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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A precomposition analysis of linear operators on $ l\sp p$


Author: Steve M. Hudson
Journal: Proc. Amer. Math. Soc. 111 (1991), 227-233
MSC: Primary 47B37; Secondary 47B38
MathSciNet review: 1031666
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Abstract: Given a function $ g$, the operator that sends the function $ f(x)$ to the function $ f(g(x))$ is called a precomposition operator. If $ g$ preserves measure on its domain, at least approximately, then this operator is bounded on all the $ {L^p}$ spaces. We ask which operators can be written as an average of precomposition operators. We give sufficient, almost necessary conditions for such a representation when the domain is a finite set. The class of operators studied approximate many commonly used positive operators defined on $ {L^p}$ of the real line, such as maximal operators.

A major tool is the combinatorial theorem of distinct representatives, commonly called the marriage theorem. A strong connection between this theorem and operators of weak-type 1 is demonstrated.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1031666-8
PII: S 0002-9939(1991)1031666-8
Article copyright: © Copyright 1991 American Mathematical Society