We prove some general results on the existence of partitions of unity in Sobolev type spaces on various innnite dimensional manifolds. As special cases we obtain in particular, (continuous) partitions of unity a) in the Malliavin test functions on an abstract Wiener space; b) in the rst order Sobolev space on pinned and free loop spaces; c) in the rst order Sobolev spaces associated with revers...

ANALYSIS AND PDE ON METRIC MEASURE SPACES: SOBOLEV FUNCTIONS AND VISCOSITY SOLUTIONS Xiaodan Zhou, PhD University of Pittsburgh, 2016 We study analysis and partial differential equations on metric measure spaces by investigating the properties of Sobolev functions or Sobolev mappings and studying the viscosity solutions to some partial differential equations. This manuscript consists of two par...

Let $$G:\mathbb {R}\rightarrow \mathbb {R}$$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on G such that $$\begin{aligned} \{G(f):f\in {\dot{K}}_{p,q}^{\alpha }F_{\beta }^{s}\}\subset {\dot{K}} _{p,q}^{\alpha }^{s} \end{aligned}$$ holds. Here $${\dot{K}}_{p,q}^{\alpha }^{s}$$ are Herz-type Triebel–Lizorkin spaces. These spaces unify and general...

We consider the multiplication operator, M, in Sobolev spaces with respect to general measures and give a characterization for M to be bounded, in terms of sequentially dominated measures. This has important consequences for the asymptotic behaviour of Sobolev orthogonal polynomials. Also, we study properties of Sobolev spaces with respect to measures. 2001 Academic Press

We give short simple proofs of Uspenskii’s results characterizing Besov spaces as trace spaces of weighted Sobolev spaces. We generalize Uspenskii’s results and prove the optimality of these generalizations. We next show how classical results on the functional calculus in the Besov spaces can be obtained as straightforward consequences of the theory of weighted Sobolev spaces.