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Hessian determinants and averaging operators over surfaces in ${\mathbb R}^3$

Michael Greenblatt

Abstract

We prove $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$ Sobolev improvement theorems for local averaging operators over real analytic surfaces in ${\mathbb R}^3$. For most such operators, in a sense made precise in the paper, the set of $(p,s)$ for which we prove $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$ boundedness is optimal up to endpoints. Using an interpolation argument in conjunction with these $L^p({\mathbb R}^3)$ to $L^p_s({\mathbb R}^3)$ results we obtain an $L^p({\mathbb R}^3)$ to $L^q({\mathbb R}^3)$ 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.

Hessian determinants and averaging operators over surfaces in ${\mathbb R}^3$

Abstract

We prove to Sobolev improvement theorems for local averaging operators over real analytic surfaces in . For most such operators, in a sense made precise in the paper, the set of for which we prove to boundedness is optimal up to endpoints. Using an interpolation argument in conjunction with these to results we obtain an to improvement theorem, and the set of exponents 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.
Paper Structure (18 sections, 15 theorems, 60 equations)

This paper contains 18 sections, 15 theorems, 60 equations.

Table of Contents

  1. Introduction and theorem statements
  2. Definitions and key concepts
  3. $L^p$ to $L^p_s$ theorems
  4. $L^p$ to $L^q$ theorems
  5. The condition $\eta = {2\eta' \over 1 + 2\eta'}$
  6. More about Newton polygons
  7. Additional examples where $\eta \leq {2\eta' \over 1 + 2\eta'}$
  8. Resolution of singularities and some consequences
  9. The original resolution of singularities theorem
  10. Adjusting the algorithm
  11. Some useful consequences
  12. The proof of Theorem 1.1.
  13. The decomposition of the operator
  14. Defining the $V_{ijkn}^l$ and $W_{ijkn}^l$
  15. The analysis of the operators $V_{ijkn}^2$ and $W_{ijkn}^2$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose $(1.2)$ and $(1.4)$ hold and neither $S(t_1,t_2)$ nor $H(t_1,t_2)$ are identically zero. Let $\eta$ be the exponent in $(1.3)$ corresponding to $S(t_1,t_2)$, and let $\eta'$ be the exponent in $(1.3)$ corresponding to $H(t_1,t_2)$. There is a neighborhood $U$ of the origin such that if $\phi

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 2.1
  • ...and 13 more