Multilinear Square Functions with Kernels of Dini’s Type

Abstract

Let T be a multilinear square function with a kernel satisfying Dini(1) condition and let T⁎ be the corresponding multilinear maximal square function. In this paper, first, we showed that T is bounded from L1×⋯×L1 to L1/m,∞. Secondly, we obtained that if each pi>1, then T and T⁎ are bounded from Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→) and if there is pi=1, then T and T⁎ are bounded from Lp1(ω1)×⋯×Lpm(ωm) to Lp,∞(νω→), where νω→=∏i=1mωip/pi. Furthermore, we established the weighted strong and weak type boundedness for T and T⁎ on weighted Morrey type spaces, respectively.