A precomposition analysis of linear operators on $l^ p$

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.