Improved regularity of second derivatives for subharmonic functions

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.