A PRECOMPOSITION ANALYSIS OF LINEAR-OPERATORS ON LP

Given a function g, the operator that sends the function closed-integral (x) to the function closed-integral g(x)) is called a precomposition opertator. 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.