Initial and internal limits of supertemperatures (Q1819145)

Let \(D\) be an open subset of \(\mathbb{R}^{n+1}= \{(x,t): x\in \mathbb{R}^n\), \(t\in \mathbb{R}\}\) and let \(D_0\) (respectively \(D_+\)) denote the set of points \((x,t)\) in \(D\) for which \(t=0\) (respectively \(t>0\)). We suppose that \(D_0\neq \emptyset\). Let \(u\) be a nonnegative supertemperature on \(D_+\), and let \(\mu\) be the Riesz measure associated with the supertemperature \(u_0\), defined to be equal to \(u\) on \(D_+\) and equal to 0 on \(D\setminus D_+\). The following decomposition theorem is proved: if \(E\) is a bounded open set such that \(\overline{E} \subset D\) and \(E_0\neq \emptyset\), then there is a temperature \(v\) on \(E\) such that \(v=0\) on \(E_0\) and \(u= G\mu_E+ v\) on \(E_+\), where \(G\mu_E\) is the thermic potential on \(\mathbb{R}^{n+1}\) of the restriction of \(\mu\) to \(E\). This enables the author to deduce initial limit theorems for \(u\) from corresponding results for \(G\mu_E\). For example, writing \(\mu_0(S)= \mu(S\times \{0\})\) for every Borel subset \(S\) of \(\mathbb{R}^n\) such that \(S\times \{0\} \subseteq D_0\), the author shows that \(\int_B u(x,t)dx\to \mu_0(B)\) as \(t\to 0_+\) for any Borel set \(B\) such that \(\overline{B} \subset D_0\) and \(\mu_0 (\partial B)=0\). Further, with \(\lambda\) denoting \(n\)-dimensional Lebesgue measure, \(\liminf_{t\to 0+} u(x,t)= (d\mu_0/ d\lambda) (x)\) for \(\lambda\)-almost all \(x\) in \(D_0\).