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 $$\begin{aligned} T_{m_1,m_2}(f,g,h)(x):=\int _{{\mathbb {R}}^{6}}m_1(\xi ,\eta ,\zeta )m_2(\eta ,\zeta ){\hat{f}}(\xi ) {\hat{g}}(\eta ){\hat{h}}(\zeta )e^{2\pi i(\xi +\eta +\zeta )\cdot x}d\xi d\eta d\zeta , \end{aligned}$$when \(m_1,m_2\) are two bi-parameter symbols. We study Hölder type estimates: \(L^{p_1}\times L^{p_2}\times L^{p_3} \rightarrow L^r\) for \(1<p_1,p_2,p_3< \infty \) with \(1/p_1+1/p_2+1/p_3=1/r\), and \(0<r<\infty \). 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 $$\begin{aligned} T_{ab}(f,g,h)(x):=\int _{{\mathbb {R}}^6}a(x,\xi ,\eta ,\zeta )b(x,\eta ,\zeta ){\hat{f}}(\xi ){\hat{g}}(\eta ){\hat{h}}(\zeta )e^{2\pi i x(\xi +\eta +\zeta )}d\xi d\eta d\zeta , \end{aligned}$$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). Using our reduction of the flag multiplier, we also provide an alternative proof of some of the mixed norm estimates recently established by Muscalu and Zhai (Five-linear singular integrals of Brascamp-Lieb type, arXiv:2001.09064m) when the functions g and h are of tensor product forms (Theorem 1.8). Moreover, our method also allows us to establish the weighted mixed norm estimates (Theorem 1.9). 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 (Muscalu et al. in Acta Math 193:269–296, 2004; Rev Mat Iberoam 22(3):963–976, 2006). They may also be viewed as the bi-parameter trilinear variants of estimates obtained for the one-parameter flag paraproducts by Muscalu (Rev Mat Iberoam 23(2):705–742, 2007).