Related papers: The structure of ${\cal A}$-free measures with uni…

We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $\mu$ satisfying…

It is shown that given a metric space $X$ and a $\sigma$-finite positive regular Borel measure $\mu$ on $X$, there exists a bounded continuous real-valued function on $X$ that is one-to-one on the complement of a set of $\mu$ measure zero.

We establish a general structure theorem for the singular part of ${\mathscr A}$-free Radon measures, where ${\mathscr A}$ is a linear PDE operator. By applying the theorem to suitably chosen differential operators ${\mathscr A}$, we obtain…

Let $\mu$ be a positive finite measure on the unit circle. The Dirichlet type space $\mathcal{D}(\mu)$, associated to $\mu$, consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against…

We study the kernel function of the operator u $\rightarrow$ L $\mu$ u = --$\Delta$u + $\mu$ |x| 2 u in a bounded smooth domain $\Omega$ $\subset$ R N + such that 0 $\in$ $\partial$$\Omega$, where $\mu$ $\ge$ -- N 2 4 is a constant. We show…

We give a characterization of commutative semispectral measures by means of Feller and Strong Feller Markov kernels. In particular: {itemize} we show that a semispectral measure $F$ is commutative if and only if there exist a self-adjoint…

In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution $(\lambda,\phi)$ of a nonlocal operator.…

The structured singular value $\mu_E$ for a linear subspace $E$ of $M_n(\mathbb C)$ is defined by \[ \mu_E(A)=1 / \inf\{\|X\| \ : \ X \in E, \ \det(I_n-AX)=0 \} \quad (A \in M_n(\mathbb{C})), \] and $\mu_E(A)=0$ if there is no $X \in E$…

For a an arbitrary periodic Borel measure $\mu$, we prove order $O(\varepsilon)$ operator-norm resolvent estimates for the solutions to scalar elliptic problems in $L^2({\mathbb R}^d, d\mu^\varepsilon)$ with $\varepsilon$-periodic…

Let $X$ be a topological space and $\mu$ be a nonatomic finite measure on a $\sigma$-algebra $\Sigma$ containing the Borel $\sigma$-algebra of $X$. We say $\mu$ is weakly outer regular, if for every $A \in \Sigma$ and $\epsilon>0$, there…

Let $\mu,\nu$ be Radon measures on $\mathbb{R}$, with $\mu$ non-atomic and $\nu$ doubling, and write $\mu = \mu_{a} + \mu_{s}$ for the Lebesgue decomposition of $\mu$ relative to $\nu$. For an interval $I \subset \mathbb{R}$, define…

If $\mu$ is a finite complex measure in the complex plane $\C$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called reflectionless if it is continuous (has no atoms) and $C^\mu=0$ at…

Let $\mathcal{M}$ be a $W^*$-factor and let $S\left( \mathcal{M} \right) $ be the space of all measurable operators affiliated with $\mathcal{M}$. It is shown that for any self-adjoint element $a\in S(\mathcal{M})$ there exists a scalar…

Let $A$ be a vector space of real valued functions on a non-empty set $X$ and $L:A\rightarrow\mathbb{R}$ a linear functional. Given a $\sigma$-algebra $\mathcal{A}$, of subsets of $X$, we present a necessary condition for $L$ to be…

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,…

Assuming a unitarily invariant norm $|||\cdot|||$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $|||\cdot|||$ on matrix algebras $\mathcal{M}_n$ for…

Given a finite positive Borel measure $\mu$ in the open unit disc of the complex plane, we construct a bounded outer function $E$ whose boundary values have vanishing mean oscillation such that $|E| \mu$ is a vanishing Carleson measure. As…

Consider a totally irregular measure $\mu$ in $\mathbb{R}^{n+1}$, that is, the upper density $\limsup_{r\to0}\frac{\mu(B(x,r))}{(2r)^n}$ is positive $\mu$-a.e.\ in $\mathbb{R}^{n+1}$, and the lower density…

Consider a finite dimensional complex Hilbert space $\cH$, with $dim(\cH) \geq 3$, define $\bS(\cH):= \{x\in \cH \:|\: ||x||=1\}$, and let $\nu_\cH$ be the unique regular Borel positive measure invariant under the action of the unitary…

Let $\mu$ be a positive finite Borel measure on the unit circle. The associated Dirichlet space $\mathcal{D}(\mu)$ consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson…