Weighted inequalities for integral operators on Lebesgue and \(BMO^{\gamma }(\omega )\) spaces

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Abstract

We characterize the power weights \(\omega \) for which the fractional type operator \(T_{\alpha ,\beta }\) is bounded from \(L^p (\omega ^p)\) into \(L^q (\omega ^q)\) for \(1< p < n/(n- (\alpha + \beta ))\) and \(1/q = 1/p - (n- (\alpha + \beta ))/n\). If \(n/(n-(\alpha + \beta )) \le p < n/(n -(\alpha +\beta ) -1)^{+}\) we prove that \(T_{\alpha ,\beta }\) is bounded from a weighted weak \(L^p\) space into a suitable weighted \(BMO^\delta \) space for weights satisfying a doubling condition and a reverse Hölder condition. Also, we prove the boundedness of \(T_{\alpha ,\beta }\) from a weighted local space \(BMO_{0}^{\gamma }\) into a weighted \(BMO^\delta \) space, for weights satisfying a doubling condition.

Keywords

BMO spaces Lebesgue spaces Weighted inequalities Integral operators 

Mathematics Subject Classification

30H35 42B25 42B35 

Notes

Acknowledgements

The authors wish to thank Eleonor Harboure and the referee for helpful comments and suggestions.

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Copyright information

© Universitat de Barcelona 2018

Authors and Affiliations

  1. 1.CIEM (CONICET), FaMAFUniversidad Nacional de CórdobaCórdobaArgentina

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