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

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.