On multilinear square function and its applications to multilinear Littlewood–Paley operators with non-convolution type kernels

Abstract

Let m≥2, n≥1 and x∈Rn, define the multilinear square function T by T(f→)(x)=(∫0∞|∫(Rn)mKt(x,y1,…,ym)∏j=1mfj(yj)dy1⋯dym|2dtt)1/2, where the kernel K satisfies a class of integral smooth conditions which is much weaker than the standard Calderón–Zygmund kernel conditions. In this paper, we first established the Lp1(w1)×⋯×Lpm(wm)→Lp(νω→) estimate of T when each pi>1 and weak type Lp1(w1)×⋯×Lpm(wm)→Lp,∞(νω→) estimate of T when there is a pi=1, where νω→=∏i=1mωip/pi and each wi is a nonnegative function on Rn. As applications of the above results, we obtained the boundedness of multilinear Littlewood–Paley operators with non-convolution type kernels, including multilinear g-function, Marcinkiewicz integral and gλ⁎-function.