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

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 s r ( R 2 n ) , 2 ≤ r < ∞ , r s > 2 n , uniformly over all annuli. More precisely, given suc...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2019-12, Vol.190 (4), p.735-753
Main Authors: Grafakos, Loukas, Van Nguyen, Hanh
Format: Article
Language:English
Subjects:
Interpolation
Mathematics
Mathematics and Statistics
Smoothness
Sobolev space
Theorems
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Summary: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 s r ( R 2 n ) , 2 ≤ r < ∞ , r s > 2 n , 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 ( R n ) × L p 2 ( R n ) to L p ( R n ) when 1 / p = 1 / p 1 + 1 / p 2 , 1 < p 1 , p 2 < ∞ .
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-019-01300-x