A Precomposition Analysis of Linear Operators on lp

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 Lpspaces. 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 Lpof 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.