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Spingroups and spherical means. II. (English) Zbl 0651.35073
Abstract analysis, Proc. 14th Winter Sch., SrnĂ­/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 157-177 (1987).
[For the entire collection see Zbl 0627.00012.]
[For Part I see Nato ASI Ser., Ser. C 183, 149-158 (1986; Zbl 0602.35103).] Spherical means of functions or distributions are very useful for partial differential operators, invariant under the group of Euclidean motions. Together with the Radon transform, it forms a basic special topic in integral geometry [see S. Helgason, “Groups and geometric analysis. Integral geometry, invariant differential operators and spherical functors”, Pure and Applied Mathematics, Vol. 113 (1984; Zbl 0543.58001) and F. John, “Plane waves and spherical means. Applied to partial differential equations” (1981; Zbl 0464.35001)]. The classical spherical mean Pf(x,r) of a function f(x) is known to satisfy the Darboux equation, which is the second order differential equation \[ \Delta_ xPf(x,r)=(\partial ^ 2_ r+(m-1)/r\partial_ r)Pf(x,r), \] expressing the laplacian in terms of the radial derivatives. In part I, we introduce a second spherical mean operator Qf(x,r), which takes values in a Clifford algebra and which, together with the basic spherical mean Pf(x,r), satisfies the first order Darboux system \[ D_ xPf(x,r)=(\partial r+(m-1)/r)Qf(x,r),\quad D_ xQf(x,r)=-\partial rPf(x,r), \] D being the Dirac operator. We thus obtain a refinement of the classical theory of spherical means.
In this paper, we study mean values of functions over spheres of any dimension, on which special Clifford algebra valued measures are given. In this way we obtain a collection of integral transforms satisfying generalized Darboux systems. In the case of spheres of codimension 2, we obtain a Darboux equation of the simple form \((D_ x+iD_ y)f(x,y)=0,\) \(D_ x\) and \(D_ y\) being m-dimensional Dirac operators, which shows a strong connection with complex analysis.
Our methods are based on representations of the spingroups Spin(m), and involve invariant differential operators, defined on the unit sphere and on Grassmann manifolds.
Reviewer: F.Sommen

35Q05 Euler-Poisson-Darboux equations
15A66 Clifford algebras, spinors
30G35 Functions of hypercomplex variables and generalized variables
33C55 Spherical harmonics
53C65 Integral geometry
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