New families of fractional Sobolev spaces

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