Welland's type inequalities for fractional operators of convolution with kernels satisfying a Hörmander type condition and their commutators

Abstract

Let μ be a non-negative Ahlfors n-dimensional measure on Rd. In this context we shall consider convolution type operators Tαf = Kα ∗f, 0 <α< n, where the kernels Kα are supposed to satisfy certain size and regularity conditions. We prove Welland's type inequalities for the operator Tα and its commutator [b,Tα], with b ∈ BMO,that include the case Tα = Iα. As far as we know both estimates are new even in the case of the Lebesgue measure. We shall also give sufficient conditions on a pair of weights that guarantee the boundedness of [b, Tα] between two different weighted Lebesgue spaces when the underlying measure is Ahlfors n-dimensional.