Transference on certain multilinear multiplier operators (Q2726648)

Let \(\mathbb{R}^{nm}\) be the \(m\)-fold product space \(\mathbb{R}^n\times\cdots\times \mathbb{R}^n\). Define on \(\mathbb{R}^{nm}\) the multilinear operators \(T\), generated by any function \(\lambda(u_1,\dots, u_m)\in \mathbb{R}^{nm}\), \(j= 1,2,\dots,m\), by NEWLINE\[NEWLINET(f_1,\dots, f_m)(x)= \int_{\mathbb{R}^{nm}} \prod^m_{j=1} \widehat f_j(u_j) \lambda(u_1,\dots, u_m)\exp\Biggl(2\pi i \sum^m_{j=1} \langle u_j, x\rangle\Biggr) du_1\cdots du_m,NEWLINE\]NEWLINE where \(\langle u_j, x\rangle\) is the inner product of \(u_j\) and \(x\).NEWLINENEWLINENEWLINEAnalogously, define multilinear operators on the torus. The \(n\)-torus \(\mathbb{T}^n\) can be identified with \(\mathbb{R}^n\setminus \Lambda\), where \(\Lambda\) is the unit lattice which is an additive group of points in \(\mathbb{R}^n\) having integer coordinates. Let \(\mathbb{T}^{nm}\) be the \(m\)-fold product space \(\mathbb{T}^n\times\cdots\times \mathbb{T}^n\). The multilinear operators \(\widehat T_\varepsilon\), \(\varepsilon> 0\), on \(\mathbb{T}^{nm}\) associated with the function \(\lambda\) are defined by NEWLINE\[NEWLINE\widetilde T_\varepsilon(\widetilde f_1,\dots,\widetilde f_m)(x)= \sum_{k_1\in \Lambda}\cdots\sum_{k_m\in \Lambda} \lambda(\varepsilon k_1,\dots, \varepsilon k_m) a_{k_1}\cdots a_{k_m}\exp \Biggl(2\pi i\sum^m_{j=1} \langle k_j, x\rangle\Biggr)NEWLINE\]NEWLINE for all \(C^\infty(\mathbb{T}^n)\) functions NEWLINE\[NEWLINE\widetilde f_j(x)= \sum_{k_j\in\Lambda} a_{k_j} \exp(2\pi i\langle k_j,x\rangle),\quad j= 1,2,\dots, m.NEWLINE\]NEWLINE Denote by \(\widetilde T= \widetilde T_\varepsilon\) if \(\varepsilon= 1\).NEWLINENEWLINENEWLINEThe authors prove de Leeuw type theorems [\textit{K. de Leeuw}, Ann. Math. (2) 81, 364-379 (1965; Zbl 0171.11803)] for \(\widetilde T_\varepsilon\), \(\widetilde T\) and \(T\) on the Lebesgue spaces, and on the Hardy spaces by using the atomic characterization. Furthermore, on the torus they provide an analog of Lacey-Thiele's theorem [\textit{M. Lacey} and \textit{C. Thiele}, Ann. Math. (2) 146, No. 3, 693-724 (1997; Zbl 0914.46034)] on the bilinear Hilbert transform, and analogies of recent works on multilinear singular integrals by \textit{C. E. Kenig} and \textit{E. M. Stein} [Math. Res. Lett. 6, No. 1, 1-15 (1999; Zbl 0952.42005)] and by Grafakos-Torres.