How to recognize polynomials in higher order Sobolev spaces

TL;DR

This paper extends the characterization of Sobolev spaces to higher orders, providing new integral conditions that identify polynomials through Taylor remainders and exploring related Whitney jet concepts.

Contribution

It introduces higher order Sobolev space characterizations and integral conditions for polynomial recognition, expanding prior work by Bourgain, Brézis, and Mironescu.

Findings

Derived integral conditions for Taylor remainders indicating polynomial functions

Extended Sobolev space characterizations to higher order cases

Connected polynomial recognition with Whitney jets

Abstract

This paper extends characterizations of Sobolev spaces by Bourgain, Br\'{e}zis, and Mironescu to the higher order case. As a byproduct, we obtain an integral condition for the Taylor remainder term, which implies that the function is a polynomial. Similar questions are also considered in the context of Whitney jets.