×

zbMATH — the first resource for mathematics

Spaces of functions of generalized smoothness. (English) Zbl 0636.46033
Let \(f\in S'({\mathbb{R}}^ n\)) (tempered distribution) be decomposed by \(f=\sum^{\infty}_{k=1}f_ k\) with supp Ff\({}_ k\subset \{x|| x| \leq N_ k\}\), where F stands for the Fourier transform. Let \(\infty \geq p\geq 1\) and \(\infty \geq q\geq 1\), then the spaces under consideration are characterized by the norms \[ (\sum^{\infty}_{k=1}\alpha \quad q_ k\| f_ k| L_ p({\mathbb{R}}\quad n)\| \quad q)^{1/q}\quad and\quad \| (\sum^{\infty}_{k=1}\alpha \quad q_ k| f_ k(\cdot)| \quad q)^{1/q}| L_ p({\mathbb{R}}\quad n)\|, \] where \(\alpha_ k\) are positive numbers. If \(N_ k=2\) k and \(\alpha_ k=2^{sk}\), \(s>0\), then one obtains the two scales \(B\) \(s_{pq}({\mathbb{R}}^ n\)) and \(F\) \(s_{pq}({\mathbb{R}}^ n\)) which attracted much attention in the last two or three decades, and which cover (together with their extensions to \(p<1\), \(q<1\), \(s\leq 0)\) Hölder-Zymund, (fractional) Sobolev, Hardy, Nikol’skij-Besov spaces etc. The authors consider the above indicated generalizations of these two scales: \(N_ k\) is an increasing sequence of positive numbers, \(N_ k\to \infty\) if \(k\to \infty\), \(\alpha_ k\) are positive numbers. The paper is a survey about recent results mostly obtained in the Soviet Union in the last 10 or 15 years about this subject. Here are some key words: Fourier-analytical characterizations; descriptions via differences, means of differences, derivatives, thermic extensions; embeddings; trace of hyperplanes, capacity, spaces on domains and the extension problem.
Reviewer: H.Triebel

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Russian Language Ignored 1976
[2] Russian Language Ignored 1970, 203–262
[3] Russian Language Ignored CCCP 273 pp 1294– (1983)
[4] Russian Language Ignored CCCP 275 pp 1036– (1984)
[5] Russian Language Ignored 1975
[6] Russian Language Ignored CCCP 161 pp 3– (1983)
[7] Russian Language Ignored 1976
[8] Russian Language Ignored CCCP 56 pp 170– (1976)
[9] Russian Language Ignored CCCP 172 pp 60– (1985)
[10] Russian Language Ignored CCCP 140 pp 27– (1976)
[11] Russian Language Ignored CCCP 150 pp 24– (1979)
[12] Russian Language Ignored CCCP 56 pp 229– (1947)
[13] Russian Language Ignored 1 pp 3– (1965)
[14] Russian Language Ignored 1983
[15] Russian Language Ignored CCCP 233 pp 273– (1977)
[16] Russian Language Ignored CCP 156 pp 47– (1980)
[17] Russian Language Ignored CCCP 170 pp 86– (1984)
[18] Russian Language Ignored CCCP 172 pp 128– (1985)
[19] Russian Language Ignored 1983
[20] Russian Language Ignored 34 pp 17– (1979)
[21] Russian Language Ignored CCP 21 pp 10– (1965)
[22] Fefferman, Amer. J. Math. 93 pp 107– (1971)
[23] Banach spaces of distributions of Wiener’s type and interpolation. Proc. Conf. ”Funct. Anal. and Approximation” Oberwolfach 1980, Birkhäuser Verlag, Basel – Boston – Stuttgart, (1981) 153–165
[24] Russian Language Ignored CCCP 124 pp 164– (1983)
[25] Russian Language Ignored CCCP 227 pp 284– (1976)
[26] Russian Language Ignored CCCP 140 pp 169– (1976)
[27] Russian Language Ignored CCCP 236 pp 1056– (1977)
[28] Russian Language Ignored CCCP 232 pp 1245– (1977)
[29] Russian Language Ignored 104 pp 42– (1977)
[30] Russian Language Ignored AH CCCP, cep. mat. 41 pp 1138– (1977)
[31] Russian Language Ignored AH CCCP, cep. 42 pp 305– (1978)
[32] Russian Language Ignored CCCP 150 pp 160– (1979)
[33] Russian Language Ignored CCCP 251 pp 25– (1980)
[34] Russian Language Ignored CCCP 251 pp 274– (1980)
[35] Russian Language Ignored CCCP 156 pp 82– (1980)
[36] Russian Language Ignored 30 pp 517– (1981)
[37] Russian Language Ignored CCCP 161 pp 111– (1983)
[38] Russian Language Ignored CCCP 271 pp 795– (1983)
[39] Russian Language Ignored CCCP 172 pp 173– (1985)
[40] Russian Language Ignored CCCP 239 pp 42– (1978)
[41] Russian Language Ignored CCCP 161 pp 125– (1983)
[42] Russian Language Ignored 36 pp 304– (1984)
[43] Russian Language Ignored M. (1972) 259–265
[44] Russian Language Ignored CCCP 202 pp 21– (1972)
[45] Russian Language Ignored 63 (1964)
[46] Russian Language Ignored CCCP 170 pp 508– (1966)
[47] Russian Language Ignored CCCP 89 pp 214– (1967)
[48] Russian Language Ignored 81 pp 79– (1970)
[49] Russian Language Ignored 1971. M. 1972, 135–139
[50] Russian Language Ignored CCCP 117 pp 212– (1972)
[51] Russian Language Ignored CCCP 131 pp 158– (1974)
[52] Analysis Math. 9 pp 207– (1983)
[53] Russian Language Ignored 1977
[54] Russian Language Ignored CCCP: I 77 (1965)
[55] Russian Language Ignored CCCP: II 89 (1967)
[56] Russian Language Ignored CCCP: III 105 (1969)
[57] Russian Language Ignored CCCP: IV 117 (1972)
[58] Russian Language Ignored CCCP: V 131 (1974)
[59] Russian Language Ignored CCCP: VI 140 (1977)
[60] Russian Language Ignored CCCP: VII 150 (1979)
[61] Russian Language Ignored CCCP: VIII 156 (1980)
[62] Russian Language Ignored CCCP: IX 161 (1983)
[63] Russian Language Ignored CCCP: X 170 (1984)
[64] Russian Language Ignored CCCP: XI 172 (1985)
[65] Russian Language Ignored CCCP 38 pp 244– (1951)
[66] Russian Language Ignored 1950
[67] Shapiro, Ark. for Mat. 9 pp 91– (1971)
[68] Singular integrals and Differentiability Properties of Functions. Princeton 1970 · Zbl 0207.13501
[69] Stöckert, Math. Nachr. 89 pp 257– (1979)
[70] Strichartz, J. Math. Mech. 16 pp 1031– (1967)
[71] Russian Language Ignored 38 pp 23– (1983)
[72] Russian Language Ignored 15 pp 492– (1979)
[73] Russian Language Ignored 14 pp 83– (1980)
[74] Theory of function spaces. Leipzig, Akad.-Verl. 1983 · Zbl 1235.46002
[75] Interpolation Theory, Function Spaces, Differential Operators, Berlin 1978
[76] Triebel, Comm. Part. Diff. Equat. 5 pp 245– (1980)
[77] Triebel, J. Approx. Th. 35 pp 275– (1982)
[78] Triebel, Ann. Mat. Pura Appl. 113 pp 33– (1977)
[79] Russian Language Ignored 1 pp 405– (1967)
[80] Russian Language Ignored 32 pp 649– (1968)
[81] Russian Language Ignored 81 pp 104– (1970)
[82] Trigonometric series. 2-nd edition, Cambridge 1968
[83] Russian Language Ignored 13 pp 1109– (1972)
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.