Maximal functions of multilinear multipliers

Let $m_j$ be Fourier multipliers on $\bbR^{2d}$ that satisfy $$ |\partial^\alpha m_j(\xi_1,\xi_2)|\leq A_\alpha(|\xi_1|+|\xi_2|)^{-|\alpha|} $$ for sufficiently large $\alpha$ uniformly in $j$, for $j=1,2,\dots , N$. We study the maximal operator of two variables $$ \fM(f,g)(x)=\sup_{1\leq j\leq N} |T_{m_j}(f,g)(x)|, $$ where $T_{m_j}$ are the associated bilinear operators $$ T_{m_j}(f,g)(x)=\int_{\bbR^{2d}}m(\xi_1,\xi_2)\widehat f(\xi_1)\widehat g(\xi_2)e^{2\pi i (\xi_1+\xi_2)\cdot x}{\rm d}\xi_1{\rm d}\xi_2. $$ We prove that $\fM$ maps $L^{p_1}(\bbR^{d})\times L^{p_2}(\bbR^{d})$ to $L^{p}(\bbR^{d})$ with norm at most a constant multiple $\sqrt{\log (N+2)}$. We also provide an example to indicate the sharpness of this result.

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Published 1 January 2009