×

zbMATH — the first resource for mathematics

Atomic and subatomic decompositions in anisotropic function spaces. (English) Zbl 0954.46021
Let \(1< p< \infty\) and \(\overline s= (s_1,\dots, s_n)\in \mathbb{N}^n\). Then the anisotropic Sobolev space \(W^{\overline s}_p(\mathbb{R}^n)\) in \(\mathbb{R}^n\) is the collection of all \(f\in L_p(\mathbb{R}^n)\) such that \({\partial^{s_k}f\over\partial x^{s_k}_k}\in L_p(\mathbb{R}^n)\), where \(k= 1,\dots, n\). Anisotropic function spaces such as classical and fractional Sobolev spaces \(H^{\overline s}_p(\mathbb{R}^n)\) and Besov spaces \(B^{\overline s}_{pq}(\mathbb{R}^n)\) have been treated, especially by the Russian school, since the early sixties parallel to the isotropic spaces. In the eighties and nineties new far-reaching tools for isotropic spaces \(B^s_{pq}(\mathbb{R}^n)\) and \(F^s_{pq}(\mathbb{R}^n)\), now for all \(s\in\mathbb{R}\), \(0< p\leq\infty\), \(0< q\leq\infty\), have been developed. The key words are local means, atomic and, most recently, quarkonial decompositions. The aim of the paper is to raise the theory of anisotropic spaces at the same level as it is now available for isotropic spaces.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B25 Maximal functions, Littlewood-Paley theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] and : Function Spaces and Potential Theory, Springer-Verlag, Berlin, Heidelberg, New York, 1996
[2] Berkolaiko, Trudy Mat. Inst. Steklov 204 pp 35– (1993)
[3] Berkolaiko, Trudy Mat. Inst. Steklov 210 pp 5– (1995)
[4] : Il’in, V. P., and Nikol’skij, S. M.: Integral Representations of Functions and Embedding Theorems (Russian), Nauka, Moskva, 1975
[5] Dintelmann, Math. Nachr. 173 pp 115– (1995)
[6] : On Fourier Multipliers between Anisotropic Weighted Function Spaces (German), Ph. D. Thesis, TH Darmstadt, 1995
[7] and : Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, 1996
[8] Fefferman, Amer. Journ. Math. 93 pp 107– (1971)
[9] Frazier, Indiana Univ. Math. J. 34 pp 777– (1985)
[10] Frazier, J. Funct. Anal. 93 pp 34– (1990)
[11] Frazier, CBMS-AMS Regional Conf. Ser. 79 (1991) · doi:10.1090/cbms/079
[12] Johnsen, Math. Nachr. 175 pp 85– (1995)
[13] Leopold, Z. Anal. Anw. 5 pp 409– (1986)
[14] : Approximation of Functions of Several Variables and Embedding Theorems (Russian), Second ed., Nauka, Moskva, 1977. English translation of the first edition: Springer-Verlag, Berlin, Heidelberg, New York, 1975
[15] Peetre, Ark. Math. 13 pp 123– (1975)
[16] Peetre, Ark. Math. 14 pp 299– (1976)
[17] and : Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, de Gruyter, 1996
[18] and : Topics in Fourier Analysis and Function Spaces, Geest & Portig, Leipzig, 1987, Wiley, Chichester, 1987 · Zbl 0661.46025
[19] Seeger, Banach Centre Publ. 22 pp 391– (1989)
[20] Sickel, Forum Math. 2 pp 451– (1990)
[21] Soardi, Lect. Notes in Math. 992 pp 115– (1993)
[22] Stein, Bull. Amer. Math. Soc. 84 pp 1239– (1978)
[23] Stöckert, Math. Nachr. 89 pp 247– (1979)
[24] Triebel, Teubner Texte Math. 7 (1977)
[25] : Interpolation Theory, Function Spaces, Differential Operators, North Holland Publ. Comp. 1978, Amsterdam, New York, Oxford; VEB Deutscher Verlag der Wissenschaften, Berlin, 1978
[26] : Theory of Function Spaces, Geest & Portig, Leipzig, 1983
[27] : Theory of Function Spaces II, Birkhäuser, 1992
[28] : Fractals and Spectra, Birkhäuser, 1997
[29] Triebel, Math. Z. 221 pp 647– (1996)
[30] Triebel, Studia Math. 121 pp 149– (1996)
[31] Yamazaki, J. Fac. Sci. Univ. Tokyo, Sect. I. A, Math. 33 pp 131– (1986)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.