## Multilinear Fourier multipliers with minimal Sobolev regularity. II.(English)Zbl 1372.42007

Let $$m\in\mathbb{N}$$. For the Schwartz function $$f$$ on $$\mathbb{R}^n$$, the Fourier transform $$\widehat{f}$$ of $$f$$, is defined by $\widehat{f}(\xi):=\int_{\mathbb{R}^n}f(x)e^{-2\pi ix\cdot\xi}\,dx.$ Let $$\sigma$$ be a bounded function on $$\mathbb{R}^{nm}:=\mathbb{R}^n\times\cdots\times\mathbb{R}^n$$. Then, the $$m$$-linear Fourier multiplier operator $$T_\sigma$$ is defined by, for any $$x\in\mathbb{R}^n$$, $T_\sigma(f_1,\ldots,f_n)(x):=\int_{\mathbb{R}^{nm}}e^{2\pi ix\cdot(\xi_1+\cdots+\xi_m)} \sigma(\xi_1,\ldots,\xi_m)\widehat{f}_1(\xi_1)\cdots\widehat{f}_m(\xi_m)\,d\vec{\xi},$ where $$d\vec{\xi}:=d\xi_1\cdots d\xi_m$$. For $$p\in(0,\infty]$$, denote by $$H^p(\mathbb{R}^n)$$ the Hardy space on $$\mathbb{R}^n$$. When $$p\in(1,\infty]$$, the Hardy space $$H^p(\mathbb{R}^n)$$ is just the Lebesgue space $$L^p(\mathbb{R}^n)$$.
Let $$p_1,\,p_2,\,\ldots,\,p_m\in(0,\infty]$$ and $$p\in(0,\infty)$$. Assume further that $\frac1{p_1}+\frac1{p_2}+\cdots+\frac1{p_m}=\frac1p.$ In this paper, the authors obtained the optimal condition for the multiplier $$\sigma$$, in terms of local $$L^2$$-Sobolev space estimates, such that the $$m$$-linear Fourier multiplier operator $$T_\sigma$$ is bounded from $$H^{p_1}(\mathbb{R}^n)\times\cdots\times H^{p_m}(\mathbb{R}^n)$$ to $$L^p(\mathbb{R}^n)$$.
For part I, see [L. Grafakos and H. van Nguyen, Colloq. Math. 144, No. 1, 1–30 (2016; Zbl 1339.42012)].

### MSC:

 42B15 Multipliers for harmonic analysis in several variables 42B30 $$H^p$$-spaces

### Keywords:

multiplier theory; multilinear operators; Hardy spaces

Zbl 1339.42012
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