Variable weak Hardy spaces and their applications

Let p(⋅):Rn→(0,∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the authors first introduce the variable weak Hardy space on RnRn, WHp(⋅)(Rn)WHp(⋅)(Rn), via the radial grand maximal function, and then establish its radial or non-tangential maximal function characterizations. Moreover, the authors also obtain various equivalent characterizations of WHp(⋅)(Rn)WHp(⋅)(Rn), respectively, by means of atoms, molecules, the Lusin area function, the Littlewood–Paley g  -function or gλ⁎-function. As an application, the authors establish the boundedness of convolutional δ-type and non-convolutional γ  -order Calderón–Zygmund operators from Hp(⋅)(Rn)Hp(⋅)(Rn) to WHp(⋅)(Rn)WHp(⋅)(Rn) including the critical case when p−=n/(n+δ)p−=n/(n+δ) or when p−=n/(n+γ)p−=n/(n+γ), where p−:=essinfx∈Rnp(x).