$C^m$-extension of subharmonic functions

P. V. Paramonov

Izvestiya. Mathematics , Tome 69 (2005) no. 6, pp. 1211-1223

Résumé

Given $m\in(1,3)$ and any (Jordan) $B$-domain $D$ in $\mathbb R^2$, we prove that any function of class $C^m(\,\overline D\,)$ that is subharmonic in $D$ can be extended to a function of class $C^m$ that is subharmonic on the whole $\mathbb R^2$ and give an estimate of the $C^{m-1}$-norm of its gradient. The corresponding assertion for $m\in[0,1)\cup[3,+\infty)$ is false even for discs. These results also hold for balls $D$ in $\mathbb R^N$, $N\in\{3,4,\dots\}$. We also obtain some corollaries, including the corresponding assertions on the $\operatorname{Lip}^m$-extension of subharmonic functions.