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2013 WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING REINFORCED OFF-DIAGONAL ESTIMATES
The Anh Bui, Jun Cao, Luong Dang Ky, Dachun Yang, Sibei Yang
Taiwanese J. Math. 17(4): 1127-1166 (2013). DOI: 10.11650/tjm.17.2013.2719

Abstract

Let $L$ be a nonnegative self-adjoint operator on $L^2(\mathbb{R}^n)$ satisfying the reinforced $(p_L, p_L')$ off-diagonal estimates, where $p_L\in[1,2)$ and $p_L'$ denotes its conjugate exponent. Assume that $p\in(0,1]$ and the weight $w$ satisfies the reverse Hölder inequality of order $(p'_L/p)'$. In particular, if the heat kernels of the semigroups $\{e^{-tL}\}_{t\gt 0}$ satisfy the Gaussian upper bounds, then $p_L=1$ and hence $w\in A_\infty({\mathbb R}^n)$. In this paper, the authors introduce the weighted Hardy spaces $H^p_{L,\,w}(\mathbb{R}^n)$ associated with the operator $L$, via the Lusin area function associated with the heat semigroup generated by $L$. Characterizations of $H^p_{L,\,w}(\mathbb{R}^n)$, in terms of the atom and the molecule, are obtained. As applications, the boundedness of singular integrals such as spectral multipliers, square functions and Riesz transforms on weighted Hardy spaces $H^p_{L,\,w}(\mathbb{R}^n)$ are investigated. Even for the Schrödinger operator $-\Delta+V$ with $0\le V\in L_{\rm{loc}}^1 (\mathbb{R}^n)$, the obtained results in this paper essentially improve the known results by extending the narrow range of the weights into the whole $A_\infty(\mathbb{R}^n)$ weights.

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The Anh Bui. Jun Cao. Luong Dang Ky. Dachun Yang. Sibei Yang. "WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING REINFORCED OFF-DIAGONAL ESTIMATES." Taiwanese J. Math. 17 (4) 1127 - 1166, 2013. https://doi.org/10.11650/tjm.17.2013.2719

Information

Published: 2013
First available in Project Euclid: 10 July 2017

zbMATH: 1284.42066
MathSciNet: MR3085503
Digital Object Identifier: 10.11650/tjm.17.2013.2719

Subjects:
Primary: 42B30
Secondary: 42B15, 42B20, 42B25, 42B35, 47B06

Rights: Copyright © 2013 The Mathematical Society of the Republic of China

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Vol.17 • No. 4 • 2013
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