Marcinkiewicz Type Theorems for Interpolation of Operators Acting on M{\"u}ntz Spaces

TL;DR

This paper extends Marcinkiewicz-type interpolation theorems to M"untz spaces, which are not dense in $L^p$, by leveraging strong decoupling of $L^p$ norms, enabling new interpolation results.

Contribution

It establishes Marcinkiewicz-type interpolation theorems for operators on M"untz spaces, overcoming the challenge of non-density in $L^p$ spaces.

Findings

Proves interpolation results for M"untz spaces.

Utilizes strong decoupling of $L^p$ norms.

Extends classical interpolation theory to non-dense subspaces.

Abstract

We prove interpolation results in the spirit of the Marcinkiewicz theorem. The operators considered in this article are defined on M\"untz spaces, which are not dense subspaces of $L^p$, and for which the classical interpolation theory cannot be applied directly. Our proofs crucially rely on strong decoupling of $L^p$ norms, a that was first observed by Gurariy-Macaev and later generalized.