Journal of the Mathematical Society of Japan

Multilinear Fourier multipliers with minimal Sobolev regularity, II

Loukas GRAFAKOS, Akihiko MIYACHI, Hanh VAN NGUYEN, and Naohito TOMITA

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We provide characterizations for boundedness of multilinear Fourier multiplier operators on Hardy or Lebesgue spaces with symbols locally in Sobolev spaces. Let $H^q(\mathbb R^n)$ denote the Hardy space when $0 \lt q \le 1$ and the Lebesgue space $L^q(\mathbb R^n)$ when $1 \lt q \le \infty$. We find optimal conditions on $m$-linear Fourier multiplier operators to be bounded from $H^{p_1}\times \cdots \times H^{p_m}$ to $L^p$ when $1/p=1/p_1+\cdots +1/p_m$ in terms of local $L^2$-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderón and Torchinsky [1] and of the bilinear results of Miyachi and Tomita [17]. The extension to general $m$ is significantly more complicated both technically and combinatorially; the optimal Sobolev space smoothness required of the symbol depends on the Hardy–Lebesgue exponents and is constant on various convex simplices formed by configurations of $m2^{m-1}+1$ points in $[0,\infty)^m$.

Article information

J. Math. Soc. Japan, Volume 69, Number 2 (2017), 529-562.

First available in Project Euclid: 20 April 2017

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Zentralblatt MATH identifier

Primary: 42B15: Multipliers 42B30: $H^p$-spaces

multiplier theory multilinear operators Hardy spaces


GRAFAKOS, Loukas; MIYACHI, Akihiko; VAN NGUYEN, Hanh; TOMITA, Naohito. Multilinear Fourier multipliers with minimal Sobolev regularity, II. J. Math. Soc. Japan 69 (2017), no. 2, 529--562. doi:10.2969/jmsj/06920529.

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