Variable Hardy–Lorentz spaces associated to operators satisfying Davies–Gaffney estimates

Abstract

Let $L$ be a one-to-one operator of type $w$ with $w\in [0,\pi /2)$, which satisfies the Davies–Gaffney estimates and has a bounded holomorphic calculus, and let $p(\cdot )$ be a measurable function on ${\mathbb{R}}^n$ with $0<p_{-}:=ess{inf\hspace{0.17em}}_{x\in {\mathbb{R}}^n}p(x)\le ess{sup\hspace{0.17em}}_{x\in {\mathbb{R}}^n}p(x)=:p_{+}<\infty$. Under the assumption that $p(\cdot )$ satisfies the global log-Hölder condition, we introduce the variable Hardy–Lorentz space $H_L^{p(\cdot ),q}({\mathbb{R}}^n)$ for $0<q<\infty$ and construct its molecular decomposition. Furthermore, we investigate the dual spaces of the variable Hardy–Lorentz space $H_L^{p(\cdot ),q}({\mathbb{R}}^n)$ with $0<p_{-}\le p_{+}\le 1$ and $0<q<\infty$. These results are new even when $p(\cdot )$ is a constant.