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Density and representation theorems for multipliers of type (p, q)

Published online by Cambridge University Press:  09 April 2009

Alessandro Figà-Talamanca  and
G. I. Gaudry
Alessandro Figà-Talamanca
Affiliation:
Massachusetts Institute of Technology
G. I. Gaudry
Affiliation:
Australian National University
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Let G be a locally compact Abelian Hausdorff group (abbreviated LCA group); let X be its character group and dx, dx be the elements of the normalised Haar measures on G and X respectively. If 1 < p, q < ∞, and Lp(G) and Lq(G) are the usual Lebesgue spaces, of index p and q respectively, with respect to dx, a multiplier of type (p, q) is defined as a bounded linear operator T from Lp(G) to Lq(G) which commutes with translations, i.e. τxT = Tτx for all x ∈ G, where τxf(y) = f(x+y). The space of multipliers of type (p, q) will be denoted by Lqp. Already, much attention has been devoted to this important class of operators (see, for example, [3], [4], [7]).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 7 , Issue 1 , February 1967 , pp. 1 - 6
Copyright
Copyright © Australian Mathematical Society 1967

References

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