Geodesic
On two-sided and asymptotic estimates for the norms of embedding operators of $\mathring W_2^n(-1,1)$ into $L_q(d\mu)$
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 169-175.

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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 $(-d^2/dx^2)^nu(x)=\lambda|x|^{\beta-1}u(x)$, $x\in(-1,1)$, $u^{(k)}(\pm1)=0$, $k\in\{0,1,\dots ,n-1\}$. 
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     title = {On two-sided and asymptotic estimates for the norms of embedding operators of $\mathring W_2^n(-1,1)$ into $L_q(d\mu)$},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     year = {2014},
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     url = {http://geodesic.mathdoc.fr/item/TM_2014_284_a10/}
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G. A. Kalyabin. On two-sided and asymptotic estimates for the norms of embedding operators of $\mathring W_2^n(-1,1)$ into $L_q(d\mu)$. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces and related problems of analysis, Tome 284 (2014), pp. 169-175. http://geodesic.mathdoc.fr/item/TM_2014_284_a10/

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