A comment on an $L^\frac{2n}{n+2}-L^\frac{2n}{n-2}$ Carleman inequality in relation to "the determination of an unbounded potential from Cauchy data"

TL;DR

This paper corrects a previously published proof related to a specific Carleman inequality by providing a new proof under additional hypotheses, and rectifies related proofs that used the incorrect argument.

Contribution

It offers a corrected proof of a key proposition involving a specific Carleman inequality, addressing prior inaccuracies and clarifying the conditions needed.

Findings

Provides a new proof requiring an extra hypothesis.

Corrects the proof of a related proposition in another work.

Addresses inaccuracies in previous arguments.

Abstract

The proof of \cite[Proposition 2.1]{DKS}[arXiv:1104.0232] is partially incorrect. In this short note, we provide a new proof, which requires an additional hypothesis. A modification of this new proof also corrects the proof of \cite[Proposition 2.1]{Ch}[arXiv:2310.17456], where the incorrect argument of \cite{DKS} has been repeated.