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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Harmonic analysis for functors on categories of Banach spaces of distributions


Author: Thomas Donaldson
Journal: Trans. Amer. Math. Soc. 185 (1973), 1-82
MSC: Primary 46F99; Secondary 46M15
DOI: https://doi.org/10.1090/S0002-9947-1973-0440365-6
MathSciNet review: 0440365
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0440365-6
Keywords: General Banach function space, $ {L^p}$ space, Orlicz space, Lorentz space, O'Neil space, Sobolev space, Lipschitz spaces, Fourier series, singular integrals, approximation, inverse theorems, multipliers, Calderón-Zygmund operators, interpolation functors, Hölder functor, Banach space categories
Article copyright: © Copyright 1973 American Mathematical Society