We consider the operator, f(Δ) for Δ the Laplacian, on spaces of measures on the sphere in Rd, show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for the L2-norm of f(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure on Rd with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.
This research was done while the first author enjoyed the hospitality of the Department of Mathematics of Göteborg University and Chalmers Institute of Technology. It was supported in part by NSERC and the Swedish natural sciences research council.
"Multipliers of spherical harmonics and energy of measures on the sphere." Ark. Mat. 41 (2) 281 - 294, October 2003. https://doi.org/10.1007/BF02390816