Abstract
We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear Hörmander multiplier theorem concerning symbols that lie in the Sobolev space \(L^r_s({\mathbb {R}}^{2n})\), \(2\le r<\infty \), \(rs>2n\), uniformly over all annuli. More precisely, given such a symbol with smoothness index s, we find the largest open set of indices \((1/p_1,1/p_2 )\) for which we have boundedness for the associated bilinear multiplier operator from \(L^{p_1}({\mathbb {R}}^{ n})\times L^{p_2} ({\mathbb {R}}^{ n})\) to \( L^p({\mathbb {R}}^{ n})\) when \(1/p=1/p_1+1/p_2\), \(1<p_1,p_2<\infty \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Calderón, A.P., Torchinsky, A.: Parabolic maximal functions associated with a distribution, II. Adv. Math. 24, 101–171 (1977)
Fujita, M., Tomita, N.: Weighted norm inequalities for multilinear Fourier multipliers. Trans. Am. Math. Soc. 364, 6335–6353 (2012)
Grafakos, L.: Classical Fourier Analysis, Graduate Texts in Mathematics, GTM 249, 3rd edn. Springer, New York (2014)
Grafakos, L.: Modern Fourier Analysis, Graduate Texts in Mathematics, GTM 250, 3rd edn. Springer, New York (2014)
Grafakos, L., He, D., Honzík, P., Nguyen, H.V.: The Hörmander multiplier theorem I: the linear case. Ill. J. Math. 61, 25–35 (2017)
Grafakos, L., He, D., Honzík, P.: The Hörmander multiplier theorem II: the local \(L^2\) case. Math. Zeit. 289, 875–887 (2018)
Grafakos, L., Miyachi, A., Nguyen, H.V., Tomita, N.: Multilinear Fourier multipliers with minimal Sobolev regularity, II. J. Math. Soc. Japan 69, 529–562 (2017)
Grafakos, L., Miyachi, A., Tomita, N.: On multilinear Fourier multipliers of limited smoothness. Can. J. Math. 65, 299–330 (2013)
Grafakos, L., Nguyen, H.V.: Multilinear Fourier multipliers with minimal Sobolev regularity, I. Colloq. Math. 144, 1–30 (2016)
Grafakos, L., Oh, S.: The Kato–Ponce inequality. Comm. PDE 39, 1128–1157 (2014)
Grafakos, L., Si, Z.: The Hörmander multiplier theorem for multilinear operators. J. Reine Angew. Math. 668, 133–147 (2012)
Hirschman Jr., I.I.: On multiplier transformations. Duke Math. J. 26, 221–242 (1959)
Hörmander, L.: Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104, 93–139 (1960)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)
Mikhlin, S.G.: On the multipliers of Fourier integrals. Dokl. Akad. Nauk SSSR (N.S.) 109, 701–703 (1956)
Miyachi, A.: On some Fourier multipliers for \(H^p({\mathbb{R}}^{n})\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 157–179 (1980)
Miyachi, A., Tomita, N.: Minimal smoothness conditions for bilinear Fourier multipliers. Rev. Mat. Iberoam. 29, 495–530 (2013)
Miyachi, A., Tomita, N.: Boundedness criterion for bilinear Fourier multiplier operators. Tohoku Math. J. 66, 55–76 (2014)
Slavíková, L.: On the failure of the Hörmander multiplier theorem in a limiting case. Rev. Mat. Iberoamer (to appear)
Stein, E.M., Weiss, G.: On the interpolation of analytic families of operators acting on \(H^{p}\)-spaces. Tohoku Math. J. 9, 318–339 (1957)
Tomita, N.: A Hörmander type multiplier theorem for multilinear operators. J. Funct. Anal. 259, 2028–2044 (2010)
Wainger, S.: Special trigonometric series in k-dimensions. Mem. Am. Math. Soc. 59, 1–102 (1965)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Teschl.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author would like to thank the Simons Foundation.
Rights and permissions
About this article
Cite this article
Grafakos, L., Van Nguyen, H. The Hörmander multiplier theorem, III: the complete bilinear case via interpolation. Monatsh Math 190, 735–753 (2019). https://doi.org/10.1007/s00605-019-01300-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-019-01300-x