Abstract
Let
{p(\,\cdot\,)\colon\mathbb{R}^{n}\to[0,1]}
be a variable exponent function satisfying the globally log-Hölder continuous condition, and L a one-to-one operator of type ω 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.
In this article, we introduce the variable weak Hardy space
{\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})}
, associated with L via the corresponding square function.
Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space
{\mathrm{WT}^{p(\,\cdot\,)}(\mathbb{R}_{+}^{n+1})}
, which is also obtained in this article.
In particular, when L is non-negative and self-adjoint, we obtain the
atomic characterization of
{\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})}
.
As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, we prove that the associated Riesz transform
{\nabla L^{-1/2}}
is bounded from
{\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})}
to the variable weak Hardy space
{\mathrm{WH}^{p(\,\cdot\,)}(\mathbb{R}^{n})}
.
Moreover, when L is non-negative and self-adjoint with the kernels of
{\{e^{-tL}\}_{t>0}}
satisfying the Gaussian upper bound estimates, the atomic characterization of
{\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})}
is further used to characterize this space via non-tangential maximal functions.
The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
