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$L^p - L^q$ resolvent restriction estimates for submanifolds

Matthew D. Blair, Chamsol Park

Abstract

We consider restriction analogues on hypersurfaces of the uniform Sobolev inequalities in Kenig, Ruiz, and Sogge and the resolvent estimates in Dos Santos Ferreira, Kenig, and Salo.

$L^p - L^q$ resolvent restriction estimates for submanifolds

Abstract

We consider restriction analogues on hypersurfaces of the uniform Sobolev inequalities in Kenig, Ruiz, and Sogge and the resolvent estimates in Dos Santos Ferreira, Kenig, and Salo.

Paper Structure

This paper contains 3 sections, 12 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Thm Euclidean resolvent restriction estimates for hypersurfaces']}
  3. Proof of Theorem \ref{['Thm Manifold resolvent restriction estimates for hypersurface']}

Key Result

Theorem 1.1

Let $n\geq 3$. Suppose $\Sigma$ is a smooth hypersurface in $\mathbb{R}^n$. If then, given $\delta>0$, for all $z\in \mathbb{C}$ with $|z|\geq \delta>0$, where $\|u\|_{L^q(\Sigma)}$ is computed as in Restriction Lq u computation.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3: BlairPark2024RestrictionOfSchrodingerEigenfunctions
  • Lemma 2.4
  • Lemma 2.5
  • ...and 4 more