  We prove fractional Leibniz rules and related commutator estimates in the
settings of weighted and variable Lebesgue spaces. Our main tools are uniform
weighted estimates for sequences of square-function-type operators and a
bilinear extrapolation theorem. We also give applications of the extrapolation
theorem to the boundedness on variable Lebesgue spaces of certain bilinear
multiplier operators and singular integrals.

Analysis of PDEs