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Letp  ⁢   (  ⋅  )    :    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 inL  2   ⁢   (   ℝ  n   ){L^{2}({\mathbb{R}}^{n})}   , withω  ∈   [  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 spaceWH  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 spaceWT   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 ofWH  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 fromWH  L   p  ⁢   (  ⋅  )⁢   (   ℝ  n   ){\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})}    to the variable weak Hardy spaceWH   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  &gt;  0{\{e^{-tL}\}_{t&gt;0}}    satisfying the Gaussian upper bound estimates, the atomic characterization ofWH  L   p  ⁢   (  ⋅  )⁢   (   ℝ  n   ){\mathrm{WH}^{{p(\,\cdot\,)}}_{L}(\mathbb{R}^{n})}    is further used to characterize this space via non-tangential maximal functions.

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