The One Radius Theorem for the Bessel Convolution Operator and its Applications

It is well known that any function f ∈ C(Rn) (n - 2) which has zero integrals over all balls and spheres with fixed radius r is identically zero. In this paper, we study a similar phenomenon for ball and spherical means with respect to the α-convolution of Bessel. Let α ∈ (−1/2,+∞), let L1,loc ♮,α (−R,R) be the class of even locally summable functions with respect to the measure dμα(x) = |x|2α+1dx on the interval (−R,R), and let f α⋆ g be the Bessel convolution of a function f ∈ L1,loc ♮,α (−R,R) and an even distribution g on R with support in (−R,R). The main result of the article provides a solution to the problem of injectivity of the operator f → (f α⋆ χr, f α⋆ δr), f ∈ L1,loc ♮,α (−R,R), 0 < r < R, where χr is the indicator of the segment [−r, r] and δr is an even measure that maps an even continuous function ϕ on R to the number ϕ(r). Based on the technique associated with classical orthogonal polynomials and recent research by the authors it is shown that for R - 2r, the kernel of the specified operator is zero. For r < R < 2r, it consist of functions f ∈ L1,loc ♮,α (−R,R) that are zero in the interval (2r − R,R) and have a zero integral (with respect to the measure dμα) over the interval (0, 2r − R). This result allowed us to obtain a new criterion of the closure for the system of generalized Bessel shifts of segment indicators in the space Lp ♮,α(−R,R), 1 6 p < ∞, as well as a new uniqueness theorem for solutions of the Cauchy problem for the generalized Euler–Poisson–Darboux equation.