On the Bilinear Square Fourier Multiplier Operators Associated With Function

Abstract

This paper will be devoted to study a class of bilinear square-function Fourier multiplier operator associated with a symbol $m$ defined by $$\begin{eqnarray}\displaystyle & & \displaystyle \mathfrak{T}_{\unicode[STIX]{x1D706},m}(f_{1},f_{2})(x)\nonumber\\ \displaystyle & & \displaystyle \quad =\Big(\iint _{\mathbb{R}_{+}^{n+1}}\Big(\frac{t}{|x-z|+t}\Big)^{n\unicode[STIX]{x1D706}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\bigg|\int _{(\mathbb{R}^{n})^{2}}e^{2\unicode[STIX]{x1D70B}ix\cdot (\unicode[STIX]{x1D709}_{1}+\unicode[STIX]{x1D709}_{2})}m(t\unicode[STIX]{x1D709}_{1},t\unicode[STIX]{x1D709}_{2})\hat{f}_{1}(\unicode[STIX]{x1D709}_{1})\hat{f}_{2}(\unicode[STIX]{x1D709}_{2})\,d\unicode[STIX]{x1D709}_{1}\,d\unicode[STIX]{x1D709}_{2}\bigg|^{2}\frac{dz\,dt}{t^{n+1}}\Big)^{1/2}.\nonumber\end{eqnarray}$$ A basic fact about $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ is that it is closely associated with the multilinear Littlewood–Paley $g_{\unicode[STIX]{x1D706}}^{\ast }$ function. In this paper we first investigate the boundedness of $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ on products of weighted Lebesgue spaces. Then, the weighted endpoint $L\log L$ type estimate and strong estimate for the commutators of $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ will be demonstrated.