Carleman inequalities and unique continuation for the polyharmonic operators

TL;DR

This paper extends Carleman inequalities to polyharmonic operators, enabling new unique continuation results for higher order Schrödinger equations by relaxing previous integrability constraints.

Contribution

It provides a complete characterization of $L^p-L^q$ Carleman estimates for polyharmonic operators, generalizing earlier results for the Laplacian.

Findings

Extended Carleman inequalities to polyharmonic operators.

Established new unique continuation properties for higher order Schrödinger equations.

Relaxed integrability assumptions on solutions.

Abstract

We obtain a complete characterization of $L^p-L^q$ Carleman estimates with weight $e^{v\cdot x}$ for the polyharmonic operators. Our result extends the Carleman inequalities for the Laplacian due to Kenig--Ruiz--Sogge. Consequently, we obtain new unique continuation properties of higher order Schr\"odinger equations relaxing the integrability assumption on the solution spaces.