We derive sharp weighted norm estimates for positive kernel operators on spaces of homogeneous type. Similar problems are studied for one-sided fractional integrals. Bounds of weighted norms are of mixed type. The problems are studied using the two-weight theory of positive kernel operators. As special cases, we derive sharp weighted estimates in terms of Muckenhoupt characteristics.
"Sharp weighted bounds for fractional integrals via the two-weight theory." Banach J. Math. Anal. 12 (3) 673 - 692, July 2018. https://doi.org/10.1215/17358787-2017-0063