Journal of the Mathematical Society of Japan
Online ISSN : 1881-1167
Print ISSN : 0025-5645
ISSN-L : 0025-5645
Equivalence of Littlewoodโ€“Paley square function and area function characterizations of weighted product Hardy spaces associated to operators
Xuan Thinh DuongGuorong HuJi Li
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2019 Volume 71 Issue 1 Pages 91-115

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Abstract

Let ๐ฟ1 and ๐ฟ2 be nonnegative self-adjoint operators acting on ๐ฟ2(๐‘‹1) and ๐ฟ2(๐‘‹2), respectively, where ๐‘‹1 and ๐‘‹2 are spaces of homogeneous type. Assume that ๐ฟ1 and ๐ฟ2 have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces $๐ป^{๐‘}_{๐‘ค, ๐ฟ_1, ๐ฟ_2}(๐‘ฅ_1 ร— ๐‘ฅ_2)$ associated to ๐ฟ1 and ๐ฟ2, for ๐‘ โˆˆ (0, โˆž) and the weight ๐‘ค belongs to the product Muckenhoupt class ๐ดโˆž(๐‘‹1 ร— ๐‘‹2). Our main result is that the spaces $๐ป^{๐‘}_{๐‘ค, ๐ฟ_1, ๐ฟ_2}(๐‘ฅ_1 ร— ๐‘ฅ_2)$ introduced via area functions can be equivalently characterized by the Littlewoodโ€“Paley ๐‘”-functions and $๐‘”^{\ast}_{๐œ†_1, ๐œ†_2}$-functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of ๐ฟ1 and ๐ฟ2. Our results are new even in the unweighted product setting.

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