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 operatorsMathematics Subject Classification
30H35 42B25 42B35Notes
Acknowledgements
The authors wish to thank Eleonor Harboure and the referee for helpful comments and suggestions.
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