Let $L$ be a one-to-one operator of type $\omega$ in $L^2(\mathbb{ R}^n)$, with $\omega\in[0,\,\pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(\cdot):\ \mathbb{ R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally log-H\"{ o}lder continuous condition. In this article, the authors introduce the variable Hardy space $H^{ p(\cdot)}_L(\mathbb{ R}^n)$ associated with $L$. By means of variable tent spaces, the authors establish the molecular characterization of $H^{ p(\cdot)}_L(\mathbb{ R}^n)$. Then the authors show that the dual space of $H^{ p(\cdot)}_L(\mathbb{ R}^n)$ is the BMO-type space ${ \rm BMO}_{ p(\cdot),\,L^\ast}(\mathbb{ R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$. In particular, when $L$ is the second order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H^{ p(\cdot)}_L(\mathbb{ R}^n)$ and show that the fractional integral $L^{ -\alpha}$ for $\alpha\in(0,\,\frac12]$ is bounded from $H_L^{ p(\cdot)}(\mathbb{ R}^n)$ to $H_L^{ q(\cdot)}(\mathbb{ R}^n)$ with $\frac1{ p(\cdot)}-\frac1{ q(\cdot)}=\frac{ 2\alpha}{ n}$ and the Riesz transform $\nabla L^{ -1/2}$ is bounded from $H^{ p(\cdot)}_L(\mathbb{ R}^n)$ to the variable Hardy space $H^{ p(\cdot)}(\mathbb{ R}^n)$.
Categories: math.CA math.FA