Published Paper

Inserted: 7 oct 2021
Last Updated: 31 aug 2023

Journal: Proc. of the AMS
Year: 2021
Doi: https://doi.org/10.1090/proc/16490

Abstract:

In this note, we prove that if a subharmonic function $\Delta u\ge 0$ has pure second derivatives $\partial_{ii} u$ that are signed measures, then their negative part $(\partial_{ii} u)_-$ belongs to $L^1$ (in particular, it is not singular). We then show that this improvement of regularity cannot be upgraded to $L^p$ for any $p > 1$. We finally relate this problem to a natural question on the one-sided regularity of solutions to the obstacle problem with rough obstacles.