Let \(T_m\) be the \(N\)-linear Fourier multiplier operator: \[ T_m(f_1,\dots,f_N)(x)=\int_{\mathbb{R}^{Nn}}e^{ix(\xi_1+\cdots+\xi_N)}m(\xi_1,\dots,\xi_N)\widehat{ f_1}(\xi_1)\cdots\widehat{f_N}(\xi_N)\,d\xi_1\cdots d\xi_N, \] let \(\Psi\in\mathcal{S}(\mathbb R^{Nn})\) satisfy that \[ \text{supp}\Psi\subset\{\xi\in \mathbb R^{Nn}:1/2\leq|\xi|\leq2\}\quad \text{and }\quad \sum_{k\in\mathbb Z}\Psi(\xi/2^k)=1,\quad\text{for\;all}\;\xi\in\mathbb R^{Nn}\setminus\{0\}, \] and let \[ m_j(\xi)=m(2^j\xi_1,\dots,2^j\xi_N)\Psi(\xi_1,\dots,\xi_N). \]
The author proves that, if \(1\leq r<2\), \(r<p_1,\dots,p_N<\infty\), \(1/p_1+\cdots+1/p_N=1/p\) and \(m\in L^\infty(\mathbb R^{Nn})\) satisfies that \[ \sup_{j\in\mathbb Z}\|m_j\|_{B_{Nn/r}^{r,1}(\mathbb R^{Nn})}<\infty, \] then \(T_m\) is bounded from \(L^{p_1}(\mathbb R^n)\times\cdots\times L^{p_N}(\mathbb R^n)\) to \(L^p(\mathbb R^n)\), where \(B_{Nn/r}^{r,1}(\mathbb R^{Nn})\) is the \(L^r\)-based Besov space.
This result was previously proved in [L. Grafakos and Z. Si, J. Reine Angew. Math. 668, 133–147 (2012; Zbl 1254.42017)] (which also includes the case \(r=2\)), under the stronger assumption that \[ \sup_{j\in\mathbb Z}\|m_j\|_{H_{s}^{r}(\mathbb R^{Nn})}<\infty, \quad\text{if } \;s>Nn/r, \] since, in this range, \(H_{s}^{r}(\mathbb R^{Nn})\subset B_{Nn/r}^{r,1}(\mathbb R^{Nn}).\)