Lipschitz functions and fractional Sobolev spaces
The action of Lipschitz functions on fractional Sobolev spaces \(\mathbb{D}^\alpha_p\) corresponding to a sub-Markovian semigroup \((P_t)_{t\geq 0}\) is investigated. The semigroup defines the analytic strongly continuous semigroup of contractions in \(L^p(\mu)\), \(1<p<\infty\). The spaces \(\mathbb{D}^\alpha_p\), \(0<\alpha\leq 1\), are defined as Bessel potential spaces corresponding to the infinitesimal generator \(A_p\) of the semigroup in \(L^p(\mu)\). The examples are the heat semigroup with the classical Bessel potential spaces on \(\mathbb{R}^n\) and the Ornstein-Uhlenbeck semigroup on an abstract Wiener space with the Sobolev space involved in the Malliavin calculus. The author looks for conditions on the semigroup that guarantees that Lipschitz functions operate on the spaces by compositions. In particular, it is proved that for \(2\leq p<\infty\) the following assumptions imply the action of Lipschitz functions on \(\mathbb{D}^\alpha_p\): (0) the semigroup comes from a transition semigroup of a Markovian process, (1) measure \(\mu\) is a reference measure for the process, (2) there exists an algebra \({\mathcal D}\) of bounded functions contained in \(D(A_p)\) and dense in \(L^p\), (3) \(A_p\Gamma(f, f)- \Gamma(f, A_pf)- \Gamma(A_pf,f)\geq 0\) with \(\Gamma(f,g)= A_p(fg)- fA_pg- gAf\), \(f,g\in{\mathcal D}\). The assumptions are fulfilled by the both examples mentioned above.
action of Lipschitz functions on fractional Sobolev spaces
analytic strongly continuous semigroup of contractions
scientific article; zbMATH DE number 1396420