All functions are (locally) $s$-harmonic (up to a small error) - and applications

TL;DR

This paper discusses how functions can be locally approximated by functions with nearly zero fractional Laplacian, highlighting structural similarities and differences between classical and fractional Laplacians, with applications.

Contribution

It provides an exposition of the result that any function can be locally approximated by functions with vanishing fractional Laplacian and explores its implications.

Findings

Functions can be locally approximated by functions with near-zero fractional Laplacian.

Structural differences between classical and fractional Laplacians are significant.

The result has various mathematical applications.

Abstract

The classical and the fractional Laplacians exhibit a number of similarities, but also some rather striking, and sometimes surprising, structural differences. A quite important example of these differences is that any function (regardless of its shape) can be locally approximated by functions with locally vanishing fractional Laplacian, as it was recently proved by Serena Dipierro, Ovidiu Savin and myself. This informal note is an exposition of this result and of some of its consequences.