Related papers: On $L^1$ extremal problem for entire functions

A geometric setup for constrained variational calculus is presented. The analysis deals with the study of the extremals of an action functional defined on piecewise differentiable curves, subject to differentiable, non-holonomic…

We provide a sharp monotonicity theorem about the distribution of subharmonic functions on manifolds, which can be regarded as a new, measure theoretic form of the uncertainty principle. As an illustration of the scope of this result, we…

Given the supremal functional $E_\infty(u,\Omega')=ess\,\sup_{\Omega'} H(\cdot,D u)$ defined on $W^{1,\infty}_{loc}(\Omega,\mathbb{R}^N)$, $\Omega' \Subset \Omega\subseteq \mathbb{R}^n$, we identify a class of vectorial rank-one Absolute…

Real continuous submodular functions, as a generalization of the corresponding discrete notion to the continuous domain, gained considerable attention recently. The analog notion for entropy functions requires additional properties: a real…

We show that various functionals related to the supremum of a real function defined on an arbitrary set or a measure space are Hadamard directionally differentiable. We specifically consider the supremum norm, the supremum, the infimum, and…

We consider Dirichlet exterior value problems related to a class of non-local Schr\"odinger operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian. We prove elliptic and parabolic…

We consider transcendental entire functions of finite order for which the zeros and $1$-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that…

For a graph $H$ let $c(H)$ denote the supremum of $|E(G)|/|V(G)|$ taken over all non-null graphs $G$ not containing $H$ as a minor. We show that $$c(H) \leq \frac{|V(H)|+\mathrm{comp}(H)}{2}-1,$$ when $H$ is a union of cycles, verifying…

In this paper, we prove the second-order Sobolev inequalities on Cayley graphs of groups of polynomial growth. We use the discrete Concentration-Compactness principle to prove the existence of extremal functions for best constants in…

The Landau-Kolmogorov problem consists of finding the upper bound $M_k$ for the norm of intermediate derivative $|f^{(k)}|$, when the bounds $|f| \le M_0$ and $|f^{(n)}| \le M_n$, for the norms of the function and of its higher derivative,…

We prove an analogue of Sadullaev's theorem concerning the size of the set where a maximal totally real manifold can meet a pluripolar set. The manifold has to be of class C-1 only. This readily leads to a version of Shcherbina's theorem…

Motivated by the Maximum Theorem for convex functions (in the setting of linear spaces) and for subadditive functions (in the setting of Abelian semigroups), we establish a Maximum Theorem for the class of generalized convex functions,…

This paper is devoted to Markov's extremal problems of the form $M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\le k\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree at most $n$ and $X$ is a…

We investigate the question whether the $L^1(\mathbb R)$-norm of the second derivative of the uncentered Hardy-Littlewood maximal function can be bounded by a constant times the $L^1(\mathbb R)$-norm of the function itself. We give a…

In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement…

In this paper we find lower bounds on higher moments of the error term in the Chebotarev density theorem. Inspired by the work of Bella\''{\i}che, we consider general class functions and prove bounds which depend on norms associated to…

This paper addresses the statistical problem of estimating the infinite-norm deviation from the empirical mean to the distribution mean for high-dimensional distributions on $\{0,1\}^d$, potentially with $d=\infty$. Unlike traditional…

In this paper we find extremal one-sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination. These approximations optimize the $L^1(\mathbb{R}, |E(x)|^{-2}dx)$-error, where…

This paper is devoted to various applications of Hardy-Sobolev type inequalities. We derive a new $L^2$ estimate for the $\bar{\partial}-$equation on ${\mathbb C}^n$ which yields a quantitative generalization of the Hartogs extension…

Let $\mathbb C$ be the complex plane, $E$ be a measurable subset in a segment $[0, R]$ of the positive semiaxis $\mathbb R^+$, $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$. The main result of this article is an upper…