×

Atomic decompositions of \(F^ s_{pq}\) spaces. Applications to exotic pseudodifferential and Fourier integral operators. (English) Zbl 0766.46020

The author describes an atomic decomposition for the spaces \(F^ s_{pq}\) \((-\infty<s<\infty,\;0<p\leq 1<q\leq\infty)\). Some times earlier, another atomic decomposition for these spaces was obtained by M. Frazier and B. Jawerth [Indiana Univ. Math. J. 34, 777-799 (1985; Zbl 0551.46018)] which was based on the same atoms. The decomposition is applied to several topics: traces of functions from \(F^ s_{pq}(\mathbb{R}^ n)\) on \(\mathbb{R}^{n-1}\), action of some operators (pseudo-differential, Fourier integral, singular integral) on \(F^ s_{pq}\). Precise formulations are quite lengthy.
Reviewer: N.M.Zobin (Haifa)

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47G30 Pseudodifferential operators
47G10 Integral operators

Citations:

Zbl 0551.46018
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Beals, Mem. Amer. Math. Soc. 38 pp 264– (1982)
[2] Bui, Math. Nachr. 132 pp 301– (1987)
[3] Burenkov, Inst. Steklov 150 pp 31– (1979)
[4] Coifman, Studia Math. 51 pp 269– (1974)
[5] Coifman, Some new function spaces and their applications to harmonic analysis J. Functional Analysis 62 pp 304– (1985) · Zbl 0569.42016
[6] Fefferman, Acta Math. 129 pp 137– (1972)
[7] Franke, Math. Nachr. 125 pp 29– (1986)
[8] Frazier, Indiana Univ. Math. J. 34 pp 777– (1985)
[9] , The -transform and applications to distribution spaces. In Proc. Seminar Function Spaces and Applications, Lund 1986. Lect. Notes Math. 1302, Springer, Berlin, 223–246
[10] Frazier, A discrete transform and decompositions of distribution spaces · Zbl 0716.46031
[11] Gol’dman, Mat. Zametki 25 pp 513– (1979)
[12] The analysis of linear partial differential operators. IV. Springer, Berlin 1935
[13] Jawerth, Math. Scand. 40 pp 94– (1977) · Zbl 0358.46023
[14] Latter, Studia Math. 62 pp 93– (1978)
[15] Remarques sur un théorème de J. M. Bony. Suppl. Rend. Circ. Mat. Palermo, Atti Sem. Analizi Armonica, Pisa 1980, Sér. II, n. 1, 1981
[16] Miyachi, Math. Nachr. 133 pp 135– (1987)
[17] Miyachi, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 pp 81– (1987)
[18] Approximation of functions of several variables and imbedding theorems. Springer, Berlin, Heidelberg, New York 1975
[19] Päivärinta, Z. Analysis Anwendungen 2 pp 235– (1983)
[20] Päivärinta, Math. Nachr. 138 pp 145– (1988)
[21] The trace of Besov space–a limiting case. Technical Report, Lund 1975
[22] Lp estimates for the wave equation. In Proc. Symp. ”Harmonic Analysis in Euclidean Spaces”, Amer. Math. Soc. Providence 1979, 171–174
[23] Qiu, Acta Math. Scientia 5 pp 167– (1985)
[24] Qiu, Scientia Sinica, Ser. A 29 pp 350– (1986)
[25] Runst, Annals Global Analysis Geometry 3 pp 13– (1985)
[26] Sjölin, Math. Z. 165 pp 231– (1979)
[27] Singular integrals and differentiability properties of functions. Princeton Univ. Press, Princeton 1970 · Zbl 0207.13501
[28] Taibleson, Astépisque 77 pp 67– (1980)
[29] Introduction to pseudodifferential operators and Fourier integral operators, II. Plenum Press, New York, London 1980
[30] Theory of function spaces. Birkhäuser, Boston 1983;
[31] and Akad. Verlagsgesellschaft Geest & Portig, Leipzig 1983
[32] Triebel, Jber. Deutsch. Math.-Verein 89 pp 149– (1987)
[33] Triebel, J. Approximation Theory 52 pp 162– (1988)
[34] Triebel, Z. Analysis Anwendungen 6 pp 143– (1987)
[35] Triebel, Teubner-Text Math. 103 pp 75– (1988)
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.