A Local Version of Hardy-type Spaces Associated with Ball Quasi-Banach Spaces and Non-negative Self-adjoint Operators on Spaces of Homogeneous Type and Their Applications

Let $(\mathbb{X},d,\mu )$ be a space of homogeneous type in the sense of Coifman and Weiss, $X$ be a ball quasi-Banach function space on $\mathbb{X}$, $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{X})$, and assume that, for all $t>0$, the semigroup $e^{-tL}$ has an integral representation whose kernel satisfies a Gaussian upper bound condition. In this paper, we first study a local version of Hardy space $h_L^X(\mathbb{X})$ associated with ball quasi-Banach space $X$ and non-negative self-adjoint operator $L$, which is an extension of Goldberg’s result [Duke Math. J. {\bf46} (1979), no. 1, 27-42; MR0523600]. Even in the case of Euclidean space (that is, $\mathbb{X}={\mathbb{R}}^d$), all of these results are still new.