Journal of the Mathematical Society of Japan

Boundedness of sublinear operators on product Hardy spaces and its application

Der-Chen CHANG, Dachun YANG, and Yuan ZHOU

Full-text: Open access

Abstract

Let p ( 0 , 1 ] . In this paper, the authors prove that a sublinear operator T (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spaces H p ( R n × R m ) to some quasi-Banach space B if and only if T maps all ( p , 2 , s 1 , s 2 ) -atoms into uniformly bounded elements of B . Here s 1 n ( 1 / p - 1 ) and s 2 m ( 1 / p - 1 ) . As usual, n ( 1 / p - 1 ) denotes the maximal integer no more than n ( 1 / p - 1 ) . Applying this result, the authors establish the boundedness of the commutators generated by Calderón-Zygmund operators and Lipschitz functions from the Lebesgue space L p ( R n × R m ) with some p > 1 or the Hardy space H p ( R n × R m ) with some p 1 but near 1 to the Lebesgue space L q ( R n × R m ) with some q > 1 .

Article information

Source
J. Math. Soc. Japan, Volume 62, Number 1 (2010), 321-353.

Dates
First available in Project Euclid: 5 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1265380433

Digital Object Identifier
doi:10.2969/jmsj/06210321

Mathematical Reviews number (MathSciNet)
MR2648225

Zentralblatt MATH identifier
1195.42060

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B30: $H^p$-spaces 42B25: Maximal functions, Littlewood-Paley theory 47B47: Commutators, derivations, elementary operators, etc.

Keywords
product space Hardy space Lebesgue space sublinear operator commutator Calderón-Zygmund operator Lipschitz function

Citation

CHANG, Der-Chen; YANG, Dachun; ZHOU, Yuan. Boundedness of sublinear operators on product Hardy spaces and its application. J. Math. Soc. Japan 62 (2010), no. 1, 321--353. doi:10.2969/jmsj/06210321. https://projecteuclid.org/euclid.jmsj/1265380433


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