相关论文: Beurling's criterion and extremal metrics for Fugl…

We introduce the notion of modulus of families of walks on graphs. We show how Beurling's famous criterion for extremality, that was formulated in the continuous case, can be interpreted on graphs as an instance of the Karush-Kuhn-Tucker…

We study Fuglede's $p$-module of systems of measures in condensers in Euclidean spaces and on polarizable Carnot groups. We apply and generalize a result by Rodin, which provides an explicit method for finding the extremal function and the…

Motivated by recent developments on calculus in metric measure spaces $(X,\mathsf d,\mathfrak m)$, we prove a general duality principle between Fuglede's notion of $p$-modulus for families of finite Borel measures in $(X,\mathsf d)$ and…

We give a necessary and sufficient condition for extremality of a supermodular function based on its min-representation by means of (vertices of) the corresponding core polytope. The condition leads to solving a certain simple linear…

We consider the PDE $-\Delta_pu=\rho$, where $\rho$ is a signed Borel measure on $\mathbb{R}^n$. For each $p>n$, we characterize solutions as extremals of a generalized Morrey inequality determined by $\rho$.

We give a characterization of the invariant measures for the exclusion process on the integers with certain reversible transition kernels. Some examples include all nearest-neighbor kernels with asymptotic mean zero. One tool used is a…

Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In ``Extremal length and functional completion'', Fuglede initiated an abstract theory of extremal length which has…

We identify two sufficient conditions for locally finite Borel measures on $\mathbb{R}^n$ to give full mass to a countable family of Lipschitz images of $\mathbb{R}^m$. The first condition, extending a prior result of Pajot, is a sufficient…

We develop explicit variational formulas for the $p(\cdot)$-modulus of curve families in symmetric domains of $\mathbb{R}^n$, under a log-H\"older continuous exponent $p\colon\Omega\to(1,\infty)$, where $\Omega$ is an open set. For annuli…

We present an elementary proof of the classical Beurling sampling theorem which gives a sufficient condition for sampling of multi-dimensional band-limited functions.

We look for pointwise bounds on a plurisubharmonic function near its singularity point, given the value of its generalized Lelong number with respect to a plurisubharmonic weight. To this end, an extremal problem is considered. In certain…

Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is trivial for…

Fuglede's conjecture in $\mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $\mathbb{Q}_p$ is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation.

Given a compact Kahler manifold with an extremal metric (M,\omega), we give sufficient conditions on finite sets points p_1,...,p_n and weights a_1,...a_n for which the blow up of M at p_1,...,p_n has an extremal metric in the Kahler class…

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\Bbb{R}^n$, $n\geq 2$. To each locally finite Borel measure $\mu$, we associate a function $\widetilde J_2(\mu, x)$…

If $x_1,\dots,x_m$ are finitely many points in $\mathbb{R}^d$, let $E_\epsilon=\cup_{i=1}^m\,x_i+Q_\epsilon$, where $Q_\epsilon=\{x\in \mathbb{R}^d,\,\,|x_i|\le \epsilon/2, \, i=1,...,d\}$ and let $\hat f$ denote the Fourier transform of…

We establish generalizations of the Nyman-Beurling and B\'aez-Duarte criteria concerning lack of zeros of Dirichlet $L$-functions in the semi-plane $\Re(s) >1/p$ for $p\in (1,2]$. We pose and solve a natural extremal problem for Dirichlet…

The paper is devoted to multidimensional $(0,1)$-matrices extremal with respect to containing a polydiagonal (a fractional generalization of a diagonal). Every extremal matrix is a threshold matrix, i.e., an entry belongs to its support…

The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval $[0,R]$ is almost surely an orthogonal polynomial ensemble. In this article, we show that if $R$ tends to…

We investigate when a Borel graph admits a (Borel or measurable) orientation with outdegree bounded by $k$ for various cardinals $k$. We show that for a p.m.p. graph $G$, a measurable orientation can be found when $k$ is larger than the…