Title: The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces,
Abstract: We prove boundedness results for integral operators of fractional type and
their higher order commutators between weighted spaces, including $L^p$-$L^q$,
$L^p$-$BMO$ and $L^p$-Lipschitz estimates. The kernels of such operators
satisfy certain size condition and a Lipschitz type regularity, and the symbol
of the commutator belongs to a Lipschitz class. We also deal with commutators
of fractional type operators with less regular kernels satisfying a
Hörmander's type inequality. As far as we know, these last results are new
even in the unweighted case. Moreover, we give a characterization result
involving symbols of the commutators and continuity results for extreme values
of $p$.