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 
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/
@article{TM_2014_284_a10,
     author = {G. A. Kalyabin},
     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},
     pages = {169--175},
     year = {2014},
     volume = {284},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2014_284_a10/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $(-d^2/dx^2)^nu(x)=\lambda|x|^{\beta-1}u(x)$, $x\in(-1,1)$, $u^{(k)}(\pm1)=0$, $k\in\{0,1,\dots ,n-1\}$.

[1] Magaril-Ilyaev G.G., Tikhomirov V.M., Vypuklyi analiz i ego prilozheniya, Editorial URSS, M., 2000

[2] Kalyabin G.A., “Nekotorye zadachi dlya prostranstv Soboleva na polupryamoi”, Tr. MIAN, 255, 2006, 161–169 | MR

[3] Kalyabin G.A., “Tochnye otsenki dlya proizvodnykh funktsii iz klassov Soboleva $\mathring W_2^r(-1,1)$”, Tr. MIAN, 269, 2010, 143–149 | MR | Zbl

[4] Kolmogorov A.N., Fomin S.V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR

[5] Polia G., Segë G., Zadachi i teoremy iz analiza, Ch. 1, Nauka, M., 1978

[6] Prudnikov A.P., Brychkov Yu.A., Marichev O.I., Integraly i ryady: Elementarnye funktsii, Nauka, M., 1981 | MR | Zbl

[7] Natanson I.P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[8] Böttcher A., Widom H., “From Toeplitz eigenvalues through Green's kernels to higher-order Wirtinger–Sobolev inequalities”, The extended field of operator theory, Oper. Theory Adv. Appl., 171, Birkhäuser, Basel, 2007, 73–87 | DOI | MR