Hardy factorization in terms of fractional commutators in Lorentz spaces

Abstract

We provide a constructive proof of H^1\left({\text{ℝ}}^d\right) (the classical Hardy space) factorization in terms of fractional commutators in Lorentz spaces. As a direct application, we obtain a characterization of functions in BMO space. Furthermore, we also obtain a Lorentz compactness characterization of fractional commutators.

Key words： BMO; CMO; Lorentz space; commutator; Hardy space; Riesz potential

References

1 Arai H, Mizuhara T. Morrey spaces on spaces of homogeneous type and estimates for □_b and the Cauchy-Szegö projection. Math Nachr, 1997, 186: 5–20

2 Beatrous F, Li S Y. Boundedness and compactness of operators of Hankel type. J Funct Anal, 1993, 111: 350–379

3 Bramanti M, Cerutti M C. Commutators of singular integrals on homogeneous spaces. Boll Unione Mat Ital B (7), 1996, 10(4): 843–883

4 Brudnyi Y. Compactness criteria for spaces of measurable functions. St Petersburg Math J, 2015, 26: 49–68

5 Calderón A P, Zygmund A. Singular integral operators and differential equations. Amer J Math, 1957, 79: 901–921

6 Chanillo S. A note on commutators. Indiana Univ Math J, 1982, 31: 7–16

7 Chen Y, Ding Y, Wang X. Compactness of commutators of Riesz potential on Morrey spaces. Potential Anal, 2009, 30: 301–313

8 Chiarenza F, Frasca M, Longo P. Solvability of the Dirichlet problem for non divergence elliptic equations with VMO coefficients. Trans Amer Math Soc, 1993, 336: 841–853

9 Coifman R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann of Math, 1976, 103(3): 611–635

10 Dafni G. Local VMO and weak convergence in h¹. Canad Math Bull, 2002, 45: 46–59

11 Dao N A. Morrey boundedness and compactness characterizations of integral commutators with singular kernel on strictly pseudoconvex domains in ℂn. J Math Anal Appl, 2020, 492: 124483

12 Dao N A, Duong X T, Ha L K. Commutators of Cauchy-Fantappiè type integrals on generalized Morrey spaces on domains of complex ellipsoids. J Geom Anal (to appear)

13 Dao N A, Krantz S G. Lorentz boundedness and compactness characterization of integral commutators on spaces of homogeneous type. Nonlinear Anal, 2021, 203: 112162

14 Di Fazio G, Ragusa M A. Commutators and Morrey spaces. Boll Unione Mat Ital A (7), 1991, 5(3): 323–332

15 Hedberg L. On certain convolution inequalities. Proc Amer Math Soc, 1972, 36: 505–510

16 Iwaniec T, Sboedone C. Riesz transform and elliptic PDEs with VMO-coefficients. J Anal Math, 1998, 74: 183–212

17 Komori Y, Mizuhara T. Factorization of functions in H1(ℝn) and generalized Morrey spaces. Math Nachr, 2006, 279(5-6): 619–624

18 Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282(2): 219–231

19 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

20 O’Neil R. Convolution operators and L(p; q) spaces. Duke Math J, 1963, 30: 129–142

21 Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton Math Ser, No 30. Princeton: Princeton Univ Press, 1970

22 Tao J, Yang D C, Yang D Y. Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces. Math Methods Appl Sci, 2019, 42(5): 1631–1651

23 Uchiyama A. On the compactness of operators of Hankel type. Tohoku Math J, 1978, 30: 163–171