Multipliers of spherical harmonics and energy of measures on the sphere

We consider the operator,f(Δ) for Δ the Laplacian, on spaces of measures on the sphere inR ^d , show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for theL²-norm off(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 onR ^d 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.