## X. Fernández-Real - R. Tione

# Improved regularity of second derivatives for subharmonic functions

created by tione on 07 Oct 2021

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BibTeX]

*Submitted Paper*

**Inserted:** 7 oct 2021

**Last Updated:** 7 oct 2021

**Year:** 2021

**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.