Cobos Díaz, Fernando and Kühn, Thomas and Schonbek, Tomas (2006) Compact embeddings of Brezis-Wainger type. Revista Matemática Iberoamericana, 22 (1). pp. 305-322. ISSN 0213-2230
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Official URL: http://projecteuclid.org/euclid.rmi/1148492184
Abstract
Let Ω be a bounded domain in Rn and denote by idΩ the restriction operator from the Besov space B1+n/p pq (Rn) into the generalized Lipschitz space Lip(1,−α)(Ω). We study the sequence of entropy numbers of this operator and prove that, up to logarithmic factors, it behaves asymptotically like ek(idΩ) ∼ k−1/p if α > max (1 + 2/p −1/q, 1/p). Our estimates improve previous results by Edmunds and Haroske.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Entropy Numbers; Banach-Spaces; Operators; Compact embeddings; Besov spaces; Lipschitz spaces; Mathematics |
| Subjects: | Sciences > Mathematics > Mathematical analysis |
| ID Code: | 15042 |
| References: | [1] Bergh, J. and L¨ofstr¨om, J.: Interpolation Spaces. An Introduction. Springer Verlag, Berlin-Heidelberg-New York, 1976. [2] Br´ezis, H. and Wainger, S.: A note on limiting cases of Sobolev embeddings and convolution inequalities. Comm. Partial Differential Equations 5 (1980), 773-789. [3] Carl, B.: Entropy numbers, s-numbers and eigenvalue problems. J. Funct. Anal. 41 (1981), 290-306. [4] Carl, B. and Stephani, I.: Entropy, compactness and the approximation of operators. Cambridge Tracts in Mathematics 98. Cambridge Univ. Press, Cambridge, 1990. [5] Carl, B. and Triebel H.: Inequalities between eigenvalues, entropy numbers and related quantities of compact operators in Banach spaces. Math. Ann. 251 (1980), 129-133. [6] Cobos, F.: On the type of interpolation spaces and Sp,q. Math. Nachr. 113 (1983), 59-64. [7] Cobos, F., K¨uhn, T.: Entropy numbers of embeddings of Besov spaces in generalized Lipschitz spaces. J. Approx. Theory 112 (2001), 73–92. [8] Edmunds, D.E. and Haroske, D.: Spaces of Lipschitz type, embeddings and entropy numbers. Dissertationes Math. 380 (1999), 1-43. 322 F. Cobos, T. K¨uhn and T. Schonbek [9] Edmunds, D.E. and Haroske, D.D.: Embeddings in spaces of Lipschitz type, entropy and approximation numbers, and applications. J. Approx. Theory 104 (2000), 226-271. [10] Edmunds, D.E. and Triebel, H.: Function spaces, entropy numbers, differential Operators. Cambridge Tracts in Mathematics 120. Cambridge University Press, Cambridge, 1996. [11] K¨onig, H.: Eigenvalue distribution of compact operators. Operator Theory: Advances and Applications 16. Birkh¨auser Verlag, Basel, 1986. [12] K¨uhn, T.: A lower estimate for entropy numbers. J. Approx. Theory 110 (2001), 120-124. [13] K¨uhn, T.: Entropy numbers of diagonal operators of logarithmic type. Georgian Math. J. 8 (2001), 307-318. [14] Pietsch, A.: Operator ideals. North-Holland Mathematical Library 20. North-Holland Publishing Co., Amsterdam-New York, 1980. [15] Sch¨utt, C.: Entropy numbers of diagonal operators between symmetric Banach spaces. J. Approx. Theory 40 (1984), 121-128. [16] Triebel, H.: Interpolation theory, function spaces, differential operators. North-Holland Math. Library 18. North-Holland, Amsterdam, 1978. [17] Triebel, H.: Fractals and spectra. Related to Fourier analysis and function spaces. Monographs in Mathematics 91. Birkh¨auser, Basel, 1997. |
| Deposited On: | 27 Apr 2012 11:14 |
| Last Modified: | 27 Apr 2012 11:14 |
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