Adaptive estimation of Sobolev-type energy functionals on the sphere

We study the estimation of quadratic Sobolev-type integral functionals of an unknown density on the unit sphere. The functional is defined through fractional powers of the Laplace--Beltrami operator and provides a global measure of smoothness and spectral energy. Our approach relies on spherical needlet frames, which yield a localized multiscale decomposition while preserving tight frame properties in the natural square-integrable function space on the sphere. We construct unbiased estimators of suitably truncated versions of the functional and derive sharp oracle risk bounds through an explicit bias--variance analysis. When the smoothness of the density is unknown, we propose a Lepski-type data-driven selection of the resolution level. The resulting adaptive estimator achieves minimax-optimal rates over Sobolev classes, without resorting to nonlinear or sparsity-based methods.