The very interesting paper under review deals with weighted norm inequalities for fractional powers of elliptic operators and their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian domination by the classical heat operator. The authors’ method relies mainly upon good- technique which do not uses any size or smoothness estimates for the kernels.
The model example considered is the fractional power of an elliptic operator over given formally by
where with an elliptic, matrix of complex and -value coefficients. The authors obtain sufficient conditions for the weighted norm boundedness of Namely, if and then turns out to be bounded from into for where and are the standard Muckenhoupt and reverse Hölder classes, respectively. Moreover, estimates for the -th order commutators
with BMO functions are obtained. Precisely, if and then given and one has