Inversion of a convolution operator associated with spherical means

An obvious property of an arbitrary nonzero smooth antiperiodic function is that its derivative has no corresponding period. In other words, if r is a fixed positive number, f(x+r)+f(x-r)=0 and f′(x+r)-f′(x-r)=0 on the real axis, then f=0. This fact admits non-trivial generalizations to multidimensional spaces. One general method for such generalizations is the following Brown-Schreiber-Taylor theorem on spectral analysis: any non-zero subspace U in C(Rn) invariant under all motions of Rn contains for some λ∈C, the radial function (λ|x|)1-n2Jn2-1(λ|x|), where Jν is the Bessel function of the first kind of order ν. In particular, if a function f∈C1(Rn) and its normal derivative have zero integrals over all spheres of fixed radius r in Rn, then f=0. In terms of convolution, this means that the operator Pf=(f∗Δχr,f∗σr), f∈C(Rn), is injective, where Δ is the Laplace operator, χr is the indicator of the ball Br={x∈Rn:|x|