A Difference Characterization of Besov and Triebel-Lizorkin Spaces on Rd-Spaces

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A DIFFERENCE CHARACTERIZATION OF BESOV AND TRIEBEL-LIZORKIN SPACES ON RD-SPACES

DETLEF MULLER¨ AND DACHUN YANG

Abstract. An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in

X , or equivalently, that there exists a constant a0 > 1 such that for all x ∈ X and 0 < r < diam (X )/a0, the annulus B(x, a0r) \ B(x, r) is nonempty, where diam (X ) denotes the diameter of the metric space (X , d). An important class of RD-spaces is provided by Carnot-Carath´eodory spaces with a doubling measure. In this paper, the authors introduce some spaces of Lipschitz type on RD-spaces, and discuss their relations with known Besov and Triebel-Lizorkin spaces and various Sobolev spaces. As an application, a diﬀerence characterization of Besov and Triebel-Lizorkin spaces on RD-spaces is obtained.

1 Introduction

Metric spaces play a prominent role in many ﬁelds of mathematics. In particular, they constitute natural generalizations of manifolds admitting all kinds of singularities and still providing rich geometric structure; see [45, 46, 32]. Analysis on metric measure spaces has been studied quite intensively in recent years; see, for example, Semmes’s survey [41] for a more detailed discussion and references. Of particular interest is the study of functional inequalities, like Sobolev and Poincar´einequalities, on metric measure spaces; see, for example, [29, 15, 20, 12, 11, 27]. Also the theory of function spaces on metric measure spaces has seen a rapid development in recent years. Since HajÃlasz in [13] introduced Sobolev spaces on any metric measure space, a series of papers has been devoted to the construction and investigation of Sobolev spaces of various types on metric measure spaces; see, for example, [15, 30, 8, 10, 31, 42, 20, 21, 28, 5, 14, 9, 49]. Recently, a theory of Besov and Triebel-Lizorkin spaces on RD-spaces was developed in [17], which is a generalization

2000 Mathematics Subject Classiﬁcation: Primary 42B35; Secondary 46E35, 43A99. Key words and phrases: RD-space, space of Lipschitz type, Besov space, Triebel-Lizorkin space, Sobolev space, diﬀerence characterization. Dachun Yang (the corresponding author) is supported by National Science Foundation for Distinguished Young Scholars (No. 10425106) and NCET (No. 04-0142) of Ministry of Education of China.

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2 Detlef M¨ullerand Dachun Yang

of the corresponding theory of function spaces on Rn (see [47, 48, 49]) respectively Ahlfors n-regular metric measure spaces (see [18, 19]). An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in X , or equivalently,

that there exists a constant a0 > 1 such that for all x ∈ X and 0 < r < diam (X )/a0, the annulus B(x, a0r) \ B(x, r) is nonempty, where diam (X ) denotes the diameter of the metric space (X , d); see [17]. An important class of RD-spaces is provided by Carnot- Carathéodory spaces with a doubling measure. In this paper, we introduce spaces of Lipschitz type on RD-spaces, and discuss their relations with Besov and Triebel-Lizorkin spaces in [17] and various Sobolev spaces in [29, 13]. As an application, a difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces in [17] is obtained. Spaces of Lipschitz type considered in this paper originate in [43, 23, 24]; see also [25, 26, 11, 12, 54]. These function spaces are defined via differences, respectively local means of differences; see [48]. The results in this paper apply in a wide range of settings, for instance, to Ahlfors n-regular metric measure spaces (see [20]), Lie groups of polynomial volume growth (see [50, 51, 40, 34, 1]), compact Carnot-Carathéodory (also called sub-Riemannian) manifolds (see [40, 36, 37]) and to boundaries of certain unbounded model domains of polynomial type in CN appearing in the work of Nagel and Stein (see [39, 40, 36, 37]). The outline of this paper is as follows. In Section 2, we recall the notion of RD-spaces, the notion of approximations to the identity and spaces of test functions and the notion of Besov and Triebel-Lizorkin spaces on RD-spaces. In Section 3, we introduce spaces of Lipschitz type. In Section 4, we establish some relations between spaces of Lipschitz type and Besov and Triebel-Lizorkin spaces. Finally, some relations between spaces of Lipschitz type and various Sobolev spaces are presented in Section 5. We make the following convention. 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; A . B and B & A mean that the ratio A/B is bounded by a constant independent of the relevant variables. For any p ∈ [1, ∞], we denote by p0 its conjugate index, namely, 1/p + 1/p0 = 1. We also denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. Constants with

subscripts, such as C0, do not change in diﬀerent occurrences. If E is a subset of a metric space (X , d), we denote by χE the characteristic function of E and deﬁne

diam E = sup d(x, y). x, y∈E

We also set N ≡ {1, 2, · · ·} and Z+ ≡ N ∪ {0}. For any a, b ∈ R, we denote min{a, b}, max{a, b}, and max{a, 0} by a ∧ b, a ∨ b and a+, respectively. Throughout the whole paper, for ² ∈ (0, 1] and |s| < ², we set p(s, ²) ≡ max{n/(n + ²), n/(n + ² + s)}. 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 3

2 Preliminaries

We ﬁrst recall the notion of a space of homogeneous type in the sense of Coifman and Weiss [3, 4] and the notion of RD-spaces in [17].

Definition 2.1 Let (X , d) be a metric space with a regular Borel measure µ, which means that µ is a nonnegative countably subadditive set function defined on all subsets of X , open sets are measurable and every set is contained in a Borel set with the same measure, such that all balls defined by d have finite and positive measure. For any x ∈ X and r > 0, set B(x, r) = {y ∈ X : d(x, y) < r}. (i) The triple (X , d, µ) is called a space of homogeneous type if there exists a constant

C0 ≥ 1 such that for all x ∈ X and r > 0,

(2.1) µ(B(x, 2r)) ≤ C0µ(B(x, r)) (doubling property). (ii) Let 0 < κ ≤ n. The triple (X , d, µ) is called a (κ, n)-space if there exist constants

0 < C1 ≤ 1 and C2 ≥ 1 such that for all x ∈ X , 0 < r < diam (X )/2 and 1 ≤ λ < diam (X )/(2r),

κ n (2.2) C1λ µ(B(x, r)) ≤ µ(B(x, λr)) ≤ C2λ µ(B(x, r)), where diam (X ) = supx, y∈X d(x, y). A space of homogeneous type will be called an RD-space, if it is a (κ, n)-space for some 0 < κ ≤ n, i. e., if some “reverse” doubling condition holds.

Clearly, any Ahlfors n-regular metric measure space (X , d, µ) (which means that there exists some n > 0 such that µ(B(x, r)) ∼ rn for all x ∈ X and 0 < r < diam (X )/2) is an (n, n)-space.

Remark 2.1 (i) Obviously, any (k, n)-space is a space of homogeneous type with C0 = n C22 . Conversely, any space of homogeneous type satisﬁes the second inequality of (2.2) with C2 = C0 and n = log2 C0. Comparing with spaces of homogeneous type, the only additional restriction in (k, n)-spaces is the ﬁrst inequality of (2.2). (ii) It was proved in [17] that X is an RD-space if and only if X is a space of homogeneous type with the additional property that there exists a constant a0 > 1 such that for all x ∈ X and 0 < r < diam (X )/a0, B(x, a0r) \ B(x, r) 6= ∅. (iii) From (ii), it is obvious that an RD-space has no isolated points. (iv) It is proved in [45, 46, 32] that some curvature-dimension condition on metric measure spaces implies the doubling property of the considered measure.

Remark 2.2 Notice that all results in this paper are true if d is a quasi-metric and d has some regularity, and that all results are invariant for equivalent quasi-metrics. Thus, by the result of Mac´ıasand Segovia [33], we know that all results in this paper are still true if d on X is only known to be a quasi-metric. 中国科技论文在线

http://www.paper.edu.cn 4 Detlef M¨ullerand Dachun Yang

Remark 2.3 If (X , d, µ) is a space of homogeneous type, we also introduce the volume

functions Vδ(x) = µ(B(x, δ)) and V (x, y) = µ(B(x, d(x, y))) for all x, y ∈ X and δ > 0. By (2.1), it is easy to see that V (x, y) ∼ V (y, x); see also [37]. We shall use this fact without further mentioning.

We now recall the notion of of an approximation to the identity on X .

Definition 2.2 Let ²1 ∈ (0, 1], ²2 > 0 and ²3 > 0. 2 (I) A sequence {Sk}k∈Z of bounded linear integral operators on L (X ) is said to be an approximation to the identity of order (²1, ²2, ²3), if there exists a constant C3 > 0 such 0 0 that for all k ∈ Z and all x, x , y and y ∈ X ,Sk(x, y), the integral kernel of Sk is a measurable function from X × X into C satisfying

1 2−k²2 (i) |Sk(x, y)| ≤ C3 −k ²2 ; V2−k (x)+V2−k (y)+V (x,y) (2 +d(x,y))

³ 0 ´² 0 d(x,x ) 1 1 2−k²2 (ii) |Sk(x, y) − Sk(x , y)| ≤ C3 −k −k ²2 for 2 +d(x,y) V2−k (x)+V2−k (y)+V (x,y) (2 +d(x,y)) d(x, x0) ≤ (2−k + d(x, y))/2;

(iii) Property (ii) also holds with x and y interchanged; ³ ´² ³ ´² 0 0 0 0 d(x,x0) 1 d(y,y0) 1 (iv) |[Sk(x, y) − Sk(x, y )] − [Sk(x , y) − Sk(x , y )]| ≤ C3 2−k+d(x,y) 2−k+d(x,y) 1 2−k²3 0 −k 0 × −k ²3 for d(x, x ) ≤ (2 + d(x, y))/3 and d(y, y ) ≤ V2−k (x)+V2−k (y)+V (x,y) (2 +d(x,y)) (2−k + d(x, y))/3; R (v) X Sk(x, y) dµ(y) = 1; R (vi) X Sk(x, y) dµ(x) = 1.

(II) A sequence {Sk}k∈Z+ of linear operators is said to be an inhomogeneous approxi-

mation to the identity of order (²1 , ²2 , ²3 ), if Sk for k ∈ Z+ satisﬁes (I).

It was proved in Theorem 2.1 in [17] that approximations to the identity of order

(²1, ²2, ²3) always exists on X , for any ²1 ∈ (0, 1], ²2 > 0 and ²3 > 0, whenever X is an RD-space. We also need suitable spaces of test functions on X in [17].

Definition 2.3 Let x1 ∈ X , r > 0, 0 < β ≤ 1 and γ > 0. A function f on X is said to be a test function of type (x1, r, β, γ) if there exists a constant C ≥ 0 such that ³ ´γ (i) |f(x)| ≤ C 1 r for all x ∈ X ; Vr(x1)+V (x1,x) r+d(x1,x)

³ ´β ³ ´γ (ii) |f(x) − f(y)| ≤ C d(x,y) 1 r for all x, y ∈ X satisfying r+d(x1,x) Vr(x1)+V (x1,x) r+d(x1,x) that d(x, y) ≤ (r + d(x1, x))/2. 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 5

Moreover, we denote by G(x1, r, β, γ) the space of all test functions of type (x1, r, β, γ),

endowed with the norm kfkG(x1,r,β,γ) ≡ inf {C :(i) and (ii) hold} .

Now fix x1 ∈ X and let G(β, γ) = G(x1, 1, β, γ). It is easy to see that G(x0, r, β, γ) = G(β, γ) with the equivalent norms for all x0 ∈ X and r > 0. Furthermore, it is easy to check that G(β, γ) is a Banach space with respect to the norm in G(β, γ). n It is well-known that even when X = R , G(β1, γ) is not dense in G(β2, γ) if β1 > β2. ² To overcome this defect, in what follows, for given ² ∈ (0, 1], let G0(β, γ) be the completion ² of the space G(², ²) in G(β, γ) when 0 < β, γ ≤ ². Obviously, G0(², ²) = G(², ²). Moreover, ² f ∈ G0(β, γ) if and only if f ∈ G(β, γ) and there exist {fn}n∈N ⊂ G(², ²) such that ² kf − f k → 0 as n → ∞. If f ∈ G (β, γ), we then define kfk ² = kfk . n G(β,γ) 0 G0(β,γ) G(β,γ) ² Then, obviously, G (β, γ) is a Banach space and we also have kfkG²(β,γ) = lim kfnkG(β,γ) 0 0 n→∞ for the above chosen {fn}n∈N. ² 0 ² We define the dual space (G0(β, γ)) to be the set of all linear functionals L from G0(β, γ) ² to C with the property that there exists C ≥ 0 such that for all f ∈ G0(β, γ),

|L(f)| ≤ Ckfk ² . G0(β,γ)

² 0 ² We denote by hh, fi the natural pairing of elements h ∈ (G0(β, γ)) and f ∈ G0(β, γ). ² 0 ² Clearly, for all h ∈ (G0(β, γ)) , hh, fi is well deﬁned for all f ∈ G0(x1, r, β, γ) with x1 ∈ X and r > 0. In the sequel, we deﬁne ½ Z ¾ G˚(x1, r, β, γ) = f ∈ G(x1, r, β, γ): f(x) dµ(x) = 0 , X

˚² which is called to be the space of test functions with mean zero. The space G0(β, γ) is defined to be the completion of the space G˚(², ²) in G˚(β, γ) when 0 < β, γ < ². Moreover, ˚² if f ∈ G0(β, γ), we then define kfk˚² = kfkG(β,γ). G0(β,γ) Now we recall the notion of homogeneous Besov and Triebel-Lizorkin spaces from [17]. Such spaces are well-defined only when µ(X ) = ∞.

Definition 2.4 Let ²1 ∈ (0, 1], ²2 > 0, ²3 > 0, ² ∈ (0, ²1 ∧ ²2 ) and let {Sk}k∈Z be an approximation to the identity of order (²1, ²2, ²3). For k ∈ Z, set Dk = Sk − Sk−1. ˙ s (i) Let |s| < ², p(s, ²) < p ≤ ∞ and 0 < q ≤ ∞. The homogeneous Besov space Bp, q(X ) ³ ´0 ˚² is deﬁned to be the space of all f ∈ G0(β, γ) , for some β, γ satisfying

(2.3) max{s, 0, −s + n(1/p − 1)+} < β < ²,

max{s − κ/p, n(1/p − 1)+, −s + n(1/p − 1)+ − κ(1 − 1/p)+} < γ < ² 中国科技论文在线 http://www.paper.edu.cn

6 Detlef M¨ullerand Dachun Yang

such that ( ) X∞ 1/q ksq q kfk ˙ s = 2 kDk(f)k p < ∞, Bp, q(X ) L (X ) k=−∞ with the usual modifications made when p = ∞ or q = ∞. (ii) Let |s| < ², p(s, ²) < p < ∞ and p(s, ²) < q ≤ ∞. The homogeneous Triebel-Lizorkin ³ ´0 ˙ s ˚² space Fp, q(X ) is defined to be the space of all f ∈ G0(β, γ) for some β, γ satisfying (2.3) such that ° ° °( )1/q° ° X∞ ° ° ksq q ° kfk ˙ s = 2 |Dk(f)| < ∞, Fp, q(X ) ° ° ° k=−∞ ° Lp(X ) with the usual modification made when q = ∞.

˙ s ˙ s It was proved in [17] that the definitions of the spaces Bp, q(X ) and Fp, q(X ) are indepen- ³ ´0 ˚² dent of the choice of approximation to the identity and the distribution space G0(β, γ) , with β, γ as in (2.3). s ˙ s To recall the definition of Besov spaces Bp, q(X ) and Triebel-Lizorkin spaces F∞, q(X ) s and Fp, q(X ) in [17], we need first to recall the following construction given by Christ in [2], which provides an analogue of the set of Euclidean dyadic cubes on spaces of homogeneous type.

Lemma 2.1 Let X be a space of homogeneous type. Then there exists a collection k {Qα ⊂ X : k ∈ Z, α ∈ Ik} of open subsets, where Ik is some index set, and constants δ ∈ (0, 1) and C4,C5 > 0 such that k k k (i) µ(X \ ∪αQα) = 0 for each ﬁxed k and Qα ∩ Qβ = ∅ if α 6= β; l k l k (ii) for any α, β, k, l with l ≥ k, either Qβ ⊂ Qα or Qβ ∩ Qα = ∅; k l (iii) for each (k, α) and each l < k there is a unique β such that Qα ⊂ Qβ; k k (iv) diam (Qα) ≤ C4δ ; k k k k (v) each Qα contains some ball B(zα,C5δ ), where zα ∈ X .

k k In fact, we can think of Qα as being a dyadic cube with diameter rough δ and centered k at zα. In what follows, to simplify our presentation, we always suppose δ = 1/2; otherwise, we need to replace 2−k in the deﬁnition of our approximations to the identity by δk and some other changes are also necessary; see [18, pp. 96-98] for more details. k,ν In the following, for k ∈ Z and τ ∈ Ik, we denote by Qτ , ν = 1, 2, ··· ,N(k, τ), the k+j k k set of all cubes Qτ 0 ⊂ Qτ , where Qτ is the dyadic cube as in Lemma 2.1 and j is a ﬁxed −j k,ν k,ν positive large integer such that 2 C4 < 1/3. Denote by zτ the “center” of Qτ as in k,ν k,ν Lemma 2.1 and by yτ a point in Qτ . We next recall the notion of inhomogeneous Besov and Triebel-Lizorkin spaces from [17]. 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 7

Definition 2.5 Let ²1 ∈ (0, 1], ²2 > 0, ²3 > 0, ² ∈ (0, ²1 ∧ ²2 ) and let {Sk}k∈Z+ be an inhomogeneous approximation to the identity of order (²1, ²2, ²3). Set D0 = S0 and 0,ν Dk = Sk − Sk−1 for k ∈ N. Let {Qτ : τ ∈ I0, ν = 1, ··· ,N(0, τ)} with a ﬁxed large j ∈ N be dyadic cubes as above. s (i) Let |s| < ², p(s, ²) < p ≤ ∞ and 0 < q ≤ ∞. The Besov space Bp, q(X ) is deﬁned to ² 0 be the space of all f ∈ (G0(β, γ)) , for some β, γ satisfying

(2.4) max{s, 0, −s + n(1/p − 1)+} < β < ² and n(1/p − 1)+ < γ < ²

such that

 1/p N(0,τ) X X h ip 0,ν kfkBs (X ) = µ(Qτ ) m 0,ν (|D0(f)|) p, q  Qτ  τ∈I0 ν=1 ( ) X∞ 1/q ksq q + 2 kDk(f)kLp(X ) < ∞, k=1

with the usual modiﬁcations made when p = ∞ or q = ∞. (ii) Let |s| < ², p(s, ²) < p < ∞ and p(s, ²) < q ≤ ∞. The Triebel-Lizorkin space s ² 0 Fp, q(X ) is deﬁned to be the space of all f ∈ (G0(β, γ)) for some β, γ satisfying (2.4) such that

 1/p N(0,τ) X X h ip 0,ν kfkF s (X ) = µ(Qτ ) m 0,ν (|D0(f)|) p, q  Qτ  τ∈I ν=1 ° 0 ° °( )1/q° ° X∞ ° ° ksq q ° + ° 2 |Dk(f)| ° < ∞, ° k=1 ° Lp(X )

with the usual modiﬁcation made when q = ∞.

s s It was proved in [17] that the deﬁnitions of the spaces Bp, q(X ) and Fp, q(X ) are independent of the choice of inhomogeneous approximation to the identity and the distribution ² 0 space, (G0(β, γ)) , with β, γ as in (2.4).

Definition 2.6 Let ²1 ∈ (0, 1], ²2 > 0, ²3 > 0 and ² ∈ (0, ²1 ∧ ²2 ). (i) Assume that µ(X ) = ∞, and let {Sk}k∈Z be an approximation to the identity of order (²1, ²2, ²3). For k ∈ Z, set Dk = Sk − Sk−1. Let |s| < ² and p(s, ²) < q ≤ ∞. The ³ ´0 ˙ s ˚² Triebel-Lizorkin space F∞, q(X ) is deﬁned to be the space of all f ∈ G0(β, γ) , for some β, γ satisfying

(2.5) |s| < β < ², max{s, 0, −s − κ} < γ < ² 中国科技论文在线 http://www.paper.edu.cn

8 Detlef M¨ullerand Dachun Yang

such that ( ) Z X∞ 1/q 1 ksq q kfkF˙ s (X ) = sup sup l 2 |Dk(f)(x)| dµ(x) < ∞, ∞, q α∈I µ(Q ) l l∈Z l α Qα k=l where the supremum is taken over all dyadic cubes as in Lemma 2.1 and the usual modi- ﬁcation is made when q = ∞.

(ii) Let {Sk}k∈Z be an inhomogeneous approximation to the identity of order (²1, ²2, ²3). 0,ν Set Dk = Sk − Sk−1 for k ∈ N and D0 = S0. Let {Qτ : τ ∈ I0, ν = 1, ··· ,N(0, τ)} with a ﬁxed large j ∈ N be dyadic cubes as above. Let |s| < ² and p(s, ²) < q ≤ ∞. The s ² 0 Triebel-Lizorkin space F∞, q(X ) is deﬁned to be the space of all f ∈ (G0(β, γ)) , for some |s| < β < ² and 0 < γ < ², such that   kfk s = max sup m 0,ν (|D0(f)|), F∞, q(X ) Qτ  τ∈I0 ν=1,···,N(0,τ) " #  Z X∞ 1/q 1 ksq q sup sup l 2 |Dk(f)(x)| dµ(x) < ∞, α∈I µ(Q ) l  l∈N l α Qα k=l

where the supremum is taken over all dyadic cubes as in Lemma 2.1, and the usual modiﬁcation is made when q = ∞.

˙ s s It was also proved in [17] that the deﬁnitions of the spaces F∞, q(X ) and F∞, q(X ) are independent of the choice of the approximation to the identity and the distribution space, ³ ´0 ˚² G0(β, γ) , with β, γ as in (2.5), respectively, the inhomogeneous approximation to the ² 0 identity and the distribution space, (G0(β, γ)) , with |s| < β < ² and 0 < γ < ².

3 Spaces of Lipschitz type

Let us ﬁrst introduce the notion of spaces of Lipschitz type on arbitrary spaces of homogeneous type, which originates in [43, 23, 24]; see also [25, 26, 11, 12, 54].

Definition 3.1 Let 0 < p, q ≤ ∞, s > 0 and C6 > 0 be a constant. p ˙ (i) A function f ∈ L loc (X ) is said to belong to the Lipschitz-type space L(s, p, q; X ) if and only if ( µZ X∞ 1 kfk = 2νsq L˙ (s, p, q; X ) µ(B(x, C 2−ν)) ν=−∞ X 6 ! 1/q Z q/p × |f(x) − f(y)|p dµ(y) dµ(x) < ∞, −ν B(x,C62 )  中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 9

with the usual modiﬁcations made when p = ∞ or q = ∞. A function f ∈ Lp(X ) is said to belong to the Lipschitz-type space L(s, p, q; X ) if and

only if kfkL˙ (s, p, q; X ) < ∞. Moreover, we deﬁne

kfkL(s, p, q; X ) = kfkLp(X ) + kfkL˙ (s, p, q; X ).

1 ˙ (ii) A function f ∈ L loc (X ) is said to belong to the Lipschitz-type space Lb(s, p, q; X ) if and only if ( Ã " X∞ Z Z νsq 1 kfkL˙ (s, p, q; X ) = 2 −ν |f(x) b µ(B(x, C 2 )) −ν ν=−∞ X 6 B(x,C62 ) # ! 1/q p q/p −f(y)| dµ(y) dµ(x) < ∞, 

with the usual modiﬁcations made when p = ∞ or q = ∞. p A function f ∈ L (X ) is said to belong to the Lipschitz-type space Lb(s, p, q; X ) if and only if kfk ˙ < ∞. Moreover, we deﬁne Lb(s, p, q; X )

kfk = kfk p + kfk ˙ . Lb(s, p, q; X ) L (X ) Lb(s, p, q; X )

The Lipschitz space L(s, p, q; X ) when X is a d-set of Rn was introduced in [25, 26]; see also [11, 12] for the case p = 2 and q = ∞. When X is an Ahlfors n-regular metric measure space, these spaces are introduced in [54]. We also point out that when X is a metric measure space, these spaces may be non-trivial even when s > 1; a such example can be found in [54], when X is a self-similar fractal in R2.

It is easy to see that the deﬁnitions of L˙ (s, p, q; X ), L(s, p, q; X ), L˙ b(s, p, q; X ) and Lb(s, p, q; X ) are independent of the choice of C6 > 0; see also [25]; see also [54]. On these spaces, we have the following basic properties, whose proofs are easy and we omit the details; see [54]. We denote here and in the sequel by N ≡ C the space of all constant functions on X .

Proposition 3.1 Let s > 0.

(i) If 0 < p, q ≤ ∞, then the spaces L˙ (s, p, q; X )/N , L(s, p, q; X ), L˙ b(s, p, q; X )/N and Lb(s, p, q; X ) are quasi-Banach spaces. (ii) If 1 ≤ p ≤ ∞ and 0 < q ≤ ∞, then L˙ (s, p, q; X ) ⊂ L˙ b(s, p, q; X ) and

L(s, p, q; X ) ⊂ Lb(s, p, q; X ).

(iii) If 0 < q0 ≤ q1 ≤ ∞ and 0 < p ≤ ∞, then L˙ (s, p, q0; X ) ⊂ L˙ (s, p, q1; X ),

L˙ b(s, p, q0; X ) ⊂ L˙ b(s, p, q1; X ),L(s, p, q0; X ) ⊂ L(s, p, q1; X ), 中国科技论文在线 http://www.paper.edu.cn

10 Detlef M¨ullerand Dachun Yang

and Lb(s, p, q0; X ) ⊂ Lb(s, p, q1; X ). (iv) If 0 < q0, q1 ≤ ∞, 0 < p ≤ ∞ and ² > 0, then L(s + ², p, q0; X ) ⊂ L(s, p, q1; X ) and Lb(s + ², p, q0; X ) ⊂ Lb(s, p, q1; X ). (v) L˙ (s, ∞, ∞; X ) = C˙ s(X ) and L(s, ∞, ∞; X ) = Cs(X ), where C˙ s(X ) and Cs(X ) are respectively the homogeneous H¨olderspace and the inhomogeneous one.

The following Lipschitz-type spaces L˙ t(s, p, q; X ) and Lt(s, p, q; X ) when X is an Ahlfors n-regular metric measure space are also introduced in [54].

Definition 3.2 Let 0 < p, q ≤ ∞, s > 0 and C7 > 0 be a constant. A function 1 ˙ f ∈ L loc (X ) is said to belong to the Lipschitz-type space Lt(s, p, q; X ) if and only if

kfk ˙ Lt°(s, p, q; X ) ° °( " Z #q)1/q° ° X∞ ° ° ksq 1 ° = ° 2 −k |f(y) − f(·)| dµ(y) ° µ(B(·,C72 )) −k ° k=−∞ B(·,C72 ) ° Lp(X ) < ∞, with the usual modiﬁcations made when p = ∞ or q = ∞. p A function f ∈ L (X ) is said to belong to the Lipschitz-type space Lt(s, p, q; X ) if and only if kfk < ∞. Moreover, we deﬁne L˙ t(s, p, q; X )

kfk = kfk p + kfk . Lt(s, p, q; X ) L (X ) L˙ t(s, p, q; X )

The following basic properties of L˙ t(s, p, q; X ) and Lt(s, p, q; X ) can be proved in a similar way as Proposition 3.5 in [54]. We omit the details.

Proposition 3.2 Let s > 0.

(i) If 0 < p, q ≤ ∞, then L˙ t(s, p, q; X )/N and Lt(s, p, q; X ) are quasi-Banach spaces. (ii) If 0 < q0 ≤ q1 ≤ ∞ and 0 < p ≤ ∞, then L˙ t(s, p, q0; X ) ⊂ Lt(s, p, q1; X ) and Lt(s, p, q0; X ) ⊂ Lt(s, p, q1; X ). (iii) If 0 < q0, q1 ≤ ∞, 0 < p ≤ ∞ and ² > 0, then Lt(s+², p, q0; X ) ⊂ Lt(s, p, q1; X ). (iv) If 0 < p, q ≤ ∞, then

L˙ b(s, p, min(p, q); X ) ⊂ L˙ t(s, p, q; X ) ⊂ L˙ b(s, p, max(p, q); X )

and Lb(s, p, min(p, q); X ) ⊂ Lt(s, p, q; X ) ⊂ Lb(s, p, max(p, q); X ). (v) If 0 < p = q ≤ ∞, then L˙ b(s, p, q; X ) = L˙ t(s, p, q; X ) and Lb(s, p, q; X ) = Lt(s, p, q; X ).

Finally, we establish some basic characterizations of L(s, p, q; X ), Lb(s, p, q; X ) and Lt(s, p, q; X ). To this end, we ﬁrst recall some basic estimates, which will be used throughout the whole paper, and whose proof can be found in [37, 16]. Throughout the whole 1 paper, we denote by Mf the Hardy-Littlewood maximal function on X for any f ∈ L loc (X ). It is well-known that M is bounded on Lp(X ) with p ∈ (1, ∞]; see [3, 37]. 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 11

Lemma 3.1 Let δ > 0, a > 0, r > 0 and θ ∈ (0, 1). Then

R a R d(x,y) a 1 δa (i) d(x,y)≤δ V (x,y) dµ(y) ≤ Cδ and d(x,y)≥δ V (x,y) d(x,y)a dµ(y) ≤ C uniformly in x ∈ X and δ > 0. R 1 δa η η (ii) a d(x, y) dµ(y) ≤ Cδ uniformly in x ∈ X and δ > 0, if X Vδ(x)+V (x,y) (δ+d(x,y)) a > η ≥ 0. R 1 1 δa (iii) For all f ∈ L loc (X ) and all x ∈ X , d(x,y)>δ V (x,y) d(x,y)a |f(y)| dµ(y) ≤ CM(f)(x) 1 uniformly in δ > 0, f ∈ L loc (X ) and x ∈ X . (iv) For any ﬁxed α > 0, if d(x, y) ≤ αr, then Vr(x) ∼ Vαr(x) ∼ Vαr(y) ∼ Vr(y) uniformly in x, y ∈ X and r > 0.

Proposition 3.3 Let s > 0. (i) If 0 < p, q ≤ ∞, then f ∈ L(s, p, q; X ) if and only if f ∈ Lp(X ) and ( µZ X∞ 1 (3.1) 2νsq µ(B(x, C 2−ν)) ν=0 X 6 ! 1/q Z q/p × |f(x) − f(y)|p dµ(y) dµ(x) < ∞. −ν B(x,C62 ) 

Moreover, in this case,

kfkL(s, p, q; X ) ∼ kfkLp(X ) + Left − hand side of (3.1).

p (ii) If 1 ≤ p ≤ ∞ and 0 < q ≤ ∞, then f ∈ Lb(s, p, q; X ) if and only if f ∈ L (X ) and ( µZ · X∞ 1 (3.2) 2νsq µ(B(x, C 2−ν)) ν=0 X 6 # ! 1/q Z p q/p × |f(x) − f(y)| dµ(y) dµ(x) < ∞. −ν B(x,C62 ) 

Moreover, in this case,

p kfkLb(s, p, q; X ) ∼ kfkL (X ) + Left − hand side of (3.2).

p (iii) If 1 ≤ p ≤ ∞ and 0 < q ≤ ∞, then f ∈ Lt(s, p, q; X ) if and only if f ∈ L (X ) and ° ° °( " Z #q)1/q° ° X∞ ° ° ksq 1 ° (3.3) ° 2 −k |f(y) − f(·)| dµ(y) ° µ(B(·,C72 )) −k ° k=0 B(·,C72 ) ° Lp(X ) < ∞. 中国科技论文在线 http://www.paper.edu.cn

12 Detlef M¨ullerand Dachun Yang

Moreover, in this case,

p kfkLt(s, p, q; X ) ∼ kfkL (X ) + Left − hand side of (3.3).

Proof. To see (i), since s > 0, by Lemma 3.1 (iv), we then have ( µZ X−1 1 2νsq µ(B(x, C 2−ν)) ν=−∞ X 6 ! 1/q Z q/p × |f(x) − f(y)|p dµ(y) dµ(x) −ν B(x,C62 )  ( µZ X−1 1 . 2νsq µ(B(x, C 2−ν)) ν=−∞ X 6 ! 1/q Z q/p × [|f(x)|p + |f(y)|p] dµ(y) dµ(x) −ν B(x,C62 ) 

. kfkLp(X ),

which implies (i). To see (ii), since s > 0 and 1 ≤ p ≤ ∞, by H¨older’sinequality and Lemma 3.1 (iv), we obtain ( µZ · X−1 1 2νsq µ(B(x, C 2−ν)) ν=−∞ X 6 # ! 1/q Z p q/p × |f(x) − f(y)| dµ(y) dµ(x) −ν B(x,C62 ) 

( )1/q ( X−1 X−1 µZ · νsq νsq p 1 . 2 kfk p + 2 |f(y)| L (X ) µ(B(y, C 2−ν)) ν=−∞ ν=−∞ X 6 # ! 1/q Z q/p × dµ(x) dµ(y) −ν B(y,C62 ) 

. kfkLp(X ),

which also proves (ii). We ﬁrst verify (iii) in the case 1 < p ≤ ∞. In this case, by the Lp(X )-boundedness of the Hardy-Littlewood maximal operator and s > 0, we have ° ° °( " Z #q)1/q° ° X−1 ° ° ksq 1 ° ° 2 −k |f(y) − f(·)| dµ(y) ° µ(B(·,C72 )) −k ° k=−∞ B(·,C72 ) ° Lp(X ) 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 13

( )1/q X−1 ksq © ª . 2 kfkLp(X ) + kMfkLp(X ) k=−∞

. kfkLp(X ), which implies (iii) when 1 < p ≤ ∞.

Recall the following well-known inequality that for all r ∈ (0, 1] and all aj ∈ C with j in some countable set of indices,  r X X r (3.4)  |aj| ≤ |aj| . j j If p = 1 and 1 ≤ q ≤ ∞, by (3.4), s > 0 and Lemma 3.1 (iv), we obtain ° ° °( " Z #q)1/q° ° X−1 ° ° ksq 1 ° ° 2 −k |f(y) − f(·)| dµ(y) ° µ(B(·,C72 )) −k ° k=−∞ B(·,C72 ) ° L1(X ) ( )1/q X−1 X−1 Z ksq ks . 2 kfkL1(X ) + 2 |f(y)| k=−∞ k=−∞ X ½Z ¾ 1 × χ −k (x) dµ(x) dµ(y) −k B(y,C72 ) X µ(B(y, C72 )) . kfkL1(X ); while when 0 < q < 1, by Minkowski’s inequality, we have ° ° °( " Z #q)1/q° ° X−1 ° ° ksq 1 ° ° 2 −k |f(y) − f(·)| dµ(y) ° µ(B(·,C72 )) −k ° k=−∞ B(·,C72 ) ° L1(X ) ( )1/q X−1 ksq . 2 kfkL1(X ) k=−∞ ° " # ° ° −1 Z q°1/q ° X 1 ° + ° 2ksq |f(y)| dµ(y) ° −k −k ° µ(B(·,C72 )) B(·,C72 ) ° k=−∞ L1/q(X )

. kfkL1(X ), which completes the proof of (iii) in the case p = 1, and hence the proof of Proposition 3.3. ¤

4 Relations with Besov and Triebel-Lizorkin spaces

From now on till the end of this paper, we work on RD-spaces. In this section, we clarify the relations between spaces of Lipschitz type and Besov spaces or Triebel-Lizorkin 中国科技论文在线 http://www.paper.edu.cn

14 Detlef M¨ullerand Dachun Yang

spaces on RD-spaces. It may be useful to notice that the subsequent results will hold for any ² ∈ (0, 1).

Proposition 4.1 Let ² be as in Deﬁnition 2.4, 0 < s < ² and 0 < q ≤ ∞. Then, ˙ ˙ s (i) If n/(n + ²) < p ≤ ∞, then Lb(s, p, q; X ) ⊂ Bp, q(X ); s (ii) If 1 ≤ p ≤ ∞, then Lb(s, p, q; X ) ⊂ Bp, q(X ).

³ ´0 ˙ ˚² Proof. To see (i), we ﬁrst prove that if f ∈ Lb(s, p, ∞; X ), then f ∈ G0(β, γ) with 0 < β < ² and s < γ < ². In fact, for any g ∈ G˚(β, γ) with 0 < β < ² and s < γ < ² and

x ∈ B(x1, 1), by Lemma 3.1 (iv), we have ¯Z ¯ ¯ ¯ ¯ ¯ ¯ f(y)g(y) dµ(y)¯ X ¯Z ¯ ¯ ¯ ¯ ¯ = ¯ [f(y) − f(x)] g(y) dµ(y)¯ X Z µ ¶ 1 1 γ . kgkG(β,γ) |f(y) − f(x)| dµ(y) X V1(x1) + V (x1, y) 1 + d(x1, y) Z X∞ 1 1 . kgkG(β,γ) jγ j |f(y) − f(x)| dµ(y). 2 µ(B(x, C 2 )) j j=0 6 B(x,C62 )

When p ≤ 1, from this and (3.4), it follows that ¯Z ¯ ¯ ¯ ¯ ¯ ¯ f(y)g(y) dµ(y)¯ X ( Z ¯Z ¯ )1/p ¯ ¯p 1 ¯ ¯ = ¯ [f(y) − f(x)]g(y) dµ(y)¯ dµ(x) µ(B(x1, 1)) B(x1,1) X

. kgkG(β,γ)   Z " Z #p 1/p X∞ 1 1  × jγp j |f(y) − f(x)| dµ(y) dµ(x)  2 µ(B(x, C 2 )) j  j=0 X 6 B(x,C62 )

. kgk kfk ˙ ; G(β,γ) Lb(s, p, ∞; X )

while when 1 < p ≤ ∞, by Minkowski’s inequality, we have ¯Z ¯ ¯ ¯ ¯ ¯ ¯ f(y)g(y) dµ(y)¯ X (Z " Z #p )1/p X∞ 1 1 . kgkG(β,γ) jγ j |f(y) − f(x)| dµ(y) dµ(x) 2 µ(B(x, C 2 )) j j=0 X 6 B(x,C62 )

. kgk kfk ˙ , G(β,γ) Lb(s, p, ∞; X ) 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 15

³ ´0 ˚² by γ > s. Thus, f ∈ G0(β, γ) with 0 < β < ² and s < γ < ². By this and Proposition ³ ´0 ˚² 3.1 (iii), we know that Lb(s, p, q; X ) ⊂ G0(β, γ) , with 0 < β < ² and s < γ < ², when 0 < s < ² and 0 < p, q ≤ ∞.

Now, let {Dk}k∈Z be as in Deﬁnition 2.4. For any k ∈ Z and x ∈ X , ¯Z ¯ ¯ ¯ ¯ ¯ (4.1) |Dk(f)(x)| = ¯ Dk(x, y)[f(y) − f(x)] dµ(y)¯ X Z −k² 1 2 2 . −k ² |f(y) − f(x)| dµ(y) V −k (x) + V −k (y) + V (x, y) (2 + d(x, y)) 2 X 2 2 Z X∞ 1 1 . j² j−k |f(y) − f(x)| dµ(y). 2 2 µ(B(x, C 2 )) j−k j=0 6 B(x,C62 ) If p ≤ 1, by (3.4), we have  Z · X∞ 1 1 kDk(f)kLp(X ) . j² p j−k  2 2 µ(B(x, C 2 )) j=0 X 6 Z #p )1/p × |f(y) − f(x)| dµ(y) dµ(x) . j−k B(x,C62 ) If p ≤ 1 and q/p ≤ 1, by (3.4) again, ( ) X∞ 1/q ksq q kfk ˙ s = 2 kDk(f)k p Bp, q(X ) L (X ) k=−∞  " X∞ X∞ µZ · 1 ksq 1 . j² q 2 j−k  2 2 µ(B(x, C62 )) j=0 k=−∞ X # ! 1/p Z p q/p  × |f(y) − f(x)| dµ(y) dµ(x)  j−k B(x,C62 ) 

. kfk ˙ ; Lb(s, p, q; X ) while when p ≤ 1 and q/p > 1, by Minkowski’s inequality, ( ) X∞ 1/q ksq q kfk ˙ s = 2 kDk(f)k p Bp, q(X ) L (X ) k=−∞  ( X∞ X∞ µZ ·  1 ksq 1 . j² p 2 j−k 2 2 µ(B(x, C62 )) j=0 k=−∞ X  # ! p/q 1/p Z p q/p  × |f(y) − f(x)| dµ(y) dµ(x)  j−k B(x,C62 )  中国科技论文在线 http://www.paper.edu.cn

16 Detlef M¨ullerand Dachun Yang

. kfk ˙ . Lb(s, p, q; X )

If 1 < p ≤ ∞, then by Minkowski’s inequality,

kDk(f)kLp(X ) ÃZ " Z #p !1/p X∞ 1 1 . j² j−k |f(y) − f(x)| dµ(y) dµ(x) ; 2 2 µ(B(x, C 2 )) j−k j=0 X 6 B(x,C62 )

When q ≤ 1, by (3.4), we further have  Ã X∞ X∞ ½Z · 1 ksq 1 kfkB˙ s (X ) . j² q 2 j−k p, q  2 2 µ(B(x, C62 )) j=0 k=−∞ X # ) 1/q Z p q/p  × |f(y) − f(x)| dµ(y) dµ(x)  j−k B(x,C62 ) 

. kfk ˙ ; Lb(s, p, q; X )

while when 1 < q ≤ ∞, by Minkowski’s inequality, ( X∞ X∞ µZ · 1 ksq 1 kfkB˙ s (X ) . j² 2 j−k p, q 2 2 µ(B(x, C62 )) j=0 k=−∞ X # ! 1/q Z p q/p × |f(y) − f(x)| dµ(y) dµ(x) j−k B(x,C62 ) 

. kfk ˙ , Lb(s, p, q; X )

which completes the proof of (i). ³ ´0 To see (ii), since 1 ≤ p ≤ ∞ and f ∈ Lp(X ), it is easy to see f ∈ G˚(β, γ) for any 0 < β, γ < ².

Let now {Dk}k∈Z+ be as in Deﬁnition 2.5. By Lemma 3.1 (ii) and (i) of Deﬁnition 2.2 (I) together with Fubini’s theorem, it is easy to see that

(4.2) kD0(f)kL1(X ) = kS0(f)kL1(X ) . kfkL1(X ).

Moreover, by (i) of Deﬁnition 2.2 (I) and Lemma 3.1 (iii), for all x ∈ X , we have p |S0(f)(x)| . M(f)(x), which together with the L (X )-boundedness of M for p ∈ (1, ∞] shows that

(4.3) kD0(f)kLp(X ) . kfkLp(X ). 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 17

From (4.2), (4.3), (4.1) and Minkowski’s inequality, it follows that ( ) X∞ 1/q ksq q kfk s = 2 kD (f)k Bp, q(X ) k Lp(X ) k=0   X∞ X∞ µZ · ksq  1 1 . kfkLp(X ) + 2 j² j−k  2 2 µ(B(x, C62 )) k=1 j=0 X # ! q1/q Z p 1/p  × |f(y) − f(x)| dµ(y) dµ(x)  . j−k B(x,C62 ) 

If q ≤ 1, by (3.4),  " X∞ X∞ µZ · 1 ksq 1 kfk s . kfk p + 2 Bp, q(X ) L (X ) j² q j−k  2 2 µ(B(x, C62 )) j=0 k=1 X # ! 1/q Z p q/p  × |f(y) − f(x)| dµ(y) dµ(x)  j−k B(x,C62 ) 

. kfk p + kfk ˙ L (X ) Lb(s, p, q; X )

∼ kfkLb(s, p, q; X );

while if 1 < q ≤ ∞, by Minkowski’s inequality, ( X∞ X∞ µZ · 1 ksq 1 kfk s . kfk p + 2 Bp, q(X ) L (X ) j² j−k 2 2 µ(B(x, C62 )) j=0 k=1 X # ! 1/q Z p q/p × |f(y) − f(x)| dµ(y) dµ(x) j−k B(x,C62 ) 

. kfk p + kfk ˙ L (X ) Lb(s, p, q; X )

∼ kfkLb(s, p, q; X ),

which completes the proof of Proposition 4.1. ¤

To establish the converse to Proposition 4.1, we need to recall the discrete Calder´on reproducing formulae from [17].

∞ Lemma 4.1 Let ²1 ∈ (0, 1], ²2 > 0, ²3 > 0, ² ∈ (0, ²1 ∧ ²2 ) and let {Sk}k=−∞ be an approximation to the identity of order (²1, ²2, ²3). Set Dk = Sk − Sk−1 for any k ∈ Z. Then, for any ﬁxed j ∈ N large enough, there exists a family of linear operators {Dek}k∈Z 中国科技论文在线 http://www.paper.edu.cn

18 Detlef M¨ullerand Dachun Yang

k,ν k,ν such that for any ﬁxed yτ ∈ Qτ with k ∈ Z, τ ∈ Ik and ν = 1, ··· ,N(k, τ), and all ³ ´0 ˚² f ∈ G0(β, γ) with 0 < β, γ < ²,

X∞ X NX(k,τ) k,ν e k,ν k,ν f(x) = µ(Qτ )Dk(x, yτ )Dk(f)(yτ ), k=−∞ τ∈Ik ν=1

³ ´0 ˚² e where the series converge in G0(β, γ) . Moreover, the kernels of the operators Dk satisfy the conditions (i) and (ii) of Deﬁnition 2.2 (I) with ² and ² replaced by ²0 ∈ (², ² ∧ ² ), R R 1 2 1 2 e e and X Dk(x, y) dµ(y) = X Dk(x, y) dµ(x) = 0.

Lemma 4.2 Let ²1 ∈ (0, 1], ²2 > 0, ²3 > 0, ² ∈ (0, ²1 ∧ ²2 ) and let {Sk}k∈Z+ be an inhomogeneous approximation to the identity of order (²1, ²2, ²3). Set D0 = S0 and Dk = Sk − Sk−1 for k ∈ N. Then for any ﬁxed j large enough, there exists a family e k,ν k,ν of functions {Dk(x, y)}k∈Z+ such that for any ﬁxed yτ ∈ Qτ with k ∈ N, τ ∈ Ik and ² 0 ν = 1, ··· ,N(k, τ) and all f ∈ (G0(β, γ)) with 0 < β, γ < ²,

X NX(0,τ) Z 0,ν f(x) = De0(x, y) dµ(y)D (f) 0,ν τ,1 Qτ τ∈I0 ν=1 X∞ X NX(k,τ) k,ν e k,ν k,ν + µ(Qτ )Dk(x, yτ )Dk(f)(yτ ), k=1 τ∈Ik ν=1

² 0 0,ν where the series convergence in (G0(β, γ)) , and Dτ,1 for τ ∈ I0 and ν = 1, ··· ,N(0, τ) is the corresponding integral operator with the kernel Z 0,ν 1 D (z) = D0(u, z) dµ(u). τ,1 0,ν 0,ν µ(Qτ ) Qτ e Moreover, Dk for k ∈ N satisfies the conditions (i) and (ii) of Definition 2.2 (I) with ²1 0 0 and ²2 replaced by ² ∈ (², ²1 ∧ ²2 ); and there exists a constant C > 0 depending on ² such that the function De0(x, y) satisfies that e 1 1 (i) for all x, y ∈ X , |D0(x, y)| ≤ C 0 , V1(x)+V1(y)+V (x,y) (1+d(x,y))² (ii) for all x, x0, y ∈ X with d(x, x0) ≤ (1 + d(x, y))/2,

µ 0 ¶²0 e e 0 d(x, x ) 1 1 |D0(x, y) − D0(x , y)| ≤ C ²0 ; 1 + d(x, y) V1(x) + V1(y) + V (x, y) (1 + d(x, y)) R R e e and X Dk(x, y) dµ(x) = X Dk(x, y) dµ(y) = 1 when k = 0; = 0 when k ∈ N.

We also need the following technical lemma in [17]. 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 19

0 k,ν k,ν Lemma 4.3 Let ² > 0, k , k ∈ Z, and yτ be any point in Qτ for τ ∈ Ik and ν = 1, ··· ,N(k, τ). If n/(n + ²) < p ≤ ∞, then for any x ∈ X ,

N(k,τ) " #(p∧1) " #²(p∧1) X X 1 2−(k∧k0) µ(Qk,ν) τ k,ν −(k∧k0) k,ν V −(k0∧k) (x) + V (x, yτ ) 2 + d(x, yτ ) τ∈Ik ν=1 2 £ ¤1−(p∧1) ≤ C V2−(k0∧k) (x) ,

where C > 0 is independent of x ∈ X , k, k0, τ and ν.

Proposition 4.2 Let ² be as in Deﬁnition 2.4. If 0 < s < ², n/(n + ²) < p ≤ ∞ and 0 < q ≤ ∞, then ˙ s ˙ (i) Bp, q(X ) ⊂ L(s, p, q; X ), and s (ii) Bp, q(X ) ⊂ L(s, p, q; X ).

˙ s Proof. To verify (i), we ﬁrst prove that if f ∈ Bp, q(X ) with 0 < s < ², n/(n+²) < p ≤ p ˙ s ˙ s ∞ and 0 < q ≤ ∞, then f ∈ L loc (X ). Since Bp, q(X ) ⊂ Bp, ∞(X ) when p(s, ²) < q < ∞, we only need to verify this for q = ∞. To see this, for any l0 ∈ Z, we now prove that

(Z )1/p p −l0s (4.4) |f(x)| dµ(x) . 2 kfkB˙ s (X ). −l p, ∞ B(x1,2 0 )

˙ s Let {Dk}k∈Z be as in Deﬁnition 2.4. Since f ∈ Bp, ∞(X ), by Deﬁnition 2.4, we ³ ´0 know that f ∈ G˚²(β, γ) with β, γ as in (2.3). Thus, for any g ∈ G˚²(β, γ), since R 0 0 X g(x) dµ(x) = 0, by Lemma 4.1, we have

lX0−1 X NX(k,τ) D E k,ν k,ν e k,ν e k,ν hf, gi = µ(Qτ )Dk(f)(yτ ) Dk(·, yτ ) − Dk(x1, yτ ), g k=−∞ τ∈Ik ν=1 X∞ X NX(k,τ) D E k,ν k,ν e k,ν + µ(Qτ )Dk(f)(yτ ) Dk(·, yτ ), g . k=l0 τ∈Ik ν=1

³ ´0 ˚² Thus, f is given by the (in G0(β, γ) with β, γ as in (2.3)) convergent series of functions

lX0−1 X NX(k,τ) h i k,ν k,ν e k,ν e k,ν (4.5) µ(Qτ )Dk(f)(yτ ) Dk(·, yτ ) − Dk(x1, yτ ) k=−∞ τ∈Ik ν=1 X∞ X NX(k,τ) k,ν k,ν e k,ν + µ(Qτ )Dk(f)(yτ )Dk(·, yτ ) k=l0 τ∈Ik ν=1

≡ Y1 + Y2. 中国科技论文在线 http://www.paper.edu.cn

20 Detlef M¨ullerand Dachun Yang

We now verify that (4.4) holds for f. By the regularity of Dek, we have

Ã !² lX0−1 X NX(k,τ) ¯ ¯ k,ν ¯ k,ν ¯ d(x, x1) |Y1| . µ(Q ) ¯Dk(f)(y )¯ τ τ −k k,ν 2 + d(x1, yτ ) k=−∞ τ∈Ik ν=1 Ã !² 1 2−k × . k,ν k,ν −k k,ν V2−k (x1) + V2−k (yτ ) + V (x1, yτ ) 2 + d(x1, yτ )

k,ν k,ν If n/(n + ²) < p ≤ 1, by using (3.4), s < ² and µ(Qτ ) . V2−k (yτ ), we have

l −1 N(k,τ) X0 X X h ¯ ¯ip p ²p k²p k,ν ¯ k,ν ¯ |Y1| . d(x, x1) 2 µ(Qτ ) ¯Dk(f)(yτ )¯ k=−∞ τ∈Ik ν=1 " #p 1 × k,ν k,ν V −k (x1) + V −k (yτ ) + V (x1, yτ ) 2 2   l −1 N(k,τ) ²p X0 X X ¯ ¯p d(x, x1) k²p  k,ν ¯ k,ν ¯  . 2 µ(Qτ ) ¯Dk(f)(yτ )¯ V2−l0 (x1) k=−∞ τ∈Ik ν=1 d(x, x )²p . kfkp 1 2l0(²−s)p; B˙ s (X ) p, ∞ V2−l0 (x1) and therefore, Z p −l0sp p |Y1| dµ(x) . 2 kfk ˙ s . −l Bp, ∞(X ) B(x1,2 0 ) If 1 < p ≤ ∞, by H¨older’sinequality and Lemma 4.3,  l −1 N(k,τ) X0 X X ¯ ¯p ² k²  k,ν ¯ k,ν ¯ |Y1| . d(x, x1) 2 µ(Qτ ) ¯Dk(f)(yτ )¯ k=−∞ τ∈Ik ν=1 Ã !²#1/p 1 2−k × k,ν k,ν −k k,ν V −k (x1) + V −k (yτ ) + V (x1, yτ ) 2 + d(x1, yτ ) 2 2 X NX(k,τ)  k,ν 1 × µ(Qτ ) k,ν k,ν V −k (x1) + V −k (yτ ) + V (x1, yτ ) τ∈Ik ν=1 2 2 Ã !²#1/p0 2−k × −k k,ν 2 + d(x1, yτ ) d(x, x )² 1 l0(²−s) . kfk ˙ s 2 ; Bp, ∞(X ) 1/p [V2−l0 (x1)] and therefore, (Z )1/p p −l0s |Y1| dµ(x) . 2 kfkB˙ s (X ). −l p, ∞ B(x1,2 0 ) 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 21

By the size condition on Dek, we estimate Y2 by

X∞ X NX(k,τ) ¯ ¯ k,ν ¯ k,ν ¯ |Y2| . µ(Qτ ) ¯Dk(f)(yτ )¯ k=l0 τ∈Ik ν=1 Ã !² 1 2−k × . k,ν k,ν −k k,ν V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ )

If n/(n + ²) < p ≤ 1, by Lemma 4.3 and the symmetry, we have

Z " #p Ã !²p 1 2−k h i1−p (4.6) dµ(x) . µ(Qk,ν) , k,ν k,ν −k k,ν τ X V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ )

which together with (3.4) gives   Z ∞ N(k,τ) X X X ¯ ¯p p  k,ν ¯ k,ν ¯  |Y2| dµ(x) . µ(Qτ ) ¯Dk(f)(yτ )¯ −l B(x1,2 0 ) k=l0 τ∈Ik ν=1

−l0sp p . 2 kfk ˙ s ; Bp, ∞(X )

while when 1 < p ≤ ∞, H¨older’sinequality and Lemma 4.3 yield that  ∞ N(k,τ) X X X ¯ ¯p  k,ν ¯ k,ν ¯ |Y2| . µ(Qτ ) ¯Dk(f)(yτ )¯ k=l0 τ∈Ik ν=1 Ã !²#1/p 1 2−k × , k,ν k,ν −k k,ν V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ )

and Minkowski’s inequality together with Lemma 4.3 further gives that

(Z )1/p p |Y2| dµ(x) B(x ,2−l0 ) 1  ∞  N(k,τ) ¯ ¯ X X X ¯ ¯p . µ(Qk,ν) ¯D (f)(yk,ν)¯  τ k τ k=l0 τ∈Ik ν=1 Z Ã !² )1/p 1 2−k × dµ(x) k,ν k,ν −k k,ν X V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ )  1/p ∞ N(k,τ) X X X ¯ ¯p  k,ν ¯ k,ν ¯  . µ(Qτ ) ¯Dk(f)(yτ )¯ k=l0 τ∈Ik ν=1

−l0s . 2 kfk ˙ s . Bp, ∞(X ) 中国科技论文在线 http://www.paper.edu.cn

22 Detlef M¨ullerand Dachun Yang

In combination, these estimations show that the series in (4.5) converge locally in Lp(X ), p so that indeed f ∈ L loc (X ), and that estimation (4.4) holds true. We now verify that

kfk ˙ . kfk ˙ s . L(s, p, q; X ) Bp, q(X ) −l −1 −l −l In fact, let 2 0 < C6 ≤ 2 0 for some l0 ∈ Z, and for y ∈ B(x, C62 ),

l+Xl0−1 X NX(k,τ) h i k,ν k,ν e k,ν e k,ν f(x) − f(y) = µ(Qτ )Dk(f)(yτ ) Dk(x, yτ ) − Dk(y, yτ ) k=−∞ τ∈Ik ν=1 X∞ X NX(k,τ) + ···

k=l+l0 τ∈Ik ν=1 ≡ Z1 + Z2.

By the regularity of Dek, we have

Ã !² l+Xl0−1 X NX(k,τ) ¯ ¯ k,ν ¯ k,ν ¯ d(x, y) |Z1| . µ(Q ) ¯Dk(f)(y )¯ τ τ −k k,ν 2 + d(x, yτ ) k=−∞ τ∈Ik ν=1 Ã !² 1 2−k × . k,ν k,ν −k k,ν V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ ) If n/(n + ²) < p ≤ 1, by (3.4) and (4.6),

 Ã ! 1/q  X∞ Z Z q/p lsq 1 p 2 −l |Z1| dµ(y) dµ(x)  µ(B(x, C62 )) −l  l=−∞ X B(x,C62 )    q/p 1/q  ∞ l+l −1 N(k,τ)   X X0 X X ¯ ¯p  −l(²−s)q  k²p k,ν ¯ k,ν ¯  . 2 2 µ(Qτ ) ¯Dk(f)(yτ )¯ .   l=−∞ k=−∞ τ∈Ik ν=1

If q/p ≤ 1, by (3.4), the last quantity can be controlled by     q/p 1/q  ∞ l+l −1 N(k,τ)   X X0 X X ¯ ¯p  −l(²−s)q  k²q  k,ν ¯ k,ν ¯   2  2 µ(Qτ ) ¯Dk(f)(yτ )¯    l=−∞ k=−∞ τ∈Ik ν=1

. kfk ˙ s ; Bp, q(X ) while when 1 < q/p < ∞, by H¨older’sinequality, the last quantity can be controlled by     q/p  ∞ l+l −1 N(k,τ)  X X0 X X ¯ ¯p −l(²−s)q  k(²−s)q/2  ksp k,ν ¯ k,ν ¯   2  2 2 µ(Qτ ) ¯Dk(f)(yτ )¯   l=−∞ k=−∞ τ∈Ik ν=1 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 23

" # 1 q 1/q l+l0−1 (q/p)0 p  X 0 × 2k(²−s)p(q/p) /2  k=−∞

. kfk ˙ s . Bp, q(X )

−l If 1 < p ≤ ∞, by H¨older’sinequality and Lemma 4.3, for d(x, y) ≤ C62 , we have  l+l −1 N(k,τ) X0 X X ¯ ¯p −l² k²  k,ν ¯ k,ν ¯ |Z1| . 2 2 µ(Qτ ) ¯Dk(f)(yτ )¯ k=−∞ τ∈Ik ν=1 Ã !²#1/p 1 2−k × , k,ν k,ν −k k,ν V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ ) and by Minkowski’s inequality and Lemma 3.1 (ii),

ÃZ Z !1/p 1 p |Z1| dµ(y) dµ(x) −l −l X µ(B(x, C62 )) B(x,C62 )  1/p l+l −1 N(k,τ) X0 X X ¯ ¯p −l² k²  k,ν ¯ k,ν ¯  . 2 2 µ(Qτ ) ¯Dk(f)(yτ )¯ k=−∞ τ∈Ik ν=1    q/p 1/q l+l −1 N(k,τ)   X0 X X ¯ ¯p  −l²0 k²0q  k,ν ¯ k,ν ¯  . 2 2 µ(Qτ ) ¯Dk(f)(yτ )¯ ,   k=−∞ τ∈Ik ν=1

where ²0 = ² when q ≤ 1 by using (3.4) or ²0 ∈ (s, ²) when q > 1 by using H¨older’s inequality. Thus, when 1 < p ≤ ∞,

 Ã ! 1/q  X∞ Z Z q/p lsq 1 p 2 −l |Z1| dµ(y) dµ(x)  µ(B(x, C62 )) −l  l=−∞ X B(x,C62 )

. kfk ˙ s , Bp, q(X )

which completes the estimate for Z1. We now estimate Z2. If n/(n + 1) < p ≤ 1, by (3.4) and Lemma 3.1 (iv) together with (4.6), we ﬁrst have

ÃZ Z !1/p 1 p |Z2| dµ(y) dµ(x) −l −l X µ(B(x, C62 )) B(x,C62 )  1/p ∞ N(k,τ) Z  X X X h ip ¯ ¯p  k,ν k,ν ¯ e k,ν ¯ . µ(Qτ )|Dk(f)(yτ )| ¯Dk(x, yτ )¯ dµ(x)  X  k=l+l0 τ∈Ik ν=1 中国科技论文在线 http://www.paper.edu.cn

24 Detlef M¨ullerand Dachun Yang

 1/p  X∞ X NX(k,τ)  . µ(Qk,ν)|D (f)(yk,ν)|p .  τ k τ  k=l+l0 τ∈Ik ν=1

Now, if q/p ≤ 1, then by (3.4),

 Ã ! 1/q  X∞ Z Z q/p lsq 1 p 2 −l |Z2| dµ(y) dµ(x)  µ(B(x, C62 )) −l  l=−∞ X B(x,C62 )

. kfk ˙ s ; Bp, q(X )

while if 1 < q/p ≤ ∞,

ÃZ Z !q/p 1 p −l |Z2| dµ(y) dµ(x) X µ(B(x, C62 )) B(x,C 2−l)  6   q/p  ∞ N(k,τ)   X X X ¯ ¯p  −lsq/2 ksq/2  k,ν ¯ k,ν ¯  . 2 2 µ(Qτ ) ¯Dk(f)(yτ )¯ ,   k=l+l0 τ∈Ik ν=1

and therefore,

 Ã ! 1/q  X∞ Z Z q/p lsq 1 p 2 −l |Z2| dµ(y) dµ(x)  µ(B(x, C62 )) −l  l=−∞ X B(x,C62 )

. kfk ˙ s . Bp, q(X )

If 1 < p ≤ ∞, by H¨older’sinequality and Lemma 4.3, we have  X∞ X NX(k,τ) −ls/2  ksp/2 k,ν k,ν p |Z2| . 2 2 µ(Qτ )|Dk(f)(yτ )| k=l+l0 τ∈Ik ν=1 ! h¯ ¯ ¯ ¯i 1/p ¯ e k,ν ¯ ¯ e k,ν ¯ × ¯Dk(x, yτ )¯ + ¯Dk(y, yτ )¯ ,

which together with Lemma 3.1 (iv) and (ii) gives

ÃZ Z !q/p 1 p |Z2| dµ(y) dµ(x) −l −l X µ(B(x, C62 )) B(x,C62 )  q/p X∞ X NX(k,τ) −lsq/2  ksp/2 k,ν k,ν p . 2 2 µ(Qτ )|Dk(f)(yτ )| . k=l+l0 τ∈Ik ν=1 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 25

Now if q/p ≤ 1, then by (3.4),

 Ã ! 1/q  X∞ Z Z q/p lsq 1 p 2 −l |Z2| dµ(y) dµ(x)  µ(B(x, C62 )) −l  l=−∞ X B(x,C62 )     1/q  q/p   X∞ X∞ X NX(k,τ)  lsq/2  ksq/2  k,ν k,ν p  . 2  2 µ(Qτ )|Dk(f)(yτ )|    l=−∞ k=l+l0 τ∈Ik ν=1

. kfk ˙ s ; Bp, q(X ) while if 1 < q/p ≤ ∞, by H¨older’sinequality,

 Ã ! 1/q  X∞ Z Z q/p lsq 1 p 2 −l |Z2| dµ(y) dµ(x)  µ(B(x, C62 )) −l  l=−∞ X B(x,C62 )     q/p  X∞ X∞ X NX(k,τ) lsq/2  ksq/4  k,ν k,ν p  . 2  2 µ(Qτ )|Dk(f)(yτ )|   l=−∞ k=l+l0 τ∈Ik ν=1

  1 q 1/q ∞ (q/p)0 p  X 0 ×  2ksp(q/p) /4  k=l+l0

. kfk ˙ s , Bp, q(X ) which completes the proof of (i). We now prove (ii). Clearly, if µ(X ) = ∞ and 1 ≤ p, q, then (ii) is a direct consequence of (i) and Proposition 5.12 in [17]. The general case, however, requires some extra work. s p Let f ∈ Bp, q(X ). We ﬁrst verify that f ∈ L (X ). In fact, if 1 < p < ∞, since s > 0, by Proposition 5.9 (ii), (iii) and (iv) in [17], we have

s s/2 s/2 0 p Bp, q(X ) ⊂ Bp, min(p, q)(X ) ⊂ Fp, q (X ) ⊂ Fp, 2(X ) = L (X )

and kfk p . kfk s . We now verify that this is also true when p = ∞ or n/(n+1) < L (X ) Bp, q(X ) s p ≤ 1. By Definition 2.5 and the independence of the definition of the spaces Bp, q(X ) of ² 0 the space of test functions, we may assume that f ∈ (G0(β, γ)) with β and γ as in (2.4). e Let {Dk}k∈Z+ be as in Definition 2.5 and {Dk}k∈Z+ be as in Lemma 4.2. From Lemma ² 0 4.2, it follows that in (G0(β, γ)) with β and γ as in (2.4), X NX(0,τ) Z f(x) = m 0,ν (D0(f)) De0(x, y) dµ(y) Qτ 0,ν Qτ τ∈I0 ν=1 X∞ X NX(k,τ) k,ν e k,ν k,ν + µ(Qτ )Dk(x, yτ )Dk(f)(yτ ). k=1 τ∈Ik ν=1 中国科技论文在线 http://www.paper.edu.cn

26 Detlef M¨ullerand Dachun Yang

We prove that when p = ∞ or n/(n + ²) < p ≤ 1, the right-hand-side of the above formula is a function in L∞(X ) or in Lp(X ). In an abuse of the notation, we still denote it by f. In fact, if p = ∞, by Lemma 4.3, we have that for all x ∈ X ,

N(0,τ) Z ¯ ¯ X X ¯ ¯ |f(x)| . m 0,ν (|D0(f)|) ¯De0(x, y)¯ dµ(y) Qτ 0,ν Qτ τ∈I0 ν=1 X∞ X NX(k,τ) k,ν 1 + kDk(f)kL∞(X ) µ(Qτ ) k,ν k,ν V −k (x) + V −k (yτ ) + V (x, yτ ) k=1 τ∈Ik ν=1 2 2 Ã !² 2−k × −k k,ν 2 + d(x, yτ )   Z ¯ ¯ ∞ ¯ ¯ X .  sup m 0,ν (|D0(f)|) ¯De0(x, y)¯ dµ(y) + kDk(f)k ∞ Qτ L (X ) τ∈I0 X ν=1,···,N(0,τ) k=1

. kfk s , B∞, q(X )

where we used H¨older’sinequality when 1 < q ≤ ∞ or (3.4) when q ≤ 1. Thus, f ∈ L∞(X )

and kfk ∞ . kfk s . L (X ) B∞, q(X ) We now assume that n/(n + ²) < p ≤ 1. By (4.6), we have

N(0,τ) X X h ip p 0,ν kfk p . µ(Q )m 0,ν (|D0(f)|) L (X ) τ Qτ τ∈I0 ν=1 Z " #p Ã !²p 1 1 × 0,ν 0,ν 0,ν dµ(x) X V1(x) + V1(yτ ) + V (x, yτ ) 1 + d(x, yτ ) ∞ N(k,τ) X X X h ¯ ¯ip k,ν ¯ k,ν ¯ + µ(Qτ ) ¯Dk(f)(yτ )¯ k=1 τ∈Ik ν=1 Z " #p Ã !²p 1 2−k × dµ(x) k,ν k,ν −k k,ν X V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ ) N(0,τ) ∞ N(k,τ) X X X X X ¯ ¯p 0,ν p k,ν ¯ k,ν ¯ . µ(Q )m 0,ν (|D0(f)|) + µ(Q ) ¯Dk(f)(y )¯ τ Qτ τ τ τ∈I0 ν=1 k=1 τ∈Ik ν=1 p . kfk s , Bp, q(X )

where we used (3.4) when q/p ≤ 1 or H¨older’sinequality when 1 < q/p ≤ ∞. Thus, p f ∈ L (X ) and kfk p . kfk s . L (X ) Bp, q(X ) To ﬁnish the proof of (ii), by Proposition 3.3 (i), we now estimate the left hand side −l −1 −l term in (3.1). Without loss of generality, we may assume that 2 0 < C6 ≤ 2 0 for some l0 ∈ Z and l0 ≤ 1. (The case l0 ≥ 2 can be dealt with in a way similar to the proof 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 27

of Proposition 4.4 (ii) below.) We then write  Ã ! 1/q X∞ Z Z q/p lsq 1 p 2 −l |f(x) − f(y)| dµ(y) dµ(x)  µ(B(x, C62 )) −l  l=0 X B(x,C62 )    Z Z N(0,τ) X∞ 1 X X . 2lsq   µ(Q0,ν) −l−l0 τ  X µ(B(x, 2 )) B(x,2−l−l0 ) l=0 τ∈I0 ν=1 # ! 1/q ¯ ¯ p q/p ¯ e e ¯ ×mQ0,ν (|D0(f)|) sup ¯D0(x, z) − D0(y, z)¯ dµ(y) dµ(x) τ 0,ν  z∈Qτ    2−l Z Z N(k,τ) X0 1 X∞ X X + 2lsq   µ(Qk,ν) −l−l0 τ  X µ(B(x, 2 )) B(x,2−l−l0 ) l=0 k=1 τ∈Ik ν=1 ¾ ¯ ¯ip ´q/p 1/q k,ν ¯ e k,ν e k,ν ¯ ×|Dk(f)(yτ )| ¯Dk(x, yτ ) − Dk(y, yτ )¯ dµ(y) dµ(x)    Z Z l+l −1 N(k,τ)  X∞ 1 X0 X X + 2lsq   µ(Qk,ν) −l−l0 τ  X µ(B(x, 2 )) B(x,2−l−l0 ) l=3−l0 k=1 τ∈Ik ν=1 ¾ ¯ ¯ip ´q/p 1/q k,ν ¯ e k,ν e k,ν ¯ ×|Dk(f)(yτ )| ¯Dk(x, yτ ) − Dk(y, yτ )¯ dµ(y) dµ(x)    Z Z N(k,τ)  X∞ 1 X∞ X X + 2lsq   µ(Qk,ν) −l−l0 τ  X µ(B(x, 2 )) B(x,2−l−l0 ) l=3−l0 k=l+l0 τ∈Ik ν=1 ¾ ¯ ¯ip ´q/p 1/q k,ν ¯ e k,ν e k,ν ¯ ×|Dk(f)(yτ )| ¯Dk(x, yτ ) − Dk(y, yτ )¯ dµ(y) dµ(x)

≡ Y1 + Y2 + Y3 + Y4.

We ﬁrst estimate Y1 by further dividing it into    1X−l0 Z Z X NX(0,τ) lsq 1 0,ν Y1 . 2   µ(Q ) −l−l0 τ  X µ(B(x, 2 )) B(x,2−l−l0 ) l=0 τ∈I0 ν=1 Ã !# ! 1/q ¯ ¯ ¯ ¯ p q/p ¯ e ¯ ¯ e ¯ ×mQ0,ν (|D0(f)|) sup ¯D0(x, z)¯ + sup ¯D0(y, z)¯ dµ(y) dµ(x) τ 0,ν 0,ν  z∈Qτ z∈Qτ    Z Z N(0,τ)  X∞ 1 X X + 2lsq   µ(Q0,ν) −l−l0 τ  X µ(B(x, 2 )) B(x,2−l−l0 ) l=2−l0 τ∈I0 ν=1 # ! 1/q ¯ ¯ p q/p ¯ e e ¯ ×mQ0,ν (|D0(f)|) sup ¯D0(x, z) − D0(y, z)¯ dµ(y) dµ(x) τ 0,ν  z∈Qτ 中国科技论文在线 http://www.paper.edu.cn

28 Detlef M¨ullerand Dachun Yang

≡ Y1,1 + Y1,2.

If p ≤ 1, by (3.4), Lemma 3.1 (iv) and (4.6),  N(0,τ) X X h ip 0,ν Y1,1 . µ(Qτ )m 0,ν (|D0(f)|)  Qτ τ∈I0 ν=1 Z µ ¶ µ ¶ ¾ 1 p 1 ²p 1/p × dµ(x) X V1(x) + V1(z) + V (x, z) 1 + d(x, z) . kfk s ; Bp, q(X )

if 1 < p ≤ ∞, by H¨older’sinequality and Lemma 4.3, we ﬁrst have Ã ! X NX(0,τ) ¯ ¯ ¯ ¯ 0,ν ¯ e ¯ ¯ e ¯ µ(Qτ )mQ0,ν (|D0(f)|) sup ¯D0(x, z)¯ + sup ¯D0(y, z)¯ τ 0,ν 0,ν τ∈I0 ν=1 z∈Qτ z∈Qτ  1/p N(0,τ) Ã ! X X h ip ¯ ¯ ¯ ¯  0,ν ¯ e ¯ ¯ e ¯ . µ(Qτ ) mQ0,ν (|D0(f)|) sup ¯D0(x, z)¯ + sup ¯D0(y, z)¯ ,  τ 0,ν 0,ν  τ∈I0 ν=1 z∈Qτ z∈Qτ

and therefore, by Lemma 3.1 (ii),  N(0,τ) X X h ip 0,ν Y1,1 . µ(Qτ ) m 0,ν (|D0(f)|)  Qτ τ∈I0 ν=1 Z µ ¶ ¾ 1 1 ² 1/p × dµ(z) X V1(x) + V1(z) + V (x, z) 1 + d(x, z) . kfk s . Bp, q(X )

To estimate Y1,2, by the regularity of De0, 0 < s < ² and n/(n + ²) < p ≤ ∞, we have    µ ¶  X∞ Z X NX(0,τ) −l ² lsq 0,ν 2 Y1,2 . 2   µ(Q )m 0,ν (|D0(f)|) τ Qτ  X 1 + d(x, z) l=2−l0 τ∈I0 ν=1 µ ¶ ¸ ¶ )1/q 1 1 ² p q/p × dµ(x) V1(x) + V1(z) + V (x, z) 1 + d(x, z)  1/q    X∞  Z X NX(0,τ) −l(²−s)q 0,ν . 2   µ(Q )m 0,ν (|D0(f)|) τ Qτ   X l=2−l0 τ∈I0 ν=1 µ ¶ #p !1/p 1 1 2² × dµ(x) V1(x) + V1(z) + V (x, z) 1 + d(x, z)

. kfk s , Bp, q(X ) 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 29

where in the last step, we omit some computations similar to those used in the estimate

for Y1,1. The estimates for Y2 and Y4 are similar and we only give some details for Y4. If n/(n + ²) < p ≤ 1, by (3.4), Lemma 3.1 (iv) and (4.6), we obtain    ∞ ∞ N(k,τ) h ¯ ¯i X X X X ¯ ¯ p Y . 2lsq  µ(Qk,ν) ¯D (f)(yk,ν)¯ 4  τ k τ l=3−l0 k=l+l0 τ∈Ik ν=1  Z " #p Ã !²p !q/p 1/q 1 2−k  × dµ(x) k,ν k,ν −k k,ν  X V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ )    q/p 1/q  ∞ ∞ N(k,τ)   X X X X ¯ ¯p  lsq  k,ν ¯ k,ν ¯  . 2 µ(Qτ ) ¯Dk(f)(yτ )¯   l=3−l0 k=l+l0 τ∈Ik ν=1      q/p 1/q  ∞ k−l N(k,τ)  X X0 X X ¯ ¯p  ksqa  lsq(1−a)  k,ν ¯ k,ν ¯  . 2 2 µ(Qτ ) ¯Dk(f)(yτ )¯   k=3 l=3−l0 τ∈Ik ν=1

. kfk s , Bp, q(X )

where a = 0 when q ≤ 1 by using (3.4) and a ∈ (0, 1) when q ∈ (1, ∞] by using H¨older’s inequality.

To estimate Y3, using the regularity of Dek yields    Ã ! ∞ Z l+l −1 N(k,τ) ²  X X0 X X ¯ ¯ −l−l0 lsq k,ν ¯ k,ν ¯ 2 Y3 . 2   µ(Q ) ¯Dk(f)(y )¯ τ τ −k k,ν  X 2 + d(x, yτ ) l=3−l0 k=1 τ∈Ik ν=1  Ã !²#p !q/p 1/q 1 2−k  × dµ(x) k,ν k,ν −k k,ν  V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ )     X∞ Z l+Xl0−1 X NX(k,τ) ¯ ¯ −l(²−s)q   k² k,ν ¯ k,ν ¯ . 2 2 µ(Qτ ) ¯Dk(f)(yτ )¯  X l=3−l0 k=1 τ∈Ik ν=1  Ã !²#p !q/p 1/q 1 2−k  × dµ(x) . k,ν k,ν −k k,ν  V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ )

If n/(n + ²) < p ≤ 1, by (3.4) and (4.6), we have     q/p 1/q  ∞ l+l −1 N(k,τ)   X X0 X X ¯ ¯p  −l(²−s)q  k²p  k,ν ¯ k,ν ¯  Y3 . 2 2 µ(Qτ ) ¯Dk(f)(yτ )¯ .   l=3−l0 k=1 τ∈Ik ν=1 中国科技论文在线 http://www.paper.edu.cn

30 Detlef M¨ullerand Dachun Yang

If q/p ≤ 1, by (3.4) again, we have      q/p 1/q  ∞ ∞ N(k,τ)  X X X X ¯ ¯p  k²q  −l(²−s)q  k,ν ¯ k,ν ¯  Y3 . 2 2 µ(Qτ ) ¯Dk(f)(yτ )¯   k=1 l=k−l0+1 τ∈Ik ν=1

. kfk s . Bp, q(X ) If 1 < q/p ≤ ∞, by H¨older’sinequality,     q/p  ∞ l+l −1 N(k,τ)  X X0 X X ¯ ¯p −l(²−s)q  k(²+s)q/2  k,ν ¯ k,ν ¯   Y3 . 2  2 µ(Qτ ) ¯Dk(f)(yτ )¯   l=3−l0 k=1 τ∈Ik ν=1

" # 1 q 1/q l+l0−1 (q/p)0 p  X 0 × 2k(²−s)p(q/p) /2  k=1

. kfk s , Bp, q(X ) which completes the proof of Proposition 4.2. ¤

Combining Proposition 3.1, Proposition 4.1 and Proposition 4.2 yields the following diﬀerence characterization of Besov spaces.

Theorem 4.1 Let ² be as in Deﬁnition 2.4, 0 < s < ², 1 ≤ p ≤ ∞ and 0 < q ≤ ∞. ˙ ˙ ˙ s Then, L(s, p, q; X ) = Lb(s, p, q; X ) = Bp, q(X ) and L(s, p, q; X ) = Lb(s, p, q; X ) = s Bp, q(X ), with equivalent norms.

We now turn to the diﬀerence characterization of Triebel-Lizorkin spaces. We ﬁrst have the following proposition.

Proposition 4.3 Let ² be as in Deﬁnition 2.4, 0 < s < ² and n/(n + ²) < q ≤ ∞. Then ˙ ˙ s (i) If n/(n + ²) < p ≤ ∞, then Lt(s, p, q; X ) ⊂ Fp, q(X ); s (ii) If 1 ≤ p ≤ ∞, then Lt(s, p, q; X ) ⊂ Fp, q(X ).

Proof. To see (i), by Proposition 3.2 (iv), we know that

L˙ t(s, p, q; X ) ⊂ L˙ b(s, p, max(p, q); X ). ³ ´0 ˙ ˚² Thus, by the proof of Proposition 4.1 (i), we have Lt(s, p, q; X ) ⊂ G0(β, γ) with 0 < β < ² and s < γ < ².

Let now f ∈ L˙ t(s, p, q; X ) and {Dk}k∈Z be as in Deﬁnition 2.4. Since Z

Dk(x, y) dµ(y) = 0 X 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 31

If q ≤ 1, by (3.4) and ²2 > ², we obtain

( " Z #q)1/q X∞ 1 1 (4.7) |Dk(f)(x)| . l²q l−k |f(y) − f(x)| dµ(y) , 2 µ(B(x, C72 )) l−k l=0 B(x,C72 )

while when 1 < q ≤ ∞, by H¨older’sinequality, we have

Now if p < ∞ and p/q ≤ 1, by (4.7), (4.8) and (3.4) again, we obtain ° ° °( )1/q° ° X∞ ° ° ksq q ° kfk ˙ s = 2 |Dk(f)| Fp, q(X ) ° ° ° k=−∞ ° Lp(X ) ( °( X∞ ° X∞ · 1 ° ksq 1 . l²p ° 2 l−k 2 ° µ(B(·,C72 )) l=0 k=−∞ ° 1/p Z #q)1/q°p  °  ° × |f(y) − f(·)| dµ(y) ° l−k  B(·,C72 ) °  Lp(X ) . kfk ; L˙ t(s, p, q; X )

while when p < ∞ and 1 < p/q < ∞, by (4.7), (4.8) and Minkowski’s inequality, we have ( °( X∞ ° X∞ · 1 ° ksq 1 kfkF˙ s (X ) . l²q ° 2 l−k p, q 2 ° µ(B(·,C72 )) l=0 k=−∞ 中国科技论文在线 http://www.paper.edu.cn

32 Detlef M¨ullerand Dachun Yang

° 1/q Z #q)1/q°q  °  ° × |f(y) − f(·)| dµ(y) ° l−k  B(·,C72 ) °  Lp(X ) . kfk , L˙ t(s, p, q; X ) which completes the proof of (i) in the case n/(n + ²) < p < ∞. j Now if p = ∞, then for any j ∈ Z, α ∈ Ij and dyadic cube Qα, by (4.7) and (4.8), we have X∞ ksq q (4.9) 2 |Dk(f)(x)| k=j  " #  X∞ X∞ Z q 1 ksq 1 . l²q 2 l−k |f(y) − f(x)| dµ(y) 2  µ(B(x, C72 )) l−k  l=0 k=j B(x,C72 )

. kfkq , L˙ t(s, ∞, q; X ) which implies that Z X∞ 1 ksq q q 2 |Dk(f)(x)| dµ(x) . kfk . j j L˙ t(s, ∞, q; X ) µ(Qα) Qα k=j

˙ s Thus, f ∈ F (X ) and kfk ˙ s . kfk ˙ , which completes the proof of (i). ∞, q F∞, q(X ) Lt(s, ∞, q; X ) p ² 0 To verify (ii), since f ∈ L (X ) with 1 ≤ p ≤ ∞, it is easy to see that f ∈ (G0(β, γ)) for any 0 < β, γ < ². When 1 ≤ p < ∞, by H¨older’sinequality, (4.2), (4.3), and the above estimate, we have ° ° °( )1/q° ° X∞ ° ksq q kfk s . kS (f)k p + ° 2 |D (f)| ° ∼ kfk . Fp, q(X ) 0 L (X ) ° k ° Lt(s, p, q; X ) ° k=1 ° Lp(X ) Let now p = ∞. By Lemma 3.1 (ii), we have ¯Z ¯ ¯ ¯ ¯ ¯ |S0(f)(x)| = ¯ S0(x, y)f(y) dµ(y)¯ . kfkL∞(X ), X

and for τ ∈ I0 and ν = 1, ··· ,N(0, τ), m 0,ν (|S0(f)|) . kfk ∞ . Now, for any j ∈ N, Qτ L (X ) j α ∈ Ij and dyadic cube Qα, by (4.9), we have Z X∞ 1 ksq q q 2 |Dk(f)(x)| dµ(x) . kfk . j j L˙ t(s, ∞, q; X ) µ(Qα) Qα k=j

s Thus, f ∈ F (X ) and kfk s . kfk ∞ + kfk = kfk , which ∞, q F∞, q(X ) L (X ) L˙ t(s, ∞, q; X ) Lt(s, ∞, q; X ) completes the proof of Proposition 4.3. ¤ 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 33

To establish a converse to Proposition 4.3, we need two technical lemmas. The ﬁrst one is the following Feﬀerman-Stein vector-valued maximal function inequality in [7]; see also (2.11) in [37] and [44, Chapter II, Section 1].

Lemma 4.4 Let 1 < p < ∞, 1 < q ≤ ∞ and M be the Hardy-Littlewood maximal p operator on X . Let {fk}k∈Z ⊂ L (X ) be a sequence of measurable functions on X . Then ° ° ° ° °( )1/q° °( )1/q° ° X∞ ° ° X∞ ° ° q ° ° q ° ° [M(fk)] ° ≤ C ° |fk| ° , ° k=−∞ ° ° k=−∞ ° Lp(X ) Lp(X )

where C is independent of {fk}k∈Z.

The following lemma is established in [17].

0 k,ν k,ν Lemma 4.5 Let ² > 0, k , k ∈ Z, and yτ be any point in Qτ for τ ∈ Ik and ν = 1, ··· ,N(k, τ). If n/(n + ²) < r ≤ 1, then there exists a constant C > 0 depending k,ν on r such that for all aτ ∈ C and all x ∈ X ,

X NX(k,τ) −(k∧k0)² k,ν 1 2 k,ν µ(Qτ ) ³ ´² |aτ | k,ν 0 k,ν V −(k0∧k) (x) + V (x, yτ ) −(k∧k ) τ∈Ik ν=1 2 2 + d(x, yτ )    1/r  X NX(k,τ)  [(k∧k0)−k]n(1−1/r)  k,ν r  ≤ C2 M |aτ | χ k,ν (x) ,  Qτ  τ∈Ik ν=1

where C > 0 is also independent of k, k0, τ and ν.

Proposition 4.4 Let ² be as in Deﬁnition 2.4, 0 < s < ², 1 < p < ∞ and 1 < q ≤ ∞. Then ˙ s ˙ (i) Fp, q(X ) ⊂ Lt(s, p, q; X ), and s (ii) Fp, q(X ) ⊂ Lt(s, p, q; X ).

Proof. We ﬁrst verify (i). By Proposition 5.4 (ii) in [17], we have

˙ s ˙ s Fp, q(X ) ⊂ Bp, max(p, q)(X ).

˙ s By the proof of Proposition 4.2 (i), we then know that when n/(n+²) < p ≤ ∞, Fp, q(X ) ⊂ p ˙ s 1 L loc (X ), which shows that when 1 < p < ∞, Fp, q(X ) ⊂ L loc (X ). ˙ s We now verify that for all f ∈ F (X ), kfk ˙ . kfk ˙ s . Let {Dk}k∈Z and p, q Lt(s, p, q; X ) Fp, q(X ) {Dek}k∈Z be as in Deﬁnition 2.4 and Lemma 4.1, respectively. By Deﬁnition 2.4, we know ³ ´0 ˚² f ∈ G0(β, γ) with β and γ as in (2.3). Without loss of generality, we may assume that 中国科技论文在线 http://www.paper.edu.cn

34 Detlef M¨ullerand Dachun Yang

³ ´0 −l0−1 −l0 ˚² 2 < C7 ≤ 2 for some l0 ∈ Z. For any x, y ∈ X , using Lemma 4.1, in G0(β, γ) with β and γ as in (2.3), we write

l+Xl0−1 X NX(k,τ) h i k,ν k,ν e k,ν e k,ν f(x) − f(y) = µ(Qτ )Dk(f)(yτ ) Dk(x, yτ ) − Dk(y, yτ ) k=−∞ τ∈Ik ν=1 X∞ X NX(k,τ) + ···

k=l+l0 τ∈Ik ν=1 ≡ J1 + J2.

−l−l0 By the regularity of Dek, we know that for all d(x, y) < 2 ,

Ã !² l+Xl0−1 X NX(k,τ) ¯ ¯ k,ν ¯ k,ν ¯ d(x, y) |J1| . µ(Q ) ¯Dk(f)(y )¯ τ τ −k k,ν 2 + d(x, yτ ) k=−∞ τ∈Ik ν=1 Ã !² 1 2−k × k,ν k,ν −k k,ν V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ ) l+Xl0−1 X NX(k,τ) ¯ ¯ (k−l)² k,ν ¯ k,ν ¯ . 2 µ(Qτ ) ¯Dk(f)(yτ )¯ k=−∞ τ∈Ik ν=1 Ã !² 1 2−k × , k,ν k,ν −k k,ν V2−k (x) + V2−k (yτ ) + V (x, yτ ) 2 + d(x, yτ )

and therefore, by Lemma 4.5 and H¨older’sinequality, we have ( " # ) X∞ Z q 1/q lsq 1 2 −l |J1| dµ(y) µ(B(x, C72 )) −l l=−∞ B(x,C72 )     q1/q  X∞ X NX(k,τ) ¯ ¯  ksq   ¯ k,ν ¯   . 2 M ¯Dk(f)(yτ )¯ χ k,ν (x) .  Qτ  k=−∞ τ∈Ik ν=1

From this and Lemma 4.4 together with 1 < p < ∞ and 1 < q ≤ ∞, it follows that ° ° °( " Z #q)1/q° ° X∞ ° lsq 1 ° 2 |J1| dµ(y) ° ° −l −l ° ° µ(B(·,C72 )) B(·,C72 ) ° l=−∞ p ° L (X ) ° °   q1/q° ° X∞ X NX(k,τ) ¯ ¯  ° ° ksq   ¯ k,ν ¯  ° . ° 2 M ¯Dk(f)(yτ )¯ χ k,ν ° ° Qτ  ° ° k=−∞ τ∈Ik ν=1 ° Lp(X )

. kfk ˙ s . Fp, q(X ) 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 35

To estimate J2, by Lemma 4.5 again, we have

X∞ X NX(k,τ) ¯ ¯ h¯ ¯ ¯ ¯i k,ν ¯ k,ν ¯ ¯ e k,ν ¯ ¯ e k,ν ¯ |J2| . µ(Qτ ) ¯Dk(f)(yτ )¯ ¯Dk(x, yτ )¯ + ¯Dk(y, yτ )¯ k=l+l τ∈I ν=1 0  k   X∞  X NX(k,τ) ¯ ¯  ¯ k,ν ¯  . M ¯Dk(f)(yτ )¯ χ k,ν (x)  Qτ k=l+l τ∈I ν=1 0 k   X NX(k,τ) ¯ ¯   ¯ k,ν ¯  +M ¯Dk(f)(yτ )¯ χ k,ν (y) , Qτ  τ∈Ik ν=1

and by H¨older’sinequality, ( " # ) X∞ Z q 1/q lsq 1 2 −l |J2| dµ(y) µ(B(x, C72 )) −l l=−∞ B(x, C72 )       X∞ X∞  X NX(k,τ) ¯ ¯ lsq   ¯ k,ν ¯  . 2 M ¯Dk(f)(yτ )¯ χ k,ν (x)   Qτ l=−∞ k=l+l0 τ∈Ik ν=1   q1/q X NX(k,τ) ¯ ¯   2  ¯ k,ν ¯   +M ¯Dk(f)(yτ )¯ χ k,ν (x) Qτ   τ∈Ik ν=1     q1/q  X∞ X NX(k,τ) ¯ ¯  ksq  2  ¯ k,ν ¯   . 2 M ¯Dk(f)(yτ )¯ χ k,ν (x) ,  Qτ  k=−∞ τ∈Ik ν=1

where M = M ◦ M. Then applying Lemma 4.4 twice gives that ° ° °( " Z #q)1/q° ° X∞ ° ° lsq 1 ° ° 2 −l |J2| dµ(y) ° . kfkF˙ s (X ). µ(B(·,C72 )) −l p, q ° l=−∞ B(·,C72 ) ° Lp(X )

˙ Thus, when 1 < p < ∞, f ∈ Lt(s, p, q; X ) and kfk ˙ . kfk ˙ s , which com- Lt(s, p, q; X ) Fp, q(X ) pletes the proof of (i). To see (ii), by Proposition 5.9 (i) and (vi) in [17], we have that when 0 < s < ² and s 0 p s 1 < p < ∞, Fp, q(X ) ⊂ Fp, 2(X ) = L (X ). Thus, if f ∈ Fp, q(X ) with 1 < p < ∞, then p f ∈ L (X ) and kfk p . kfk s . Let now {D } be as in Deﬁnition 2.5 and L (X ) Fp, q(X ) k k∈Z e −l0−1 −l0 {Dk}k∈Z+ be as in Lemma 4.2. Let also 2 < C7 ≤ 2 for some l0 ∈ Z. Without loss of generality, we may assume that l0 ≥ 2. (The case l0 ≤ 1 can be dealt with in a way similar to the proof of Proposition 4.2 (ii) above.) By Deﬁnition 2.5, we know that ² 0 f ∈ (G0(β, γ)) with s < β < ² and 0 < γ < ². By Lemma 4.2, we then have that for any 中国科技论文在线 http://www.paper.edu.cn

36 Detlef M¨ullerand Dachun Yang

² 0 l ∈ Z+, in (G0(β, γ)) with s < β < ² and 0 < γ < ²,

X NX(0,τ) Z h i 0,ν f(x) − f(y) = De0(x, z) − De0(y, z) dµ(z)D (f) 0,ν τ,1 Qτ τ∈I0 ν=1 l+Xl0−1 X NX(k,τ) h i k,ν e k,ν e k,ν k,ν + µ(Qτ ) Dk(x, yτ ) − Dk(y, yτ ) Dk(f)(yτ ) k=1 τ∈Ik ν=1 X∞ X NX(k,τ) + ···

k=l+l0 τ∈Ik ν=1 ≡ Y1 + Y2 + Y3.

Obviously, ¯ ¯ ¯ ¯ ¯ Z ¯ ¯ 0,ν ¯ ¯ 1 ¯ ¯Dτ,1 (f)¯ = ¯ D0(f)(u) dµ(u)¯ . m 0,ν (|D0(f)|). ¯ 0,ν 0,ν ¯ Qτ µ(Qτ ) Qτ

−l For y ∈ B(x, C72 ), by the regularity of De0, Lemma 2.1, Lemma 3.1 (iii) and Lemma 4.5, we further have

N(0,τ) Z µ ¶ µ ¶ X X d(x, y) ² 1 1 ² |Y1| . dµ(z) 0,ν Qτ 1 + d(x, z) V1(x) + V1(z) + V (x, z) 1 + d(x, z) τ∈I0 ν=1 ×m 0,ν (|D0(f)|) Qτ   X NX(0,τ) −l² . 2 M  m 0,ν (|D0(f)|)χ 0,ν  (x). Qτ Qτ τ∈I0 ν=1

−l Similarly, for y ∈ B(x, C72 ),   l+Xl0−1 X NX(k,τ) (k−l)² k,ν |Y2| . 2 M  |Dk(f)(y )|χ k,ν  (x). τ Qτ k=1 τ∈Ik ν=1

From these estimates, H¨older’sinequality, the Lp(X )-boundedness of M with p ∈ (1, ∞) and Lemma 4.4, it follows that ° ° °( " Z #q)1/q° ° X∞ ° ° lsq 1 ° ° 2 −l |Y1 + Y2| dµ(y) ° µ(B(·,C72 )) −l ° l=0 B(·,C72 ) ° Lp(X ) °  ° ° ° ° X NX(0,τ) ° . °M  m 0,ν (|D0(f)|)χ 0,ν ° ° Qτ Qτ ° ° τ∈I ν=1 ° 0 Lp(X ) 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 37

° ° °   q1/q° °X∞ X NX(k,τ)  ° ° ksq   k,ν  ° + ° 2 M |Dk(f)(yτ )|χ k,ν ° ° Qτ  ° ° k=1 τ∈Ik ν=1 ° Lp(X )

. kfk s . Fp, q(X ) −l For y ∈ B(x, C72 ), by the size condition on Dek for k ∈ Z+ and Lemma 4.5, we have    X∞ X NX(k,τ) k,ν |Y3| . M  |Dk(f)(y )|χ k,ν  (x) τ Qτ k=l+l τ∈I ν=1 0 k   X NX(k,τ) k,ν +M  |Dk(f)(y )|χ k,ν  (y) , τ Qτ τ∈Ik ν=1 and therefore, by H¨older’sinequality, we further have Z 1 −l |Y3| dµ(y) µ(B(x, C72 )) B(x, C 2−l)  7     X∞ X NX(k,τ) −ls/2 ksq/2   k,ν  . 2 2 M |Dk(f)(yτ )|χ k,ν (x)  Qτ k=l+l0 τ∈Ik ν=1   q1/q X NX(k,τ)  2  k,ν   +M |Dk(f)(yτ )|χ k,ν (x) . Qτ  τ∈Ik ν=1 Applying Lemma 4.4 twice, we ﬁnally obtain ° ° °( " Z #q)1/q° ° X∞ ° ° lsq 1 ° ° 2 −l |Y3| dµ(y) ° µ(B(·,C72 )) −l ° l=0 B(·,C72 ) ° Lp(X ) °    ° °X∞ X NX(k,τ) ° ksq   k,ν  . 2 M |Dk(f)(yτ )|χ k,ν ° Qτ ° k=0 τ∈I ν=1 k °  q1/q° X NX(k,τ)  ° 2  k,ν  ° +M |Dk(f)(yτ )|χ k,ν ° Qτ  ° τ∈Ik ν=1 ° Lp(X )

. kfk s . Fp, q(X )

Thus, by Proposition 3.3 (iii), we know that f ∈ Lt(s, p, q; X ) and kfkLt(s, p, q; X ) . kfk s , which completes the proof of Proposition 4.4. ¤ Fp, q(X ) Combining Proposition 4.3 and Proposition 4.4 immediately leads to the following diﬀerence characterization of Triebel-Lizorkin spaces. Theorem 4.2 Let ² be as in Deﬁnition 2.4, 0 < s < ², 1 < p < ∞ and 1 < q ≤ ∞. ˙ s ˙ s Then Fp, q(X ) = Lt(s, p, q; X ) and Fp, q(X ) = Lt(s, p, q; X ) with equivalent norms. 中国科技论文在线 http://www.paper.edu.cn

38 Detlef M¨ullerand Dachun Yang

5 Relations with other Sobolev spaces

We ﬁrst recall the so-called HajÃlasz-Sobolev spaces on metric spaces. s Definition 5.1 Let 1 ≤ p ≤ ∞ and s > 0. The HajÃlasz-Sobolev space Wp (X ) is the space of all functions f ∈ Lp(X ) for which there exists a non-negative function g ∈ Lp(X ) such that |f(x) − f(y)| ≤ d(x, y)s[g(x) + g(y)] for almost all x, y ∈ X . Moreover, we deﬁne

kfkW s(X ) = kfkLp(X ) + inf kgkLp(X ), p {g} where the inﬁmum is taken over all functions g as above.

1 The Sobolev space Wp (X ) was first introduced by HajÃlaszin [13]; for further discussions 1 and applications, see [15, 21, 30, 42, 31, 53]. A Poincarécharacterization of Wp (X ) can be found in [8]. Fractional Sobolev spaces on fractals which are subsets of Rn were first s introduced in [22]. The space Wp (X ) was first studied in [52]. The following result is a generalization of Theorem 6.1 in [19] on RD-spaces.

Proposition 5.1 Let ² be as in Definition 2.5, 0 < s < s1 < ² and 1 ≤ p ≤ ∞. Then s1 s (i) Wp (X ) ⊂ Bp, q(X ) for 0 < q ≤ ∞, and s1 s (ii) Wp (X ) ⊂ Fp, q(X ) for n/(n + ²) < q ≤ ∞. Proof. Property (ii) is a simple corollary of Property (i), Proposition 5.9 (iii) in [17] and Proposition 6.9 (iii) in [17]. Thus, we only need to verify (i). s1 p Let {Dk}k∈Z+ be as in Definition 2.5. Let f ∈ Wp (X ). Since f ∈ L (X ) and 1 ≤ p ≤ ² 0 ∞, then f ∈ (G0(β, γ)) for all 0 < β, γ < ². Moreover, by Hölder’sinequality, (4.2) and (4.3), we have ( ) X∞ 1/q ksq q kfk s . kD (f)k p + 2 kD (f)k Bp, q(X ) 0 L (X ) k Lp(X ) k=1 ( ) X∞ 1/q ksq q . kfkLp(X ) + 2 kDk(f)kLp(X ) . k=1 R For k ∈ N, by X Dk(x, y) dµ(y) = 0, and Lemma 3.1 (ii) and (iii) together with s1 < ², we then have ¯Z ¯ ¯ ¯ ¯ ¯ (5.1) |Dk(f)(x)| = ¯ Dk(x, y)[f(y) − f(x)] dµ(y)¯ Z X s1 . |Dk(x, y)|d(x, y) [g(x) + g(y)] dµ(y) X . 2−ks1 [g(x) + Mg(x)]. 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 39

If 1 < p ≤ ∞, by (5.1), Minkowski’s inequality and the Lp(X )-boundedness of M, we then have £ ¤ −ks1 −ks1 kDk(f)kLp(X ) . 2 kgkLp(X ) + kMgkLp(X ) . 2 kgkLp(X ),

which together with s1 > s implies that ( ) ( ) X∞ 1/q X∞ 1/q ksq q −k(s1−s)q 2 kDk(f)kLp(X ) . 2 kgkLp(X ) k=1 k=1 . kgkLp(X ).

Thus, in this case, kfk s . kfk p + kgk p , which implies that kfk s . Bp, q(X ) L (X ) L (X ) Bp, q(X ) kfk s1 . Wp (X ) If p = 1, by (5.1), Fubini’s theorem and Lemma 3.1 (ii), we have Z ½Z ¾ −ks1 s1 kDk(f)kL1(X ) . 2 kgkL1(X ) + |Dk(x, y)|d(x, y) dµ(x) g(y) dµ(y) X X −ks1 . 2 kgkL1(X ),

and therefore, by s1 > s, ( ) X∞ 1/q ksq q kfk s . kfk 1 + 2 kD (f)k B1, q(X ) L (X ) k L1(X ) k=1 ( ) X∞ 1/q −k(s1−s)q . kfkL1(X ) + 2 kgkL1(X ) k=1 . kfkL1(X ) + kgkL1(X ),

which also shows (i) in the case p = 1. This ﬁnishes the proof of Proposition 5.1. ¤

s The following proposition shows the relations between Wp (X ) and L(s, p, q; X ).

Proposition 5.2 Let 0 < s < s1, 1 ≤ p ≤ ∞ and 0 < q ≤ ∞. Then s1 (i) Wp (X ) ⊂ L(s, p, q; X ); s1 (ii) Wp (X ) ⊂ Lb(s, p, q; X ); s1 (iii) Wp (X ) ⊂ Lt(s, p, q; X ).

Proof. Property (ii) is a simple corollary of (i) and Proposition 3.1 (ii); while (iii)

is a simple corollary of (ii) and Proposition 3.2 (iv). Moreover, if s1 < ², then (i) is a simple corollary of Proposition 5.1 (i) and Proposition 4.2 (ii). Thus, only when ² ≤ s1, Property (i) is new. We now give a simple proof of (i) by their deﬁnitions. In fact, letting 中国科技论文在线 http://www.paper.edu.cn

40 Detlef M¨ullerand Dachun Yang

s1 s1 f ∈ Wp (X ), for ν ∈ Z+, by the deﬁnition of Wp (X ), Minkowski’s inequality, Fubini’s theorem and Lemma 3.1 (iv), we obtain

(Z Z )1/p 1 |f(x) − f(y)|p dµ(y) dµ(x) −ν −ν X µ(B(x, C62 )) B(x,C62 ) (Z Z )1/p 1 . d(x, y)s1p (|g(x)| + |g(y)|)p dµ(y) dµ(x) −ν −ν X µ(B(x, C62 )) B(x,C62 ) (

−νs1 . 2 kgkLp(X )  "Z Z #1/p 1  + d(x, y)s1p|g(y)|p dµ(y) dµ(x) −ν −ν X µ(B(x, C62 )) B(x,C62 ) 

−νs1 . 2 kgkLp(X ),

which proves that

 Ã ! 1/q X∞ Z Z q/p νsq 1 p 2 −ν |f(x) − f(y)| dµ(y) dµ(x)  µ(B(x, C 2 )) −ν  ν=0 X 6 B(x,C62 ) ( ) X∞ 1/q ν(s−s1)q . 2 kgkLp(X ) ν=0 . kgkLp(X ).

This together with Proposition 3.3 (i) shows that f ∈ L(s, p, q; X ) and

kfkL(s, p, q; X ) . kfkLp(X ) + kgkLp(X ),

which implies (i), and hence completes the proof of Proposition 5.2. ¤

In [29], Korevaar and Schoen introduced a class of Sobolev maps f : X → Y, where X is a Riemannian domain and Y is a complete metric space. Later, Koskela and Macmanus [30] considered the Korevaar-Schoen definition in the case when X is an abstract metric measure space (see also [21]). We now recall this definition here. Let f ∈ Lp(X ) with 1 ≤ p ≤ ∞. For any δ > 0 and x, y ∈ X , define

|f(x) − f(y)| e (x, y; f) = δ δ and ( Z )1/p 1 p eδ(x; f) = [eδ(x, y; f)] dµ(y) . µ(B(x, δ)) B(x,δ) 中国科技论文在线 http://www.paper.edu.cn

A diﬀerence characterization 41

Then, deﬁne ½Z ¾1/p p Ep(f) = sup lim sup [eδ(x; f)] dµ(x) , B δ→0 B where the supremum is taken over all metric balls B in X . Thus, f ∈ Lp(X ) with 1,p 1 ≤ p ≤ ∞ is said to be in the Korevaar-Schoen Sobolev space KS (X ) if Ep(f) < ∞. Moreover, we deﬁne

kfkKS1,p(X ) = kfkLp(X ) + Ep(f).

Proposition 5.3 Let 1 ≤ p ≤ ∞. Then, L(1, p, ∞; X ) ⊂ KS1,p(X ).

Proof. In fact,

kfkKS1,p(X ) = kfkLp(X ) + Ep(f) (Z Z )1/p 1 |f(x) − f(y)|p . kfkLp(X ) + lim sup p dµ(y) dµ(x) δ→0 X µ(B(x, δ)) B(x,δ) δ

. kfkLp(X ) (Z Z )1/p 1 |f(x) − f(y)|p + sup p dµ(y) dµ(x) δ>0 X µ(B(x, C6δ)) B(x,C6δ) δ

. kfkLp(X ) (Z Z )1/p 1 + sup 2ν |f(x) − f(y)|p dµ(y) dµ(x) −ν −ν ν∈Z X µ(B(x, C62 )) B(x,C62 )

. kfkL(1, p, ∞; X ),

which completes the proof of Proposition 5.3. ¤

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Detlef Muller,¨ Mathematisches Seminar, Christian-Albrechts-Universi- tat¨ Kiel, Ludewig-Meyn Strasse 4, D-24098 Kiel, Germany E-mail: [email protected] Dachun Yang (Corresponding author), School of Mathematical Sciences, Bei- jing Normal University, Beijing 100875, People’s Republic of China E-mail address: [email protected]