Improved regularity of second derivatives for subharmonic functions

Abstract

In this note, we prove that if a subharmonic function Δ u ≥ 0 \Delta u\ge 0 has pure second derivatives ∂ i i u \partial _{ii} u that are signed measures, then their negative part ( ∂ i i u ) − (\partial _{ii} u)_- belongs to L 1 L^1 (in particular, it is not singular). We then show that this improvement of regularity cannot be upgraded to L p L^p for any p > 1 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.