Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces

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1 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 1/45 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces Dachun Yang (Joint work) School of Mathematical Sciences Beijing Normal University January 10, 2018 / Institute for Advanced Study in Mathematics, Harbin Institute of Technology

2 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 2/45 Outline I Motivations I Bilinear Decompositions of Products of Hardy and Campanato Spaces I Bilinear Decompositions of Products of Local Hardy and Lipschitz or BMO Spaces IV Further Remarks

3 I. Motivations Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 3/45

4 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 4/45 Hardy Spaces H p (R n ) / I Let p (0, 1]. The Hardy space H p (R n ) is defined to be the collection of all Schwartz distributions f S (R n ) such that their quasi-norms f Hp (R n ) := f Lp (R n ) := 1 where ϕ t f(x) := f, ϕ 0. R n t nϕ( x sup t (0, ) (ϕ t f) Lp (R n ) t ) with ϕ S(R n ) and <, It is known that H p (R n ) is independent of the choice of ϕ.

5 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 5/45 Campanato SpacesC α,q,s (R n ) / I Let α [0, ), q [1, ] and s Z + such that s nα. The Campanato space C α,q,s (R n ) is defined to be the set of all locally integrable functions g such that g Cα,q,s (R n ) := sup B α B R n 1 <, { 1 B B g(x) P B,s (g)(x) q dx } 1/q where P B,s g denotes the minimizing polynomial of g on B with degree s. P B,s g is minimizing: for any polynomial Q with degree s, [g P B,s g]q = 0. B

6 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 6/45 Hardy Spaces and Their Dual Spaces / I It is well known: For any p (0, 1], q [1, ) and s Z + [ n( p 1 1), ), (H p (R n )) = C 1/p 1,q,s (R n ) =: C 1/p 1 (R n ). If α = 0, then C 0 (R n ) = BMO(R n ). If α (0, 1 n ), then C α(r n ) = Λ nα (R n ) with the homogeneous Lipschitz norm g Λnα (R n ) := sup x,y R n,x y g(x) g(y) x y nα.

7 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 7/45 Questions / I For any p (0,1] and α = 1/p 1, find the smallest linear vector space Y so that H p (R n ) C α (R n ) has the following bilinear decomposition: H p (R n ) C α (R n ) L 1 (R n )+Y, namely, S : H p (R n ) C α (R n ) L 1 (R n ) and T : H p (R n ) C α (R n ) Y, which are bilinear and bounded, such that f g = S(f,g)+T(f,g), (f,g) H p (R n ) C α (R n ). It is well known that H 1 (R n ) BMO(R n ) L 1 loc (Rn ). What can we do? Why are they important?

8 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 8/45 Jacobian / I S. Müller, Higher integrability of determinants and weak convergence in L 1, J. Reine Angew. Math. 412 (1990), R. R. Coifman & L. Grafakos, Hardy space estimates for multilinear operators. I, Rev. Mat. Iberoamericana 8 (1992), L. Grafakos, Hardy space estimates for multilinear operators. II, Rev. Mat. Iberoamericana 8 (1992), L. Grafakos, H 1 boundedness of determinants of vector fields, Proc. Amer. Math. Soc. 125 (1997), [Bijz07] A. Bonami, T. Iwaniec, P. Jones & M. Zinsmeister, On the product of functions in BMO and H 1, Ann. Inst. Fourier (Grenoble) 57 (2007),

9 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 9/45 Commutators / I L. D. Ky, Bilinear decompositions and commutators of singular integral operators, Trans. Amer. Math. Soc. 365 (2013),

10 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 10/45 Div-Curl Lemma (1) / I Div-Curl lemma. Let F := (F 1,..., F n ) X with curlf 0 and G := (G 1,..., G n ) Y with divg 0. Here, for any i, j {1,..., n}, ( Fj curlf := F ) i = (DF) T (DF) x i x j and divg := i,j n j=1 G j x j. Find suitable function space Z such that n F G:= F j G j F X G Y. Z j=1

11 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 11/45 Div-Curl Lemma (2) / I If p, q ( n n+1, ) with 1 p + 1 q = 1 r < n+1 n, then F G Hr (R n ) F Hp (R n ) G Hq (R n ). R. R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993),

12 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 12/45 Div-Curl Lemma (3) / I The endpoint case r = n 1 p + 1 q = 1 r = n+1 n, then n+1 : If p, q (1, ) with F G H r, (R n ) F Hp (R n ) G Hq (R n ), where H r, (R n ) denotes the weak Hardy space. T. Miyakawa, Hardy spaces of solenoidal vector fields, with applications to the Navier-Stokes equations, Kyushu J. Math. 50 (1996), 1-64.

13 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 13/45 Div-Curl Lemma (4): Endpoint Case q = (1) / I If p = 1 and q =, then F G H log (R n ) F H1 (R n ) G BMO(Rn ), where H log (R n ) denotes the Musielak-Orlicz-Hardy space related to (see [K14]). θ(x, t) := t log(e+t)+log(e+ x ) Theorem ([Bgk12]). The product space H 1 (R n ) BMO(R n ) has the following sharp bilinear decomposition: H 1 (R n ) BMO(R n ) L 1 (R n )+H log (R n ).

14 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 14/45 Div-Curl Lemma (4): Endpoint Case q = (2) / I [K14] L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of subilinear operators, Integral Equations Operator Theory 78 (2014), [Bgk12] A. Bonami, S. Grellier and L. D. Ky, Paraproducts and products of functions in BMO(R n ) and H 1 (R n ) through wavelets, J. Math. Pures Appl. (9) 97 (2012), How about the casep (0, 1)? [Bck17] A. Bonami, J. Cao, L. D. Ky, L. Liu, D. Yang and W. Yuan, A complete solution to bilinear decompositions of products of Hardy and Campanato spaces, In Progress.

15 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 15/45 Difficulties / I Restriction from the method of wavelets; There exist more complicated structures of the space C α (R n ). Theorem ([Bck17]) Let α (0, ). Then, for any g C α (R n ) & ball B := B(c B, r B ) of R n, with c B R n and r B (0, ), [ g(x) + PB,s g(x) ] sup x B { [1+ c B +r B ] nα g C + α (R n ) if nα / N, [1+ c B +r B ] nα log(e+ c B +r B ) g C + α (R n ) if nα N, where g C + α (R n ) := g C α (R n ) + 1 B( 0 n, 1) B( 0 n,1) g(x) dx.

16 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 16/45 II. Bilinear Decompositions of Products of Hardy and Campanato Spaces

17 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 17/45 Definition of Products: Case p = 1 / II Let f H 1 (R n ) and g BMO(R n ). The product f g is defined to be a Schwartz distribution in S (R n ) such that, for any Schwartz function φ S(R n ), f g, φ := φg, f, where the last bracket denotes the dual pair between BMO(R n ) and H 1 (R n ). [Recall (H 1 (R n )) = BMO(R n ).] [Bijz07] Bonami, Iwaniec, Jones and Zinsmeister, Theorem ([Ny85]) Every φ S(R n ) is a pointwise multiplier of BMO(R n ). The above product can be extended as a distribution on the class of all pointwise multipliers of BMO(R n ).

18 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 18/45 Pointwise Multipliers on BMO(R n ) / II Theorem ([Ny85]) A locally integrable function g is a pointwise multiplier of BMO(R n ) iff g BMO log (R n ) L (R n ), where f BMO log (R n ) iff f BMOlog (R n ) := sup B(a,r) with f B(a,r) := 1 B(a,r) logr +log(e+ a ) B(a, r) B(a,r) f(x) f B(a,r) dx < B(a,r) f, here a Rn and r (0, ). [Ny85] E. Nakai and K. Yabuta, Pointwise multipliers for functions of bounded mean oscillation, J. Math. Soc. Japan 37 (1985),

19 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 19/45 Duality Between H log (R n ) & BMO log (R n ) / II Theorem ([K14]) (H log (R n )) = BMO log (R n ). H 1 (R n ) BMO(R n ) L 1 (R n )+H log (R n ) ([Bck17]) (L 1 (R n )+H log (R n )) = L (R n ) BMO log (R n ) (Sharp). f g, φ := φg, f Sharp: Assume that H 1 (R n ) BMO(R n ) L 1 (R n )+Y and Y is smallest. Then L (R n ) BMO log (R n ) = (L 1 (R n )+H log (R n )) (L 1 (R n )+Y) = all pointwise multipliers of BMO(R n ) = L (R n ) BMO log (R n ). ([Ny85])

20 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 20/45 Bake to Case p (0, 1) / II Letα := 1/p 1. Find a suitable function space X such that H p (R n ) C α (R n ) L 1 (R n )+X. (L 1 (R n )+X) = L (R n ) (X) characterizes the class of all pointwise multipliers ofc α (R n ). The space X turns out to be the Musielak-Orlicz Hardy space H Φ p (R n ) associated with the Musielak-Orlicz growth function Φ p defined by setting, for any x R n and t [0, ), Φ p (x, t):= t log(e+t)+[t(1+ x ) n ] 1 p if nα / Z +, t log(e+t)+[t(1+ x ) n ] 1 p [log(e+ x )] p if nα Z +. Φ 1 (x, t) = θ(x, t).

21 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 21/45 M-O Lebesgue & Hardy Spaces Associated with Φ p / II Let p (0, 1]. The Musielak-Orlicz Lebesgue spacel Φ p (R n ) consists of all measurable functions f on R n such that } f L Φp(R n ) {λ := inf (0, ) : Φ p (x, f(x) /λ)dx 1 R n <. The Musielak-Orlicz Hardy spaceh Φ p (R n ) consists of all f S (R n ) such that f belongs to L Φ p (R n ) and is equipped with the quasi-norm f H Φp(R n ) := f L Φp(R n ), where f (x) := sup t (0, ) f, 1 t nϕ( x t ) for any x R n with ϕ S(R n ) and R n ϕ 0.

22 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 22/45 M-O Campanato Spaces Associated with Φ p / II Let p (0, 1] and s Z + [ n(1/p 1), ). The Musielak-Orlicz Campanato spacec Φp (R n ) consists of all measurable functions g on R n such that 1 g CΦ p(r n ) := sup B R n χ B L Φp(R n ) <, B g(x) PB,s (g)(x) dx where P B,s g denotes the minimizing polynomial of g on B with degree s.

23 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 23/45 Pointwise Multipliers of C α (R n )/ I Theorem ([Bck17]) Let p (0, 1] and α := 1/p 1. A function g is a pointwise multiplier of C α (R n ) iff g L (R n ) C Φp (R n ). Proof: need to show that g L (R n ) C Φp (R n ) iff for any f C α (R n ), it holds true that fg C α (R n ). Sufficiency: f(x)g(x) PB,s f(x)p B,s g(x) f(x) PB,s f(x) g(x) + PB,s f(x) g(x) PB,s g(x). Necessity: find some subtle examples of functions in C α (R n ). (Difficult)

24 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 24/45 Facts on Pointwise Multipliers ofc α (R n )/ II L (R n ) C Φp (R n ) characterizes the class of all pointwise multipliers of C 1/p 1 (R n ). (L 1 (R n )+H Φ p (R n )) = L (R n ) (H Φ p (R n )) = L (R n ) C Φp (R n ). This predicts the following sharp bilinear decomposition: for any p (0, 1), H p (R n ) C 1/p 1 (R n ) L 1 (R n )+H Φ p (R n ).

25 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 25/45 Wavelet Representation of Functions / II n times {}}{ Let E := {0, 1} n \{( 0,..., 0)} and D be the set of all dyadic cubes. For any I D and λ E, let φ I and ψi λ be, respectively, the father and the mother wavelets satisfying Support condition: supp φ I mi, supp ψi λ being a positive constant. mi with m Cancelation condition: φ R n I (x)dx = (2π) 1/2 and there exists s Z + such that, for any multi-index α satisfying α s, x α ψ λ R n I (x)dx = 0. Then, for any f L 2 (R n ), f = I D λ E f, ψλ I ψλ I.

26 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 26/45 Dobyinsky s Renormalizatoin (1) / II Let f g L 2 (R n ) L 2 (R n ). Then f g = Π 1 (f, g)+π 2 (f, g)+π 3 (f, g)+π 4 (f, g), where Π 1 (f, g) := I,I D I = I Π 2 (f, g) := I,I D I = I Π 3 (f, g) := I,I D I = I Π 4 (f, g) := I D f, φ I g, ψi λ φ I ψi λ, λ E f, ψi λ g, φ I ψi λ φ I, λ E λ,λ E (I,λ) (I,λ ) f, ψ λ I 2. f, ψi λ g, ψλ I ( ψi) λ λ E g, ψλ I ψ λ I ψλ I,

27 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 27/45 Dobyinsky s Renormalizatoin (2) / II S. Dobyinsky, La version ondelettes" du théoréme du Jacobien, Rev. Mat. Iberoam. 11 (1995),

28 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 28/45 Boundedness of Operators {Π i } 4 i=1 / II f g = Π 1 (f, g)+π 2 (f, g)+π 3 (f, g)+π 4 (f, g). Let p (0, 1) and α := 1/p 1. Π 1 and Π 3 can be extended to bilinear operators bounded from H p (R n ) C α (R n ) to H 1 (R n ). Π 4 can be extended to a bilinear operator bounded from H p (R n ) C α (R n ) to L 1 (R n ). Atomic decomposition of H p (R n ); Simple Hardy and Lebesgue estimates for Π i ; Wavelet characterization of H p (R n ) and C α (R n ); Π 2 can be extended to a bilinear operator bounded from H p (R n ) C α (R n ) to H Φ p (R n ).

29 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 29/45 Bilinear Decomposition / II Let p (0, 1). For any f H p (R n ) and g C 1/p 1 (R n ), it holds true that f g = S(f, g)+t(f, g) with S(f, g) := Π 4 (f, g) L 1 (R n ). T(f, g) := Π 1 (f, g)+π 2 (f, g)+π 3 (f, g) H Φ p (R n ). Theorem ([Bck17]) Let p (0, 1]. Then the space H p (R n ) C 1/p 1 (R n ) has the following sharp bilinear decomposition: H p (R n ) C 1/p 1 (R n ) L 1 (R n )+H Φ p (R n ).

30 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 30/45 An Observation / II Indeed, Π 2 : H p (R n ) C α (R n ) H 1 (R n )+H p W p (R n ) and H 1 (R n )+H p W p (R n ) H Φ p (R n ), where H p W p (R n ) is the weighted Hardy space associated with the weight W p defined by setting, for any x R n, W p (x) := 1 (1+ x ) n(1 p) if n[1/p 1] / Z +, 1 (1+ x ) n(1 p) [log(e+ x )] p if n[1/p 1] Z +. What is the relationship between H p W p (R n ) andh Φ p (R n )?

31 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 31/45 Φ p (x, t) := Intrinsic Structure of H Φ p (Rn ) (1) / II Let α := 1 p 1. Musielak-Orlicz growth function: for any (x,t) R n [0, ), t log(e+t)+[t(1+ x ) n ] 1 p if nα / Z +, t log(e+t)+[t(1+ x ) n ] 1 p [log(e+ x )] p if nα Z +, Weight function: for any x R n, 1 W p (x) := (1+ x ) n(1 p) if nα / Z +, 1 (1+ x ) n(1 p) [log(e+ x )] p if nα Z +.

32 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 32/45 Intrinsic Structure of H Φ p (Rn ) (2) / II Orlicz function: φ 0 (t) := t log(e+t), t [0, ). (Φ p (x, t)) 1 = (φ 0 (t)) 1 +(t p W p (x)) 1, x R n, t (0, ). Theorem ([clyy]) for p (0, 1], H Φ p (R n ) = H φ 0 (R n )+H p W p (R n ); for p (0, 1), H Φ p (R n ) = H 1 (R n )+H p W p (R n ). H p W p (R n ) H Φ p (R n ) with when p (0, 1). [ ] H p W p (R n [ ) = H Φ p (R n ) ]

33 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 33/45 Intrinsic Structure of H Φ p (Rn ) (3) / II Lemma ([clyy]) Let B be a ball in R n. Then for p = 1, χ B 1 L Φ 1(R n ) χ B 1 L φ 0(R n ) + χ B 1 L 1 W 1 (R n ) ; for p (0, 1), χ B 1 L Φ p(r n ) χ B 1 L p Wp (Rn ). [clyy] J. Cao, L. Liu, D. Yang and W. Yuan, Intrinsic structures of certain Musielak-Orlicz-Hardy spaces, J. Geom. Anal. (to appear).

34 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 34/45 III. Bilinear Decompositions of Products of Local Hardy and Lipschitz or BMO Spaces

35 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 35/45 Local Hardy Spaces h p (R n ) / III The local Hardy space h p (R n ) for p (0, 1] is defined to be the set of all f S (R n ) such that their quasi-norms f hp (R n ) := f loc L p (R n ) := sup (ϕ t f) <, t (0, 1) Lp (R n ) where ϕ t f(x) := f, 1 t nϕ( x t ) with ϕ S(R n ).

36 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 36/45 Local BMO Spaces / II The local BMO space bmo(r n ) is defined via the norm { } 1 g bmo(rn ) := sup g(x) g B dx B <1 B B { } 1 + sup f(x) dx, B 1 B where g B := 1 B B g. B

37 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 37/45 Local Hardy Spaces and Their Dual Spaces / III The inhomogeneous Lipschitz space Λ α (R n ) with α (0, 1) is defined via the norm g Λα (R n ) := sup x,y R n x y g(x) g(y) x y α + g L (R n ). It is well known that { [h p (R n )] bmo(r n ) when p = 1, = Λ α (R n ) when p ( n+1 n, 1) with α := n( 1 p 1).

38 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 38/45 An variant of local Orlicz space/ III Let Q be an cube with side length 1. For any measurable function g on Q, define the Orlicz space L φ 0 (Q) on Q by { ( ) } g(x) g L φ 0 (Q) := inf λ (0, ) : φ 0 dx 1 λ Q with φ 0 (t) := t log(e+t), t [0, ). A generalized Hölder s inequality: fg L φ 0 (Q) f L1 (Q) g bmo(rn ).

39 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 39/45 A Variant of Local Orlicz-Hardy Space / III Let φ 0 (t) := t log(e+t) measurable function g, let for any t [0, ). For any g L φ 0 (R n ) := j Z n g L φ 0 (Q j ) with j := (j 1,..., j n ), Q j := [j 1, j 1 +1) [j n, j n +1). The local Orlicz-Hardy space h φ 0 (R n ) is defined by setting h φ 0 (R n ) := {f S (R n ) : f h φ 0 (R n ) := f loc L φ 0 (R n ) < }.

40 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 40/45 Bilinear Decompositions (1) / III For any f h 1 (R n ) and g bmo(r n ), f g = S(f, g)+t(f, g) with S(f, g) := Π 4 (f, g) L 1 (R n ). T(f, g) := Π 1 (f, g)+π 2 (f, g)+π 3 (f, g) h φ 0 (R n ). [cky17] J. Cao, L. D. Ky & D. Yang, Bilinear decompositions of products of local Hardy and Lipschitz or BMO spaces through wavelets, Commun. Contemp. Math. (to appear). Theorem ([cky17]). h 1 (R n ) bmo(r n ) has the following bilinear decomposition: h 1 (R n ) bmo(r n ) L 1 (R n )+h φ 0 (R n ). The sharpness of this decomposition is still unknown.

41 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 41/45 Bilinear Decompositions (2) / III Theorem ([cky17]) For any p ( n n+1, 1) and α = 1 p 1, h p (R n ) Λ α (R n ) has the following sharp bilinear decomposition: h p (R n ) Λ α (R n ) L 1 (R n )+h p (R n ). Theorem ([cky17]) Let F h 1 (R n ; R n ) with curlf 0 and G bmo(r n ; R n ) with divg 0. Then F G h Φ (R n ) with F G h φ 0 (R n ) F h 1 (R n ) G bmo(rn ). The last theorem when p ( n+1 n, 1) is still unknown.

42 IV. Further Remarks Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 42/45

43 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 43/45 Spaces of Homogeneous Type A quasi-metric space(x, d) equipped with a nonnegative measure µ is called a space of homogeneous type if µ satisfies the following measure doubling condition: C (X) [1, ) such that, for any ball B(x,r) := {y X : d(x,y) < r} with x X and r (0, ), µ(b(x,2r)) C (X) µ(b(x,r)). R. R. Coifman & G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, Lecture Notes in Math. 242, Springer-Verlag, Berlin-New York, R. R. Coifman & G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977),

44 Spaces of Homogeneous Type X. Fu & D. Yang, Wavelet characterizations of the atomic Hardy space H 1 on spaces of homogeneous type, Appl. Comput. Harmon. Anal. 44 (2018), X. Fu, D. Yang & Y. Liang, Products of functions in BMO(X) and Hat 1 (X) via wavelets over spaces of homogeneous type, J. Fourier Anal. Appl. 23 (2017), L. Liu, D.-C. Chang, X. Fu & D. Yang, Endpoint boundedness of commutators on spaces of homogeneous type, Appl. Anal. 96 (2017), L. Liu, D. Yang & W. Yuan, Bilinear decompositions for products of Hardy and Lipschitz spaces on spaces of homogeneous type, Submitted. X. Fu & D. Yang, Products of functions in H 1 ρ(x) and BMO ρ (X) over RD-spaces and applications to Schrödinger operators, J. Geom. Anal. 27 (2017), Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 44/45

45 Sharp Bilinear Decompositions of Products of Hardy Spaces and Their Dual Spaces p. 45/45 Lecture Notes in Math. 2182, 2017 / IV Thank you for your attention.