Remarks on the Bernstein inequality for higher order operators and related results

TL;DR

This paper investigates Bernstein inequalities for higher order operators, providing counterexamples, analyzing heat semi-group properties, and establishing some positive results within specific ranges.

Contribution

It offers new counterexamples for Bernstein inequalities and analyzes the maximum principle failure for higher order Laplacian powers.

Findings

Counterexamples for frequency-localized Bernstein inequalities

Heat semi-group of higher powers does not satisfy the maximum principle

Positive results established in certain parameter ranges

Abstract

This note is devoted to several results about frequency localized functions and associated Bernstein inequalities for higher order operators. In particular, we construct some counterexamples for the frequency-localized Bernstein inequalities for higher order Laplacians. We show also that the heat semi-group associated to powers larger than one of the laplacian does not satisfy the strict maximum principle in general. Finally, in a suitable range we provide several positive results.