Injectivity of the Spherical Mean Operator on the Conical Manifolds of Spheres

Let f be a continuous function on ⁿ . If f has zero integral over every sphere intersecting a given subset A of ⁿ and A lies in no affine plane of dimension n -2, then f vanishes identically. The condition on the dimension of A is sharp.     

LOCALIZATION OPERATORS ASSOCIATED WITH THE SPHERICAL MEAN OPERATOR

In this paper, we define the localization operator associated with the spherical mean operator, and show that the localization operator is not only bounded, but also in Schatten-Von Neumann class. We also give a trace formula when the symbol function is a nonnegative function.