Bilateral estimates for the least norm of extension operators from plane convex domains for Sobolev spaces (Q2714712)

Two-sided estimates of norm of the extension operator \(T_G\), NEWLINE\[NEWLINE T_G: W^{1,p}(G)\to W^{1,p}(\mathbb R^2), NEWLINE\]NEWLINE are obtained. Here \(W^{1,p}(G)\), or \(W^{1,p}(\mathbb R^2)\), \(1<p<\infty\), denotes the classical Sobolev space on \(G\), or on \(\mathbb R^2\), respectively, \(G\) is a convex domain in \(\mathbb R^2\). More precisely, the author proves that the least possible norm \(\tau_p(G)\) of such an operator \(T_G\) satisfies the estimates NEWLINE\[NEWLINE c_p^{-1}\big(|G|^{-1}\gamma_p(\delta)\big)^{1/p} \leq\tau_p(G)\leq c_p\big((|G|^{-1}\gamma_p(\delta)\big)^{1/p} NEWLINE\]NEWLINE if \(\delta=\delta(G):=\operatorname {diam}G\leq 1\) (\(\gamma_p(\delta):=\begin{cases} \delta^{2-p},&1<p<2,\\ \frac 1{\ln(2/\delta)},&p=2,\\ 1,&2<p<\infty.\\ \end{cases}\), and NEWLINE\[NEWLINE c_p^{-1}\Big(\inf_{x\in G}\big|G\cap B_x|^{-1/p}\Big)\leq \tau_p(G)\leq c_p\Big(\inf_{x\in G}\big|G\cap B_x)|^{-1/p}\Big) NEWLINE\]NEWLINE if \(\delta=\delta(G)>1\) (here \(c_p\geq 1\) is a suitable constant which may depend only on \(p\), \(|M|\) denotes the \(2\)-dimensional Lebesgue measure of a set \(M\), \(B_x\) is the unit ball in \(\mathbb R^2\) centered in \(x\)).NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].