A note on the Blaschke-Petkantschin formula, Riesz distributions, and drury's identity

The Blaschke-Petkantschin formula is a variant of the polar decomposition of the k-fold Lebesgue measure on R ⁿ in terms of the corresponding measures on k-dimensional linear subspaces of R ⁿ . We suggest a new elementary proof of this famous formula and discuss its connection with Riesz distributions associated with fractional powers of the Cayley-Laplace operator on matrix spaces. Another application of our proof is the celebrated Drury identity that plays a key role in the study of mapping properties of the Radon-John k-plane transforms. Our proof gives precise meaning to the constants in Drury's identity and to the class of admissible functions.