Weighted Estimates for Bilinear Fourier Multiplier Operators with Multiple Weights

Abstract

In the weighted theory of multilinear operators, the weights class which usually has been considered is the product of $A_p$ weights. However, it is known that $\prod _{k=1}^2A_{p_k}({\mathbb {R}}^n)\varsubsetneq A_{\vec {p}}({\mathbb {R}}^{2n})$, and $\vec {w}=(w_1,\,w_2)\in A_{\vec {p}}({\mathbb {R}}^{2n})$ does not imply that $w_k\in L^1_{\mathrm{loc}}({\mathbb {R}}^n)$ for $k=1,\,2$. Therefore, it is very interesting to study the weighted theory of multilinear operators with the weights in $A_{\vec {p}}({\mathbb {R}}^{2n})$. In this paper, we consider the weights class $A_{\vec {p}/\vec {r}}({\mathbb {R}}^{2n})$, which is more general than $A_{\vec {p}}({\mathbb {R}}^{2n})$. If $\vec {w}=(w_1,\,w_2)\in A_{\vec {p}/\vec {r}}({\mathbb {R}}^{2n})$, we show that the bilinear Fourier multiplier operator $T_{\sigma }$ is bounded from $L^{p_1}(w_1)\times L^{p_2}(w_2)$ to $L^p(\nu _{\vec {w}})$ when the symbol $\sigma$ satisfies the Sobolev regularity that $\sup _{\kappa \in {\mathbb {Z}}}\Vert \sigma _k\Vert _{W^{s_1,s_2}({\mathbb {R}}^{2n})} <\infty$ with $s_1,s_2\in (\frac{n}{2},\,n].$