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We consider the operator,$f$(Δ) for Δ the Laplacian, on spaces of measures on the sphere in$R$^{$d$}, 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$f$(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$R$^{$d$}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.

We give the following representation theorem for a class containing quasianalytic ultradistributions and all the non-quasianalytic ultradistributions: Every ultradistribution in this class can be written as $$u = P(\Delta )g(x) + h(x)$$ where$g(x)$is a bounded continuous function,$h(x)$is a bounded real analytic function and$P(d/dt)$is an ultradifferential operator. Also, we show that the boundary value of every heat function with some exponential growth condition determines an ultradistribution in this class. These results generalize the theorem of Matsuzawa [M] for the above class of quasianalytic ultradistributions and partially solve a question of A. Kaneko [Ka]. Our interest lies in the quasianalytic case, although the theorems do not exclude non-quasianalytic classes.

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