Variable weak Hardy spaces WH L p(·)(ℝ n ) associated with operators satisfying Davies–Gaffney estimates

Abstract

Abstract Let p ⁢ ( ⋅ ) : ℝ n → [ 0 , 1 ] {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 ⁢ ( ℝ n ) {L^{2}({\mathbb{R}}^{n})} , with ω ∈ [ 0 , π / 2 ) {\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 WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\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 WT p ⁢ ( ⋅ ) ⁢ ( ℝ + n + 1 ) {\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 WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\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 ∇ ⁡ L - 1 / 2 {\nabla L^{-1/2}} is bounded from WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} to the variable weak Hardy space WH p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{p(\,\cdot\,)}(\mathbb{R}^{n})} . Moreover, when L is non-negative and self-adjoint with the kernels of { e - t ⁢ L } t > 0 {\{e^{-tL}\}_{t>0}} satisfying the Gaussian upper bound estimates, the atomic characterization of WH L p ⁢ ( ⋅ ) ⁢ ( ℝ n ) {\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})} is further used to characterize this space via non-tangential maximal functions.