Harmonic analysis for functors on categories of Banach spaces of distributions

Donaldson, Thomas

doi:10.1090/S0002-9947-1973-0440365-6

Harmonic analysis for functors on categories of Banach spaces of distributions
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Trans. Amer. Math. Soc. 185 (1973), 1-82 Request permission

Abstract:

This paper develops a theory of harmonic analysis (Fourier series, approximation, convolution, and singular integrals) for a general class of Banach function or distribution spaces. Continuity of singular convolution operators and convergence of trigonometric series is shown with respect to the norms of all the spaces in this class; the maximal class supporting a theory of the type developed in this paper is characterized (for other classes other theories exist). Theorems are formulated in category language throughout. However only elementary category theory is needed, and for most results the notions of functor and natural mapping are sufficient.

References

A. Benedek, A.-P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365. MR 133653, DOI 10.1073/pnas.48.3.356
R. G. Bartle, A general bilinear vector integral, Studia Math. 15 (1956), 337–352. MR 80721, DOI 10.4064/sm-15-3-337-352
A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 167830, DOI 10.4064/sm-24-2-113-190
A.-P. Calderón and A. Zygmund, Singular integral operators and differential equations, Amer. J. Math. 79 (1957), 901–921. MR 100768, DOI 10.2307/2372441
Clifford V. Comisky, Multipliers of Banach modules, Nederl. Akad. Wetensch. Proc. Ser. A 74=Indag. Math. 33 (1971), 32–38. MR 0279585
C. Corduneanu, Admissibility with respect to an integral operator and applications, Advances in differential and integral equations (Conf. Qualitative Theory of Nonlinear Differential and Integral Equations, Univ. Wisconsin, Madison, Wis., 1968; in memoriam Rudolph E. Langer (1894–1968)), Studies in Appl. Math., No. 5, Soc. Indust. Appl. Math., Philadelphia, Pa., 1969, pp. 55–63. MR 0377600

Singular integrals in Banach function spaces

Jean Dieudonné, Sur les espaces de Köthe, J. Analyse Math. 1 (1951), 81–115 (French). MR 41347, DOI 10.1007/BF02790084
N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
R. E. Edwards, Fourier series: A modern introduction. Vol. I, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1967. MR 0216227
Alessandro Figà-Talamanca and G. I. Gaudry, Density and representation theorems for multipliers of type $(p,\,q)$, J. Austral. Math. Soc. 7 (1967), 1–6. MR 0209770
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 75539
Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
J. J. Kohn and L. Nirenberg, An algebra of pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 269–305. MR 176362, DOI 10.1002/cpa.3160180121
M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron. MR 0126722
G. G. Lorentz, Bernstein polynomials, Mathematical Expositions, No. 8, University of Toronto Press, Toronto, 1953. MR 0057370
G. G. Lorentz, Some new functional spaces, Ann. of Math. (2) 51 (1950), 37–55. MR 33449, DOI 10.2307/1969496
G. G. Lorentz, On the theory of spaces $\Lambda$, Pacific J. Math. 1 (1951), 411–429. MR 44740
Wilhelmus Anthonius Josephus Luxemburg, Banach function spaces, Technische Hogeschool te Delft, Delft, 1955. Thesis. MR 0072440
José Luis Massera and Juan Jorge Schäffer, Linear differential equations and function spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York-London, 1966. MR 0212324
S. G. Mikhlin, Multidimensional singular integrals and integral equations, Pergamon Press, Oxford-New York-Paris, 1965. Translated from the Russian by W. J. A. Whyte; Translation edited by I. N. Sneddon. MR 0185399
B. S. Mitjagin and A. S. Švarc, Functors in categories of Banach spaces, Uspehi Mat. Nauk 19 (1964), no. 2 (116), 65–130 (Russian). MR 0166593
Richard O’Neil, Fractional integration in Orlicz spaces. I, Trans. Amer. Math. Soc. 115 (1965), 300–328. MR 194881, DOI 10.1090/S0002-9947-1965-0194881-0
Richard O’Neil, Integral transforms and tensor products on Orlicz spaces and $L(p,\,q)$ spaces, J. Analyse Math. 21 (1968), 1–276. MR 626853, DOI 10.1007/bf02787670
Jaak Peetre, On the theory of ${\cal L}_{p},\,_{\lambda }$ spaces, J. Functional Analysis 4 (1969), 71–87. MR 0241965, DOI 10.1016/0022-1236(69)90022-6
Jaak Peetre, On convolution operators leaving $L^{p,}\,^{\lambda }$ spaces invariant, Ann. Mat. Pura Appl. (4) 72 (1966), 295–304. MR 209917, DOI 10.1007/BF02414340
W. Pogorzelski, Integral equations and their applications. Vol. I, International Series of Monographs in Pure and Applied Mathematics, Vol. 88, Pergamon Press, Oxford-New York-Frankfurt; PWN—Polish Scientific Publishers, Warsaw, 1966. Translated from the Polish by Jacques J. Schorr-Con, A. Kacner and Z. Olesiak. MR 0201934
Marc A. Rieffel, Induced Banach representations of Banach algebras and locally compact groups, J. Functional Analysis 1 (1967), 443–491. MR 0223496, DOI 10.1016/0022-1236(67)90012-2
Marc A. Rieffel, Multipliers and tensor products of $L^{p}$-spaces of locally compact groups, Studia Math. 33 (1969), 71–82. MR 244764, DOI 10.4064/sm-33-1-71-82
Juan Jorge Schäffer, Function spaces with translations, Math. Ann. 137 (1959), 209–262. MR 101476, DOI 10.1007/BF01343353
Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
J. Schwartz, A remark on inequalities of Calderon-Zygmund type for vector-valued functions, Comm. Pure Appl. Math. 14 (1961), 785–799. MR 143031, DOI 10.1002/cpa.3160140410
L. Schwartz, Théorie des distributions. Tome I, Publ. Inst. Math. Univ. Strasbourg, vol. 9, Hermann & Cie, Paris, 1950 (French). MR 0035918
Laurent Schwartz, Théorie des distributions à valeurs vectorielles. I, Ann. Inst. Fourier (Grenoble) 7 (1957), 1–141 (French). MR 107812
R. T. Seeley, Singular integrals and boundary value problems, Amer. J. Math. 88 (1966), 781–809. MR 209915, DOI 10.2307/2373078
Harold S. Shapiro, A Tauberian theorem related to approximation theory, Acta Math. 120 (1968), 279–292. MR 239332, DOI 10.1007/BF02394612
Victor L. Shapiro, Fourier series in several variables, Bull. Amer. Math. Soc. 70 (1964), 48–93. MR 158222, DOI 10.1090/S0002-9904-1964-11026-0
A. S. Švarc, Duality of functors, Dokl. Akad. Nauk SSSR 148 (1963), 288–291 (Russian). MR 0168629
Mitchell H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean $n$-space. I. Principal properties, J. Math. Mech. 13 (1964), 407–479. MR 0163159
Mitchell H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean $n$-space. II. Translation invariant operators, duality, and interpolation, J. Math. Mech. 14 (1965), 821–839. MR 0180857
A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47–76. MR 12691
A. Zygmund, On the preservation of classes of functions, J. Math. Mech. 8 (1959), 889-895; erratum 9 (1959), 663. MR 0117498, DOI 10.1512/iumj.1960.9.59040

Trigonometric series

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