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$L^p$ estimates for the Laplacian via blow-up

Jan Lewenstein-Sanpera, Xavier Ros-Oton

TL;DR

The paper establishes $W^{2,p}$ Calderón–Zygmund estimates for the elliptic equation $\Delta u=f$ and the parabolic analogue $\partial_t u-\Delta u=f$ in suitable domains via a blow-up/contradiction compactness argument. A central step is a pointwise bound for the sharp maximal function of $D^2 u$ using the 2-sharp maximal function $\mathcal{M}_2^{\#}$; the argument reduces to limits that are harmonic or caloric, producing a contradiction unless the bound holds. The authors implement this with a purely $L^p$-based pathway that avoids interpolation theorems and then extend to general non-divergence form operators through the standard freezing coefficients. The results unify elliptic and parabolic regularity in a simple framework and highlight the efficacy of sharp maximal function techniques in obtaining $L^p$ estimates.

Abstract

In this note we provide a new proof of the $W^{2,p}$ Calderón-Zygmund regularity estimates for the Laplacian, i.e., $Δu=f$ and its parabolic counterpart $\partial_t u-Δu=f$. Our proof is an adaptation of a contradiction and compactness argument that so far had been only used to prove estimates in Hölder spaces. This new approach is simpler than previous ones, and avoids the use of any interpolation theorem.

$L^p$ estimates for the Laplacian via blow-up

TL;DR

The paper establishes Calderón–Zygmund estimates for the elliptic equation and the parabolic analogue in suitable domains via a blow-up/contradiction compactness argument. A central step is a pointwise bound for the sharp maximal function of using the 2-sharp maximal function ; the argument reduces to limits that are harmonic or caloric, producing a contradiction unless the bound holds. The authors implement this with a purely -based pathway that avoids interpolation theorems and then extend to general non-divergence form operators through the standard freezing coefficients. The results unify elliptic and parabolic regularity in a simple framework and highlight the efficacy of sharp maximal function techniques in obtaining estimates.

Abstract

In this note we provide a new proof of the Calderón-Zygmund regularity estimates for the Laplacian, i.e., and its parabolic counterpart . Our proof is an adaptation of a contradiction and compactness argument that so far had been only used to prove estimates in Hölder spaces. This new approach is simpler than previous ones, and avoids the use of any interpolation theorem.
Paper Structure (6 sections, 6 theorems, 62 equations)

This paper contains 6 sections, 6 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Elliptic equations
  4. Sharp maximal function
  5. Regularity estimates
  6. Parabolic Equations

Key Result

theorem 1

Let $1<p<\infty$, and $u\in H^1(B_{1})$ be any weak solution to: with $f\in L^{p}(B_{1})$. Then u is in $W^{2,p}$ inside $B_{1}$ and the following estimate holds: The constant $C$ depends only on $n$ and $p$.

Theorems & Definitions (13)

  • theorem 1: CZ52
  • proposition 2
  • lemma 3
  • proof
  • proof : Proof of Proposition \ref{['prop: elliptic_estimate']}
  • proof : Proof of \ref{['th: calderon-zygmund theorem']}
  • remark 4
  • theorem 5
  • proposition 6
  • proof
  • ...and 3 more