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

Abstract

Abstract. For kernels ν which are positive and integrable we show that the operator g → Jν g = ν(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) = x 0 ν(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 Jν "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 functions−1 /Γ(s) ds, the latter being relevant for instance in the analysis of the Schrödinger equation with concentrated nonlinearities in R 2 .

Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one

“…This entails that "solving the nonlinear analogue of (10)" means searching for a solution of (19) such that (21) satisfies (16). is the unique solution of (16) (where the former is meant as an equality in L 2 pRq).…”

“…and hence, if ν is sufficiently small, then Epu ν q ă 0. Before stating the second result, we recall that a standing wave is a function ψ ω pt, xq " e ıωt u ω pxq, with ω P R and 0 ı u ω P DpH n s q, that satisfies (16), namely such that…”

Solvable models of quantum beating

“…The problem was extensively discussed in [4], where it was proved that, if ψ(t, ±a) are solutions of (2.7), then the function (2.6) is the unique solution of (2.5) (see [6,7,9] and [1,2] for d=2 and d=3).…”

Fundamental solutions and decay of fully non-local problems

“…We point out that the kernels of type (PC) are divisors of the unit with respect to the temporal convolution. These kernels are also called Sonine kernels and they have been successfully used to study integral equations of first kind in the spaces of Hölder continuous, Lebesgue and Sobolev functions, see [6].…”