Spherical associated homogeneous distributions on Rn
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A structure theorem for spherically symmetric associated homogeneous distributions (SAHDs) based on Rn is given. It is shown that any SAHD is the pullback, along the function |x|λ,\ λ∈C, of an associated homogeneous distribution (AHD) on R. The pullback operator is found not to be injective and its kernel is derived (for λ=1). Special attention is given to the basis SAHDs, Dmz|x|z, which become singular when their degree of homogeneity z=−n−2p, ∀p∈N. It is shown that (Dmz|x|z)z=−n−2p are partial distributions which can be non-uniquely extended to distributions ((Dmz|x|z)e)z=−n−2p and explicit expressions for their evaluation are derived. These results serve to rigorously justify distributional potential theory in Rn.
CitationFranssens, G.R. (2010). Spherical associated homogeneous distributions on Rn. , Bulletin of the Belgian Mathematical Society - Simon Stevin, Vol. 17, Issue 5 SUPPL., 781-806,