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, Kp,µ (D) and Kq,ν (D). Alongside the boundedness statement itself, we pin down an explicit rate at which the operator norm grows with the diameter of 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 ρ → 0+, carrying a logarithmic kernel (log v σ)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.

In this work, we establish several new variants of the Hermite–Hadamard integral inequality, together with various product-type extensions, by employing the Prabhakar fractional integral operator with three-parameter Mittag-Leffler kernels for mappings defined on harmonic sets. This unified operator generalizes several classical fractional operators, including Riemann–Liouville, Erdélyi–Kober, and Weyl, which are recovered as special cases. The results obtained herein constitute natural generalizations and significant improvements of existing results developed using classical integral operators. To demonstrate the validity and effectiveness of the obtained results, we present nontrivial numerical examples supported by graphical illustrations and tabulated comparisons for different choices of the fractional parameters. Furthermore, as an application of our main results, we derive explicit error estimates for the trapezoidal rule on harmonic intervals.