Poisson modification of subharmonic function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T02:38:35Z http://mathoverflow.net/feeds/question/104593 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104593/poisson-modification-of-subharmonic-function Poisson modification of subharmonic function hardy 2012-08-13T03:26:37Z 2012-08-13T07:28:25Z <p>Let $u\in C^2(\Omega)$ be such that $\Delta u \ge 0$ on $\Omega\supset \overline{B(a,r)}$. We consider the Poisson modification $U$ of $u$ for the ball $B(a,r),$ that is $U$ equals $u$ on $\Omega-B(a,r)$ and that on $B=B(a,r)$ equals the solution to Direchlet problem with boundary data $u|_{\partial B}$, which is given by the Poisson kernel classically denoted by $P(x,y)$. It is known that $U$ is subharmonic in the sense that it verifies an inequality mean property. My question is : Do we have $U\in H^2(\Omega)?$.</p> http://mathoverflow.net/questions/104593/poisson-modification-of-subharmonic-function/104597#104597 Answer by Alexandre Eremenko for Poisson modification of subharmonic function Alexandre Eremenko 2012-08-13T07:03:13Z 2012-08-13T07:03:13Z <p>Subharmonicity of this modification is true and easy to prove just from the definition of a subharmonic function. Condition $C^2$ is redundant. Of course the modification will not be in $C^2$. Now what is $H(\Omega)$ ?</p> http://mathoverflow.net/questions/104593/poisson-modification-of-subharmonic-function/104600#104600 Answer by Pietro Majer for Poisson modification of subharmonic function Pietro Majer 2012-08-13T07:28:25Z 2012-08-13T07:28:25Z <p>Note that in your assumption $u$ itself need not be in $H^2(\Omega)$, nor even $L^2(\Omega)$, even if it is harmonic (so $u=U$). You are possibly interested on the local behaviour, that is whether $U\in H^2_{loc}$. But note that for $n=1$ any smooth, convex function is subharmonic; the function $U$ is affine on $(a-r,a+r)$, and in general not in $C^1$ (unless $u$ was already harmonic ), hence not $H^2_{loc}$. However, I think in your assumptions it is true that $U\in H^1_{loc}$.</p>