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

TL;DR

This paper presents a simpler proof for $W^{2,p}$ regularity estimates of the Laplacian and its parabolic counterpart, using a contradiction and compactness argument previously applied only to H"older spaces.

Contribution

It introduces a novel, simplified proof method for Calderón-Zygmund estimates that avoids interpolation theorems, extending techniques to $L^p$ spaces.

Findings

New proof simplifies existing regularity estimates

Avoids use of interpolation theorems

Extends techniques from H"older to $L^p$ spaces

Abstract

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