Local Hardy Spaces Associated with Ball Quasi-Banach Function 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--Weiss, let $X$ be a ball quasi-Banach function space on $\mathbb X$ under suitable maximal-function and associate-space assumptions, and let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb X)$. Assume that, for every $t>0$, the semigroup $e^{-tL}$ admits an integral kernel satisfying a Gaussian upper bound. In this paper, we introduce and systematically study the local Hardy space $h_L^X(\mathbb X)$ associated with $X$ and $L$, defined in terms of a local Lusin area function together with an appropriate low-frequency term. As applications of this theory, we establish the boundedness of the local Riesz transform $\nabla(L+\kappa I)^{-1/2}$ from $h_L^X(\mathbb R^d)$ into the corresponding $X$-valued vector function space for second-order divergence-form elliptic operators. We also obtain a H\"{o}rmander-type spectral multiplier theorem for $F(L+I)$ on $h_L^X(\mathbb X)$. Finally, the abstract results are applied to local Orlicz-Hardy spaces, local variable Hardy spaces, and local mixed-norm Hardy spaces. This theory develops Goldberg's original local Hardy space theory [Duke Math. J. {\bf 46} (1979), 27-42; MR0523600] to the setting of ball quasi-Banach function spaces and non-negative self-adjoint operators on spaces of homogeneous type. To the best of our knowledge, several of the results obtained in this paper are new even in the Euclidean setting $\mathbb X:=\mathbb R^d$.