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Zawartość
Pełne teksty:
Warianty tytułu
Abstrakty
CONTENTS
Introduction.........................................................................................5
1. Non-limiting embeddings - a short review........................................8
2. The spaces Lₚ(log L)ₐ on ℝⁿ.........................................................12
2.1. The spaces Lₚ(log L)ₐ(Ω) and $L_{p,q}(log L)ₐ(Ω)$................12
2.2. The spaces $Lₚ(log L)_{-a}(ℝⁿ)$, a > 0...................................20
2.3. The spaces Lₚ(log L)ₐ(ℝⁿ), a > 0.............................................27
2.4. Hölder inequalities...................................................................32
2.5. Examples.................................................................................35
3. Entropy numbers, limiting embeddings.........................................37
4. Applications..................................................................................47
4.1. Eigenvalue distribution............................................................47
4.2. Negative spectrum...................................................................53
5. Homogeneous spaces..................................................................55
References.......................................................................................58
Introduction.........................................................................................5
1. Non-limiting embeddings - a short review........................................8
2. The spaces Lₚ(log L)ₐ on ℝⁿ.........................................................12
2.1. The spaces Lₚ(log L)ₐ(Ω) and $L_{p,q}(log L)ₐ(Ω)$................12
2.2. The spaces $Lₚ(log L)_{-a}(ℝⁿ)$, a > 0...................................20
2.3. The spaces Lₚ(log L)ₐ(ℝⁿ), a > 0.............................................27
2.4. Hölder inequalities...................................................................32
2.5. Examples.................................................................................35
3. Entropy numbers, limiting embeddings.........................................37
4. Applications..................................................................................47
4.1. Eigenvalue distribution............................................................47
4.2. Negative spectrum...................................................................53
5. Homogeneous spaces..................................................................55
References.......................................................................................58
Słowa kluczowe
Tematy
Kategoryzacja MSC:
- 35P15: Estimation of eigenvalues, upper and lower bounds
- 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
- 35J70: Degenerate elliptic equations
- 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
- 35P20: Asymptotic distribution of eigenvalues and eigenfunctions
- 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
373
Liczba stron
59
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXXIII
Daty
wydano
1998
otrzymano
1996-10-26
poprawiono
1997-07-08
Twórcy
autor
- Mathematisches Institut, Friedrich-Schiller-Universität Jena, D-07740 Jena, Germany
Bibliografia
- [1] C. Bennett and K. Rudnick, On Lorentz-Zygmund spaces, Dissertationes Math. 175 (1980).
- [2] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston, 1988.
- [3] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976.
- [4] J. Bourgain, A. Pajor, S. J. Szarek and N. Tomczak-Jaegermann, On the duality problem for entropy numbers of operators, in: J. Lindenstrauss and V. D. Milman (eds.), Israel Seminar, GAFA 1987/88, Lecture Notes in Math. 1376, Springer, 1989, 50-63.
- [5] B. Carl, Entropy numbers, s-numbers and eigenvalue problems, J. Funct. Anal. 41 (1981), 290-306.
- [6] B. Carl and I. Stephani, Entropy, Compactness and the Approximation of Operators, Cambridge Univ. Press, Cambridge, 1990.
- [7] B. Carl and H. Triebel, Inequalities between eigenvalues, entropy numbers and related quantities in Banach spaces, Math. Ann. 251 (1980), 129-133.
- [8] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
- [9] D. E. Edmunds and H. Triebel, Entropy numbers and approximation numbers in function spaces, Proc. London Math. Soc. 58 (1989), 137-152.
- [10] D. E. Edmunds and H. Triebel, Entropy numbers and approximation numbers in function spaces II, Proc. London Math. Soc. 64 (1992), 153-169.
- [11] D. E. Edmunds and H. Triebel, Eigenvalue distributions of some degenerate elliptic operators: an approach via entropy numbers, Math. Ann. 299 (1994), 311-340.
- [12] D. E. Edmunds and H. Triebel, Logarithmic Sobolev spaces and their applications to spectral theory, Proc. London Math. Soc. 71 (1995), 333-371.
- [13] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, Cambridge, 1996.
- [14] D. E. Edmunds and H. Triebel, Logarithmic spaces and related trace problems, preprint, Brighton & Jena, 1997.
- [15] Y. S. Han and E. T. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc. 530 (1994).
- [16] D. Haroske, Entropy numbers and approximation numbers in weighted function spaces of type $B^s_{p,q}$ and $F^s_{p,q}$, eigenvalue distributions of some degenerate pseudodifferential operators, PhD thesis, Friedrich-Schiller-Universität Jena, Germany, 1995.
- [17] D. Haroske, Some limiting embeddings in weighted function spaces and related entropy numbers, Forschungsergebnisse Math/Inf/97/04, Universität Jena, 1997.
- [18] 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.
- [19] D. Haroske and H. Triebel, Entropy numbers in weighted function spaces and eigenvalue distribution of some degenerate pseudodifferential operators II, Math. Nachr. 168 (1994), 109-137.
- [20] H. Heuser, Functional Analysis, Wiley, Chichester, 1982.
- [21] H. König, Eigenvalue Distribution of Compact Operators, Birkhäuser, Basel, 1986.
- [22] R. A. Macias and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. Math. 33 (1979), 257-270.
- [23] L. Päivärinta, Pseudodifferential operators in Hardy-Triebel spaces, Z. Anal. Anwendungen 2 (1983), 235-242.
- [24] T. Runst, Pseudo-differential operators of the 'exotic' class $L^0_{1,1}$ in spaces of Besov and Triebel-Lizorkin type, Ann. Global Anal. Geom. 3 (1985), 13-28.
- [25] M. Schechter, Spectra of Partial Differential Operators, 2nd ed., North-Holland, Amsterdam, 1986.
- [26] H.-J. Schmeißer and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley, Chichester, 1987.
- [27] R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities, Indiana Univ. Math. J. 21 (1972), 841-842.
- [28] R. H. Torres, Continuity properties of pseudodifferential operators of type 1,1, Comm. Partial Differential Equations 15 (1990), 1313-1328.
- [29] R. H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 442 (1991).
- [30] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
- [31] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
- [32] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992.
- [33] H. Triebel, Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces, Proc. London Math. Soc. 66 (1993), 589-618.
- [34] H. Triebel, A localization property for $B^s_{p,q}$ and $F^s_{p,q}$ spaces, Studia Math. 109 (1994), 183-195.
- [35] N. S. Trudinger, On embeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483.
Języki publikacji
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Uwagi
1991 Mathematics Subject Classification: 46E35, 46E30, 41A46, 35P15, 35P20, 35J70.
Identyfikator YADDA
bwmeta1.element.zamlynska-51e7c367-d257-4b20-a083-ceddb21ff33c
Identyfikatory
ISSN
0012-3862
Kolekcja
DML-PL
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