The action of Volterra integral operators with highly singular kernels on Holder continuous, Lebesgue and Sobolev functions

For kernels zi which are positive and integrable we show that the operator g bar right arrow J(v)g = integral(x)(0) v(x-s)g(s)ds on a finite time interval enjoys a regularizing effect when applied to Holder continuous and Lebesgue functions and a "contractive" effect when applied to Sobolev functions. For Holder 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) = integral(x)(0) v(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. "shrinks" the norm of the argument by a factor that, as in the Holder 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 Z(x) = mu(x,0, -1) = integral(infinity)(0)x(s-1)/Gamma(s)ds, the latter being relevant for instance in the analysis of the Schrodinger equation with concentrated nonlinearities in R-2. (C) 2017 Elsevier Inc. All rights reserved.