Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extension and boundedness of operators on Morrey spaces from
  extrapolation techniques and embeddings   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.

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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio