## Taiwanese Journal of Mathematics

### WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING REINFORCED OFF-DIAGONAL ESTIMATES

#### 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.

#### Article information

Source
Taiwanese J. Math., Volume 17, Number 4 (2013), 1127-1166.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706109

Digital Object Identifier
doi:10.11650/tjm.17.2013.2719

Mathematical Reviews number (MathSciNet)
MR3085503

Zentralblatt MATH identifier
1284.42066

#### Citation

Bui, The Anh; Cao, Jun; Ky, Luong Dang; Yang, Dachun; Yang, Sibei. WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING REINFORCED OFF-DIAGONAL ESTIMATES. Taiwanese J. Math. 17 (2013), no. 4, 1127--1166. doi:10.11650/tjm.17.2013.2719. https://projecteuclid.org/euclid.twjm/1499706109

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