LOUKAS GRAFAKOS, LIGUANG LIU, DIEGO MALDONADO and DACHUN YANG. Multilinear analysis on metric spaces

Transcription

1 LOUKAS GRAFAKOS, LIGUANG LIU, DIEGO MALDONADO and DACHUN YANG Multilinear analysis on metric spaces

2 Loukas Grafakos Department of Mathematics University of Missouri Columbia, MO 652, USA Liguang Liu Department of Mathematics School of Information Renmin University of China Beijing 00872, People s Republic of China liuliguang@ruc.edu.cn Diego Maldonado Department of Mathematics Kansas State University Manhattan, KS 66506, USA dmaldona@math.ksu.edu Dachun Yang (corresponding author School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 00875, People s Republic of China dcyang@bnu.edu.cn

3 Contents. Introduction Real analysis on spaces of homogeneous type Spaces of homogeneous type and RD-spaces Dyadic cubes, covering lemmas, and the Calderón Zygmund decomposition Space of test functions Approximations of the identity Singular integrals on spaces of homogeneous type Multilinear Calderón Zygmund theory Multilinear Calderón Zygmund operators Multilinear weak type estimates Weighted multilinear Calderón Zygmund theory Multiple weights Weighted estimates for the multi-sublinear maximal function Weighted estimates for multilinear Calderón Zygmund operators Weighted estimates for maximal multilinear singular integrals A multilinear T -theorem on Lebesgue spaces Some lemmas on multilinear Calderón Zygmund operators BMO-boundedness of multilinear singular integrals A multilinear T -theorem Bilinear T -theorems on Triebel Lizorkin and Besov spaces Bilinear weak boundedness property Bilinear molecules Bilinear T -theorem on Triebel Lizorkin spaces Bilinear T -theorem on Besov spaces Multilinear vector-valued T type theorems A multilinear off-diagonal estimate Quadratic T type theorems on Lebesgue spaces Quadratic T type theorems on Besov and Triebel Lizorkin spaces Paraproducts as bilinear Calderón Zygmund operators Paraproducts Almost diagonal estimates Paraproducts as bilinear Calderón Zygmund operators Boundedness of paraproducts on Triebel Lizorkin and Besov spaces Bilinear multiplier operators on Triebel Lizorkin and Besov spaces Bilinear multiplier operators Off-diagonal estimates for bilinear multiplier operators Boundedness of bilinear multiplier operators References ]

4 Abstract The multilinear Calderón Zygmund theory is developed in the setting of RD-spaces which are spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón Zygmund theory in this context is also developed in this work. The bilinear T -theorems for Besov and Triebel Lizorkin spaces in the full range of exponents are among the main results obtained. Multilinear vector-valued T type theorems on Lebesgue spaces, Besov spaces, and Triebel Lizorkin spaces are also proved. Applications include the boundedness of paraproducts and bilinear multiplier operators on products of Besov and Triebel Lizorkin spaces. Acknowledgements. Loukas Grafakos is supported by grant DMS of the National Science Foundation of the USA. Liguang Liu is supported by the National Natural Science Foundation of China (grant No Diego Maldonado is supported by grant DMS of the National Science Foundation of the USA. Dachun Yang (the corresponding author is supported by the National Natural Science Foundation of China (grant nos & and the Specialized Research Fund for the Doctoral Program of Higher Education of China (grant No All authors would like to thank the copy editor, Jerzy Trzeciak, for his valuable remarks which made this article more readable. 200 Mathematics Subject Classification: Primary 42B20, 42B25, 42B35; Secondary 35S50, 42C5, 47G30, 30L99. Key words and phrases: space of homogeneous type, multilinear Calderón Zygmund operator, multilinear weighted estimate, paraproduct, bilinear T -theorem, quadratic T type theorem, bilinear multiplier, Besov space, Triebel-Lizorkin space. 4]

5 . Introduction In this work we develop the theory of multilinear analysis related to the Calderón Zygmund program within the framework of metric spaces. The impetus created by the recent developments in the theory of multilinear operators has naturally led us to consider its extension to the setting of metric spaces. Since the techniques involved in the proofs transcend the algebraic and differential structures of the underlying spaces, it is appropriate to undertake this study in a unified way. This setting is quite general and it includes graphs, fractals, Riemannian manifolds, Carnot-Carathéodory groups, anisotropic structures in R n, Ahlfors spaces, etc. As a consequence of this work, previously disconnected topics concerning multilinear operators are integrated and streamlined. These topics include bilinear Calderón Zygmund operators, vector-valued bilinear operators (e. g., square-function-like operators, paraproducts, and Coifman Meyer multipliers on Lebesgue spaces, Besov, and Triebel Lizorkin function spaces. One of the first examples of multilinear operators in Euclidean harmonic analysis are the commutators of Calderón which appear in a series representation of the Cauchy integral along Lipschitz curves. The sharpest possible (endpoint results for the m- commutators of Calderón were obtained by Calderón 5] when m =, Coifman and Meyer 9] when m {, 2} and Duong, Grafakos, and Yan 29] for m 3. In particular, the article of Coifman and Meyer 9] not only established delicate estimates for the commutators but also set a solid foundation for a comprehensive study of general multilinear operators; this work, together with 20], 2], has been both fundamental and pioneering in this subject and certainly inspiring in our own work. Another important example of a bilinear operator is the paraproduct of Bony 2], which has been studied extensively and has experienced remarkable development in recent years, in view of its important connections with partial differential equations. Section 8 is devoted to paraproducts and its introduction contains recent advances in the theory. Among the main motivations for the multilinear analysis in this work, we mention the m-linear versions of the fractional Leibniz-type rules; namely, inequalities of the type m m m f j C D α (f j L p j (R n f k L p k (Rn, (. Dα L p (R n where C is a positive constant independent of {f j } m, the indices obey the Hölder scaling /p = /p + +/p m < and each p j,. Indeed, inequalities like (. are based on mapping properties of bilinear Coifman-Meyer multipliers, which, in turn, follow from paraproduct decompositions and mapping properties for such paraproducts. In 5], Bényi, 5] k= k j

6 6. Introduction Maldonado, Nahmod, and Torres proved that paraproducts can be realized as bilinear singular integrals of Calderón Zygmund type. Consequently, inequalities (. are now best understood via the use of the powerful multilinear Calderón Zygmund theory that was systematically developed by Grafakos and Torres 53] (see, for example, Grafakos 42] for a survey of these techniques. In addition, it turns out that there is a rich weighted-norm theory for multilinear operators. In particular, multilinear Calderón Zygmund operators obey vector-valued and weighted estimates, with respect to certain classes of weights. Very natural classes of multilinear weights surfaced in the work of Lerner, Ombrosi, Pérez, Torres, and Trujillo- González 72]. These weights are intrinsically multilinear and they have brought into fruition a rich weighted theory for multilinear operators analogous to that of the classical A p weighted theory for linear operators in Euclidean spaces. The metric-space implementation of this class of weights is carried out in the present work. Besov spaces were originally introduced by Besov 0, ] as the trace spaces of Sobolev spaces, and were later generalized by Taibleson 96, 97, 98]. These spaces also arise as the real interpolation intermediate spaces of Sobolev spaces. Around 970, Triebel 00] and Lizorkin 73, 74] started to investigate the scale F s p,q, nowadays known as the Triebel Lizorkin spaces. The scales of Besov and Triebel Lizorkin spaces include fundamental function spaces such as Lebesgue spaces, Sobolev spaces, Hardy spaces, and the space BMO of functions with bounded mean oscillation. We refer to Frazier and Jawerth 33] for a survey of the theory of Besov and Triebel Lizorkin spaces and to Triebel s books 0, 02, 03] for a more comprehensive study. Over the last few decades, Besov and Triebel Lizorkin spaces have consistently appeared in prominent parts of the literature and their usefulness has been exposed in different areas of mathematics and physics, such as partial differential equations, potential theory, the approximation theory, and fluid dynamics. The complete framework of the classical Besov and Triebel Lizorkin theory was extended to the content of RD-spaces by Han, Müller and Yang 60, 84]. In this work, we establish the bilinear T -theorems for Besov and Triebel Lizorkin spaces, in the full range of indices. Moreover, we obtain multilinear vector-valued T type theorems on Lebesgue spaces, Besov spaces, and Triebel Lizorkin spaces. As an application, we deduce the boundedness of paraproducts and bilinear multiplier operators on products of Besov and Triebel Lizorkin spaces. Some of our results, for example these contained in Sections 6, 7, 9 and Subsection 8.4, are new even in the Euclidean case. In particular, in Section 6 we establish T theorems for bilinear Calderón Zygmund operators on Triebel Lizorkin spaces F p,q( s and Besov spaces Ḃp,q( s for the full admissible range of s, p, q, successfully answering an open problem posed by Grafakos and Torres 55, p. 85]. Appropriate contextual descriptions as well as references are included at the beginning of each section. As a whole, our results complement, from the Littlewood Paley and real-analysis side, the recent advances in analysis on metric spaces related to first-order calculus (e. g. Sobolev functions, see Haj lasz and Koskela 56], Koskela and Saksman 69], Shanmugalingam 93] and the references therein, and the (weighted and unweighted multilinear theory of potential operators in Grafakos and Kalton 43], Kenig and Stein, 68], Moen

7 . Introduction 7 83], and the references therein. Notation. Let N := {, 2,... }, Z + := N {0} and R + := 0,. For any p, ], we denote by p the conjugate index, namely, /p + /p = ; if p =, then p = and, if p =, then p =. For any a, b R, let (a b := min{a, b} and (a b := max{a, b}. For any ball B and κ > 0, denote by κb the ball contained in with the same center as B but radius dilated by the factor κ. Let C b ( be the set of all continuous functions on with bounded support (namely, contained in a ball of (, d. Let L b ( be the set of all bounded functions on with bounded support. We use L loc ( to represent the collection of all locally integrable functions on (, d, µ. Moreover, for q (0,, Let where n N, ɛ > 0 and s R. For any set E of, we define L q loc ( := {f : f q L loc ( }. p(s, ɛ := max{n/(n + ɛ, n/(n + s + ɛ}, d(x, E := inf{d(x, y : y E}. We use T Y to denote the operator norm of T : Y. Denote by C a positive constant independent of main parameters involved, which may vary at different occurrences. Constants with subscripts do not change through the whole paper. Occasionally we use C α,β,... or C(α, β,... to indicate that the positive constant C depends only on parameters α, β,.... Denote f Cg and f Cg by f g and f g, respectively. If f g f, we then write f g.

8 2. Real analysis on spaces of homogeneous type Spaces of homogeneous type provide a general framework where the real-variable approach in the study of singular integrals of Calderón and Zygmund can be carried out. It turns out that classical analysis topics such as the Littlewood Paley theory and function spaces can be introduced and developed in this context without resorting to the differential or algebraic structure of the underlying space. This section provides necessary notions and results related to the spaces of homogeneous type and the so-called RD-spaces; see 22, 23, 75, 76, 27, 64, 6, 59, 60, 65] and the references therein. The readers who are familiar with these basic knowledge can directly proceed to the next section. 2.. Spaces of homogeneous type and RD-spaces. Let be a set. A function d : R + is called a quasi-metric if (i d(x, y = d(y, x for all x, y ; (ii d(x, y = 0 if and only if x = y; and (iii there exists a constant K, such that d(x, y Kd(x, z + d(z, y] for all x, y, z. In this case we call (, d a quasi-metric space. In particular, when K =, we call d a metric and (, d a metric space. For all x and r > 0, set B(x, r := {y : d(x, y < r}. Next we recall the notions of spaces of homogeneous type in the sense of Coifman and Weiss 22, 23] and of RD-spaces introduced in 59, 60]. Definition 2.. Let (, d be a metric space and the balls {B(x, r : r > 0} form a basis of open neighborhoods of the point x. Suppose that µ is a regular Borel measure defined on a σ-algebra which contains all Borel sets induced by the open balls {B(x, r : x, r > 0}, and that 0 < µ(b(x, r < for all x and r > 0. The triple (, d, µ is called a space of homogeneous type if there exists a constant C, such that, for all x and r > 0, µ(b(x, 2r C µ(b(x, r (doubling condition. (2. The triple (, d, µ is called an RD-space if it is a space of homogeneous type and there exist constants κ (0, and C 2 (0, ] such that, for all x, 0 < r < 2 diam ( and λ < 2 diam ( /r, C 2 λ κ µ(b(x, r µ(b(x, λr, (2.2 8]

9 2.. Spaces of homogeneous type and RD-spaces 9 where above and in what follows, diam ( := sup d(x, y. x, y RD-spaces have become the underlying context in numerous areas of analysis and PDEs, we refer the reader to 22, 23], 59, 60, 84], 70, 7, 39], 04, 06, 05] and references therein. Remark 2.2. (i For a space of homogeneous type, by (2., there exist C 3, and n (0, such that, for all x, r > 0 and λ, µ(b(x, λr C 3 λ n µ(b(x, r. Indeed, we can choose C 3 := C and n := log 2 C. In some sense, n measures the dimension of. When is an RD-space, we obviously have n κ,. (ii For a space of homogeneous type (, d, µ, if µ( <, then there exists a positive constant R 0 such that = B(x, R 0 for all x ; see Nakai and Yabuta 86, Lemma 5.]. It follows that µ( < if and only if diam ( <. (iii is an RD-space if and only if is a space of homogeneous type with the additional property that there is a constant a 0 > such that for all x and 0 < r < diam ( /a 0, B(x, a 0 r \ B(x, r ; see, for instance, 60, Remark.], 04] and 86]. Consequently, any connected space of homogeneous type is an RD-space. (iv For any space of homogeneous type (, d, µ, the set Atom(, d, µ := {x : µ({x} > 0} is countable and, for every x Atom(, d, µ, there exists r > 0 such that B(x, r = {x}; see Macías and Segovia 75, Theorem ]. From (iii or (2.2, it follows that any RD-space (, d, µ is non-atomic, i. e., µ({x} = 0 for all x. (v Throughout this paper, we always assume that (, d, µ is an RD-space and µ( =, unless it is clearly stated that (, d, µ is a space of homogeneous type. Remark 2.3. For any quasi-metric space (, d, Macías and Segovia 75, Theorem 2] proved that there exists an equivalent quasi-metric ρ such that all balls corresponding to ρ are open in the topology induced by ρ, and there exist constants C > 0 and θ (0, such that, for all x, y, z, ρ(x, z ρ(y, z Cρ(x, y] θ ρ(x, z + ρ(y, z] θ. If the metric d in Definition 2. is replaced by ρ, then all results in this paper have corresponding analogues on (, ρ, µ. In order to simplify the presentation, in this work we always assume that d is a metric and the balls corresponding to d are open sets in the topology induced by d. Set V δ (x := µ(b(x, δ and V (x, y := µ(b(x, d(x, y for all x, y and δ > 0. It follows from (2. that V (x, y V (y, x. Here we present some estimates regarding the space of homogeneous type; see, for example, 60, Lemma 2.] or 59].

10 0 2. Real analysis on spaces of homogeneous type Lemma 2.4. Let (, d, µ be a space of homogeneous type, r > 0, δ > 0, α > 0, η 0 and γ (0,. (a For all x, y and all r > 0, it holds true that µ(b(x, r + d(x, y µ(b(y, r + d(x, y V r (x + V (x, y V r (y + V (x, y V r (x + V r (y + V (x, y, with implicit constants depending only on C. (b If x, x, x satisfy d(x, x γ(r + d(x, x, then r + d(x, x r + d(x, x and µ(b(x, r + d(x, x µ(b(x, r + d(x, x, with implicit constants depending only on γ and C. (c There exists a positive constant C, depending only on C and α, such that, for all x and δ > 0, d(x,y δ d(x, y α dµ(y Cδα and V (x, y d(x,y δ dµ(y C. V (x, y d(x, y α (d If α > η 0, then there exists a positive constant C, depending only on C, α and η, such that, for all x and δ > 0, ] α δ d(x, y η dµ(y Cδ η. V δ (x + V (x, y δ + d(x, y 2.2. Dyadic cubes, covering lemmas, and the Calderón Zygmund decomposition. Throughout this subsection we always assume that (, d, µ is a space of homogeneous type. Recall that in R n the dyadic cubes are defined, for all k Z and l = (l,..., l n Z n, as Q k,l := {x = (x,..., x n R n : 2 k l i x i < 2 k (l i +, i {,..., n}}. Most of their properties are retained in the case of abstract spaces of homogeneous type. Indeed, a construction due to Christ (see 6] allows the following version of the Euclidean dyadic cubes in a general space of homogeneous type. It should be remarked that recently Hytönen et. al., 66, 67] constructed a randomized dyadic structure by only assuming that the underlying metric space is geometrically doubling. Lemma 2.5. Let be a space of homogeneous type. Then there exists a collection Q = {Q k α : k Z, α I k } of open subsets, where I k is some index set, and constants δ (0, and C 5, C 6 > 0 such that (i for each fixed k Z, µ( \ α Q k α = 0 and Q k α Q k β = if α β; (ii for any α, β, k, l with l k, either Q l β Qk α or Q l β Qk α = ; (iii for each (k, α and each l < k, there is a unique β such that Q k α Q l β ; (iv diam (Q k α C 5 δ k and each Q k α contains some ball B(zα, k C 6 δ k, where zα k. One can think of Q k α as being a dyadic cube centered at z k α with diameter roughly δ k. In what follows, for simplicity, we always assume that δ = /2; see, for example, Han and Sawyer 6, pp ] on how to remove this restriction. δ α

11 2.2. Dyadic cubes, covering lemmas, and the Calderón Zygmund decomposition Regarding covering lemmas, we begin with the so-called basic covering lemma (see, for instance, Heinonen 64, p. 2] on metric spaces, which is particularly useful. Lemma 2.6. Every family F of balls of uniformly bounded diameter in a metric space (, d contains a subfamily G of pairwise disjoint balls such that 5B. B F B B G Moreover, every ball B from F meets a ball from G with radius at least half that of B. Two geometric facts about spaces of homogeneous type, the Vitali Wiener type covering lemma and Whitney type covering lemma, play fundamental roles in establishing the Calderón Zygmund theory on (, d, µ; see 22, 23] as well as 76]. Lemma 2.7. (Vitali Wiener type covering lemma Let E be a bounded set (namely, contained in a ball. Consider any covering of E of the form {B(x, r x : x E}. Then there exists a sequence of points x j E such that {B(x j, r xj } j are pairwise disjoint and {B(x j, C 0 r xj } j is a covering of E. Here C 0 depends only on the doubling and quasitriangle constants. We remark that, when Ω is an open bounded set, the following Lemma 2.8 was proved in 22, pp. 70 7] and 23, Theorem 3.2]. The current version was claimed in 22, p. 70] without a proof; see also 76, p. 277] for another variant that Ω is assumed to be an open set of finite measure strictly contained in. In fact, a detailed proof of Lemma 2.8 can be given by borrowing some ideas from 94, pp. 5 6]; see also 47]. Lemma 2.8. (Whitney type covering lemma Let Ω be an open proper subset of. For x define d(x := dist (x, Ω. For any given c, let r(x := d(x/(2c. Then there exist a positive number M, which depends only on c and C but independent of Ω, and a sequence {x k } k such that, if denote r(x k by r k, then (i {B(x k, r k /4} k are pairwise disjoint and k B(x k, r k = Ω; (ii for every given k, B(x k, cr k Ω; (iii for every given k, x B(x k, cr k implies that cr k < d(x < 3cr k ; (iv for every given k, there exists a y k / Ω such that d(x k, y k < 3cr k ; (v for every given k, the number of balls B(x i, cr i intersecting with the ball B(x k, cr k is at most M. For any f L loc (, the Hardy Littlewood maximal function Mf is defined by Mf(x := sup f(y dµ(y, x, (2.3 B x µ(b B where the supremum is taken over all balls B containing x. It is easy to see that the function Mf is lower semi-continuous (hence µ-measurable for every f L loc (. Using the Vitali Wiener type covering lemma, Coifman and Weiss 22] proved that M is bounded from L ( to L, ( and bounded on L p ( for all p (, ]. Also, by an argument similar to Grafakos 40, Exercise 2..3], we know that M is bounded on L p, ( for p (,. The operator norms M L ( L, (, M L p ( L p ( and M L p, ( L p, (

12 2 2. Real analysis on spaces of homogeneous type all depend only on C and p. Denote by C b ( the space of all continuous functions on with bounded supports (namely, contained in a ball of (, d. As in Definition 2. we are assuming that µ is a regular Borel measure on the metric space (, d, which means that µ has the outer and inner regularity (see Heinonen 64], so C b ( is dense in L p ( for all p,. This, combined with the weak type (, boundedness of M and a standard argument (see, for instance, 64, pp. 2 3], implies the differentiation theorem for integrals: for all f L loc ( and almost every x, and lim B x, µ(b 0 lim B x, µ(b 0 µ(b B f(y dµ(y = f(x f(y f(x dµ(y = 0. µ(b B A consequence of the current Whitney covering lemma and the differentiation theorem for integrals on (, d, µ as well as the weak-(, boundedness of M is the celebrated Calderón Zygmund decomposition process for integrable functions; see Coifman and Weiss 22, 23]. Lemma 2.9. Let f L (. Then, for every λ > f L ( /µ(, there exists a sequence of balls, {B k } k I, where I is some index set, such that (i { 4 B k} k I are pairwise disjoint; (ii f(x Cλ for almost every x \ k I B k ; (iii for any k I, f dµ > Cλ; µ(b k B k (iv µ(b k C f L ( /λ, k I where the positive constant C depends only on C. Lemma 2.0. Let f L (. For every λ > f L ( /µ(, let {B k } k I be the sequence of balls provided by Lemma 2.9. Then, there exist functions g and {b k } k I such that (i (ii g L ( Cλ; (iii for every k I, (iv for every k I, supp b k B k ; (v g L ( C f L ( f = g + k I b k ; b k dµ = 0; and b k L ( C f L (, k I where C is a positive constant depending only on C.

13 2.2. Dyadic cubes, covering lemmas, and the Calderón Zygmund decomposition 3 For any η (0, ], let C η ( be the set of all functions f : C such that f Ċη ( f(x f(y := sup x y d(x, y η <. Denote by supp f the closure of the set {x : f(x 0} in. Define C η b ( := {f Cη ( : f has bounded support}. Then C η b ( L ( and the norm on C η b ( is given by f C η ( := f L ( + f Ċη (. In what follows, we endow C η b ( with the strict inductive limit topology (see 76, p. 273] arising from the sequence of spaces, (C η b (B n, C η (, where {B n } n Z is any given increasing sequence of balls with the same center such that = n N B n and C η b (B n := {f C η b ( : supp f B n}. Denote by (C η b ( the dual space of C η b (, namely, the collection of all continuous linear functionals on C η b (. The space (Cη b ( is endowed with the weak -topology. For functions in C η b (, we have a more elaborate version of the Calderón Zygmund decomposition lemma when µ( =. Lemma 2.. Let µ( =, η (0, ] and f C η b (. For any λ > 0, f has the decomposition f = g + b k, k I where g, {b k } k I and k I b k are functions in C η b ( satisfying (i-(v of Lemma 2.0. Proof. Suppose that f C η b ( with η (0, ] and λ > 0. Consider the level set Ω := {x : Mf(x > λ}. Without loss of generality, we may assume that supp f B(y 0, R, where y 0 and R > 0. Moreover, since µ( =, we can choose R sufficiently large so that f, λ and B(y 0, R satisfy the assumptions of 23, p. 625, Lemma (3.9]. From this, we deduce that Ω is contained in some ball. Then, we cover Ω by using balls {B k } k I := {B(x k, r k } k I which satisfy (i-(v of Lemma 2.8. Take a radial function h Cc (R such that 0 h, h(t = when t, and h(t = 0 when t 2. For every B k, define ( d(x, xk φ k (x := h, x. r k It is easy to show that every φ k C b ( and φ k(x = when x B k. Moreover, if we take the constant c in Lemma 2.8 sufficiently large (for example, c > 2, then we have supp φ k 2B k Ω and {2B k } k I has the bounded overlapping property. If we let Φ k := φ k j φ, k I, j then {Φ k } k I forms a partition of unity of Ω with every Φ k Cb (. Now let b k := fφ k fφ k dµ Φ k dµ Φ k, k I,

14 4 2. Real analysis on spaces of homogeneous type and g := f k I b k. Then, it is a standard procedure to show that g and {b k } k I satisfy (i-(v of Lemma 2.0. By f, Φ k C η b (, we see that every b k C η b ( and supp b k 2B k. Since supp f and Ω are both bounded sets, so k I b k and g have bounded supports. The finite overlap property of {2B k } k I implies that k I b k(x has only finitely many terms for any fixed x. From this and each b k C η b (, it follows that k I b k C η b (, so does g. This finishes the proof of the lemma Space of test functions. We recall the notion of the space of test functions on the RD-space (, d, µ used in 59, 60]. Definition 2.2. Let x, r (0,, β (0, ] and γ (0,. A function ϕ on is called a test function of type (x, r, β, γ if there exists a positive constant C such that (i for all x, ] γ r ϕ(x C ; V r (x + V r (x + V (x, x r + d(x, x (ii for all x, y satisfying that d(x, y r + d(x, x]/2, ] β d(x, y ϕ(x ϕ(y C r + d(x, x V r (x + V r (x + V (x, x ] γ r. r + d(x, x Denote by G(x, r, β, γ the set of all test functions of type (x, r, β, γ. If ϕ G(x, r, β, γ, its norm is defined by ϕ G(x, r, β, γ := inf{c : (i and (ii hold}. The space G(x, r, β, γ is called the space of test functions. Set { } G(x, r, β, γ := ϕ G(x, r, β, γ : ϕ(x dµ(x = 0. The space G(x, r, β, γ is called the space of test functions with mean zero. It should be remarked that the prototype of such test functions on R n first appeared in the work of Meyer 80], where Definition 2.2(ii is replaced by ( x x β ( γ ( γ ] r r ϕ(x ϕ(y C +. (2.4 r r + x x r + y x Instead of imposing the condition that (2.4 holds for all x, y R n, Han 57] only required (2.4 for the points x, y satisfying x y (r + x x /2. The above definitions of G(x, r, β, γ and G(x, r, β, γ for general RD-spaces were first introduced in 59, 60]. Following 60], fix x and let G(β, γ := G(x,, β, γ.

15 2.4. Approximations of the identity 5 It is easy to see that, for any x 2 and r > 0, we have G(x 2, r, β, γ = G(β, γ with equivalent norms (but with constants depending on x, x 2 and r. The space G(β, γ is a Banach space. For any given ɛ (0, ], let G ɛ 0(β, γ be the completion of the space G(ɛ, ɛ in G(β, γ when β, γ (0, ɛ]. Then, ϕ G ɛ 0(β, γ if and only if ϕ G(β, γ and there exists {φ j } j N G(ɛ, ɛ such that lim j ϕ φ j G(β,γ = 0. If ϕ G ɛ 0(β, γ, we define For the above chosen {φ j } j N, we have ϕ G ɛ 0 (β,γ := ϕ G(β,γ. ϕ G ɛ 0 (β,γ = lim j φ j G(β,γ. Similarly, the space G ɛ 0(β, γ is defined to be the completion of G(ɛ, ɛ in G(β, γ when β, γ (0, ɛ] and, for any ϕ G ɛ 0(β, γ, we define ϕ Gɛ 0 (β,γ := ϕ G(β,γ. Both G ɛ 0(β, γ and G ɛ 0(β, γ are Banach spaces. Denote by (G ɛ 0(β, γ and ( G ɛ b,0 (β, γ, respectively, the sets of all bounded linear functionals on G0(β, ɛ γ and G b,0 ɛ (β, γ. Define f, ϕ to be the natural pairing of elements f (G0(β, ɛ γ and ϕ G0(β, ɛ γ, or f ( G 0(β, ɛ γ and ϕ G 0(β, ɛ γ. The space G0(β, ɛ γ plays the same role as that the Schwartz class S(R n and the space of all compactly supported functions with infinite differentiability do on R n. Obviously, any function f C η b ( with η (0, ] is a test function of type (x 0, r, η, γ for all x 0, r > 0 and γ > 0; moreover, there exists a positive constant C, depending only on C, supp f, x, β and γ, such that f G(x0,r,η,γ C f Cη (. Conversely, if f G(x,, β, γ for some β (0, ] and γ > 0, then f C β (. Moreover, by the size condition of f (see Definition 2.2(i, we see that f L ( V (x f G(x,,β,γ and, by Definition 2.2(ii when d(x, y /2 and Definition 2.2(i when d(x, y > /2, { } f Ċβ ( = max f(x f(y f(x f(y sup x y, d(x,y /2 d(x, y β, sup x y, d(x,y>/2 d(x, y β 2β+ V (x f G(x,,β,γ Approximations of the identity. Approximations of the identity on Ahlfors - regular metric measure spaces (, d, µ satisfying with µ( = and µ({x} = 0 for all x first appeared in David, Journé and Semmes 26, Lemma 2.2] and Han 58] (see also 57, 6]. Also based on the ideas in 26], the corresponding versions in the context of RD-spaces were proved in 60, Definition 2.2]. The following definition is from 59, 60]. Definition 2.3. Let ɛ (0, ], ɛ 2 > 0, and ɛ 3 > 0. A sequence {S k } k Z of bounded linear integral operators on L 2 ( is called an approximation of the identity of order (ɛ, ɛ 2, ɛ 3 (in short, (ɛ, ɛ 2, ɛ 3 - ATI, if there exists a positive constant C such that, for

16 6 2. Real analysis on spaces of homogeneous type all k Z and x, x, y and y, the integral kernel, S k (x, y, of S k is a measurable function, from into C, satisfying that (i 2 kɛ2 S k (x, y C V 2 k(x + V 2 k(y + V (x, y 2 k + d(x, y] ; ɛ2 (ii for all d(x, x 2 k + d(x, y]/2, S k (x, y S k (x, y 2 kɛ2 d(x, x ɛ C 2 k + d(x, y] ɛ V 2 k(x + V 2 k(y + V (x, y 2 k + d(x, y] ; ɛ2 (iii S k satisfies (ii with x and y interchanged; (iv for d(x, x 2 k + d(x, y]/3 and d(y, y 2 k + d(x, y]/3, (v S k (x, y S k (x, y ] S k (x, y S k (x, y ] d(x, x ɛ d(y, y ɛ C 2 k + d(x, y] ɛ 2 k + d(x, y] ɛ 2 kɛ3 V 2 k(x + V 2 k(y + V (x, y 2 k + d(x, y] ; ɛ3 S k (x, w dµ(w = = S k (w, y dµ(w. Remark 2.4. (i If {S k } k Z is an (ɛ, ɛ 2, ɛ 3 - ATI with bounded support, namely, there exists a positive constant C such that S k (x, y = 0 whenever d(x, y C2 k, then {S k } k Z is an (ɛ, ɛ 2, ɛ 3- ATI for all ɛ 2 > 0 and ɛ 3 > 0. Such a sequence of operators, {S k } k Z, is called an approximation of the identity of order ɛ with bounded support (for short, ɛ - ATI with bounded support. The existence of - ATI with bounded support was shown in 60, Theorem 2.6] by using the ideas of David, Journé and Semmes 26]. (ii Let {S k } k Z be an ɛ - ATI with bounded support. For any η (0, ɛ ], there exists a positive constant C such that, for all x, x, y and all k Z, d(x, x S k (x, y S k (x ] η, y C 2 k V 2 k(y. (2.5 Indeed, (2.5 follows from the regular condition of S k if d(x, x 2 k and the size condition of S k if d(x, x > 2 k. Combining (2.5 and the size condition of S k, we see that S k (, y C η b ( for all k Z and y. The same holds true for S k(y,. Classical examples of operators satisfying Definition 2.3 for the special case = R n can be built as follows. Let F bass (F stands for filter be the collection of non-negative radial functions ϕ S(R n such that supp ϕ {ξ R n : ξ 2} and ϕ(ξ = if ξ, where ϕ represents the Fourier transform of ϕ. Let F band be the collection of non-negative radial functions ψ S(R n such that supp ψ {ξ R n : 2 ξ 4} and ψ(ξ = if ξ 2. Given ϕ F bass, define S j by S j (f(x := S j (x, yf(ydy, f S(R n and x R n, R n

17 2.5. Singular integrals on spaces of homogeneous type 7 where j Z and S j (x, y := 2 jn ϕ ( 2 j (x y. Thus, S j (f(x = ϕ j f(x, where we used the following convention: Given a function g and j Z, we define the dilations g j as g j (x := 2 jn g(2 j x. Also, set D j (x, y := ψ j (x y, where ψ j := ϕ j+ ϕ j and ψ is such that ψ(ξ = ϕ(ξ/2 ϕ(ξ, ξ R n. Notice that ψ F band. The operators S j and D j are the basic tools to develop the Littlewood Paley theory. Finally, we summarize some properties concerning the size condition of such approximations of the identity; see 60, p. 6, Proposition 2.7]. Lemma 2.5. Suppose that a sequence {S k } k Z of functions defined on and taking values in C satisfies Definition 2.3. Then the following hold true: (i there exists a positive constant C such that, for all k Z and x, y, S k (x, z dµ(z C and S k (z, y dµ(z C; (ii for all p, ], there exists a positive constant C p such that, for all f L p (, (iii for all p, and f L p (, S k (f L p ( C p f L p ( ; lim S kf f Lp ( = 0; k (iv there exists a positive constant C such that, for all k Z, f L loc ( and x, S k (f(x CMf(x Singular integrals on spaces of homogeneous type. In this subsection we follow the pioneer work of Coifman and Weiss 22, 23], and more recent results provided by Han, Müller and Yang 59, 60] to present the analogs of the boundedness for singular integrals on some classical function spaces on R n for spaces of homogeneous type. The following theorem is due to Coifman Weiss 22, Theorem 2.4]. Lemma 2.6. Let T be a continuous linear operator T : C η b ( (Cη b ( such that, for all f C η b ( and x away from supp f, T (f(x = K(x, yf(y dµ(y, where the kernel K satisfies Hörmander s condition sup K(x, y K(x, y + K(y, x K(y, x ] dµ(x C K < y,y d(y,y d(x,y/2 for some positive constant C K. If T is bounded on L p ( for some p (,, then (i T can be extended to a bounded linear operator on L q ( for all q (, ; (ii T can be extended to a bounded linear operator from L ( to L, (. The norm of T in (i or (ii is at most a positive constant multiple of C K + T L p ( L p (.

18 8 2. Real analysis on spaces of homogeneous type The T -Theorem gives necessary and sufficient conditions for the continuity of singular integral operators in L 2 (. The first instance of such a theorem, in the Euclidean setting, was proved by David and Journé in 25]. The theorem also extends to RD-spaces in 59, 60]. Definition 2.7. Let δ (0, ]. A continuous complex-valued function K(x, y on Ω := {(x, y : x y} is called a Calderón Zygmund kernel of type δ if there is a positive constant C K such that, for all (x, y, (x, y Ω, and, when d(x, x d(x, y/2, K(x, y C K V (x, y K(x, y K(x, y + K(y, x K(y, x C K d(x, x d(x, y In this case, write K Ker(C K, δ. ] δ V (x, y. Definition 2.8. Let η (0, ]. A Calderón Zygmund singular integral operator is a continuous operator T : C η b ( (Cη b ( such that, for all f C η b ( and x / supp f, T (f(x = K(x, yf(y dµ(y, where the kernel K Ker(C K, δ for some C K > 0 and δ (0, ]. The transpose T of T is defined by T f, g = T g, f for all f, g C η b (. The kernel K of T is related to the one of T by K (x, y = K(y, x for all x, y. Definition 2.9. Given η (0, ], x and r > 0, a function ϕ on is called a normalized bump function for the ball B(x, r if (i ϕ C η b ( and supp ϕ B(x, r; (ii ϕ L ( ; (iii ϕ(z ϕ(y r η d(z, y η for all z, y. Denote by T (η, x, r the collection of all normalized bump functions for the ball B(x, r. Definition Let 0 < η θ. A singular operator T : C η b ( (Cη b ( is said to have the weak boundedness property or (for short, WBP(η if there exists a positive constant C such that, for all x, r > 0 and ϕ, ψ T (η, x, r, T ϕ, ψ Cµ(B(x, r. (2.6 The smallest possible constant C in (2.6 is denoted by T WBP(η. The following BMO -type spaces on spaces of homogeneous type (, d, µ were introduced by Coifman and Weiss 23].

19 2.5. Singular integrals on spaces of homogeneous type 9 Definition 2.2. Let q,. A function f L q loc ( is said to be in the space BMO q ( if where f BMO q( := { sup B /q f(x f B dµ(x} q <, µ(b B f B := f(y dµ(y. µ(b B Remark (i If f, f 2 BMO q ( and f f 2 is a constant, then we regard f and f 2 as the same element in BMO q (. (ii If q =, we write BMO ( instead of BMO ( for simplicity. (iii For any given q (,, the two spaces BMO q ( and BMO ( coincide with equivalent norms; see 23, pp ]. For the following T-theorem on spaces of homogeneous type, (i (ii is due to 60, Theorem 5.56], and the proof of 60, Theorem 5.57] implies that (i = (iii = (iv = (ii. Theorem Let ɛ (0, ], η (0, ɛ and T be a continuous linear operator from C η b ( to (Cη b ( as in Definition 2.8 associated with a kernel K Ker(C K, ɛ for some C K > 0. Then the following statements are equivalent: (i T extends to a bounded linear operator on L 2 ( ; (ii T ( BMO (, T ( BMO ( and T WBP(η; (iii there exists a positive constant C such that, for all x, R > 0 and φ T (η, x, R, (iv for all x, R > 0 and φ T (η, x, R, T (φ L 2 ( + T (φ L 2 ( < Cµ(B(x, R; T (φ BMO ( + T (φ BMO ( <.

20 3. Multilinear Calderón Zygmund theory This section is entirely devoted to the extension of the multilinear Calderón Zygmund theory in the Euclidean case, as developed by Grafakos and Torres in 53], to the context of the RD-spaces (, d, µ. This theory stems from the work of Coifman and Meyer 9, 20, 2, 82]; see also Kenig and Stein 68]. 3.. Multilinear Calderón Zygmund operators. Motivated by 53] we study the following multilinear singular integrals on (, d, µ. Definition 3.. Given m N, set Ω m := m+ \ {(y 0, y,..., y m : y 0 = y = = y m }. Suppose that K : Ω m C is locally integrable. The function K is called a Calderón Zygmund kernel if there exist constants C K (0, and δ (0, ] such that, for all (y 0, y,..., y m Ω m, and that, for all k {0,,..., m}, K(y 0, y,..., y m C K m k= V (y 0, y k ] m (3. K(y 0, y,..., y k,..., y m K(y 0, y,..., y k,... y m d(y k, y k C ] δ K max 0 k m d(y 0, y k m k= V (y 0, y k ], (3.2 m whenever d(y k, y k max 0 k m d(y 0, y k /2. In this case, write K Ker(m, C K, δ. Definition 3.2. Let η (0, ]. An m-linear Calderón Zygmund operator is a continuous operator m times {}}{ T : C η b ( Cη b ( (Cη b ( such that, for all f,..., f m C η b ( and x / m i= supp f i, m T (f,..., f m (x = K(x, y,..., y m f i (y i dµ(y dµ(y m, (3.3 m where the kernel K Ker(m, C K, δ for some C K > 0 and δ (0, ]. As an m-linear operator, T has m formal transposes. The j-th transpose T j of T is defined by i= T j (f,..., f m, g = T (f,..., f j, g, f j+,..., f m, f j 20]

21 3.2. Multilinear weak type estimates 2 for all f,..., f m, g in C η b (. The kernel K j of T j is related to the one of T by K j (x, y,..., y j, y j, y j+,..., y m = K(y j, y,..., y j, x, y j+,..., y m. To maintain uniform notation, we may occasionally denote T by T 0 and K by K Multilinear weak type estimates. We use the Calderón Zygmund decomposition to obtain the endpoint weak-type boundedness for multilinear operators; see 53] when (, d, µ is the Euclidean space. Theorem 3.3. Let T be an m-linear Calderón Zygmund operator associated with a kernel K Ker(m, C K, δ. Assume that, for some q, q 2,..., q m and some 0 < q < with m q j = q, T maps Lq ( L qm ( L q, (. Then T can be extended to a bounded m-linear operator from the m-fold product L ( L ( L /m, ( and T L ( L ( L /m, ( C ] C K + T L q ( L qm ( L q, ( for some positive constant C that depends only on C, C 2, δ and m. To show Theorem 3.3, we first establish the following lemma. Lemma 3.4. For any δ > 0, there exists positive constant C, depending only on C, δ and m, such that, for all i {,..., m} and all x, y k with k i, ] δ ] δ max k m d(x, y k m k= V (x, y k dµ(y i C. max k m, k i d(x, y k Proof. Let a := max d(x, y k. k m, k i By (a and (b of Lemma 2.4 and (2., we see that, when d(x, y i < 2a, m V (x, y k µ(b(x, a, k= which further implies that ] δ d(x,y i<2a max k m d(x, y k m k= V (x, y k dµ(y i ] δ µ(b(x, 2a max k m, k i d(x, y k µ(b(x, a ] δ. max k m, k i d(x, y k On the other hand, when d(x, y i 2a, we have max d(x, y k = d(x, y i k m d(x,y i 2a and m V (x, y k V (x, y i ; consequently, again using (2. implies that ] δ max k m d(x, y k m k= V (x, y k dµ(y i k=

22 22 3. Multilinear Calderón Zygmund theory l= 2 l a d(x,y i<2 l+ a ( δ µ(b(x, 2 l+ a 2 l a µ(b(x, 2 l a l= ] δ. max k m, k i d(x, y k This finishes the proof of the lemma. ] δ d(x, y i V (x, y i dµ(y m Applying Lemmas 2. and 3.4, we now show Theorem 3.3. Proof of Theorem 3.3. Let A := T L q ( L qm ( L q, (. Since the space C η b ( is dense in L ( for any η (0, ] (see 60, p. 22, Corollary 2.], it suffices to prove the theorem for functions {f j } m Cη b (. Fix α > 0. By homogeneity, without loss of generality, we may assume that f j L ( =. For any α (0,, let E α := {x : T (f,..., f m (x > α}. We only need to show that µ(e α (C K + A /m α /m. Let γ > 0 be a constant to be determined later. For all j {,..., m}, apply the Calderón Zygmund decomposition (Lemma 2. to f j at height (αγ /m to obtain good and bad functions g j and b j and families of balls, {B j,k } k Ij,j {,...,m} with {I j } j m being index sets, such that f j = g j + b j, where b j = k I j b j,k satisfying that, for all k I j and s, ], (i supp b j,k B j,k and (ii (iii (iv b j,k (y dµ(y = 0; b j,k L ( (αγ /m µ(b j,k ; k I j µ(b j,k (αγ /m ; g j L s ( (αγ m ( s and b j L ( k I j b j,k L (. Also, notice that g j and b j as above are functions in C η b ( by Lemma 2.. Now let E = {x : T (g, g 2,..., g m (x > α/2 m }, E 2 = {x : T (b, g 2,..., g m (x > α/2 m }, E 3 = {x : T (g, b 2,..., g m (x > α/2 m },

23 3.2. Multilinear weak type estimates 23 Since. E 2 m = {x : T (b, b 2,..., b m (x > α/2 m }. µ({x : T (f,..., f m (x > α} we only need to prove that, for all r {,..., 2 m }, 2 m r= µ(e r, µ(e r (C K + A /m α /m. (3.4 Chebychev s inequality and the L q ( L qm ( L q, ( boundedness of T give ( 2 m q A µ(e g q α L q ( g m q Lqm ( ( q A m ( q q mq (αγ j A (αγ (m q q m A q α m γ q m. α α Consider a set E r as above with 2 r 2 m. Suppose that, for some l m, we have exactly l bad functions appearing in T (h,..., h m where h j {b j, g j } and assume that the bad functions occur at the entries j,..., j l. The next step is to show that µ(e r = µ({x : T (h,..., h m (x > α/2 m } α /m γ /m + γ /m (γc K /l]. (3.5 Let r j,k and c j,k be the radius and the center of the ball B j,k, respectively. Set Since (B j,k := B(c j,k, 5r j,k. m µ (B j,k (αγ /m, k I j (3.5 is a consequence of ({ m } µ x / (B j,k : T (h,..., h m (x > α/2 m (αγ /m (γc K /l. (3.6 k I j Fix x / m k I j (B j,k. Then, T (h,..., h m (x K(x, y,..., y m k I j k l I m jl l g i (y i b ji,k i (y ji dµ(y dµ(y m i/ {j,...,j l } i= =: H k,...,k l. k I j k l I jl

24 24 3. Multilinear Calderón Zygmund theory Fix, for the moment, the balls Bj,k,..., Bj l,k l and, without loss of generality, we may suppose that Bj,k has the smallest radius among them. Notice that K(x, y,..., y m b j,k (y j dµ(y j B j,k = K(x, y,..., y j,..., y m K(x, y,..., c j,k,..., y m ] b j,k (y j dµ(y j B j,k ] δ d(y j, c j,k C K max k m d(x, y k m k= V (x, y k] b m j,k (y j dµ(y j. B j,k Integrating with respect to every y i with i / {j,..., j l } and using Lemma 3.4 m l times, we obtain K(x, y,..., y m b j,k (y j dµ(y j dµ(y i m l B j,k C K B j,k b j,k (y j i / {j,...,j l } ] δ d(y j, c j,k max m l k m d(x, y k m k= V (x, y dµ(y k] m i dµ(y j i / {j,...,j l } ] δ d(y j, c j,k C K b j,k (y j B j,k max i l d(x, y ji l i= V (x, y j i ] dµ(y l j ] δ r j,k C b j,k L ( K max i l d(x, c ji,k i l i= V (x, c j i,k i ], l where, in the last step, we used the fact that y ji B ji,k i and x / m k I j (B j,k imply that d(x, y ji d(x, c ji,k i and V (x, y ji V (x, c ji,k i. By the arithmetic-geometric mean inequality and since we are assuming that r j,k is the smallest among {r ji,k i } l i=, we have ] δ r j,k l ] δ/l max i l d(x, c ji,k i l i= V (x, c j i,k i ] r ji,k i l d(x, c ji,k i V (x, c ji,k i. Then, by the fact that, for all i {,..., m}, we have H k,...,k l m i= g i L ( (αγ /m, K(x, y,..., y m b j,k (y j dµ(y j B j,k i/ {j,...,j l } g i (y i l b ji,k i (y ji i=2 dµ(y i i {,...,m} i j

25 i= 3.2. Multilinear weak type estimates 25 C K (αγ m l m bj,k L ( l l r ji,k i d(x, c ji,k i C K (αγ m l m C K (αγ m l m l i= l b ji,k i (y ji i=2 ] δ/l V (x, c ji,k i dµ(y j 2 dµ(y j3 dµ(y jl r ji,k i d(x, c ji,k i ] δ/l b ji,k i L ( V (x, c ji,k i l ] δ/l r ji,k i (αγ /m µ(b ji,k i. d(x, c ji,k i V (x, c ji,k i i= We now bound T (h,..., h m (x as follows: for any x / m k I j (B j,k, T (h,..., h m (x C K (αγ m l m k I j l C K αγ C K αγ i= l k i I ji i= k i I ji l k l I jl i= r ji,k i d(x, c ji,k i r ji,k i d(x, c ji,k i ] δ/l µ(b ji,k i V (x, c ji,k i +δ/(nl M (χ Bji,ki (x], ] δ/l (αγ /m µ(b ji,k i V (x, c ji,k i where, in the last step, the doubling condition (2. and x / (B ji,k i imply that ] δ/l ] δ/l r ji,k i µ(b ji,k i d(x, c ji,k i V (x, c ji,k i r ji,k i µ(b ji,k i d(x, c ji,k i + r ji,k i µ(b(x, d(x, c ji,k i + r ji,k i ] +δ/(nl µ(b ji,k i µ(b(x, d(x, c ji,k i + r ji,k i +δ/(nl M (χ Bji,ki (x]. By this, the L +δ/(nl ( -boundedness of M and Hölder s inequality, we conclude that m µ x / (B j,k : T (h,..., h m (x > α/2 m k I j α /l \ m k Ij (B j,k T (h,..., h m (x /l dµ(x (C K γ /l (C K γ /l i= l i= k i I ji l k i I ji /l ] +δ/(nl M (χ Bji,ki (x dµ(x /l ] +δ/(nl M (χ Bji,ki (x dµ(x

26 26 3. Multilinear Calderón Zygmund theory (C K γ /l l µ(b ji,k i k i I ji i= (C K γ /l (αγ /m. This proves (3.6. Selecting γ = (C K + A, we see that all the sets E r satisfy (3.4, which completes the proof of Theorem 3.3. /l

27 4. Weighted multilinear Calderón Zygmund theory Weighted estimates for multilinear Calderón Zygmund operators first appear in the article of Grafakos and Torres 53]. Weighted estimates for multilinear commutators via the sharp maximal function were subsequently obtained by Pérez and Torres 9]. One of the main motivations for the results in this section comes from the article by Lerner, Ombrosi, Pérez, Torres, and Trujillo-González 72] where a very natural multiple-weight theory adapted to the multilinear Calderón Zygmund theory was developed. It should be mentioned the recent work by Bui and Duong 3], in which multiple weighted norm inequalities of the multilinear Calderón Zygmund operators on R n were studied, but with kernels satisfying some mild regularity condition which is weaker than the usual Hölder s continuity. Given a measure ρ absolutely continuous with respect to the measure µ, that is, there is a non-negative locally integrable function w such that dρ(x = w(xdµ(x for all x, then ρ is called a weighted measure with respect to µ and w is called a weight. A weight w is said to belong to the Muckenhoupt class A p for p (, if ] p w] Ap := sup w(y dµ(y w(y dµ(y] p <, B µ(b B µ(b B where the supremum is taken over all balls B contained in. When p =, a weight w is said to belong to the Muckenhoupt class A if ] w] A := sup w(y dµ(y] inf B µ(b w(x <. B B Set A := A p. p< For more details on the A p weights on spaces of homogeneous type, see for instance 95]; for the Muckenhoupt class on R n, see for example 4, 36]. As part of results in this section we extend the multiple-weight Calderón Zygmund theory of 72] to the context of spaces of homogeneous type. Multiple-weight norm inequalities for maximal truncated operators of multilinear singular integrals are also obtained. 4.. Multiple weights. In the context of RD-spaces, motivated by 72], we consider the following multiple weights. Definition 4.. For m exponents p,..., p m, write p for the exponent defined by p = + + p p m 27]

28 28 4. Weighted multilinear Calderón Zygmund theory and P := (p,..., p m. Definition 4.2. Let p,..., p m < and p (0, such that p = m p j. Suppose that ν is a weight and w := (w,..., w m with every w j a weight. We say that (ν; w satisfies the A P condition if sup B balls where, when p j =, ν(x dµ(x µ(b B ] /p m µ(b B µ(b B ] /p j w j (x p j dµ(x ] /p j w j (x p j dµ(x <, (4. is understood as (inf B w j. The expression in the left-hand side of (4. is referred to as the A P constant of (ν; w and denoted by (ν; w] A P. In particular, if ν is taken to be the weight ν w := m w p/pj j, (4.2 then (ν w ; w and (ν w ; w] A P are respectively written by w and w] A P. Proposition 4.3. Let p,..., p m <, p = p + + p m and P = (p,..., p m. For any given weights w := (w,..., w m and ν w as in (4.2, the following hold true: (i If every w j A pj, then w A P and w] A P m w j ] /pj A pj. (ii If w A P, then ν w A mp and w p j j A mp j for all j {,..., m}, where the condition w p j j A mp j in the case p j = is understood as w /m j A. Moreover, ν w ] Amp w] A P and w p j j ] Amp j w] p j A P if p j > or w /m j ] A w] /m A P if p j =. (iii If ν w A mp and w p j j A mp j for all j {,..., m}, then w A P and w] A P ν w ] Amp w p j /pj j ] /p j w /m j j m, p j> A mp j j m, p ] A, where the condition w p j j A mp j in the case p j = is understood as w /m j A. Proof. To see (i, if each w j A pj, then by p = m p j and Hölder s inequality, we have w] A P = sup ν w (x dµ(x B µ(b B ] /p m µ(b B w j (x p j dµ(x ] /p j