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Abstract
We establish the sharpness of the embedding of certain Besov and Triebel-Lizorkin spaces in spaces of Lipschitz type. In particular, this proves the sharpness of the Brézis-Wainger result concerning the "almost" Lipschitz continuity of elements of the Sobolev space $H^{1+n/p}_p(ℝⁿ)$, where 1 < p < ∞. Upper and lower estimates are obtained for the entropy numbers of related embeddings of Besov spaces on bounded domains.
We establish the sharpness of the embedding of certain Besov and Triebel-Lizorkin spaces in spaces of Lipschitz type. In particular, this proves the sharpness of the Brézis-Wainger result concerning the "almost" Lipschitz continuity of elements of the Sobolev space $H^{1+n/p}_p(ℝⁿ)$, where 1 < p < ∞. Upper and lower estimates are obtained for the entropy numbers of related embeddings of Besov spaces on bounded domains.
CONTENTS
Introduction...........................................................5
1. Preliminaries.....................................................6
Spaces on ℝⁿ......................................................6
Atomic decompositions........................................8
Spaces on domains...........................................10
Embeddings.......................................................11
Entropy numbers................................................11
2. Sharpness.......................................................13
3. Lipschitz embedding, entropy numbers...........21
4. Comparison with related results......................30
Embeddings.......................................................30
Entropy numbers...............................................36
Estimate from above..........................................37
References.........................................................42
Introduction...........................................................5
1. Preliminaries.....................................................6
Spaces on ℝⁿ......................................................6
Atomic decompositions........................................8
Spaces on domains...........................................10
Embeddings.......................................................11
Entropy numbers................................................11
2. Sharpness.......................................................13
3. Lipschitz embedding, entropy numbers...........21
4. Comparison with related results......................30
Embeddings.......................................................30
Entropy numbers...............................................36
Estimate from above..........................................37
References.........................................................42
Słowa kluczowe
Tematy
Kategoryzacja MSC:
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
380
Liczba stron
43
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXXX
Daty
wydano
1999
otrzymano
1998-04-24
poprawiono
1998-08-13
Twórcy
autor
- CMAIA, University of Sussex at Brighton, Falmer, Brighton BN1 9QH, United Kingdom , D.E.Edmunds@sussex.ac.uk
autor
- Mathematical Institute, Friedrich-Schiller-University, D-07740 Jena, Germany, haroske@minet.uni-jena.de
Bibliografia
- [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988.
- [2] J. Bourgain, A. Pajor, S. J. Szarek and N. Tomczak-Jaegermann, On the duality problem for entropy numbers of operators, in: Geometric Aspects of Functional Analysis (1987-88), J. Lindenstrauss and V. D. Milman (eds.), Lecture Notes in Math. 1376, Springer, 1989, 50-63.
- [3] H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differential Equations 5 (1980), 773-789.
- [4] B. Carl and I. Stephani, Entropy, Compactness and the Approximation of Operators, Cambridge Univ. Press, Cambridge, 1990.
- [5] R. DeVore and G. G. Lorentz, Constructive Approximation, Grundlehren Math. Wiss. 303, Springer, Berlin, 1993.
- [6] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
- [7] D. E. Edmunds, P. Gurka and B. Opic, On embeddings of logarithmic Bessel potential spaces, J. Funct. Anal. 146 (1997), 116-150.
- [8] D. E. Edmunds, P. Gurka and B. Opic, Optimality of embeddings of logarithmic Bessel potential spaces, to appear.
- [9] D. E. Edmunds and M. Krbec, Two limiting cases of Sobolev imbeddings, Houston J. Math. 21 (1995), 119-128.
- [10] D. E. Edmunds and H. Triebel, Entropy numbers and approximation numbers in function spaces, Proc. London Math. Soc. (3) 58 (1989), 137-152.
- [11] D. E. Edmunds and H. Triebel, Entropy numbers and approximation numbers in function spaces II, Proc. London Math. Soc. (3) 64 (1992), 153-169.
- [12] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 1996.
- [13] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34-170.
- [14] D. Haroske and H. Triebel, Entropy numbers in weighted function spaces and eigenvalue distribution of some degenerate pseudodifferential operators I, Math. Nachr. 167 (1994), 131-156.
- [15] H. König, Eigenvalue Distribution of Compact Operators, Birkhäuser, Basel, 1986.
- [16] H.-G. Leopold, Limiting embeddings and entropy numbers, Forschungsergebnisse Math/Inf/98/05, Universität Jena, Germany, 1998.
- [17] A. Pietsch, Eigenvalues and s-Numbers, Akad. Verlagsgesellschaft Geest & Portig, Leipzig, 1987.
- [18] W. Sickel and H. Triebel, Hölder inequalities and sharp embeddings in function spaces of $B^s_{p,q}$ and $F^s_{p,q}$ type, Z. Anal. Anwendungen 14 (1995), 105-140.
- [19] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
- [20] H. Triebel, Spaces of Besov-Hardy-Sobolev type, Teubner, Leipzig, 1978.
- [21] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
- [22] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992.
- [23] H. Triebel, Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces, Proc. London Math. Soc. (3) 66 (1993), 589-618.
- [24] H. Triebel, Fractals and Spectra, Birkhäuser, Basel, 1997.
Języki publikacji
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Uwagi
1991 Mathematics Subject Classification: 26A16, 46E35, 41A46, 46E15.
Identyfikator YADDA
bwmeta1.element.zamlynska-17126c89-a594-4107-82bf-95c4741d8313
Identyfikatory
ISSN
0012-3862
Kolekcja
DML-PL
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