Boundedness Properties of Multidimensional Fractional Hadamard-Type Operators on Campanato Spaces

Abstract

The present work aims to extend the one-dimensional Hadamard fractional operator to a general multidimensional setting and study how this newly built operator acts between two Campanato spaces, \(\mathcal{K}^{\mathrm{p},\mu}(\mathcal{D})\) and \(\mathcal{K}^{\mathrm{q},\nu}(\mathcal{D})\). Alongside the boundedness statement itself, we pin down an explicit rate at which the operator norm grows with the diameter of \(\mathcal{D}\). Two features distinguish this study from prior boundedness results for fractional-type operators. First, the Campanato scale is strictly finer than the Morrey scale on which such questions are usually posed. Second, the Hadamard operator sits inside the Katugampola family as the degenerate case \(\rho \to 0^{+}\), carrying a logarithmic kernel \(\left(\log\frac{v}{\sigma}\right)^{s-1}\) in place of the power kernels that dominate the classical theory — and it is exactly this logarithmic structure that drives the sharper estimates obtained below.