Bi-parameter trilinear Fourier multipliers and pseudo-differential operators with flag symbols

Abstract

The main purpose of this paper is to study $L^r$ Hölder type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for $f,g,h\in \mathcal{S}({\mathbb{R}}^2)$, the bi-parameter trilinear flag Fourier multiplier operators we consider are defined by $$T_{m_1,m_2}(f,g,h)(x):={\int }_{{\mathbb{R}}^6}{m}_1(\xi ,\eta ,\zeta )m_2(\eta ,\zeta )\hat{f}(\xi )\hat{g}(\eta )\hat{h}(\zeta )e^{2\pi i(\xi +\eta +\zeta )\cdot x}d\xi d\eta d\zeta ,$$ when $m_1,m_2$ are two bi-parameter symbols. We will show that our problem can be reduced to establish the $L^r$ estimate for the special multiplier $m_1({\xi }_1,{\eta }_1,{\zeta }_1)m_2({\eta }_2,{\zeta }_2)$ (see Theorem 1.7). We also study these $L^r$ estimates for the corresponding bi-parameter trilinear pseudo-differential operators defined by $$T_{ab}(f,g,h)(x):={\int }_{{\mathbb{R}}^6}a(x,\xi ,\eta ,\zeta )b(x,\eta ,\zeta )\hat{f}(\xi )\hat{g}(\eta )\hat{h}(\zeta )e^{2\pi ix(\xi +\eta +\zeta )}d\xi d\eta d\zeta ,$$ where the smooth symbols $a,b$ satisfy certain bi-parameter Hörmander conditions. We will also show that the $L^r$ estimate holds for $T_{ab}$ as long as the $L^r$ estimate for the flag multiplier operator holds when the multiplier has the special form $m_1({\xi }_1,{\eta }_1,{\zeta }_1)m_2({\eta }_2,{\zeta }_2)$ (see Theorem 1.10). The bi-parameter and trilinear flag Fourier multipliers considered in this paper do not satisfy the conditions of the classical bi-parameter trilinear Fourier multipliers considered by Muscalu, Tao, Thiele and the second author [21, 22]. They may also be viewed as the bi-parameter trilinear variants of estimates obtained for the one-parameter flag paraproducts by Muscalu [18].

This new version is of 75 pages now. We corrected a mistake in the last version and expanded in other aspects