Title: Quadratic forms and Sobolev spaces of fractional order,
Abstract: We study quadratic functionals on $L^2(\mathbb{R}^d)$ that generate seminorms
in the fractional Sobolev space $H^s(\mathbb{R}^d)$ for $0 < s < 1$. The
functionals under consideration appear in the study of Markov jump processes
and, independently, in recent research on the Boltzmann equation. The
functional measures differentiability of a function $f$ in a similar way as the
seminorm of $H^s(\mathbb{R}^d)$. The major difference is that differences $f(y)
- f(x)$ are taken into account only if $y$ lies in some double cone with apex
at $x$ or vice versa. The configuration of double cones is allowed to be
inhomogeneous without any assumption on the spatial regularity. We prove that
the resulting seminorm is comparable to the standard one of
$H^s(\mathbb{R}^d)$. The proof follows from a similar result on discrete
quadratic forms in $\mathbb{Z}^d$, which is our second main result. We
establish a general scheme for discrete approximations of nonlocal quadratic
forms. Applications to Markov jump processes are discussed.