Related papers: The trace problem for vector fields satisfying H\"…
We prove the $L^2$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves.
We consider the minimization problem corresponding to a Sobolev inequality for vector fields and show that minimizing sequences are relatively compact up to the symmetries of the problem. In particular, there is a minimizer. An ingredient…
In H\"ormander inner product spaces, we investigate initial-boundary value problems for an arbitrary second order parabolic partial differential equation and the Dirichlet or a general first-order boundary conditions. We prove that the…
An exact trace formula for the perturbation of the Laplacian by a Dirac delta potential on a compact hyperbolic Riemann surface is derived. The formula can be considered an analogue of the Selberg trace formula. The difference of perturbed…
Sobolev trace inequalities on nonhomogeneous fractional Sobolev spaces are established.
The (chordal) Loewner differential equation encodes certain curves in the half-plane (aka traces) by continuous real-valued driving functions. Not all curves are traces; the latter can be defined via a geometric condition called the local…
The problem for consistency between linear transports along paths and real bundle metrics in real vector bundles is stated. Necessary and/or sufficient conditions, as well as conditions for existence, for such consistency are derived. All…
This paper establishes trace-formulae for a class of operators defined in terms of the functional calculus for the Laplace operator on divergence-free vector fields with relative and absolute boundary conditions on Lipschitz domains in…
We prove existence, uniqueness and Sobolev regularity of weak solution of the Cauchy problem of the stochastic transport equation with drift in a large class of singular vector fields containing, in particular, the $L^d$ class, the weak…
We show that a discrete sequence $\Lambda$ of the complex plane is the union of $n$ interpolating sequences for the H\"ormander algebras $A_p$ if and only if the trace of $A_p$ on $\Lambda$ coincides with the space of functions on $\Lambda$…
We seek an analogy of the mathematical form of the alternative form of Einstein's field equations for Lovelock's field equations. We find that the price for this analogy is to accept the existence of the trace anomaly of the energy-momentum…
We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best…
Graph Laplacians on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of either $\delta$ or $\delta'$ type. In either case, an infinite series of trace formulae which…
In this paper, we obtain a regularized trace formula for a Sturm Liouville problem which has two points of discontinuity and also contains an eigenparameter in a boundary ondition.
On the Sierpinski gasket $\mathcal{SG}$, we consider Sobolev spaces $L^2_\sigma(\mathcal{SG})$ associated with the standard Laplacian $\Delta$ with order $\sigma\geq 0$. When $\sigma\in\mathbb{Z}^+$, $L^2_\sigma(\mathcal{SG})$ consists of…
For functions from the Sobolev space $H^s(\Omega)$, 1/2<s<3/2, definitions of non-unique generalised and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and…
This paper is devoted to give a simplified proof of the trace theorem for functions of bounded deformation defined on bounded Lipschitz domains of $\mathbb{R}^n$. As a consequence, the existence of one-sided Lebesgue limits on countably…
Given operators $A,B$ in some ideal $\mathcal{I}$ in the algebra $\mathcal{L}(H)$ of all bounded operators on a separable Hilbert space $H$, can we give conditions guaranteeing the existence of a trace-class operator $C$ such that $B…
The Dirichlet problem for a class of stochastic partial differential equations is studied in Sobolev spaces. The existence and uniqueness result is proved under certain compatibility conditions that ensure the finiteness of…
We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field are prescribed in an open, bounded, Sobolev-class domain, and either the normal component…