The Radon transformation on the spheres in the two-point homogenous spaces (Q2901715)

The present paper extends some earlier author's results from the space \({\mathbb R}^n\) to more wide family of spaces. The description of the class of functions having zero integrals over the balls centered at the fixed sphere in the two-point homogenous space is obtained. More detail. A two-point homogeneous space is defined as Riemann manifold with a distance \(d(\cdot, \cdot)\) and having the following property. For every pair of points \(x_1, x_2\in X\) and \(y_1, y_2\in X\) such that \(d(x_1, x_2)=d(y_1, y_2)\) there exists the isometry transforming \(x_1\) to \(y_1\) and \(x_2\) to \(y_2\). The Radon transformation on the spheres in \(X\) is defined as \((M^r f)(x)=\frac{1}{A(r)}\int\limits_{S_r(x)}f(s)d\omega(s),\) \(r\geq 0,\) where \(S_r(x)=\{y\in X: d(x, y)=r\}\) and \(A(r)\) be the area of it. Let \(B_R=\{y\in X: d(0, y)<r\},\) then for given \(0< R< \text{ diam\,} X\) and \(r\in (0, R)\) to denote by \(\nu_r(B_R)\) the class of all functions \(f\in L^{1, loc}(B_R)\) such that \((M^t f)(x)=0\) for all \(x\in S_r(0) \) at almost every \(t\in (0, R-r)\). The main authors' result is Theorem 1, which states the criterion of the belonging of the function \(f\in L^{1, loc}(B_R)\) to \(\nu_r(B_R)\) in the two-point homogeneous space through some representation of the distributions \(f^{k,m,g},\) connecting with the Fourier sums of \(f,\) by the sum of the special type.