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.