http://www.paper.edu.cn Forum Math., to appear A DIFFERENCE CHARACTERIZATION OF BESOV AND TRIEBEL-LIZORKIN SPACES ON RD-SPACES DETLEF MULLERÄ 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 2 X and 0 < r < diam (X )=a0, the annulus B(x; a0r) n 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 di®erence characterization of Besov and Triebel-Lizorkin spaces on RD-spaces is obtained. 1 Introduction Metric spaces play a prominent role in many ¯elds 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 Classi¯cation: Primary 42B35; Secondary 46E35, 43A99. Key words and phrases: RD-space, space of Lipschitz type, Besov space, Triebel-Lizorkin space, Sobolev space, di®erence 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. 转载 1 中国科技论文在线 http://www.paper.edu.cn 2 Detlef MÄullerand 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 2 X and 0 < r < diam (X )=a0, the annulus B(x; a0r) n 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- Carath¶eodory 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 di®erence 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 de¯ned via di®erences, respectively local means of di®erences; 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-Carath¶eodory (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 2 [1; 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 di®erent occurrences. If E is a subset of a metric space (X ; d), we denote by ÂE the characteristic function of E and de¯ne diam E = sup d(x; y): x; y2E We also set N ´ f1; 2; ¢ ¢ ¢g and Z+ ´ N [ f0g. For any a; b 2 R, we denote minfa; bg, maxfa; bg, and maxfa; 0g by a ^ b, a _ b and a+, respectively. Throughout the whole paper, for ² 2 (0; 1] and jsj < ², we set p(s; ²) ´ maxfn=(n + ²); n=(n + ² + s)g: 中国科技论文在线 http://www.paper.edu.cn A di®erence characterization 3 2 Preliminaries We ¯rst 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 ¹, which means that ¹ is a nonnegative countably subadditive set function de¯ned 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 de¯ned by d have ¯nite and positive measure. For any x 2 X and r > 0, set B(x; r) = fy 2 X : d(x; y) < rg: (i) The triple (X ; d; ¹) is called a space of homogeneous type if there exists a constant C0 ¸ 1 such that for all x 2 X and r > 0, (2:1) ¹(B(x; 2r)) · C0¹(B(x; r)) (doubling property): (ii) Let 0 < · · n. The triple (X ; d; ¹) is called a (·; n)-space if there exist constants 0 < C1 · 1 and C2 ¸ 1 such that for all x 2 X , 0 < r < diam (X )=2 and 1 · ¸ < diam (X )=(2r), · n (2:2) C1¸ ¹(B(x; r)) · ¹(B(x; ¸r)) · C2¸ ¹(B(x; r)); where diam (X ) = supx; y2X d(x; y). A space of homogeneous type will be called an RD-space, if it is a (·; n)-space for some 0 < · · n, i. e., if some \reverse" doubling condition holds. Clearly, any Ahlfors n-regular metric measure space (X ; d; ¹) (which means that there exists some n > 0 such that ¹(B(x; r)) » rn for all x 2 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 satis¯es 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 ¯rst 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 homo- geneous type with the additional property that there exists a constant a0 > 1 such that for all x 2 X and 0 < r < diam (X )=a0, B(x; a0r) n 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¶³asand 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 MÄullerand Dachun Yang Remark 2.3 If (X ; d; ¹) is a space of homogeneous type, we also introduce the volume functions V±(x) = ¹(B(x; ±)) and V (x; y) = ¹(B(x; d(x; y))) for all x; y 2 X and ± > 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 2 (0; 1], ²2 > 0 and ²3 > 0. 2 (I) A sequence fSkgk2Z 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 2 Z and all x; x ; y and y 2 X ;Sk(x; y); the integral kernel of Sk is a measurable function from X £ X into C satisfying 1 2¡k²2 (i) jSk(x; y)j · 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) jSk(x; y) ¡ Sk(x ; y)j · 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) j[Sk(x; y) ¡ Sk(x; y )] ¡ [Sk(x ; y) ¡ Sk(x ; y )]j · 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) d¹(y) = 1; R (vi) X Sk(x; y) d¹(x) = 1: (II) A sequence fSkgk2Z+ of linear operators is said to be an inhomogeneous approxi- mation to the identity of order (²1 ; ²2 ; ²3 ), if Sk for k 2 Z+ satis¯es (I).