Two-weight norm inequalities for Hardy operators in generalized Morrey spaces

Source abstract

Abstract We introduce the classes $$B_{p,q} (\varphi , \psi )$$ B p , q ( φ , ψ ) and $$\mathcal {B}_{p,q} (\varphi , \psi )$$ B p , q ( φ , ψ ) of pairs ( u , v ) of weights u and v on $$\mathbb {R}^n$$ R n and prove two-weight norm estimates between two local generalized Morrey spaces for the multi-dimensional Hardy operator and its adjoint for weights in these classes. Such estimates were earlier known in the one-weight case and for radial weights. The proof is based on point-wise estimates for the Hardy operator and its adjoint via the norm of the domain space. A number of theorems on inclusion of pairs of radial weights into the classed $$B_{p,q} (\varphi , \psi )$$ B p , q ( φ , ψ ) and $$\mathcal {B}_{p,q} (\varphi , \psi )$$ B p , q ( φ , ψ ) is proved. Belonging of pairs of weights to the introduced classes proves to have a form of criterion for two-weight norm estimates in the case of classical Morrey spaces and power weights. We also apply the obtained results to two-weight study of multidimensional versions of Calderón and Stieltjes operators in Morrey spaces.