
MAXIMAL FUNCTION CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH BOTH NONNEGATIVE SELFADJOINT OPERATORS SATISFYING GAUSSIAN ESTIMATES AND BALL QUASIBANACH FUNCTION SPACES
Xiaosheng LIN, Dachun YANG, Sibei YANG, Wen YUAN
Acta mathematica scientia,Series B. 2024, 44 (2):
484514.
DOI: 10.1007/s104730240207y
Assume that $L$ is a nonnegative selfadjoint operator on $L^2(\mathbb{R}^n)$ with its heat kernels satisfying the socalled Gaussian upper bound estimate and that $X$ is a ball quasiBanach function space on $\mathbb{R}^n$ satisfying some mild assumptions. Let $H_{X,\,L}(\mathbb{R}^n)$ be the Hardy space associated with both $X$ and $L,$ which is defined by the Lusin area function related to the semigroup generated by $L$. In this article, the authors establish various maximal function characterizations of the Hardy space $H_{X,\,L}(\mathbb{R}^n)$ and then apply these characterizations to obtain the solvability of the related Cauchy problem. These results have a wide range of generality and, in particular, the specific spaces $X$ to which these results can be applied include the weighted space, the variable space, the mixednorm space, the Orlicz space, the Orliczslice space, and the Morrey space. Moreover, the obtained maximal function characterizations of the mixednorm Hardy space, the Orliczslice Hardy space, and the MorreyHardy space associated with $L$ are completely new.
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