An extension problem for the fractional derivative defined by Marchaud

Abstract

We prove that the (nonlocal) Marchaud fractional derivative in $\mathbb{R}$ can be obtained from a parabolic extension problem with an extra (positive) variable, as the operator that maps the heat conduction equation to the Neumann condition. Some properties of the fractional derivative are deduced from those of the local operator. In particular we prove a Harnack principle for Marchaud-stationary functions.

17 pages

References in corpus (1)

• Local density of Caputo-stationary functions in the space of smooth functions

Cited by in corpus (2)

• Weyl and Marchaud derivatives: a forgotten history
• Extension Properties and Boundary Estimates for a Fractional Heat Operator