The Hörmander Multiplier Theorem III: The complete bilinear case via interpolation

2016/07/09 by Grafakos, Loukas, Van Nguyen, Hanh
#42B15 #42B30 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.02617

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 R²ⁿ), 2≤ r2n, uniformly over all annuli. More precisely, given a smoothness index s, we find the largest open set of indices (1/p₁,1/p₂ ) for which we have boundedness for the associated bilinear multiplier operator from L^p₁(\mathbb R ⁿ)× L^p₂ (\mathbb R ⁿ) to L^p(\mathbb R ⁿ) when 1/p=1/p₁+1/p₂, 1

Related

• The Hörmander multiplier theorem, II: The bilinear local L² case
• The Hörmander multiplier theorem I: The Linear Case
• Notes on the spaces of bilinear multipliers
• A sharp version of the Hörmander multiplier theorem
• The multilinear Hormander multiplier theorem with a Lorentz-Sobolev\n condition