ИСТИНА

"On $C^m$-extension and reflection of subharmonic functions". P.V. Paramonov, Moscow State (M.V. Lomonosov) University. In [1], Theorem~5.1, it was proved that any function $f$ subharmonic in the unit ball $B \subset {\bf R}^N$ ($N\geqslant 2$), $f\in C^1(\overline{B})$, can be extended to a function $F\in C^1({\bf R}^N)$ subharmonic on the whole space ${\bf R}^N$ with the property $\|\nabla F\|_{{\bf R}^N} \leq c\,\|\nabla f\|_{\overline{B}}\,$, where $c\in(0,+\infty)$ depends only on $N$. In [2] and [4] it is shown that for $N \geq 2$ the above formulated result still holds if, instead of a ball $B$, one takes an arbitrary Dini-Lyapunov type domain $D$ with connected complement (then $c=c(D,N)$). A wide class of Jordan $C^1$-smooth domains is constructed, for which the mentioned extension property fails. In [3] and [5] it is proved that for each $m\in (1,3)$ and Dini-Lyapunov (briefly DL-) domain $D$ in ${\bf R}^N\,,\, N \geq 2$ with connected complement, any function $f$ of the (Whitney) space $C^m(\overline D)$ that is subharmonic in $D$ can be extended to a function $F$ that is subharmonic and of the class $C^m$ on the whole ${\bf R}^N$ with appropriate estimate of the $C^{m-1}$-norm of $\nabla F$ on ${\bf R}^N$. The corresponding assertion for $m\in [0,1)\cup [3,+\infty)$ is false even for balls. A localization theorem for $C^m$-subharmonic extension ($m\in [1,3)$) from DL-domains was also obtained, as well as some other corollaries, including corresponding assertions on $Lip^m$-extension of subharmonic functions. It is planned to discuss more explicitly these results as well as theorems on $C^1$-harmonic and subharmonic reflection from DL-domains into ${\bf R}^N\,,\, N \geq 2$. {\bf Bibliography} 1. {\it Verdera J., Melnikov M.S. and Paramonov P.V.}, $C^1$-approxi\-ma\-tion and extension of subharmonic functions.// Sbornik: Mathematics. {\bf 192}:4 (2001), 37-58. 2. {\it Melnikov M.S. and Paramonov P.V.}, $C^1$-exten\-sion of subharmonic functions from closed Jordan domains in ${\bf R}^2$.// Iz\-ves\-tiya: Ma\-the\-ma\-tics. {\bf 68}:6 (2004), 1165-1178. 3. {\it Paramonov P.V.}, $C^m$-extension of subharmonic functions.// Iz\-ves\-tiya: Ma\-the\-matics. {\bf 69}:6 (2005), 1211-1223. 4. {\it Paramonov P.V.}, $C^1$-extension and $C^1$-reflection of subharmonic functions from Lyapunov-Dini domains into ${\bf R}^N$.// Sbornik: Mathematics. $\bf 199$:12 (2008), 1809-1846. 5. {\it Paramonov P.V.}, On $C^m$-extension of subharmonic functions from Lyapunov-Dini domains into ${\bf R}^N$. (Russian) Mat. Zametki. {\bf 89}:1 (2011), (Translation in Math. Notes {\bf 89}:1 (2011), ) \medskip The research was partially supported by the Russian Foundation for Basic Research (grant 09-01-12160 ofi.m) and by the Program "Leading Scientific Schools of the Russian Federation" (grant NSh - 3877.2008.1). petr.paramonov@list.ru