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Multipliers of spherical harmonics and energy of measures on the sphere. (English) Zbl 1034.43007
Let \(\Delta \) be the Laplacian on the unit sphere in \(\mathbb R^{d}(d>2)\). If \(f\) is a complex valued function on the spectrum of \(\Delta \), the action of an operator \(f(\Delta )\) can be defined on various function spaces using the spectral theorem. In this paper the authors consider the operator \( f(\Delta )\) acting on measures on the unit sphere in \(\mathbb{R}^{d}.\) They show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied and give estimates for the \(L^{2}\)-norm of \(f(\Delta )(\mu )\) in terms of the energy of the measure \(\mu\). An analog to the classical formula relating the energy of a measure on \(\mathbb{R}^{d}\) with its Fourier transform is obtained: in this setting the formula establishes the relationship between the energy of a measure on the sphere and the size of its spherical harmonics. This result is applied to estimate the size of the coefficients of pluriharmonic measures.

MSC:
43A90 Harmonic analysis and spherical functions
28A12 Contents, measures, outer measures, capacities
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