Functions with vanishing integrals over spheres centered on cones
Suppose that \({\mathbb R}^n\) is Euclidean space of dimension \(\geq 2\) with Euclidean norm \(|\cdot |\), \(d\omega_{n-1}\) is the surface measure on the sphere \({\mathbb S}^{n-1}=\{x\in{\mathbb R}^n : |x|=1\}\) induced by the Lebesgue measure, and \(\omega_{n-1}\) is the area of \({\mathbb S}^{n-1}\). By \(L^{1,loc}({\mathcal U})\) we denote the class of locally integrable functions on the domain \({\mathcal U}\subset {\mathbb R}^n\). For any function \(f\in L^{1,loc}({\mathcal U})\), any \(x\in{\mathcal U}\), and almost all \(r\in (0, dist(x, \partial{\mathcal U}))\), the spherical mean operator \[ {\mathcal R} f(x,r)=\frac1{\omega_{n-1}}\int_{{\mathbb S}^{n-1}} f(x+r\eta)\, d\omega_{n-1}(\eta) \] is defined. This operator is also known as the Radon transform on spheres. Let \(X(\mathcal U)\) be a class of locally summable functions on \(\mathcal U\). The kernel of the transform \(\mathcal R\) on the class \(X(\mathcal U)\) with respect to a set \(E\subset{\mathcal U}\) is defined as the set of functions \(f\in X(\mathcal U)\) such that \({\mathcal R} f(x,r)=0\) for all \(x\in E\) and almost all \(r\in (0, dist(x, \partial{\mathcal U}))\). The set \(E\subset{\mathcal U}\) is called the injectivity set of the transform \(\mathcal R\) if the kernel of \(\mathcal R\) with respect to \(E\) contains only the zero function. The authors study some problems connected with the spherical mean operator \(\mathcal R\) for \(X(\mathcal U)= L^{1,loc}(K_R)\) and \(E=\partial K_r\) or \(E= \partial K_{r_1}\cup \partial K_{r_2}\), where \(K_\alpha\) is the circular cone with aperture \(2\alpha\). The main results of the paper are as follows: (i) The kernel of \(\mathcal R\) with respect to \(\partial K_r\) is described for the class \(L^{1,loc}(K_R)\), and a sharp uniqueness theorem for functions from this kernel is proved. (ii) Necessary and sufficient conditions under which \(\partial K_{r_1}\cup \partial K_{r_2}\) is an injectivity set of \(\mathcal R\) for \(L^{1,loc}(K_R)\) are given, and an analogue of this result for classes of quasi-analytic functions is obtained.
scientific article; zbMATH DE number 5998394