x ∈ B,



f (x) − At f (xB )

B 



≤ g(x) atB (xB , y) d(x, y)s dμ(y) + X

satisfying (17). From AtB (1) = 1, it



at (xB , y) d(x, y)s g(y) dμ(y) B

X

= J11 + J12 .

The estimates (16) and (4), and the fact that s < 2N +  yield that for x ∈ B, 



at (xB , y) d(x, y)s dμ(y) B X

1  μ(B(xB , rB )) +

∞ 



 d(x, y)s dμ(y)

d(xB ,y)≤r0 rB



k=1 k−1 2 r0 rB
 n+2N+ rB s d(x , y) dμ(y) B d(xB , y)n+2N+

s  rB , s g(x). Similarly, the estimates (16) and (4) and the fact that which tells us that J11  rB s < 2N +  lead us that for x ∈ B,   1 s 2 J1  rB g(y) dμ(y) μ(B(xB , rB )) d(xB ,y)≤r0 rB   ∞ n+2N+  rB s d(x , y) g(y) dμ(y) + B d(xB , y)n+2N+ k=1 k−1 kr r 2 r% r
Combining the estimates for J11 and J12 tells us that for x ∈ B,



 

f (x) − At f (xB )  rs g(x) + Mg(x) , B B which together with the Lp (X)-boundedness of M implies that (20) holds and

f

s,p  HS A (X)

 f Lp (X) + g + Mg Lp (X)  f Lp (X) + g Lp (X) .

The arbitrariness of g further tells us that f

s,p  HS A (X)

 f M1,p (X) .

Conversely, for any δ > 0, there is a non-negative function g ∈ Lp (X) such that for any ball B ⊂ X and a. e. x ∈ B, (20) holds and f Lp (X) + g Lp (X) < f  + δ. s,p HSA (X)



# $ s g(x) + g(y) . By suitably From this, it follows that for a. e. x, y ∈ B, f (x) − f (y) ≤ rB

# $ choosing rB , we finally obtain that for a. e. x, y ∈ X, f (x)−f (y) ≤ d(x, y)s g(x)+g(y) . Thus, f ∈ M1,p (X) and f M1,p (X) ≤ f Lp (X) + g Lp (X) < f  + δ. Letting s,p δ → 0 gives that f M1,p (X) ≤ f

, s,p  HS A (X)

HSA (X)

which completes the proof of Theorem 3.1.  

New Sobolev spaces

147

Remark 3.2 It is easy to see that if we replace 2N in (6) by N, Theorem 3.1 still holds when 0 < s < N + . To give a new characterization in terms of the Poincaré inequality for the Hajłasz– Sobolev spaces, we first establish a theorem similar to Theorem 2.1 without assuming that A satisfies CLP, whose proof is a slight modification of the proof of Theorem 2.1. Theorem 3.2 Let 1 ≤ q < ∞ and n/q < , s > 0, A ≡ {At }t>0 be the same as in Definition 2.1 with At (1) = 1 for all t > 0. Assume that f is a locally integral function on X for which there are a non-negative function g ∈ Lq (X) and some λ ≥ 1 such that the Poincaré inequality ⎫1/q ⎧  ⎬ ⎨



s – f (x) − AtB f (xB ) dμ(x) ≤ CrB (21) – g(x)q dμ(x) ⎭ ⎩ λB

B

holds for every ball B ⊂ X. Then for a. e. x ∈ B,



 

f (x) − At f (xB ) ≤ Crs M gq (x)1/q , B B where C is independent of x, B and f . Proof Let x ∈ B be a Lebesgue point of functions f (y) − AtB f (xB ) and g(y)q . Let B = B0 and Bj = B(x, 2−j/m rB ) for j ∈ N. Then 





f (x) − At f (xB ) = lim – f (y) − At f (xB ) dμ(y) B B j→∞







≤ lim sup – f (y) − AtBj f (x) dμ(y) Bj

j→∞





+ lim sup AtBj f (x) − AtB f (xB )

Bj

j→∞

= J21 + J22 . For J21 , by (21) and the definition of the Lebesgue point, we have  1/q 1 −js/m s q J2  lim sup 2 rB – g(x) dμ(x) = 0g(x) = 0. j→∞

λBj

To estimate J22 , by At (1) = 1, (4), (5) and (21) together with an argument similar to that used in the proof of Theorem 2.1, we obtain that for a. e. x ∈ B,





At f (x) − At f (xB )

B

B1











≤ AtB /2 f (x) − AtB f (x) + AtB f (x) − AtB f (xB )





# $ =

atB /2 (x, z) f (z) − AtB f (x) dμ(z)

X 

+

X



$ atB (xB , z) f (z) − AtB f (x) dμ(z)

#

s M(gq )(x)1/q .  rB

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L. Yan, D. Yang

Similarly, we have that for a. e. x ∈ B and l = 2, . . . , j,





 





AtBl f (x) − AtBl−1 f (x) = a2tBl (x, z) f (z) − AtBl f (x) dμ(z)

X s  2−ls/m rB M(gq )(x)1/q .

Thus, J22 ≤ lim sup j→∞



∞ 

j









AtBl f (x) − AtBl−1 f (x) + AtB1 f (x) − AtB f (xB )

l=2

s 2−ls/m rB M(gq )(x)1/q

l=1 s  rB M(gq )(x)1/q .



s M(gq )(x)1/q , which completes Therefore, for a. e. x ∈ B(xB , rB ), f (x)−AtB f (xB )  rB the proof of Theorem 3.2.   Using Theorem 3.2 and Theorem 3.1, we can establish a characterization for the Hajłasz–Sobolev spaces in terms of the Poincaré-type inequality related to a generalized approximation of the identity. Theorem 3.3 Let max{1, n/} < p ≤ ∞, s > 0, and A ≡ {At }t>0 be the same as in Definition 2.1 with At (1) = 1 for all t > 0. Then f ∈ Ms,p (X) if and only if f ∈ Lp (X) and there exist some q ∈ [1, p), some λ ≥ 1 and a non-negative function g ∈ Lq (X) such that the Poincaré inequality (21) holds for every ball B ⊂ X. Moreover, if we define

f  = f Lp (X) + inf g g Lp (X) , where the infimum is taken over all functions g s,p PA (X)

satisfying (21), then f Ms,p (X) ∼ f  . s,p PA (X)

Based on Lemma 3.1 and replaced Theorem 2.1 by Theorem 3.2, we can prove Theorem 3.3 by a literal repetition of the proof of Theorem 2.2. We omit the details. s, Finally we introduce a new variant of the local sharp maximal function fγ ,A associated with a generalized approximation to the identity {At }t>0 in (13). For s ∈ R, C6 > 0, γ ∈ (0, ∞] and x ∈ X, we introduce the following local sharp maximal function   1/γ s,  fγ ,A (x) = sup t−s – |f (y) − Atm f (x)|γ dμ(y) . 0
B(x,t)

We now establish the following new characterization of Hajłasz–Sobolev spaces s, fγ ,A by using Theorem 3.1. Ms,p (X) in terms of  Theorem 3.4 Let 1 ≤ γ < p ≤ ∞ and γ > n/, s > 0, C6 > 0, and A ≡ {At }t>0 be the same as in Definition 2.1 with At (1) = 1 for all t > 0. Then f ∈ Ms,p (X) if and only if s, s,  fγ ,A ∈ Lp (X). Moreover, f Ms,p (X) ∼ f Lp (X) +  fγ ,A Lp (X) . f ∈ Lp (X) and  Proof Let f ∈ Ms,p (X). Repeating the proof of Theorem 2.3 by replacing Definition  s,  s, fγ ,A Lp (X)  2.2 there by Theorem 3.1 then tells us that  fγ ,A ∈ Lp (X) and f Lp (X) + 

f Ms,p (X) .

New Sobolev spaces

149

s, Let now f ∈ Lp (X) and  fγ ,A ∈ Lp (X). For any fixed x ∈ X and 0 < t < C6 , let y be a Lebesgue point of the function f (v) − Atm f (x). Moreover, if we use the same notation as in the proof of Theorem 2.3, then for a. e. y ∈ B(x, t),  m |f (y) − At f (x)| = lim – |f (v) − Atm f (x)| dμ(v) j→∞





≤ lim sup – f (v) − A2−j tm f (y) dμ(v) Bj

j→∞

Bj

+ lim sup A2−j tm f (y) − Atm f (x)

j→∞

= J31 + J32 . Similarly to the estimate for I41 in the proof of Theorem 2.3, by the Hölder inequality, we obtain J31 ≤ lim sup 2−js/m ts fγ ,A (y) = 0. s,

j→∞

To estimate J32 , by At (1) = 1 and an estimate similar to that for J21 , we have that for a. e. y ∈ Bj ,



Atm /2 f (y) − Atm f (x)







# $

≤ atm /2 (y, z) f (z) − Atm f (y) dμ(z)

X 

+





# $ m a (x, z) f (z) − At f (y) dμ(z)

tm

X

s, fγ ,A (y).  ts

Similarly, we have that for a. e. y ∈ B(x, t) and l = 2, . . . , j,





 









At f (y) − At a2tBl (y, z) f (z) − AtBl f (y) dμ(z)

Bl−1 f (y) =

Bl X

s,  2−ls/m ts fγ ,A (y).

Thus, by s > 0, J32 ≤ lim sup j→∞



∞ 

j





AtBl f (y) − AtBl−1 f (y) + |Atm f (y) − Atm f (x)| l=2

2−ls/m ts fγ ,A (y) s,

l=1 s,  ts fγ ,A (y).



s, fγ ,A (y). Thus, the assumption that Therefore, for a. e. y ∈ B(x, t), f (y) − Atm f (x)  ts s, fγ ,A ∈ Lp (X) together with Theorem 3.1 tells us that f ∈ Ms,p (X) and f ∈ Lp (X) and       s,   s,  fγ ,A  p fγ ,A  p .

f Ms,p (X)  f Lp (X) +   f Lp (X) +  L (X)

L (X)

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L. Yan, D. Yang

 

This finishes the proof of Theorem 3.4.

The proof of the following corollary is similar to that of Corollary 2.1 by using Theorem 3.4. We omit the details. Corollary 3.1 Let all the assumptions be the same as in Theorem 3.4. Then f ∈ Ms,p (X) s, fγ ,A,k ∈ Lp (Rn ), where for any k > 0, if and only if f ∈ Lp (Rn ) and  s,  fγ ,A,k (x) =

Moreover, f Ms,p (X)

sup

t−s

0
1/γ   . – |f (y) − Atm f (x)|γ dμ(y)

(22)

B(x,t)

 s,  ∼ f Lp (X) +  fγ ,A,k Lp (X) . s,p

4 Comparison between HSA (X) and M s,p (X) s,p

In this section, we clarify the relations between HSA (X) and Ms,p (X). 4.1 An embedding theorem s,p

In this subsection, we establish an embedding theorem between HSA (X) and Ms,p (X). Proposition 4.1 Let 1 < p ≤ ∞, 0 < s < 2N +  and A ≡ {At }t>0 be the same as in s,p Definition 2.1 with At (1) = 1 for all t > 0. Then Ms,p (X) ⊂ HSA (X). Proof The proof of this proposition is a slight modification of the proof of Theorem 3.1. Letting f ∈ Ms,p (X), if we use the same notation as in the proof of Theorem 3.1, AtB (1) = 1, (5) and (8), we then obtain that for x ∈ B,



f (x) − At f (x)

B





# $ =

atB (x, y) f (x) − f (y) dμ(y)

X ≤



# $

at (x, y) d(x, y)s g(x) + g(y) dμ(y) B

  s g(x) + Mg(x) .  rB X

s,p

Now Definition 2.2 and the Lp (X)-boundedness of M tell us that f ∈ HSA (X) and for all f ∈ Ms,p (X), f HSs,p (X)  f Ms,p (X) . This finishes the proof of Proposition 4.1. A   Remark 4.1 It is easy to see that if we replace 2N in (6) by N, Proposition 4.1 still holds when 0 < s < N + . s,p

4.2 The spaces Ms,p (Rn ) and HSA (Rn ) In this subsection, we verify that if we take some special generalized approximation s,p of the identity A on Rn , then HSA (Rn ) coincides with Ms,p (Rn ) when 0 < s ≤ 1 and 1 < p < ∞.

New Sobolev spaces

151

Let ϕ0 and ϕ be two Schwartz functions on Rn satisfying the following Tauberian conditions that |' ϕ0 (x)| > 0 and

# $

' ϕ (x) 1 − ' ϕ (2x) > 0

if |x| ≤ 2

(23)

if 1/2 ≤ |x| ≤ 2,

(24)

where ' f denotes the Fourier transform of any function f . Let φ(x) = ϕ(x) − ϕ ∗ ψ(x) with ψ(x) = 21n ϕ( x2 ) and ϕt (x) = t1n ϕ( xt ) for t > 0 and x ∈ Rn . If the generalized approximation of the identity {At }t>0 as in Definition 2.1 s,p s,p is determined by at (x, y) = ϕt (x − y), we then denote HSA (Rn ) by HSϕ (Rn ). The following is one of the main results of this subsection. Theorem 4.1 Let ϕ satisfy (24) and ' ϕ (0) = 1, 1 < p < ∞ and 0 < s < 1. Then s (Rn ) = HSs,p (Rn ) with equivalent norms. Ms,p (Rn ) = Fp,∞ ϕ By Proposition 4.1, it suffices to verify that if 0 < s < 1 and 1 < p < ∞, then s,p HSϕ (Rn ) ⊂ Ms,p (Rn ). To this end, we need the following characterization of the s (Rn ) when space Ms,p (Rn ); see Sect. 2.4.1 in [33]. We recall again that Ms,p (Rn ) = Fp,∞ 0 < s < 1 and 1 < p < ∞; see [36]. Lemma 4.1 Let ϕ0 and φ be as above, 0 < s < 1 and 1 < p < ∞. Then       −s

ϕ0 ∗ f Lp (Rn ) +  sup t |φt ∗ f | 0
is an equivalent norm in Ms,p (Rn ). Remark 4.2 From the proof of Theorem 4.1 and Theorem 2.4.1 in [33], it is easy to see that Theorem 4.1 is still true for some other suitable function ϕ satisfying (24) which is not necessary to be a Schwartz function; see Sects. 2.4.1 and 1.8.4 in [33] for more details. We also need the following technical lemma. Lemma 4.2 Let ϕ0 and ϕ be the same as in (23) and (24), 1 ≤ p ≤ ∞ and s > 0. Then (i) There exists a constant C > 0 such that for all f ∈ L1loc (Rn ) and all x ∈ Rn , |ϕ0 ∗ f (x)| ≤ CM(f )(x); (ii) Let f , g ∈

Lp (Rn )

satisfy that for all balls B ⊂ Rn and a. e. x ∈ B,



f (x) − ϕr ∗ f (x) ≤ rs g(x). B B

(25)

Then there exists a constant C > 0, independent of f and g, such that for all K > 1, all t > 0 and all x ∈ Rn , |φt ∗ f (x)| ≤ Cts M(g)(x),

(26)

|ϕt ∗ f (x) − ϕKt ∗ f (x)| ≤ C(Kt)s M(g)(x).

(27)

and

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L. Yan, D. Yang

Proof A routine argument together with the fact that ϕ0 is a Schwartz function gives (i). We omit the details. To prove (ii), for any t > 0, we choose ν such that t/4 ≤ ν ≤ t. We first estimate |ϕt ∗ f (x) − ϕt+ν ∗ f (x)|. From the property of convolution, it is easy to see that ϕt ∗ϕt+ν = ϕt+ν ∗ϕt and ϕt ∗f (x)−ϕt+ν ∗f (x) = ϕt ∗(f −ϕt+ν ∗f )(x)−ϕt+ν ∗(f −ϕt ∗f )(x). We only estimate the second term ϕt+ν ∗(f −ϕt ∗f )(x) by similarity. Since ϕ is a Schwartz function, then for any L > 0 which will be determined later, there exists a constant CL > 0 such that for all x ∈ Rn , |ϕ(x)| ≤ CL (1 + |x|)−L . From this, the assumption that t/4 ≤ ν ≤ t and (25), it follows that |ϕt+ν ∗ (f − ϕt ∗ f )(x)| ≤ |ϕt+ν (x − y)||f (y) − ϕt ∗ f (y)| dy Rn



B(x,t)





1  ts M(g)(x) + n t





 1 – |f (y) − ϕt ∗ f (y)| dy + n t

1+

|x − y| t

−L

|f (y) − ϕt ∗ f (y)| dy

B(x,t)

|x − y| 1+ t

−L

|f (y) − ϕt ∗ f (y)| dy

B(x,t)

= ts M(g)(x) + H1 ,

where B(x, t) = Rn \B(x, t). Moreover, we have

H1 =

1

 n t





∞ 1  tn

1+

k=0 B(x,2k+1 t)\B(x,2k t)  ∞  −kL

−L

|f (y) − ϕt ∗ f (y)| dy.

2

k=1

|x − y| t

|f (y) − ϕt ∗ f (y)| dy

(28)

B(x,2k t)

For any fixed k ∈ N, we consider the ball B(x, 2k t), which is contained in the cube Q[x, 2k+1 t] centered at x with side length 2k+1 t. We then divide this cube into √ √ Nk centered at xki with equal side length ([ n] + [2k+1 ([ n] + 1)]n small cubes {Qxki }i=1 √ √ √ 1)−1 t, where [ n] is the biggest integer no more than n and Nk = [2k+1 ([ n] + 1)]n . For any i = 1, 2, . . . , Nk , each of these small cubes Qxki is then contained in the corresponding ball Bki with the center xki and radius t. Thus, for any k ∈ N and any ball Nk B(x, 2k t), there exist associated balls {Bki }i=1 such that (i) (ii) (iii)

Each Bki is of the radius t; (Nk ) Bki and B(x, 2k t) Bki  = ∅ for i = 1, . . . , Nk (Otherwise, we B(x, 2k t) ⊂ i=1 remove Bki ); Each point of B(x, 2k t) is contained in at most a finite number M of the balls Nk , where M is independent of k. {Bki }i=1

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From Property (ii), it follows that x ∈ B(xki , 2k+1 t), which together with other properties, (25) and (28), yields H1 

∞ 

2

−kL

k=1

 ts

∞ 

i=1 Bki

Nk   – |f (y) − ϕt ∗ f (y)| dy i=1 B ki Nk   – g(y) dy

2−kL

k=1

 ts

∞ 

|f (y) − ϕt ∗ f (y)| dy

( Nk

k=1





∞ 1  −kL 2 n t

i=1 B ki

2−k(L−n)

k=1

 ts M(g)(x)

∞ 

Nk 

 –

i=1

,2k+1 t)

B(xki

g(y) dy

2−k(L−n) Nk

k=1

 ts M(g)(x),

where we choose L > 2n. Thus, |ϕt+ν ∗ (f − ϕt ∗ f )(x)|  ts M(g)(x). The same estimate also holds for |ϕt ∗ (f − ϕt+ν ∗ f )(x)|. In particular, we obtain (26) when ν = t. Furthermore, for all t/4 ≤ ν ≤ t and all x ∈ Rn , |ϕt ∗ f (x) − ϕt+ν ∗ f (x)|  ts M(g)(x).

(29)

If 0 < ν ≤ t/4, we write

# $ # $ ϕt ∗ f (x) − ϕt+ν ∗ f (x) = ϕt ∗ f (x) − ϕ2t ∗ f (x) − ϕt+ν ∗ f (x) − ϕ(t+ν)+(t−ν) ∗ f (x) .

Noting that (t + ν)/4 < t − ν < t + ν, we deduce (27) from (29). In general, for any K > 1, let l be the integer satisfying 2l < K ≤ 2l+1 . Thus, by (29), for all K > 1, all t > 0 and all x ∈ Rn , |ϕt ∗ f (x) − ϕKt ∗ f (x)| l−1 





ϕ k ∗ f (x) − ϕ k+1 ∗ f (x) + ϕ l ∗ f (x) − ϕKt ∗ f (x)

≤ 2 t 2t 2 t k=0 ⎧ ⎫ l−1 ⎨  k  s  s ⎬  M(g)(x) 2 t + Kt ⎩ ⎭ k=0  s  Kt M(g)(x), which completes the proof of Lemma 4.2.

 

We now return to the proof of Theorem 4.1. Proof of Theorem 4.1 As pointed above, to finish the proof of Theorem 4.1, it suffices s,p s,p to verify that if 0 < s < 1 and 1 < p < ∞, HSϕ (Rn ) ⊂ Ms,p (Rn ). Let f ∈ HSϕ (Rn ). p n By Definition 2.2, for any δ > 0, there exists a function g ∈ L (R ) such that (25) holds

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and f Lp (Rn ) + g Lp (Rn ) < f HSs,p n + δ. Lemma 4.2 and Lemma 4.1 together with ϕ (R ) s,p

the Lp (Rn )-boundedness of M for p ∈ (1, ∞) tell us that for all f ∈ HSϕ (Rn ),       −s

f Ms,p (Rn )  ϕ0 ∗ f Lp (Rn ) +  sup t |φt ∗ f | 0
 M(f ) Lp (Rn ) + M(g) Lp (Rn )  f Lp (Rn ) + g Lp (Rn )

< f HSs,p n + δ. ϕ (R ) Letting δ → 0 gives that f Ms,p (Rn )  f HSs,p n , which completes the proof of ϕ (R )  Theorem 4.1. We also recall that M1,p (Rn ) when 1 < p < ∞ coincides with the classical Sobolev space W 1,p (Rn ); see [14]. We now prove that if we suitably choose A = {At }t>0 , then 1,p the space HSA (Rn ) also coincides with the classical Sobolev space W 1,p (Rn ). Some ideas of the proof of this fact come from the proof of Theorem 1 in [14], which originate from Calderón [2]. To this end, let ϕ be a function on Rn such that for some s1 > 1, lim |x|n+s1 |ϕ(x)| = 0,

|x|→∞

(30)

 ϕ(x) dx = 1,

(31)

Rn

and for j = 1, . . . , n, if defining ξj e−|ξ | ' bj (ξ ) = , 1−' ϕ (ξ ) 2

then there exist constants C > 0 and L > 2n such that for all j = 1, . . . , n and all x ∈ Rn ,



(32) (1 + |x|)L bj (x) ≤ C. Theorem 4.2 Let 1 < p < ∞, ϕ satisfy (30), (31) and (32), and At f = ϕt ∗ f for t > 0 1,p and f ∈ Lp (Rn ). Then M1,p (Rn ) = W 1,p (Rn ) = HSϕ (Rn ) with equivalent norms. Proof By (30), (31) and Proposition 4.1, it suffices to verify that when 1 < p < ∞, 1,p 1,p HSϕ (Rn ) ⊂ W 1,p (Rn ). Let f ∈ HSϕ (Rn ). For any δ > 0, there exists a nonnegative p n function g ∈ L (R ) such that for all balls B ⊂ Rn and a. e. x ∈ B, |f (x) − ϕrB ∗ f (x)| ≤ rB g(x)

(33)

f Lp (Rn ) + g Lp (Rn ) ≤ f HS1,p (Rn ) + δ.

(34)

and ϕ

By the Riesz representation theorem and the Radon–Nikodym theorem, to prove the desired result, we only need to verify that there exists a non-negative function

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155

h ∈ Lp (Rn ) such that for all φ ∈ C0∞ (Rn ),









∂φ

f (x)

≤ |φ(x)|h(x) dx (x) dx



∂xj

n

n R

R

'(ξ ) = e−|ξ |2 for all ξ ∈ Rn . Then Let ψ    ∂φ ∂φ ∂ψt f (x) (x) dx = lim (x)ψt ∗ f (x) dx = − lim φ(x) ∗ f (x) dx. t→0 t→0 ∂xj ∂xj ∂xj Rn

Rn

Rn

By the definition of b, we further obtain 1 ∂ψt ∗ f (x) = ∂xj t



# $ (bj )t (x − y) f (y) − ϕt ∗ f (y) dy.

Rn

From this, (33) and the estimate (32) together with an argument similar to that of Lemma 4.2, it follows that





∂ψt

 M(g)(x),

∗ f (x)

∂x j and therefore,









∂φ

f (x) (x) dx

 |φ(x)| M(g)(x) dx,

∂x j

n n R

R

which together with the Hölder inequality, the boundedness of M and a dual argument ∂f ∈ Lp (Rn ) and tells us that ∂x j    ∂f     g Lp (Rn ) ;  ∂x  j Lp (Rn )

(35)

see also [19]. Thus, f ∈ W 1,p (Rn ), and by (34), (35) and the arbitrariness of δ, we obtain    ∂f    f 1,p n ,

f W 1,p (Rn ) = f Lp (Rn ) +   ∂x  HSϕ (R ) j Lp (Rn ) which completes the proof of Theorem 4.2.

 

Remark 4.3 It is not so difficult to find a function ϕ satisfying (30), (31) and (32). For ϕ (ξ ) = example, letting ν ∈ C0∞ (Rn ) with ν(ξ ) = 1 when |ξ | < 1/2 and then defining ' e−|ξ | ν(ξ ) for ξ ∈ Rn , one can verify that ϕ satisfies (30), (31) and (32). In particular, if s,p we just take At = Pt , the Poisson kernel on Rn , then we have Ms,p (Rn ) ⊂ HSA (Rn ) s,p n s,p n when 0 < s < 1 and 1 < p ≤ ∞, and HSA (R ) ⊂ M (R ) when 0 < s < 1 and 1 < p < ∞.

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4.3 An example of Ms,p (Rn )  HSN (Rn ) for 0 < s ≤ 1 and n/s < p To begin with, we recall some basic facts about the Neumann Laplacian N on Rn , which was studied in [6]. In what follows, Rn+ denotes the usual upper-half space in Rn , i.e.,   Rn+ = (x , xn ) ∈ Rn : x = (x1 , . . . , xn−1 ) ∈ Rn−1 , xn > 0 . Similarly, Rn− denotes the lower-half space in Rn . We denote by N+ (resp. N− ) the Neumann Laplacian on Rn+ (resp. on Rn− ). See page 57 of [30]. The Neumann Laplacians are self-adjoint and positive definite operators. Using the spectral theory one can define the semigroup {exp(−tN+ )}t≥0 (resp. {exp(−tN− )}t≥0 ) generated by the operator N+ (resp. N− ). For any f defined on Rn , we set f− = f |Rn− and f+ = f |Rn+ , where f |Rn+ and f |Rn− are restriction of the function f to Rn+ and Rn− , respectively. Let N be the uniquely determined unbounded operator acting on L2 (Rn ) such that (N f )+ = N+ f+ and (N f )− = N− f− for all f : Rn  → R such that f+ ∈ W 1,2 (Rn+ ) and f− ∈ W 1,2 (Rn− ). Then, N generates the conservative semigroup such that e−tN (1) = 1 for every t > 0, whose kernels at satisfy

  |2 2 2 1 n| n| − |x −y − |xn −y − |xn +y 4t 4t 4t e H(xn yn ), e + e (36) at (x, y) = n (4πt) 2 where H : R → {0, 1} is the Heaviside function, given by H(t) = 0 if t < 0 and H(t) = 1 if t ≥ 0. For more details, we refer to Sect. 2.2 of [6]. Observe that for any Lp (Rn ) function f , 1 ≤ p ≤ ∞, and x ∈ Rn , we have  At f (x) = e−tN f (x) = at (x, y)f (y)dy. Rn

s,p

s,p

We will say that the resulting HSA (Rn ) space as in Definition 2.2 is the space HSN (Rn ) associated with the semigroup {e−tN }t≥0 . The main result of this subsection is the following proposition. Proposition 4.2 Suppose that 0 < s ≤ 1 and n/s < p < ∞. Then, (i) (ii)

s,p

Ms,p (Rn ) ⊆ HSN (Rn ); We have f (x) = e−|x| χ{x: x∈Rn+ } (x) ∈ HSN (Rn ); 2

s,p

however, f  ∈ Ms,p (Rn ). s,p

As a consequence, we have that Ms,p (Rn )  HSN (Rn ). That is, Ms,p (Rn ) is a proper s,p subspace of HSN (Rn ). Proof Since for every t > 0, e−tN (1) = 1 almost everywhere; by Proposition 4.1, we s,p have that Ms,p (Rn ) ⊂ HSN (Rn ), and then (i) is proved.

Let us prove (ii). Set fe (x) = e−|x| . Using the formula (36), we can verify that  0 if x ∈ Rn− ; (37) At f (x) = −t e (fe )(x) if x ∈ Rn+ , 2

New Sobolev spaces

157

where  is the Laplacian on Rn . On the other hand, it follows from Proposition 4.1 that s,p fe ∈ Ms,p (Rn ) ⊂ HS (Rn ). By Definition 2.3, for any δ > 0, there exists a function g ∈ Lp (Rn ) such that (25) holds and

fe Lp (Rn ) + g Lp (Rn ) < fe HSs,p (Rn ) + δ. 

(38)

We now prove that f (x) = e−|x| χ{x: x∈Rn+ } (x) ∈ HSN (Rn ). By Theorem 2.3 and Corollary 2.1, it suffices to prove    s,  fγ ,N  p n  fe HSs,p (Rn ) , s,p

2

L (R )

where fγ ,N (x) = sup t−s s,

0


  1/γ

γ – f (y) − At2 f (y) dy . Q(x,t)

Here we have replaced the balls B(x, t) in (14) by cubes Q(x, t) of Rn , which is obviously s, true. Let us now estimate fγ ,N by examining the cubes Q(x, t) in several cases. Case 1 Q(x, t) ⊆ Rn− . From the estimate (37), we have that for any y ∈ Rn− , f (y) = s, At f (y) = 0, and then fγ ,N (x) = 0. Case 2 Q(x, t) ∩ Rn−  = ∅ and Q(x, t) ∩ Rn+  = ∅. In this case, let   Q+ = {y = (y , yn ) ∈ Rn : (y , 0) ∈ Q(x, t) ∩ Rn−1 × {0} , 0 ≤ yn < t}. The estimate (37), together with Proposition 4.1 and (27), yields   1/γ

γ s, fγ ,N (x) = sup t−s – f (y) − At2 f (y) dy 0
Q(x,t)

 1/γ

γ −s



– f (y) − At2 f (y) dy = sup t 0
Q+



1/γ

γ



 sup t−s – fe (y) − e−t (fe )(y) dy 0


where

Q+

s, (fe )γ ,,2 (x) + M(g)(x),

⎫1/γ ⎧ ⎪ ⎪ ⎬ ⎨ 

γ



s, – fe (y) − e−t (fe )(y) dy (fe )γ ,,2 (x) = sup t−s . ⎪ ⎪ 0
Case 3 Q(x, t) ⊆ Rn+ . In this case, it follows from (37) and Proposition 4.1 that we have ⎧ ⎫1/γ ⎪ ⎪ ⎨ 

⎬

γ s, s, fγ ,N (x) = sup t−s – f (y) − At2 f (y) dy  (fe )γ , (x). ⎪ ⎪ 0
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Combining the estimates of Cases 1, 2 and 3 above, together with Corollary 2.1 and Theorem 4.1, we have        s,   s,   s,  fγ ,N  p n  (fe )γ ,,2  p n + (fe )γ ,  p n + M(g) Lp (Rn ) L (R )

L (R )

L (R )

 fe HSs,p (Rn ) + M(g) Lp (Rn ) 

 fe HSs,p (Rn ) + δ. 

Note that fe ∈ W s,p (Rn ) (= Ms,p (Rn )). It follows from Proposition 4.1 that

fe HSs,p (Rn )  fe W s,p (Rn ) < ∞. 

Now, letting δ → 0 gives us the estimate (38), and then f HSs,p s,p HSN (Rn ).

N (R

n)

 fe HSs,p (Rn ) . 

This proves that f ∈ On the other hand, from Proposition 2.3.2/2, Theorem 2.7.1 and Proposition 2.5.7 in [32], it follows that when 0 < s ≤ 1 and n/s < p < ∞, Ms,p (Rn ) ⊂ C(Rn ). Since the above f  ∈ C(Rn ), then f ∈ / Ms,p (Rn ). Hence, (ii) is obtained and then the proof of Proposition 4.2 is complete.   Remark 4.4 (i) By a slight modification of the proof as above, we can prove that Proposition 4.2 also holds when p = ∞. We omit the details. (ii) By Lemma 3.1, we have that the space P1,∞ (X) is the Lipschitz space Lip 1 (X). However, if X = Rn and {At }t>0 is determined by a Schwartz function ϕ satisfying the conditions (A1 ) and (A2 ) as in [7], then by Theorem 2.3 and Theorem 1,∞ n 3.4 in [7], the space PA (R ) is the homogeneous Lipschitz space 1 (Rn ), which is also called the homogenous Zygmund space; see [29,32,33]. It is well-known that Lip 1 (Rn ) is a proper subspace of 1 (Rn ); see [29]. Hence, P1,∞ (Rn ) is a 1,∞ n proper subspace of the space PA (R ). Remark 4.5 Finally, we point out that throughout the paper, if we replace the assumption that d is a metric by that d is a quasi-metric together with all balls of X are open, then all the conclusions of this paper are still true. Acknowledgments Dachun Yang would like to thank Professor Hans Triebel for some helpful discussions on this topic. The authors would also like to thank the referee for his/her several remarks which make this paper more readable. The first author is supported by NNSF of China (Grant No. 10371134), and the second author is supported by National Science Foundation for Distinguished Young Scholars (Grant No. 10425106) and NCET (Grant No. 04-0142) of Ministry of Education of China.

References 1. Auscher, P., Coulhon, T., Duong, X.T., Hofmann, S.: Riesz transform on manifolds and heat kernel regularity. Ann. Sci. École Norm. Sup. (4) 37, 911–957 (2004) 2. Calderón, A.P.: Estimates for singular integral operators in terms of maximal functions. Stud. Math. 44, 563–582 (1972) 3. Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999) 4. Christ, M.: A T(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61, 601–628 (1990) 5. Coifman, R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Math. vol. 242, Springer, Berlin Heidelberg New York (1971)

New Sobolev spaces

159

6. Deng, D., Duong, X.T., Sikora, A., Yan, L.: Comparison of the classical BMO with the new BMO spaces associated with operators and applications. Preprint (2005) 7. Deng, D., Duong, X.T., Yan, L.: A characterization of the Morrey–Campanato spaces. Math. Z. 250, 641–655 (2005) 8. Duong, X.T., McIntosh, A.: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 15, 233–265 (1999) 9. Duong, X.T., Yan, L.: New function spaces of BMO type, the John–Nirenberg inequality, interpolation, and applications. Comm. Pure Appl. Math. 58, 1375–1420 (2005) 10. Duong, X.T., Yan, L.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18, 943–973 (2005) 11. Franchi, B., Hajłasz, P., Koskela, P.: Definitions of Sobolev classes on metric spaces. Ann. Inst. Fourier (Grenoble) 49, 1903–1924 (1999) 12. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin Heidelberg New York (1983) 13. Grigor’yan, A., Hu, J., Lau, K.: Heat kernels on metric measure spaces and an application to semilinear elliptic equations. Trans. Am. Math. Soc. 355, 2065–2095 (2003) 14. Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5, 403–415 (1996) 15. Hajłasz, P.: Geometric approach to Sobolev spaces and badly degenerated elliptic equations. Nonlinear analysis and applications (Warsaw, 1994), 141–168, GAKUTO Internat. Ser. Math. ¯ Sci. Appl., 7, Gakkotosho, Tokyo, 1996 16. Hajłasz, P., Kinnunen, J.: Hölder quasicontinuity of Sobolev functions on metric spaces. Rev. Mat. Iberoam. 14, 601–622 (1998) 17. Hajłasz, P., Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Ser. I Math. 320, 1211–1215 (1995) 18. Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), 1–101 (2000) 19. Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, Berlin Heidelberg New York, (2001) 20. Hofmann, S., Martell, J.M.: Lp bounds for Riesz transforms and square roots associated to second order elliptic operators. Publ. Mat. 47, 497–515 (2003) 21. Hu, J.: A note on Hajłasz–Sobolev spaces on fractals. J. Math. Anal. Appl. 280, 91–101 (2003) 22. Jonsson, A., Wallin, H.: Function Spaces on Subsets of Rn . Math. Rep. 2, Harwood Academic Publishers, London (1984) 23. Korevaar, N.J., Schoen, R.M.: Sobolev spaces and harmonic maps for metric space targets. Comm. Anal. Geom. 1, 561–659 (1993) 24. Liu, Y., Lu, G., Wheeden, R.L.: Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces. Math. Ann. 323, 157–174 (2002) 25. Martell, J.M.: Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications. Stud. Math. 161, 113–145 (2004) 26. Maz ya, V.G.: Sobolev Spaces. Springer, Berlin Heidelberg New York (1985) 27. Robinson, D.W.: Elliptic Operators and Lie Groups. Oxford University Press, Oxford (1991) 28. Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoam. 16, 243–279 (2000) 29. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970) 30. Strauss, W.A.: Partial Differential Equations: An Introduction. Wiley, New York (1992) 31. Strichartz, R.S.: Function spaces on fractals. J. Funct. Anal. 198, 43–83 (2003) 32. Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel (1983) 33. Triebel, H.: Theory of Function Spaces II. Birkhäuser Verlag, Basel (1992) 34. Triebel, H.: Fractals and Spectra. Birkhäuser Verlag, Basel (1997) 35. Triebel, H.: Theory of Function Spaces III. Birkhäuser Verlag, Basel (2006) 36. Yang, D.: New characterizations of Hajłasz-Sobolev spaces on metric spaces. Sci. China (Ser. A) 46, 675–689 (2003) 37. Yang, D.: Some function spaces relative to Morrey-Campanato spaces on metric spaces. Nagoya Math. J. 177, 1-29 (2005) 38. Yang, D., Lin, Y.: Spaces of Lipschitz type on metric spaces and their applications. Proc. Edinb. Math. Soc. (2) 47, 709–752 (2004)