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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3081–3108 | Cite as

Extension and Boundedness of Operators on Morrey Spaces from Extrapolation Techniques and Embeddings

  • Javier DuoandikoetxeaEmail author
  • Marcel Rosenthal
Article

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 \(A_1\) weighted \(L_p\) 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 (instead of using duality), one is neither restricted in the parameter range of the p’s (in particular \(p=1\) is admissible and applies to weak-type inequalities) nor the operator has to be linear. Another remarkable consequence of our results is that vector-valued inequalities in Morrey spaces are automatically deduced. On the other hand, we also obtain \(A_\infty \)-weighted inequalities with Morrey quasinorms.

Keywords

Morrey spaces Embeddings Muckenhoupt weights Extrapolation Calderón–Zygmund operators Mihlin–Hörmander multipliers Rough operators 

Mathematics Subject Classification

Primary 42B35 46E30 42B15 42B20 Secondary 42B25 

Notes

Acknowledgements

Javier Duoandikoetxea is supported by the Grants MTM2014-53850-P of the Ministerio de Economía y Competitividad (Spain) and Grant IT-641-13 of the Basque Government. Marcel Rosenthal is supported by the German Academic Exchange Service (DAAD).

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Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Departamento de Matemáticas/Matematika sailaUniversidad del País Vasco/Euskal Herriko UnibertsitateaBilbaoSpain

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