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.