Characterizations of generalized Hardy and BMO spaces via square functions

Abstract

Let $d\in \{3,4,5,\dots \}$. Consider $L=-\frac{1}{w}\mathrm{div}(A\nabla u)+\mathrm{\mu }$ over its maximal domain in $L_w^2({ℝ}^d)$. Under certain conditions on the weight w, the coefficient matrix A, and the positive Radon measure μ, we characterize the Hardy space $H_L^1({ℝ}^d)$ and BMO space ${\mathrm{BMO}}_L({ℝ}^d)$ associated with L using square functions. Other results include $H_L^1({ℝ}^d)\equiv H_{L,max}^1({ℝ}^d)$ and ${(H_L^1({ℝ}^d))}^{\ast }\equiv {\mathrm{BMO}}_L({ℝ}^d)$ as norm spaces, where $H_{L,max}^1({ℝ}^d)$ denotes the maximal Hardy space associated with L.