The effect of the smoothness of fractional type operators over their commutators with Lipschitz symbols on weighted spaces
  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 $p$.
