Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces.

This paper proves the boundedness of the generalized Riesz potentials I ρ , μ , τ ⁢ f {I_{\rho,\mu,\tau}f} of functions in the Morrey space L (1 , φ ; κ) ⁢ (G) {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is X) {X)} , as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on x ∈ X {x\in X}. An example in the end of this article convincingly explains why the modification parameter τ must be introduced. [ABSTRACT FROM AUTHOR]