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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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##### References:
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