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Convolution Estimates and Generalized de Leeuw Theorems for Multipliers of Weak Type (1,1)

Published online by Cambridge University Press:  20 November 2018

Nakhlé Asmar ,
Earl Berkson  and
T. A. Gillespie
Nakhlé Asmar
Affiliation:
Department of Mathematics University of Missouri- Columbia Columbia, Missouri 65211 U.S.A.
Earl Berkson
Affiliation:
University of Illinois Department of Mathematics1409 West Green Street Urbana, Illinois 61801 U.S.A.
T. A. Gillespie
Affiliation:
Department of Mathematics University of EdinburghJames Clerk Maxwell Building Edinburgh EH9 3JZ Scotland
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Abstract

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In the context of a locally compact abelian group, we establish maximal theorem counterparts for weak type (1,1) multipliers of the classical de Leeuw theorems for individual strong multipliers. Special methods are developed to handle the weak type (1,1) estimates involved since standard linearization methods such as Lorentz space duality do not apply to this case. In particular, our central result is a maximal theorem for convolutions with weak type (1,1) multipliers which opens avenues of approximation. These results complete a recent series of papers by the authors which extend the de Leeuw theorems to a full range of strong type and weak type maximal multiplier estimates in the abstract setting.

Keywords

43A32
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 2 , 01 April 1995 , pp. 225 - 245
Copyright
Copyright © Canadian Mathematical Society 1995

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