The Hörmander multiplier theorem for multilinear operators (Q2904013)

The authors provide a version of the Mikhin-Hörmander multiplier theorem for multilinear operators in the case where the target space is \(L^p\) for \(0<p\leq1\). Let \(\psi\in \mathcal S((\mathbb R^n)^m)\) satisfy NEWLINE\[NEWLINE\operatorname{supp}\hat\psi\subset \{\vec \xi\in (\mathbb R^n)^m ; c_1\leq |\vec \xi|\leq c_2\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sum_{j\in\mathbb Z}\hat\psi(\vec\xi/2^j)=1 \quad (\vec\xi\neq0).NEWLINE\]NEWLINE For \(\sigma\in L^\infty((\mathbb R^n)^m)\), set \(\sigma_k(\vec\xi)=\sigma(2^k\vec\xi)\), and define the \(m\)-linear Fourier multiplier \(T_\sigma\) by NEWLINE\[NEWLINE T_\sigma(f_1,f_2,\dotsc,f_m)(x) =\int_{\left(\mathbb R^n\right)^m}e^{2\pi ix\cdot (\xi_1+\dotsb+\xi_m)} \sigma(\xi_1,\dotsc,\xi_m)\hat f_1(\xi_1)\dotsm\hat f_m(\xi_m) d\xi_1\dotsm d\xi_mNEWLINE\]NEWLINE for \(f_1,f_2,\dotsc,f_m\in\mathcal S(\mathbb R^n)\). The authors' main theorem is the following.NEWLINENEWLINELet \(1<r\leq2\). Suppose a bounded function \(\sigma\) satisfies \(\sup_{k\in\mathbb Z}\|\sigma_k\hat\psi\|_{L_\gamma^r\left(\left(\mathbb R^n\right)^m\right)}<\infty\) for some \(\gamma>nm/r\). Then there is a positive constant \(\delta=\delta(mn,\gamma,r)\) satisfying \(0<\delta\leq r-1\) such that \(T_\sigma\) is bounded from \(L^{p_1}(\mathbb R^n)\times\dotsb\times L^{p_m}(\mathbb R^n)\) to \(L^p(\mathbb R^n)\), whenever \(r-\delta<p_1,\dotsc,p_m<\infty\) and \(p\) is given by \(1/p=1/p_m+\dotsb+1/p_m\), where \(L_\gamma^r((\mathbb R^n)^m)\) is the Sobolev space.NEWLINENEWLINE\textit{N. Tomita} [J. Funct. Anal. 259, No. 8, 2028--2044 (2010; Zbl 1201.42005)] showed earlier that, in the case \(r=2\) in the above, \(T_\sigma\) is bounded from \(L^{p_1}(\mathbb R^n)\times\dotsb\times L^{p_m}(\mathbb R^n)\) to \(L^p(\mathbb R^n)\), whenever \(1<p_1,\dotsc,p_m\), \(p<\infty\) and \(p\) is given by \(1/p=1/p_m+\dotsb+1/p_m\). The authors extend Tomita's result to the case \(0<p\leq1\).