Weighted local Hardy spaces associated with operators

Abstract

Let L be a self-adjoint positive operator on $L^2(\mathbb {R}^n)$. Assume that the semigroup $e^{-tL}$ generated by $-L$ satisfies the Gaussian kernel bounds on $L^2(\mathbb {R}^n)$. In this article, we study weighted local Hardy space $h_{L,w}^{1}(\mathbb {R}^n)$ associated with L in terms of the area function characterization, and prove their atomic characters. Then, we introduce the weighted local BMO space $\mathrm{bmo}_{L,w}(\mathbb {R}^n)$ and prove that the dual of $h_{L,w}^{1}(\mathbb {R}^n)$ is $\mathrm{bmo}_{L,w}(\mathbb {R}^n)$. Finally a broad class of applications of these results is described.