Weighted generalized Hölder spaces as wellposedness classes for Sonine integral equations
A special class of Volterra integral equations of the first kind \[ K \varphi := \int_0^x k(x-t) \varphi(t) \,dt = f(x),\quad x \in(0, b), \] where \(0 < b < \infty\), is studied in the case of almost decreasing kernels \(k(t)\) which are assumed to satisfy the condition that there exists another kernel \(l(x)\) such that \[ \int_0^x l(x-t) k(t)\, dt \equiv 1,\quad x>0 \] almost everywhere. Such condition being known as the Sonine condition due to the paper by \textit{N. Sonine} [Sur la generalization d'une formula d'Abel, Acta Math. IV. 171--176 (1884; JFM 16.0354.01)]. The corresponding kernel \(k(t)\) is called Sonine kernel. In the present paper, weighted estimates of the continuity moduli \(\omega (K\varphi, h)\) and \( \omega (K^{-1}f, h)\) are derived. Based on these estimates the authors demonstrate that the weighted generalized Hölder spaces \(H^{\omega}(\rho)\) and \(H^{\omega_1}(\rho)\) are suitable well-posedness classes for these integral equations of the first kind under the choice \(\omega_1(h)= h k(h) \omega(h)\).
almost decreasing and almost increasing functions
Volterra integral equations of the first kind
scientific article; zbMATH DE number 5509276
scientific article