On the bilinear square Fourier multiplier operators and related multilinear square functions

TL;DR

This paper establishes boundedness results for bilinear square Fourier multiplier operators and related multilinear square functions in weighted Lebesgue spaces, including endpoint and commutator estimates, by analyzing kernel regularity and improving key lemmas.

Contribution

It provides new boundedness criteria for bilinear square Fourier multipliers and multilinear square functions, extending previous results with refined techniques and endpoint estimates.

Findings

Boundedness of rak{T}_m in weighted L^p spaces for certain p ranges.

Weak-type endpoint estimates involving L log L spaces.

Boundedness of commutators of rak{T}_m with weighted estimates.

Abstract

Let $n\ge 1$ and $\mathfrak{T}_{m}$ be the bilinear square Fourier multiplier operator associated with a symbol $m$, which is defined by $$ \mathfrak{T}_{m}(f_1,f_2)(x) = \biggl( \int_{0}^\infty\Big|\int_{(\mathbb{R}^n)^2} e^{2\pi ix\cdot (\xi_1 +\xi_2) }m(t\xi_1,t\xi_2) \hat{f}_{1}(\xi_1)\hat{f}_{2}(\xi_2)d\xi_1 d\xi_2\Big|^2\frac{dt}{t } \biggr)^{\frac 12}. $$ Let $s$ be an integer with $s\in[n+1,2n]$ and $p_0$ be a number satisfying $2n/s\le p_0\le 2$. Suppose that $\nu_{\vec{\omega}}=\prod_{i=1}^2\omega_i^{p/ p_i}$ and each $\omega_i$ is a nonnegative function on $\mathbb{R}^n$. In this paper, we show that $\mathfrak{T}_{m}$ is bounded from $L^{p_1}(\omega_1)\times L^{p_2}(\omega_2)$ to $L^p(\nu_{\vec{\omega}})$ if $p_0< p_1, p_2<\infty$ with $1/p=1/p_1+ 1/p_2$. Moreover, if $p_0>2n/s$ and $p_1=p_0$ or $p_2=p_0$, then $\mathfrak{T}_{m}$ is bounded from $L^{p_1}(\omega_1)\times L^{p_2}(\omega_2)$ to $L^{p,\infty}(\nu_{\vec{\omega}})$. The weighted end-point $L\log L$ type estimate and strong estimate for the commutators of $\mathfrak{T}_{m}$ are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.