Spherical means, distributions and convolution operators in Clifford analysis (Q1425574)

There are considered Clifford algebra-valued distributions, that is, left or right linear functionals on the spaces of Clifford algebra-valued test functions. In a previous paper the authors introduced three families of distributions which are interrelated by the action (on the left or on the right) of the Dirac operator. For the specific values of the parameters determining those distributions, the latter turns into known kernel functions in harmonic and Clifford analysis, thus illustrating the unifying character of the approach, for instance, one of them reduces to the higher dimensional analog of the ``Principal Value'' distribution on the real line and the kernel for the Hilbert transform; the others two reduce to the fundamental solutions of the Dirac and the Laplace operators, etc. In the paper, the fourth family of distributions is constructed for which threefold motivation is given and its importance for constructing new distributions generalizing the principal value distributions is explained.