On New Families of Fractional Sobolev Spaces

TL;DR

This paper introduces three new families of fractional Sobolev spaces based on weak fractional derivatives, providing foundational theorems and relationships that support future development of fractional calculus and PDE theories.

Contribution

It develops a novel framework for fractional Sobolev spaces using weak fractional derivatives, including new one-sided and symmetric spaces with key properties and theorems.

Findings

Established density and approximation theorems

Proved extension and trace theorems

Derived embedding theorems and Sobolev inequalities

Abstract

This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the well-established integer order Sobolev spaces and theory. In particular, two new families of one-sided fractional Sobolev spaces are introduced and analyzed, they reveal more insights about another family of so-called symmetric fractional Sobolev spaces. Many key theorems/properties, such as density/approximation theorem, extension theorems, one-sided trace theorem, and various embedding theorems and Sobolev inequalities in those Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also discovered. The results of this paper lay down a solid theoretical foundation for systematically developing a fractional calculus of variations theory and a fractional PDE theory as well as their numerical solutions in subsequent works. This paper is a concise presentation of the materials of Sections 1, 4 and 5 of reference [7].