A note on the boundedness of the multidimensional Katugampola operator in Campanato spaces

Abstract

The aim of this paper is to extend the classical Katugampola fractional operator to the multidimensional setting and establish its boundedness from the Campanato space L^{p,\beta}(\Omega) to another Campanato space L^{q,\mu}(\Omega). We also derive estimates describing the dependence of the operator norm on the diameter of \Omega. Our results extend and improve existing boundedness results for fractional operators in function spaces in two directions: first, Campanato spaces provide a finer framework than Morrey spaces; second, the Katugampola fractional operator generalizes both the Riemann--Liouville operator (\rho = 1) and the Hadamard operator (\rho \to 0^{+}), leading to broader and more general results.