Some equivalent definitions of Besov spaces of generalized smoothness

Loosveldt, Laurent; Nicolay, Samuel

doi:10.1002/mana.201800111

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Some equivalent definitions of Besov spaces of generalized smoothness

Loosveldt, Laurent; Nicolay, Samuel

2019 • In Mathematische Nachrichten, 292 (10), p. 2262-2282

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https://hdl.handle.net/2268/239996

DOI
10.1002/mana.201800111

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Keywords :

Besov spaces; finite difference; generalized smoothness; real interpolation; Sobolev spaces

Abstract :

[en] In this paper, we present some alternative definitions of Besov spaces of general-ized smoothness, defined via Littlewood–Paley-type decomposition, involving weakderivatives, polynomials, convolutions and generalized interpolation spaces.

Disciplines :

Mathematics

Author, co-author :

Loosveldt, Laurent ; Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes

Nicolay, Samuel ; Université de Liège - ULiège > Département de mathématique > Analyse - Analyse fonctionnelle - Ondelettes

Language :

English

Title :

Some equivalent definitions of Besov spaces of generalized smoothness

Publication date :

October 2019

Journal title :

Mathematische Nachrichten

ISSN :

0025-584X

eISSN :

1522-2616

Publisher :

Wiley - VCH Verlag GmbH & Co., Germany

Volume :

292

Issue :

Pages :

2262-2282

Peer reviewed :

Peer Reviewed verified by ORBi

Funders :

F.R.S.-FNRS - Fonds de la Recherche Scientifique [BE]

Available on ORBi :

since 04 October 2019

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Bibliography

R. A. Adams, Sobolev spaces, Pure Appl. Math., vol. 65, Academic Press, 1975.
A. Almeida, Wavelet bases in generalized Besov spaces, J. Math. Anal. Appl. 304 (2005), 198–211.
J. Bergh and J. Löfström, Interpolation spaces: An introduction, Springer-Verlag, 1976.
O. V. Besov, On a family of function spaces: Embedding theorems and applications, Dokl. Akad. Nauk SSSR 126 (1959), 1163–1165.
O. V. Besov, On a family of functions spaces in connection with embeddings and extensions, Trudy Mat. Inst. Steklov 60 (1961), 42–81.
O. V. Besov, V. P. Il'in, and S. M. Nikol'skiĭ, Integral representations of functions and imbedding theorems, Scripta Series in Mathematics, John Wiley & Sons, 1979.
O. V. Besov, L. D. Kudrjavcev, and P. I. Lizorkin, Studies on the theory of spaces of differentiable functions of several variables, Proc. Steklov Inst. Math. 182 (1990), 73–140.
G. Boole and J. F. Moulton, A treatise on the calculus of finite differences, 2nd ed., Dover, 1960.
Ju. A. Brudnyĭ, A multidimensional analog of a theorem of Whitney, Math. USSR Sb. 11 (1970), 157–170.
V. Burenkov, Extension theorems for Sobolev spaces, The Maz'ya Anniversary Collection, Vol. 1 (J. Rossmann, P. Takáč, and G. Wildenhain, eds.), Oper. Theory Adv. Appl., vol. 109, Birkhäuser Verlag, 1999, pp. 187–200.
A. M. Caetano and S. D. Moura, Local growth envelopes of spaces of generalized smoothness: The critical case, J. Inequal. Appl. 7 (2004), 573–606.
F. Cobos and D. L. Fernandez, Hardy–Sobolev spaces and Besov spaces with a function parameter, Function Spaces and Applications (M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin, eds.), Lectures Notes in Math., vol. 1302, Springer-Verlag, 1988, pp. 158–170.
M. Cwikel and J. Peetre (eds.), Interpolation spaces and allied topics in analysis, Lectures Notes in Math., vol. 1070, Springer-Verlag, 1984.
M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin (eds.), Function spaces and applications, Lectures Notes in Math., vol. 1302, Springer-Verlag, 1988.
R. A DeVore and G. G. Lorentz, Constructive approximation, Grundlehren Math. Wiss., Springer, 1993.
D. E. Edmunds, P. Gurka, and B. Opic, On embeddings of logarithmic Bessel potential spaces, J. Funct. Anal. 146 (1997), 116–150.
D. E. Edmunds and D. Haroske, Spaces of Lipschitz type, embeddings and entropy numbers, Dissertationes Math. (Rozprawy Mat.) 380 (1999).
R. E. Edwards and G. I. Gaudry, Littlewood–Paley and Multiplier theory, Ergeb. Math. Grenzgeb., Band 90, Springer-Verlag, 1977.
W. Farkas, Function spaces of generalised smoothness and pseudo-differential operators associated to a continuous negative definite function, Habilitation thesis, Universität München, 2002.
W. Farkas and H.-G. Leopold, Characterisations of function spaces of generalised smoothness, Ann. Mat. Pura Appl. (4) 185 (2006), 1–62.
I. M. Gel'fand and G. E. Shilov, Generalized functions 1–6, AMS Chelsea Publishing, 2016.
A. Jonsson and H. Wallin, Function spaces on subsets of Rn, Math. Rep., vol. 2, Harwood Academic Publishers, 1984.
C. Jordan, Calculus of finite differences, AMS Chelsea Publishing, 3rd edition, 1965.
G. A. Kalyabin and P. I. Lizorkin, Spaces of generalized smoothness, Math. Nachr. 133 (1987), 7–32.
T. Kleyntssens and S. Nicolay, A refinement of the Sν-based multifractal formalism: Toward the detection of the law of the iterated logarithm, (submitted).
S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Exposition. Math. 1 (1983), no. 3, 193–260.
D. Kreit and S. Nicolay, Some characterizations of generalized Hölder spaces, Math. Nachr. 285 (2012), no. 17–18, 2157–2172.
D. Kreit and S. Nicolay, Characterizations of the elements of generalized Hölder–Zygmund spaces by means of their representation, J. Approx. Theory 172 (2013), 23–36.
D. Kreit and S. Nicolay, Generalized pointwise Hölder spaces defined via admissible sequences, Hindawi, Journal of Function Spaces 2018, article ID 8276258, available at https://doi.org/10.1155/2018/827658.
G. Lemarié-Reusset, Recent developments in the Navier–Stokes problem, Research Notes in Mathematics, Chapman & Hall/CRC, 2002.
H.-G. Leopold, Embeddings and entropy numbers in Besov spaces of generalized smoothness, Function Spaces (Poznań, 1998), Lectures Notes in Pure and Appl. Math., vol. 213, Dekker, 2000, pp. 323–336.
C. Merucci, Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, Interpolation Spaces and Allied Topics in Analysis (M. Cwikel and J. Peetre, eds.), Lectures Notes in Math., vol. 1070, Springer, 1984, pp. 183–201.
S. D. Moura, On some characterizations of Besov spaces of generalized smoothness, Math. Nachr. 280 (2007), no. 8–9, 1190–1199.
S. Nicolay and D. Kreit, A refined method for estimating the global Hölder exponent, Information Technology: New Generations (S. Latifi, ed.), Adv. Intell. Syst. Comput., vol. 448, Springer, 2016, pp. 949–958.
B. Opic and W. Trebels, Bessel potentials with logarithmic components and Sobolev-type embeddings, Anal. Math. 26 (2000), 299–319.
E. M. Stein, Topics in harmonic analysis related to the Littlewood–Paley theory, Ann. of Math. Stud., vol. 63, Princeton University Press, 1970.
L. Tartar, An introduction to Sobolev spaces and interpolation spaces, Lect. Notes Unione Mat. Ital., Springer Verlag, 2007.
H. Triebel, Interpolation theory, function spaces, differential operators, North Holland Publishing Company, 1978.
H. Triebel, Theory of function spaces I–III, Birkhäuser Verlag, 1983.
H. Triebel, A note on wavelet bases in function spaces, Banach Center Publ. 64 (2004), 193–206.
H. Triebel, Bases in function spaces, sampling, discrepancy, numerical integration, EMS Tracts Math., vol. 11, 2010.