## 月曜解析セミナー： Global integrability of supertemperatures

2018年 　 5月 21日 15時 00分 ～ 2018年 　 5月 21日 16時 30分

Ever since Armitage showed that every nonnegative superharmonic function on a bounded domain of bounded curvature ($=C^{1,1}$ domain) in $R^n$ is $L^p$-integrable up to the boundary for $0< p< n/(n-1)$, the global integrability of nonnegative supersolutions has attracted many mathematicians.

In this talk we consider a parabolic counterpart. We study the global integrability of nonnegative supertemperatures on the cylinder $D\times(0,T)$, where $D$ is a Lipschitz domain or a John domain. We show that the integrability depends on the lower estimate of the Green function for the Dirichlet Laplacian on $D$. In particular, if $D$ is a bounded $C^1$-domain, then every nonnegative supertemperature on $D\times(0,T)$ is $L^p$-integrable over $D\times(0,T')$ for any $0\lt T'< T$, provided $0< p< (n+2)/(n+1)$. The bound $(n+2)/(n+1)$ is sharp.

Joint work with Hara and Hirata.