Extension and boundedness of operators on Morrey spaces from extrapolation techniques and embeddings
Javier Duoandikoetxea, Marcel Rosenthal

TL;DR
This paper develops a new approach to extend and analyze operators on Morrey spaces using extrapolation and embeddings, leading to broad, unified boundedness results without relying on operator-specific formulas.
Contribution
It introduces a novel embedding-based method for extending operators to Morrey spaces, avoiding traditional pointwise bounds and allowing for more general parameter ranges and non-linear operators.
Findings
Operators satisfying extrapolation hypotheses are bounded on weighted Morrey spaces.
The method applies to unweighted cases and includes weak-type inequalities.
Vector-valued inequalities and $A_ abla$-weighted inequalities are derived.
Abstract
We prove that operators satisfying the hypotheses of the extrapolation theorem for Muckenhoupt weights are bounded on weighted Morrey spaces. As a consequence, we obtain at once a number of results that have been proved individually for many operators. On the other hand, our theorems provide a variety of new results even for the unweighted case because we do not use any representation formula or pointwise bound of the operator as was assumed by previous authors. To extend the operators to Morrey spaces we show different (continuous) embeddings of (weighted) Morrey spaces into appropriate Muckenhoupt weighted spaces, which enable us to define the operators on the considered Morrey spaces by restriction. In this way, we can avoid the delicate problem of the definition of the operator, often ignored by the authors. In dealing with the extension problem through the embeddings…
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