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An h-set is a nonempty compact subset of the Euclidean n-space which supports a finite Radon measure for which the measure of balls centered on the subset is essentially given by the image of their radius by a suitable function h. In most cases of interest such a subset has Lebesgue measure zero and has a fractal structure. Let ω be a bounded C∞ domain in \documentclass{article}\begin{document}$\mathbb R̂n $\end{document} with Γ ⊂ ω. Letwhere (-δ)-1 is the inverse of the Dirichlet Laplacian in ω and trΓ is, say, trace type operator. The operator B, acting in convenient function spaces in ω, is studied. Estimations for the eigenvalues of B are presented, and generally shown to be dependent on h, and the smoothness of the associated eigenfunctions is discussed. Some results on Besov spaces of generalised smoothness on \documentclass{article}\begin{document}${{\bb R}̂n} $\end{document} and on domains which were obtained in the course of this work are also presented, namely pointwise multipliers, the existence of a universal extension operator, interpolation with function parameter and mapping properties of the Dirichlet Laplacian.