On two-sided and asymptotic estimates for the norms of embedding operators of $\mathring W_2^n(-1,1)$ into $L_q(d\mu)$

G. A. Kalyabin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 169-175

Résumé

Explicit upper and lower estimates are given for the norms of the operators of embedding of $\mathring W_2^n(-1,1)$, $n\in\mathbb N$, in $L_q(d\mu)$, $0$. Conditions on the measure $\mu$ are obtained under which the ratio of the above estimates tends to $1$ as $n\to\infty$, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as $n\to\infty$) is established for the minimum eigenvalues $\lambda_{1,n,\beta}$, $\beta>0$, of the boundary value problems $(-d^2/dx^2)^nu(x)=\lambda|x|^{\beta-1}u(x)$, $x\in(-1,1)$, $u^{(k)}(\pm1)=0$, $k\in\{0,1,\dots ,n-1\}$.