科研之友
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摘要
In this paper we develop a harmonic analysis associated to the differential operators
Delta(theta) := 1-vertical bar x vertical bar(2)/4{(1 - vertical bar x vertical bar(2)) Sigma(n)(j=1) partial derivative(2)/partial derivative x(j)(2) -2 theta Sigma(n)(j=1) x(j) partial derivative/partial derivative x(j) + theta(2 - n - theta)}
in a parallel way to that on real hyperbolic space. We make a detailed study of the generalized Helgason-Fourier transform and the theta-spherical transform associated to these differential operators. In particular, we obtain the inversion formula and the Plancherel theorem for them. As an application, we solve the relevant heat equation.
