Related papers: Traces on General Sets in $\mathbb{R}^n$ for Funct…

We introduce a new approach to traces on the principal ideal $\mathcal L_{1,\infty}$ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J.~Dixmier,…

We survey a few trace theorems for Sobolev spaces on $N$-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries,…

For \Omega a C^{2}-smooth domain, and a positive bounded continuous map a \in C(\Omega), we prove existence of a minimizer of the functional u \mapsto $\int_{\Omega} a|Du| over the space BV(\Omega) of functions of bounded variation with…

It is a classical theorem that if a function is integrable along the boundary of the unit circle, then the function is the nontangential limit of a holomorphic function on the open disc if and only if its Fourier coefficients for…

We present other examples illustrating the operator-theoretic approach to invariant integrals on quantum homogeneous spaces developed by Kuersten and the second author. The quantum spaces are chosen such that their coordinate algebras do…

Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the…

In metric measure spaces, we study boundary traces of BV functions in domains equipped with a doubling measure and supporting a Poincar\'e inequality, but possibly having a very large and irregular boundary. We show that the trace exists in…

A complete description of traces on $\mathbb{R}^{n}$ of functions from the weighted Sobolev space $W^{l}_{1}(\mathbb{R}^{n+1},\gamma)$, $l \in \mathbb{N}$, with weight $\gamma \in A^{\rm loc}_{1}(\mathbb{R}^{n+1})$ is obtained. In the case…

We prove that given any positive integer $k$, for each open set $\Omega$ and any closed subset $D$ of its closure such that $\Omega$ is locally an (epsilon,delta)-domain near points in the boundary of $\Omega$ not contained in $D$ there…

Given a smooth manifold $M$ (with or without boundary), in this paper we study the regularisation of traces for the global pseudo-differential calculus in the context of non-harmonic analysis. Indeed, using the global pseudo-differential…

Let M be a closed manifold and let CL(M) be the algebra of classical pseudodifferential operators. The aim of this note is to classify trace functionals on the subspaces CL^a(M) of CL(M) of operators of order a. CL^a(M) is a CL^0(M)-module…

We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is \Phi_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m positive definite…

We study the approximative trace for individual elements in the Sobolev space $W^{1,p}(\Omega)$ for $1\le p\le\infty$. This notion of a trace was introduced for $p=2$ in [AtE11] in the setting of general open sets…

We prove trace theorems for weighted mixed norm Sobolev spaces in the upper-half space where the weight is a power function of the vertical variable. The results show the differentiability order of the trace functions depends only on the…

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain that can be written as $\Omega=\bigcup_{t} \Omega_t$, where $\{\Omega_t\}_{t\in\Gamma}$ is a countable collection of domains with certain properties. In this work, we develop a technique…

Let $\gp$ be a finite group acting on a compact manifold $M$ and $\maA(M)$ denote the algebra of classical complete symbols on $M$. We determine all traces on the cross-product algebra $\maA(M) \rtimes \Gamma$. These traces appear as…

It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $\Lambda$. We show that the same is true for much wider spaces of continuous functions. In particular,…

Let $\Sigma\subset\mathbb{R}^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb{R}^d$ into two domains $\Omega_\pm$. In this note self-adjoint Schr\"odinger operators with $\delta$ and…

Noncommutative functions are graded functions between sets of square matrices of all sizes over two vector spaces that respect direct sums and similarities. They possess very strong regularity properties (reminiscent of the regularity…

Schmidt characterised the class of rayless graphs by an ordinal rank function, which makes it possible to prove statements about rayless graphs by transfinite induction. Halin asked whether Schmidt's rank function can be generalised to…