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Integral operators of Abel type of order a > 0 arise naturally in a largespectrum of physical processes. Their inversion requires care since theresulting inverse problem is ill-posed. The purpose of this work is to deviseand analyse a family of appropriate Hilbert scales so that the operator isill-posed of order a in the scale. We provide weak regularity assumptions onthe kernel underlying the operator for the above to hold true. Our constructionleads to a well-defined regularisation strategy by Tikhonov regularisation inHilbert scales. We thereby generalise the results of Gorenflo and Yamamoto fora < 1 to arbitrary a > 0 and more general kernels. Thanks to tools frominterpolation theory, we also show that the a priori associated to the Hilbertscale formulates in terms of smoothness in usual Sobolev spaces up to boundaryconditions, and that the regularisation term actually amounts to penalisingderivatives. Finally, following the theoretical construction, we develop acomprehensive numerical approach, where the a priori is encoded in a singleparameter rather than in a full operator. Several numerical examples are shown,both confirming the theoretical convergence rates and showing the generalapplicability of the method.

Solving Abel integral equations by regularisation in Hilbert scales