IMPROVED REGULARITY OF SECOND DERIVATIVES FOR SUBHARMONIC FUNCTIONS

In this note, we prove that if a subharmonic function Delta u >= 0 has pure second derivatives partial derivative(ii)u that are signed measures, then their negative part (partial derivative(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.