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

Let $n\ge 1$ and $\mathfrak{T}_{m}$ be the bilinear square Fourier multiplier operator associated with a symbol $m$, which is defined by $\mathfrak{T}_{m}(f_1,f_2)(x) = \biggl( \int_{0}^\infty\Big|\int_{(\mathbb{R}^n)^2} e^{2πix\cdot (ξ_1 +ξ_2) }m(tξ_1,tξ_2) \hat{f}_{1}(ξ_1)\hat{f}_{2}(ξ_2)dξ_1 dξ_2\Big|^2\frac{dt}{t } \biggr)^{\frac 12}.$ 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 $ν_{\vecω}=\prod_{i=1}^2ω_i^{p/ p_i}$ and each $ω_i$ is a nonnegative function on $\mathbb{R}^n$. In this paper, we show that $\mathfrak{T}_{m}$ is bounded from $L^{p_1}(ω_1)\times L^{p_2}(ω_2)$ to $L^p(ν_{\vecω})$ 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 $L^{p_1}(ω_1)\times L^{p_2}(ω_2)$ to $L^{p,\infty}(ν_{\vecω})$. 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.

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