  Let $L$ be a linear operator on $L^2(\mathbb R^n)$ generating an analytic
semigroup $\{e^{-tL}\}_{t\ge0}$ 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
${\mathrm{BMO}}_{p(\cdot),L}(\mathbb R^n)$. 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
$H^{p(\cdot)}(\mathbb R^n)$.
