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

Let n > 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m, which is defined by Tm(f1,f2)(x)=(∫0∞|∫(ℝn)2e2πix⋅(ξ1+ξ2)m(tξ1,tξ2)f^1(ξ1)f^2(ξ2)dξ1dξ2|2dtt)12.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T_m}\left( {{f_1},{f_2}} \right)\left( x \right) = {\left( {\int_0^\infty {{{\left| {\int_{{{\left( {{\mathbb{R}^n}} \right)}^2}} {{e^{2\pi ix \cdot \left( {{\xi _1} + {\xi _2}} \right)}}m\left( {t{\xi _1},t{\xi _2}} \right){{\hat f}_1}\left( {{\xi _1}} \right){{\hat f}_2}\left( {{\xi _2}} \right)d{\xi _1}d{\xi _2}} } \right|}^2}\frac{{dt}}{t}} } \right)^{\frac{1}{2}}}.$$\end{document} Let s be an integer with s ∈ [n + 1, 2n] and p0 be a number satisfying 2n/s ≤ p0 ≤ 2. Suppose that νω→=Πi=12ωip/pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\nu _{\vec \omega }} = \Pi _{i = 1}^2\omega _i^{p/{p_i}}$$\end{document} and each ωi is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from Lp1(ω1) × Lp2(ω2) to Lp(νω→)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^p}\left( {{\nu _{\vec \omega }}} \right)$$\end{document} if p0 < p1, p2 < ∞ with 1/p = 1/p1 + 1/p2. Moreover, if p0 > 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from Lp1(ω1) × Lp2(ω2) to Lp,∞(νω→)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{p,\infty }}\left( {{\nu _{\vec \omega }}} \right)$$\end{document}. The weighted end-point L log L type estimate and strong estimate for the commutators of Tm 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.