2018/05/20 by Zhuo, Ciqiang, Yang, Dachun
#42B25 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 42B30 #Secondary 42B35
paper · doi:10.48550/arxiv.1805.07778
Let p(⋅): \mathbb Rn→(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 L2(\mathbb Rn), with ω∈[0, π/2), which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space W HLp(⋅)(\mathbb Rn) 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 W Tp(⋅)(\mathbb Rn) which is also obtained in this article. In particular, when L is non-negative and self-adjoint, the authors obtain the atomic characterization of W HLp(⋅)(\mathbb Rn). As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform ∇ L-1/2 is bounded from W HLp(⋅)(\mathbb Rn) to the variable weak Hardy space W Hp(⋅)(\mathbb Rn). Moreover, when L is non-negative and self-adjoint with the kernels of \e-tL\t>0 satisfying the Gauss upper bound estimates, the atomic characterization of W HLp(⋅)(\mathbb Rn) is further used to characterize the space via non-tangential maximal functions.