Commutators of maximal functions on spaces of homogeneous type and their weighted, local versions

Abstract

We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows that the local version of commutators also has an independent interest.

Key words： Maximal function; space of homogeneous type; BMO space; weight; commutator

References

1 Agcayazi M , Gogatishvili A , Koca K , Mustafayev R . A note on maximal commutators and commutators of maximal functions. J Math Soc Japan, 2015, 67 (2): 581- 593

2 Aimar H , Bernardis A , Iaffei B . Multiresolution approximations and unconditional bases on weighted Lebesgue spaces on spaces of homogeneous type. J Approx Theory, 2007, 148 (1): 12- 34

3 Auscher P , Hytönen T . Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl Comput Harmon Anal, 2013, 34 (2): 266- 296

4 Bastero J , Milman M , Ruiz F J . Commutators for the maximal and sharp functions. Proc Amer Math Soc, 2000, 128: 3329- 3334

5 Carbery A , Chang S -Y A , Garnett J . Weights and L log L. Pacific J Math, 1985, 120 (1): 33- 45

6 Chen P , Li J , Ward L A . BMO from dyadic BMO via expectations on product spaces of homogeneous type. J Funct Anal, 2013, 265: 2420- 2451

7 Chen W , Sawyer E . Endpoint estimates for commutators of singular integrals on spaces of homogeneous type. J Math Anal Appl, 2003, 282 (2): 553- 566

8 Coifman R R , Rochberg R , Weiss G . Factorization theorems for Hardy spaces in several variables. Ann of Math (2), 1976, 103: 611- 635

9 Coifman R R , Weiss G . Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. Lecture Notes in Math, Vol 242. Berlin: Springer-Verlag, 1971

10 Coifman R R , Weiss G . Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83: 569- 645

11 David G , Journé J L , Semmes S . Calderón-Zygmund operators, para-accretive functions and interpolation. Rev Mat Iberoam, 1985, 1 (4): 1- 56

12 Deng D G , Han Y S . Harmonic Analysis on Spaces of Homogeneous Type. Berlin: Springer, 2009

13 Duong X T , Lacey M , Li J , Wick B D , Wu Q Y . Commutators of Cauchy type integrals for domains in C n with minimal smoothness. Indiana Univ Math J (in press)

14 Edmunds D E , Kokilashvili V , Meskhi A . Weight inequalities for singular integrals defined on spaces of homogeneous and nonhomogeneous type. Georgian Math J, 2001, 8 (1): 33- 59

15 García-Cuerva J , Harboure E , Segovia C , Torrea J L . Weighted norm inequalities for commutators of strongly singular integrals. Indiana Univ Math J, 1991, 40: 1397- 1420

16 García-Cuerva J , Rubio de Francia J L . Weighted Norm Inequalities and Related Topics. North-Holland Math Studies, Vol 116. Amsterdam: North Holland, 1985

17 Grafakos L , Liu L G , Yang D C . Vector-valued singular integrals and maximal functions on spaces of homogeneous type. Math Scand, 2009, 104 (2): 296- 310

18 Han Y S . Calderón-type reproducing formula and the T_b theorem. Rev Mat Iberoam, 1994, 10: 51- 91

19 Han Y S . Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications. Proc Amer Math Soc, 1998, 126 (11): 3315- 3327

20 Han Y S , Li J , Lin C C . Criterions of the L² boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type. Ann Sc Norm Super Pisa Cl Sci, 2016, 16 (3): 845- 907

21 Han Y S , Sawyer E T . Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces. Mem Amer Math Soc, Vol 110, No 530. Providence: Amer Math Soc, 1994

22 Hart J , Torres R H . John-Nirenberg inequalities and weight invariant BMO spaces. J Geom Anal, 2019, 29 (2): 1608- 1648

23 He Z Y , Han Y S , Li J , Liu L G , Yang D C , Yuan W . A complete real-variable theory of Hardy spaces on spaces of homogeneous type. J Fourier Anal Appl, 2019, 25 (5): 2197- 2267

24 Hu G E , Lin H B , Yang D C . Commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. Abstr Appl Anal, 2008, 2008 (Mar): (21 pp)

25 Hu G E , Yang D C . Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type. J Math Anal Appl, 2009, 354 (1): 249- 262

26 Hytönen T , Kairema A . Systems of dyadic cubes in a doubling metric space. Colloq Math, 2012, 126 (1): 1- 33

27 Krantz S G , Li S Y . Boundedness and compactness of integral operators on spaces of homogeneous type and applications, Ⅱ. J Math Anal Appl, 2001, 258: 642- 657

28 Kronz M . Some function spaces on spaces of homogeneous type. Manuscripta Math, 2001, 106 (2): 219- 248

29 Lanzani L , Stein E M . The Cauchy–Szegö projection for domains in C n with minimal smoothness. Duke Math J, 2017, 166: 125- 176

30 Lerner A K . On weighted estimates of non-increasing rearrangements. East J Approx, 1998, 4: 277- 290

31 Macías R A , Segovia C . Lipschitz functions on spaces of homogeneous type. Adv Math, 1979, 33: 257- 270

32 Milman M , Schonbek T . Second order estimates in interpolation theory and applications. Proc Amer Math Soc, 1990, 110 (4): 961- 969

33 Nagel A , Stein E M . On the product theory of singular integrals. Rev Mat Iberoam, 2004, 20: 531- 561

34 Nagel A , Stein E M . ∂ ¯ b -complex on decoupled boundaries in C n . Ann of Math (2), 2006, 164 (2): 649- 713

35 Pradolini G , Salinas O . Commutators of singular integrals on spaces of homogeneous type. Czechoslovak Math J, 2007, 57 (1): 75- 93

36 Range R M . Holomorphic Functions and Integral Representations in Several Complex Variables. Grad Texts in Math, Vol 108. New York: Springer-Verlag, 1986

37 Rao M M , Ren Z D . Theory of Orlicz Spaces. New York: Marcel Dekker, 1991