A note on Riemann-Liouville fractional Sobolev spaces

Alessandro Carbotti; Giovanni E. Comi

doi:10.3934/cpaa.2020255

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2021, Volume 20, Issue 1: 17-54. Doi: 10.3934/cpaa.2020255

This issue Previous Article On principal eigenvalues of biharmonic systems Next Article Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters

A note on Riemann-Liouville fractional Sobolev spaces

Alessandro Carbotti^1, and
Giovanni E. Comi^2, ,

1.
Dipartimento di Matematica e Fisica, Università del Salento, Via Per Arnesano, 73100 Lecce, Italy
2.
Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

^* Corresponding author
^* Corresponding author

Received: May 31, 2020

Revised: June 30, 2020

Early access: October 2020

Published: January 2021

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Abstract

Taking inspiration from a recent paper by Bergounioux et al., we study the Riemann-Liouville fractional Sobolev space $ W^{s, p}_{RL, a+}(I) $, for $ I = (a, b) $ for some $ a, b \in \mathbb{R}, a < b $, $ s \in (0, 1) $ and $ p \in [1, \infty] $; that is, the space of functions $ u \in L^{p}(I) $ such that the left Riemann-Liouville $ (1 - s) $-fractional integral $ I_{a+}^{1 - s}[u] $ belongs to $ W^{1, p}(I) $. We prove that the space of functions of bounded variation $ BV(I) $ and the fractional Sobolev space $ W^{s, 1}(I) $ continuously embed into $ W^{s, 1}_{RL, a+}(I) $. In addition, we define the space of functions with left Riemann-Liouville $ s $-fractional bounded variation, $ BV^{s}_{RL,a+}(I) $, as the set of functions $ u \in L^{1}(I) $ such that $ I^{1 - s}_{a+}[u] \in BV(I) $, and we analyze some fine properties of these functions. Finally, we prove some fractional Sobolev-type embedding results and we analyze the case of higher order Riemann-Liouville fractional derivatives.

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Mathematics Subject Classification: Primary:26A33, 26A45, 26B30;Secondary:47B38.

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