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

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 $x\ge x_0$. 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.

20 pages. To appear in Journal of Differential Equations