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Semyanistyi's fractional integrals have come to analysis from integralgeometry. They take functions on $R^n$ to functions on hyperplanes, commutewith rotations, and have a nice behavior with respect to dilations. We obtainsharp inequalities for these integrals and the corresponding Radon transformsacting on $L^p$ spaces with a radial power weight. The operator norms areexplicitly evaluated. Similar results are obtained for fractional integralsassociated to $k$-plane transforms for any $1\le k<n$.

Weighted norm estimates for the Semyanistyi fractional integrals and Radon transforms