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

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

Digital Object 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.

Export citation


  • A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, II, Adv. in Math., 24 (1977), 101–171.
  • R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315–331.
  • R. R. Coifman and Y. Meyer, Commutateurs d' intégrales singulières et opérateurs multilinéaires, Ann. Inst. Fourier, Grenoble, 28 (1978), 177–202.
  • C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137–193.
  • M. Frazier, B. Jawerth and G. L. Weiss, Littlewood–Paley Theory and the Study of Function Spaces, CBMS Regional Conference Series, 79 (1991).
  • M. Fujita and N. Tomita, Weighted norm inequalities for multilinear Fourier multipliers, Trans. Amer. Math. Soc., 364 (2012), 6335–6353.
  • L. Grafakos and D. He, Multilinear Calderón–Zygmund operators on Hardy spaces, II, J. Math. Anal. Appl., 416 (2014), 511–521.
  • L. Grafakos, Modern Fourier Analysis, 3rd edition, GTM 250, Springer, NY 2014.
  • L. Grafakos and N. Kalton, Multilinear Calderón–Zygmund operators on Hardy spaces, Collect. Math., 52 (2001), 169–179.
  • L. Grafakos, A. Miyachi and N. Tomita, On multilinear Fourier multipliers of limited smoothness, Can. Jour. Math., 65 (2013), 299–330.
  • L. Grafakos and S. Oh, The Kato–Ponce Inequality, Comm. PDE, 39 (2014), 1128–1157.
  • L. Grafakos and Z. Si, The Hörmander multiplier theorem for multilinear operators, J. Reine Angew. Math., 668 (2012), 133–147.
  • L. Grafakos and H. V. Nguyen, Multilinear Fourier Multipliers with Minimal Sobolev Regularity, I, Colloquium Math., 144 (2016), 1–30.
  • L. Grafakos and R. Torres, Multilinear Calderón–Zygmund Theory, Adv. in Math., 165 (1999), 124–164.
  • L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93–140.
  • Y. Meyer and R. R. Coifman, Wavelets: Calderón–Zygmund and multilinear operators, Cambridge Stud. Adv. Math., 48, Cambridge Univ. Press, Cambridge, United Kingdom, 1997.
  • A. Miyachi and N. Tomita, Minimal smoothness conditions for bilinear Fourier multipliers, Rev. Mat. Iberoamer., 29 (2013), 495–530.
  • A. Miyachi and N. Tomita, Boundedness criterion for bilinear Fourier multiplier operators, Tohoku Math. J. (2), 66 (2014), 55–76.
  • S. G. Mikhlin, On the multipliers of Fourier integrals, (Russian), Dokl. Akad. Nauk SSSR (N.S.), 109 (1956), 701–703.
  • C. Pérez and R. H. Torres, Minimal regularity conditions for the end-point estimate of bilinear Calderón–Zygmund operators, Proc. Amer. Math. Soc. Ser. B, 1 (2014), 1–13.
  • E. M. Stein, Harmonic Analysis, Real-Variable Mehods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton Univ. Press, 1993.
  • N. Tomita, A Hörmander type multiplier theorem for multilinear operators, J. Funct. Anal., 259 (2010), 2028–2044.
  • H. Triebel, Theory of Function Spaces, Birkhäuser Verlag, Basel, 1983.