Product-convolution operators and mixed-norm spaces
Authors:
Robert C. Busby and Harvey A. Smith
Journal:
Trans. Amer. Math. Soc. 263 (1981), 309-341
MSC:
Primary 43A15; Secondary 44A35, 47B38
DOI:
https://doi.org/10.1090/S0002-9947-1981-0594411-4
MathSciNet review:
594411
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Abstract | References | Similar Articles | Additional Information
Abstract: Conditions for boundedness and compactness of product-convolution operators on spaces
are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces
interpolating the
spaces in a natural way
. It is then natural to study the operators acting between
spaces, where
has a compact invariant neighborhood. The theory of
is developed and boundedness and compactness conditions of a nonclassical type are obtained. It is demonstrated that the results extend easily to a somewhat broader class of integral operators. Several known results are strengthened or extended as incidental consequences of the investigation.
- [1] A. Benedek and R. Panzone, The space 𝐿^{𝑝}, with mixed norm, Duke Math. J. 28 (1961), 301–324. MR 126155
- [2] Robert C. Busby, Irwin Schochetman, and Harvey A. Smith, Integral operators and the compactness of induced representations, Trans. Amer. Math. Soc. 164 (1972), 461–477. MR 295099, https://doi.org/10.1090/S0002-9947-1972-0295099-7
- [3] Robert C. Busby and Irwin Schochetman, Compact induced representations, Canadian J. Math. 24 (1972), 5–16. MR 293418, https://doi.org/10.4153/CJM-1972-002-3
- [4] N. Dunford and J. T. Schwartz, Linear operators. I, Interscience, New York, 1958.
- [5] W. R. Emerson and F. P. Greenleaf, Covering properties and Følner conditions for locally compact groups, Math. Z. 102 (1967), 370–384. MR 220860, https://doi.org/10.1007/BF01111075
- [6] Siegfried Grosser and Martin Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1–40. MR 284541, https://doi.org/10.1515/crll.1971.246.1
- [7] E. Hewitt and K. A. Ross, Abstract harmonic analysis. I, Academic Press, New York, 1963. MR 28 #158.
- [8] Finbarr Holland, Harmonic analysis on amalgams of 𝐿^{𝑝} and 1^{𝑞}, J. London Math. Soc. (2) 10 (1975), 295–305. MR 374817, https://doi.org/10.1112/jlms/s2-10.3.295
- [9] Finbarr Holland, On the representation of functions as Fourier transforms of unbounded measures, Proc. London Math. Soc. (3) 30 (1975), 347–365. MR 0397295, https://doi.org/10.1112/plms/s3-30.3.347
- [10] C. N. Kellogg, An extension of the Hausdorff-Young theorem, Michigan Math. J. 18 (1971), 121–127. MR 280995
- [11] Konrad Knopp, Infinite sequences and series, Dover Publications, Inc., New York, 1956. Translated by Frederick Bagemihl. MR 0079110
- [12] W. A. J. Luxemburg and A. C. Zaanen, Compactness of integral operators in Banach function spaces, Math. Ann. 149 (1962/63), 150–180. MR 145374, https://doi.org/10.1007/BF01349240
- [13] Richard D. Mosak, Central functions in group algebras, Proc. Amer. Math. Soc. 29 (1971), 613–616. MR 279602, https://doi.org/10.1090/S0002-9939-1971-0279602-3
- [14] Neil W. Rickert, Convolution of 𝐿^{𝑝} functions, Proc. Amer. Math. Soc. 18 (1967), 762–763. MR 216301, https://doi.org/10.1090/S0002-9939-1967-0216301-7
- [15] Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
- [16] James Stewart, Unbounded positive definite functions, Canadian J. Math. 21 (1969), 1309–1318. MR 251461, https://doi.org/10.4153/CJM-1969-143-4
- [17] Lynn R. Williams, Generalized Hausdorff-Young inequalities and mixed norm spaces, Pacific J. Math. 38 (1971), 823–833. MR 310555
- [18] J.-P. Bertrandias, C. Datry, and C. Dupuis, Unions et intersections d’espaces 𝐿^{𝑝} invariantes par translation ou convolution, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 2, v, 53–84 (French). MR 499586
- [19] Hans G. Feichtinger, On a class of convolution algebras of functions, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 3, vi, 135–162 (English, with French summary). MR 470610
- [20] Hans G. Feichtinger, Banach convolution algebras of functions. II, Monatsh. Math. 87 (1979), no. 3, 181–207. MR 536089, https://doi.org/10.1007/BF01303075
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1981-0594411-4
Article copyright:
© Copyright 1981
American Mathematical Society