On General multilinear square function with non-smooth kernels

TL;DR

This paper establishes boundedness results for a broad class of multilinear square functions with non-smooth kernels, extending previous results and introducing a multilinear maximal square function with associated weighted estimates.

Contribution

It introduces new boundedness results for multilinear square functions with non-smooth kernels and defines a multilinear maximal square function with weighted estimates.

Findings

Boundedness of multilinear square functions with non-smooth kernels.

Introduction of a multilinear maximal square function $T^*$.

Weighted strong and weak type estimates for $T^*$.

Abstract

In this paper, we obtain some boundedness of the following general multilinear square functions $T$ with non-smooth kernels, which extend some known results significantly. $$ T(\vec{f})(x)=\big( \int_{0}^\infty \big|\int_{(\mathbb{R}^n)^m}K_v(x,y_1,\dots,y_m) \prod_{j=1}^mf_{j}(y_j)dy_1,\dots,dy_m\big|^2\frac{dv}{v}\big)^{\frac 12}. $$ The corresponding multilinear maximal square function $T^*$ was also introduced and weighted strong and weak type estimates for $T^*$ were given.