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

Abstract

ABSTRACTLet μ be a non-negative Ahlfors n-dimensional measure on . In this context we shall consider convolution type operators , , where the kernels are supposed to satisfy certain size and regularity conditions. We prove Welland's type inequalities for the operator and its commutator , with , that include the case . 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 between two different weighted Lebesgue spaces when the underlying measure is Ahlfors n-dimensional.