Summary: For kernels \(\nu\) which are positive and integrable we show that the operator \(g \mapsto J_\nu g = \int_0^x \nu(x - s) g(s) ds\) on a finite time interval enjoys a regularizing effect when applied to Hölder continuous and Lebesgue functions and a “contractive” effect when applied to Sobolev functions. For Hölder continuous functions, we establish that the improvement of the regularity of the modulus of continuity is given by the integral of the kernel, namely by the factor \(N(x) = \int_0^x \nu(s) d s\). For functions in Lebesgue spaces, we prove that an improvement always exists, and it can be expressed in terms of Orlicz integrability. Finally, for functions in Sobolev spaces, we show that the operator \(J_\nu\) “shrinks” the norm of the argument by a factor that, as in the Hölder case, depends on the function \(N\) (whereas no regularization result can be obtained).
These results can be applied, for instance, to Abel kernels and to the Volterra function \(\mathcal{I}(x) = \mu(x, 0, - 1) = \int_0^\infty x^{s - 1} /\Gamma(s) d s\), the latter being relevant for instance in the analysis of the Schrödinger equation with concentrated nonlinearities in \(\mathbb{R}^2\).