Hessian determinants and averaging operators over surfaces in \mathbb R³

2021/12/02 by Greenblatt, Michael
#42B20 44A12 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.01461

Abstract

We prove L^p(\mathbb R³) to L^p_s(\mathbb R³) Sobolev improvement theorems for local averaging operators over real analytic surfaces in \mathbb R³. For most such operators, in a sense made precise in the paper, the set of (p,s) for which we prove L^p(\mathbb R³) to L^p_s(\mathbb R³) boundedness is optimal up to endpoints. Using an interpolation argument in conjunction with these L^p(\mathbb R³) to L^p_s(\mathbb R³) results we obtain an L^p(\mathbb R³) to L^q(\mathbb R³) improvement theorem, and the set of exponents (p,q) obtained will also usually be optimal up to endpoints. The advantage the methods of this paper have over those of the author's earlier papers is that the oscillatory integral methods of the earlier papers, closely tied to the Van der Corput lemma, allow one to only prove 1/2 of a derivative of surface measure Fourier transform decay, while the methods of this paper, when combined with appropriate resolution of singularities methods, allow one to go up to the maximum possible 1 derivative. This allows us to prove the stronger sharp up to endpoints results.

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