The action of Volterra integral operators with highly singular kernels on H\"older continuous, Lebesgue and Sobolev functions

Abstract

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 H\"older continuous and Lebesgue functions and a "contractive" effect when applied to Sobolev functions. For H\"older 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 $N(x)=\int_0^x \nu(s)ds$. 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 H\"older 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 $\mathcal{I}(x) = \mu(x,0,-1) = \int_{0}^{\infty}x^{s-1}/\Gamma(s)\,ds$, the latter being relevant for instance in the analysis of the Schr\"odinger equation with concentrated nonlinearities in $\mathbb{R}^{2}$.