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The Cameron-Storvick function space integral: An L(Lp, Lp′) theory

Published online by Cambridge University Press:  22 January 2016

G. W. Johnson  and
D. L. Skoug
G. W. Johnson
Affiliation:
University of Nebraska
D. L. Skoug
Affiliation:
University of Nebraska
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In [3] Cameron and Storvick introduced a very general operator-valued function space “integral”. In [3-5, 8, 9, 11, 13-20] the existence of this integral as an operator from L2 to L2 was established for certain functions. Recently the existence of the integral as an operator from L1 to L∞, has been studied [6, 7, 21]. In this paper we study the integral as an operator from Lp to Lp′, where 1 < p ≤ 2. The resulting theorems extend the theory substantially and indicate relationships between the L2-L2 and L1-L∞ theories that were not apparent earlier. Even in the most studied case, p = p′ = 2, the results below strengthen the theory.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 60 , February 1976 , pp. 93 - 137
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1976

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