James Scott - New Characterizations of Sobolev and Bessel Vector Fields

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of “directional" difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We additionally show that the class of vector-valued Bessel potential spaces can be characterized by a Marcinkiewicz-type integral that is -- pointwise -- smaller than the classical Marcinkiewicz integral, and does not resemble other classes of potential-type integrals found in the literature. In applications, these results are used to better understand spaces of vector fields associated to a strongly coupled system of nonlocal equations related to a continuum model of peridynamics. This talk consists of joint work with Tadele Mengesha.