Maximum principles, extension problem and inversion for nonlocal one-sided equations

TL;DR

This paper investigates one-sided nonlocal equations involving fractional derivatives, establishing maximum principles, extension problems, inversion results, and unifying various fractional derivative concepts within weighted spaces.

Contribution

It introduces a novel framework connecting one-sided nonlocal equations with fractional derivatives, including maximum principles, extension problems, and inversion in weighted spaces.

Findings

Established maximum principles for one-sided fractional operators

Derived an extension problem characterization similar to Caffarelli–Silvestre

Provided inversion results and unified fractional derivative notions

Abstract

We study one-sided nonlocal equations of the form $$\int_{x_0}^\infty\frac{u(x)-u(x_0)}{(x-x_0)^{1+\alpha}} dx=f(x_0),$$ on the real line. Notice that to compute this nonlocal operator of order $0<\alpha<1$ at a point $x_0$ we need to know the values of $u(x)$ to the right of $x_0$, that is, for $x\geq x_0$. We show that the operator above corresponds to a fractional power of a one-sided first order derivative. Maximum principles and a characterization with an extension problem in the spirit of Caffarelli--Silvestre and Stinga--Torrea are proved. It is also shown that these fractional equations can be solved in the general setting of weighted one-sided spaces. In this regard we present suitable inversion results. Along the way we are able to unify and clarify several notions of fractional derivatives found in the literature.