Partial Data Calder\'on Problems for $L^{n/2}$ Potentials on Admissible Manifolds

TL;DR

This paper advances the partial data Calderón problem on CTA manifolds by establishing results for $L^{n/2}$ potentials without using Carleman estimates, leading to improved regularity and unique continuation results.

Contribution

It provides a novel approach to the Calderón problem on CTA manifolds for $L^{n/2}$ potentials by constructing a specialized Green's function without Carleman estimates.

Findings

Achieved sharp unique continuation results for $L^{n/2}$ potentials.

Constructed a Green's function satisfying boundary conditions and semiclassical estimates.

Improved regularity results for the Calderón problem on CTA manifolds.

Abstract

We solve the partial data Calder\'on problem on conformally transversallly anisotropic (CTA) manifolds with $L^{n/2}$ potentials - on par with sharp unique continuation result of \cite{JerKen}. A trivial consequence of this is the sharp regularity improvement to the result of Kenig-Sj\"ostrand-Uhlmann \cite{ksu}. This is done by constructing a "Green's function" which possesses both desirable boundary conditions {\em and} satisfies semiclassical type estimates in the suitable $L^{p}$ spaces. No Carleman estimates were used in the writing of this article which makes it starkly different from the traditional approaches based on \cite{BukUhl} and \cite{ksu}.