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

Abstract

The main purpose of this paper is to study $L^r$ H\"older 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 $$ T_{ab}(f,g,h)(x):=\int_{\mathbb{R}^6}a(x,\xi,\eta,\zeta)b(x,\eta,\zeta)\hat f(\xi)\hat g(\eta)\hat h(\zeta)e^{2\pi i x(\xi+\eta+\zeta)}d\xi d\eta d\zeta, $$ where the smooth symbols $a,b$ satisfy certain bi-parameter H\"ormander 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].