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F. Nazarov

Publications and source records attributed to F. Nazarov.

18 recordsLinked to original sources
arXiv2020

The Landis conjecture on exponential decay

Consider a solution $u$ to $Δu +Vu=0$ on $\mathbb{R}^2$, where $V$ is real-valued, measurable and $|V|\leq 1$. If $|u(x)| \leq \exp(-C |x| \log^{1/2}|x|)$, $|x|>2$, where $C$ is a sufficiently large absolute constant, then $u\equiv 0$.

math.AP↗
arXiv2018

On the Totik-Widom property for a Quasidisk

Let $K$ be a quasidisk on the complex plane. We construct a sequence of monic polynomials $p_n=p_n(\cdot,K)$ with zeros on $K$ such that $||p_n||_K \le O(1) \mathrm{cap}(K)^n$ as $n\to\infty.$

math.CV↗
arXiv2017

Generalized Grünbaum inequality

Let $f$ be an integrable log-concave function on ${\mathbb R^n}$ with the center of mass at the origin. We show that $\int\limits_0^{\infty}f(sθ)ds\ge e^{-n}\int\limits_{-\infty}^{\infty}f(sθ)ds$ for every $ θ\in S^{n-1}$, and the constant $e^{-n}$ is the best possible.

math.MG↗
arXiv2014

The $Tb$-theorem on non-homogeneous spaces that proves a conjecture of Vitushkin

This article was written in 1999, and was posted as a preprint in CRM (Barcelona) preprint series $n^0\, 519$ in 2000. However, recently CRM erased all preprints dated before 2006 from its site, and this paper became inacessible. It has certain importance though, as the reader shall see. Formally this paper is a proof of the (qualitative version of the) Vitushkin conjecture. The last section is concerned with the quantitative version. This quantitative version turns out to be very important. It allowed Xavier Tolsa to close the subject concerning Vtushkin's conjectures: namely, using the quantitative nonhomogeneous $Tb$ theorem proved in the present paper, he proved the semiadditivity of analytic capacity. Another "theorem", which is implicitly contained in this paper, is the statement that any non-vanishing $L^2$-function is accretive in the sense that if one has a finite measure $μ$ on the complex plane ${\mathbb C}$ that is Ahlfors at almost every point (i.e. for $μ$-almost every $x\in {\mathbb C}$ there exists a constant $M>0$ such that $μ(B(x,r))\le Mr$ for every $r>0$) then any one-dimensional antisymmetric Calderón-Zygmund operator $K$ (e.g. a Cauchy integral type operator) satisfies the following "all-or-nothing" princple: if there exists at least one function $ϕ\in L^2(μ)$ such that $ϕ(x)\ne 0$ for $μ$-almost every $x\in {\mathbb C}$ and such that {\it the maximal singular operator} $K^*ϕ\in L^2(μ)$, then there exists an everywhere positive weight $w(x)$, such that $K$ acts from $L^2(μ)$ to $L^2(wdμ)$.

math.AP↗
arXiv2010

The behavior of iterations of the intersection body operator in a small neighborhood of the unit ball

The intersection body of a ball is again a ball. So, the unit ball $B_d \subset \R^d$ is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach-Mazur distance.We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to $B_d$ in Banach-Mazur distance converge to $B_d$ in Banach-Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of $B_d$.

math.MG↗
arXiv2009

Asymptotics of the best polynomial approximation of $|x|^p$ and of the best Laurent polynomial approximation of $\sgn(x)$ on two symmetric intervals

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of $|x|^p$ on two symmetric intervals. Note, that in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of $\sgn(x)$ on $[-1,-a]\cup[a,1]$ by Laurent polynomials.

math.CA↗
arXiv2008

Vector-valued Riesz potentials: Cartan type estimates and related capacities

There are many interesting problems about the electrostatic potential of finitely many charges. We consider one of them concerning the intensity of the field, in other words, about the magnitude of the gradient of this potential. We want to give a sharp estimate of the size of the set of points where this gradient is large. Of course we want the estimate to be sharp in number $N$ of charges. The size will be measured by the Hausdorff content with various gauge functions. Such a setting allows us to consider a wide class of measures (not necessarily with finitely many charges). The main technique will be Calderón-Zygmund capacities and nonhomogeneous Calderón-Zygmund operators. Here we establish a relationship between various types of capacities with singular kernels (e. g. analytic capacity, lipschitz harmonic capacity, etc) and non-linear capacity from the theory of potential á la Adams, Hedberg, Havin, Maz'ya, Wolff. "Capacitary" part of the paper extends the theorem of Mateu, Prat and Verdera [J. reine und angew. Math., 578 (2005), 201--223]. "Size estimates" part of the paper extends the theorem of M. Anderson and V. Eiderman [Annals of Math., 163 (2005), 1057--1076]. The difficulty lies in the fact that we cannot use Menger's curvature anymore because we are working in spaces of dimension bigger than two.

math.AP↗
arXiv2007

Reflectionless measures with a point mass and singular continuous component

We construct mesures supported on a compact subset E of the real line having zero principal value of their Cauchy integral a.e. on E with respect to Lebesgue measure and having singular components. E is sufficiently regular (Widom property is satisfied) but not homogeneous as for homogeneous spectrum such construction is impossible. This impossibility played an important role in characterizing almost periodic Jacobi matrices with homogeneous spectrum (Sodin-Yuditskii).

math-ph↗
arXiv2007

The Jancovici - Lebowitz - Manificat law for large fluctuations of random complex zeroes

By random complex zeroes we mean the zero set of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This zero set is distribution invariant with respect to isometries of the complex plane. We study large fluctuations of random complex zeroes and show that they obey the asymptotic law that was discovered some time ago by Jancovici, Lebowitz and Manificat for charge fluctuations of a Coulomb system of particles.

math.PR↗
arXiv2005

Asymptotics of orthogonal polynomials via the Koosis theorem

The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal polynomials for which the Koosis theorem seems to be the most natural tool. Namely, we consider the case when a Szegö measure on the unit circumference is perturbed by an arbitrary measure inside the unit disk and an arbitrary Blaschke sequence of point masses outside the unit disk.

math-ph↗
arXiv2004

On generalized sum rules for Jacobi matrices

This work is in a stream initiated by a paper of Killip and Simon [Ann. of Math. (2003)]. Using methods of Functional Analysis and the classical Szegö Theorem we prove sum rule identities in a very general form. Then, we apply the result to obtain new asymptotics for orthonormal polynomials.

math.SP↗
arXiv2003

Lower bounds for quasianalytic functions, I. How to control smooth functions?

Consider a class of functions of one real variable with the following uniqueness property: if a function f(x) from the class vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a lower bound for f(x) outside a small exceptional set. Such estimates are well-known and useful for polynomials and analytic functions. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions.

math.CA↗
arXiv2001

The geometric Kannan-Lovasz-Simonovits lemma, dimension-free estimates for volumes of sublevel sets of polynomials, and distribution of zeroes of random analytic functions

The goal of this paper is to attract attention of the reader to a dimension-free geometric inequality that can be proved using the classical needle decomposition. This inequality allows us to derive sharp dimension-free estimates for the distribution of values of polynomials in n-dimensional convex bodies. Such estimates, in their turn, lead to a surprising result about the distribution of zeroes of random analytic functions; informally speaking, we show that for simple families of analytic functions, there exists a "typical" distribution of zeroes such that the portion of the family occupied by the functions whose distribution of zeroes deviates from that typical one by some fixed amount, is about Const exp{-size of the deviation}. The paper is essentially self-contained. When choosing the style, we tried to make it an enjoyable reading for both a senior undergraduate student and an expert. As to the question "What is new in the paper?", we beleive that the answer to it is a function of two variables, the first being "what is written" and the second being "who is reading". Since we have no knowledge of the value of the second variable, we can only give the range of answers with the first variable fixed. For the targeted audience it will be the standard range [Nothing, Everything] (with both endpoints included).

math.CA↗