Monatshefte für Mathematik

, Volume 190, Issue 4, pp 735–753 | Cite as

The Hörmander multiplier theorem, III: the complete bilinear case via interpolation

  • Loukas GrafakosEmail author
  • Hanh Van Nguyen


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 \(1/p=1/p_1+1/p_2\), \(1<p_1,p_2<\infty \).


Multilinear operator Multiplier operator Interpolation 

Mathematics Subject Classification

42B15 42B30 



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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of MathematicsThe University of AlabamaTuscaloosaUSA

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