The mapping properties of fractional derivatives in weighted fractional Sobolev space

TL;DR

This paper investigates how fractional derivatives, specifically Marchaud and Riemann-Liouville types, act as mappings between weighted fractional Sobolev spaces, revealing their behavior and extension properties.

Contribution

It establishes the mapping properties of fractional derivatives in weighted Sobolev spaces, including extensions and conditions for these mappings, using interpolation techniques.

Findings

Riemann-Liouville fractional derivative maps $W^{p,s}_0() to W^{p,s-}()

Marchaud fractional derivative with even extension maps $W^{p,s}((0,))$ to $W^{p,s-}( eal)$

Mapping results hold for all $0<<s<1$ with $ps\u2265 1$

Abstract

We study the mapping behavior of the Marchaud fractional derivative with different extensions in the scale of fractional weighted Sobolev spaces. In particular we show that the $\alpha$--order Riemann--Liouville fractional derivative maps $W^{p,s}_0(\Omega)$ to $W^{p,s-\alpha}(\Omega)$, for all $0<\alpha<s<1$ and the $\alpha$--order Marchaud fractional derivative with even extension maps the fractional Sobolev space $W^{p,s}((0,\infty))$ to $W^{p,s-\alpha}(\real)$ for all $0<\alpha<s<1$ and $ps\geq1$ . The proof is based on the Calder\'{o}n--Lions interpolation theorem.