Discrete and Continuous Signals: Analysis, Information and Applications

Removable sets for Newtonian space $N^{1,p}$

12 Dec 2023, 15:45

35m

St. Petersburg University, MCS faculty

Speakers

Vladimir Shlyk (Far Eastern federal university) Yuri Dymchenko (adm. G.I. Nevelskoy Maritime state university)

Description

Let $X=(X,d,\mu)$ be a complete $p$-Poincaré metric space with distance $d$ and a Borel regular doubling measure $\mu$, $1 Following Vodop'yanov and Gol'dstein, we introduce an analogue of $NC_p$-sets in the domain $\Omega$ of $X$ and give the criterion of equality $N^{1,p}(\Omega\setminus E)=N^{1,p}(\Omega)$ in terms of $E$ as an $NC_p$-set in $\Omega$. As a consequence, we obtain that the domains $\Omega_1$ and $\Omega$, $\Omega_1 \subset\Omega$, are $(1,p)$-equivalent if and only if $\Omega\setminus\Omega_1$ is an $NC_p$-set in $\Omega$. Moreover, for a quasisymmetric map $f:X\to Y$ of two complete $p$-regular, $p$-Poincaré metric spaces $X$ and $Y$, we show that $f(E)$ is an $NC_p$-set in the image $f(\Omega)$ if
and only if $E$ is an $NC_p$-set in $\Omega\subset X$.