Molecular Characterizations and Dualities of Variable Exponent Hardy Spaces Associated with Operators

Let $L$ be a linear operator on $L^2({\mathbb{R}}^n)$ generating an analytic semigroup $\{e^{-tL}{\}}_{t\ge 0}$ with kernels having pointwise upper bounds and $p(\cdot ):{\mathbb{R}}^n\to (0,1]$ be a variable exponent function satisfying the globally log-H"older continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(\cdot )}({\mathbb{R}}^n)$, and the BMO-type space ${\mathrm{B}\mathrm{M}\mathrm{O}}_{p(\cdot ),L}({\mathbb{R}}^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of $H_L^{p(\cdot )}({\mathbb{R}}^n)$ and a duality theorem between such a Hardy space and a BMO-type space. As applications, the authors study the boundedness of the fractional integral on these Hardy spaces and the coincidence between $H_L^{p(\cdot )}({\mathbb{R}}^n)$ and the variable exponent Hardy spaces $H^{p(\cdot )}({\mathbb{R}}^n)$.