arXiv subjects
Publications and source records attributed to Fedor Nazarov.
We prove that in every bipartite Cayley graph of every non-amenable group, there is a perfect matching that is obtained as a factor of independent uniform random variables. We also discuss expansion properties of factors and improve the Hoffman spectral bound on independence number of finite graphs.
We consider the question of growth of high order Sobolev norms of solutions of the conservative surface quasi-geostrophic equation. We show that if $s>0$ is large then for every given $A$ there is exist small in $H^s$ initial data such that the corresponding solution's $H^s$ norm exceeds $A$ at some time. The idea of the construction is quasilinear. We use a small perturbation of a stable shear flow. The shear flow can be shown to create small scales in the perturbation part of the flow. The control is lost once the nonlinear effects become too large.
We give a short and simple polynomial estimate of the norm of weighted dyadic shift on metric space with geometric doubling, which is linear in the norm of the weight. Combined with the existence of special probability space of dyadic lattices built in A. Reznikov, A. Volberg, "Random "dyadic" lattice in geometrically doubling metric space and $A_2$ conjecture", arXiv:1103.5246, and with decomposition of Calderón-Zygmund operators to dyadic shifts from Hytönen's "The sharp weighted bound for general Calderón-Zygmund operators", arXiv:1007.4330 (and later T. Hytönen, C. Pérez, S. Treil, A. Volberg, "A sharp estimated of weighted dyadic shifts that gives the proof of $A_2$ conjecture", arXiv 1010.0755), we will be able to have a linear (in the norm of weight) estimate of an arbitrary Calderón-Zygmund operator on a metric space with geometric doubling. This will be published separately.
Persistence and permanence are properties of dynamical systems that describe the long-term behavior of the solutions, and in particular specify whether positive solutions approach the boundary of the positive orthant. Mass-action systems (or more generally power-law systems) are very common in chemistry, biology, and engineering, and are often used to describe the dynamics in interaction networks. We prove that two-species mass-action systems derived from weakly reversible networks are both persistent and permanent, for any values of the reaction rate parameters. Moreover, we prove that a larger class of networks, called endotactic networks, also give rise to permanent systems, even if we allow the reaction rate parameters to vary in time. These results also apply to power-law systems and other nonlinear dynamical systems. In addition, ideas behind these results allow us to prove the Global Attractor Conjecture for three-species systems.
We prove strong clustering of k-point correlation functions of zeroes of Gaussian Entire Functions. In the course of the proof, we also obtain universal local bounds for k-point functions of zeroes of arbitrary nondegenerate Gaussian analytic functions. In the second part of the paper, we show that strong clustering yields the asymptotic normality of fluctuations of some linear statistics of zeroes of Gaussian Entire Functions, in particular, of the number of zeroes in measurable domains of large area. This complements our recent results from the paper "Fluctuations in random complex zeroes" (arXiv:1003.4251v1).
In these notes, we describe the recent progress in understanding the zero sets of two remarkable Gaussian random functions: the Gaussian entire function with invariant distribution of zeroes with respect to isometries of the complex plane, and Gaussian spherical harmonics on the two-dimensional sphere.
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. Extending the previous results of Sodin and Tsirelson, we compute the variance of linear statistics of random complex zeroes, and find close to optimal conditions on a test-function that yield asymptotic normality of fluctuations of the corresponding linear statistics. We also provide examples of test-functions with abnormal fluctuations of linear statistics.
This article was written in 2005 and subsequently lost (at least by the third author). Recently it resurfaced due to one of the colleagues to whom a hard copy has been sent in 2005. We consider here a problem of finding necessary and sufficient conditions for the boundedness of two weight Calderón-Zygmund operators. We give such necessary and sufficient conditions in very natural terms, if the operator is the Hilbert transform, and the weights satisfy some very natural condition. The condition on weights was lifted in a recent paper of Michael Lacey, Eric Sawyer and Ignacio Uriarte-Tuero: "A characterization of the two weight norm inequality for the Hilbert transform", arXiv:1001.4043 [math.CA] 31 January 2010. The paper of Lacey--Sawyer-Uriarte-Tuero alliviated the "pivotal" condition used in a present article and replaced it by the very interesting and correct energy condition, which, unlike the "pivotal" condition turned out to be also necessary. The paper of Lacey-Sawyer-Uriarte-Tuero used the present article in its main aspect. The thrust of the present article is to use the methods of nonhomogeneous Harmonoc Analysis together with a several paraproducts arising from a certain stopping time argument. In view of the importance of the present article for Lacey--Sawyer-Uriarte-Tuero's paper arXiv:1001.4043 [math.CA] 31 January 2010, we present it to the attention of the reader. Drawing no parallels, "Darwin spent 1838-1859 getting ready to publish "On the Origin of Species" without actually publishing it, only brooding over beaks of finches".
Given a (known) function $f:[0,1] \to (0,1)$, we consider the problem of simulating a coin with probability of heads $f(p)$ by tossing a coin with unknown heads probability $p$, as well as a fair coin, $N$ times each, where $N$ may be random. The work of Keane and O'Brien (1994) implies that such a simulation scheme with the probability $¶_p(N<\infty)$ equal to 1 exists iff $f$ is continuous. Nacu and Peres (2005) proved that $f$ is real analytic in an open set $S \subset (0,1)$ iff such a simulation scheme exists with the probability $¶_p(N>n)$ decaying exponentially in $n$ for every $p \in S$. We prove that for $α>0$ non-integer, $f$ is in the space $C^α[0,1]$ if and only if a simulation scheme as above exists with $¶_p(N>n) \le C (Δ_n(p))^α$, where $Δ_n(x)\eqbd \max \{\sqrt{x(1-x)/n},1/n \}$. The key to the proof is a new result in approximation theory: Let $\B_n$ be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree $n$. We show that a function $f:[0,1] \to (0,1)$ is in $C^α[0,1]$ if and only if $f$ has a series representation $\sum_{n=1}^\infty F_n$ with $F_n \in \B_n$ and $\sum_{k>n} F_k(x) \le C(Δ_n(x))^α$ for all $ x \in [0,1]$ and $n \ge 1$. We also provide a counterexample to a theorem stated without proof by Lorentz (1963), who claimed that if some $ϕ_n \in \B_n$ satisfy $|f(x)-ϕ_n(x)| \le C (Δ_n(x))^α$ for all $ x \in [0,1]$ and $n \ge 1$, then $f \in C^α[0,1]$.
We investigate connections between radial Fourier multipliers on $R^d$ and certain conical Fourier multipliers on $R^{d+1}$. As an application we obtain a new weak type endpoint bound for the Bochner-Riesz multipliers associated to the light cone in $R^{d+1}$, where $d\ge 4$, and results on characterizations of $L^p\to L^{p,ν}$ inequalities for convolutions with radial kernels.
We introduce a Calderon Zygmund decomposition such that the bad function has vanishing integral against a number of pure frequencies. Then we prove a variation norm variant of a maximal inequality for several frequencies due to Bourgain. To obtain the full range of Lp estimates we apply the multi frequency Calderon Zygmund decomposition.
Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur showed that a certain class of weak solutions to the drift diffusion equation with initial data in $L^2$ gain Hölder continuity provided that the BMO norm of the drift velocity is bounded uniformly in time. We show a related result: a uniform bound on BMO norm of a smooth velocity implies uniform bound on the $C^β$ norm of the solution for some $β>0.$ We use elementary tools involving control of Hölder norms using test functions. In particular, our approach offers a third proof of the global regularity for the critical surface quasi-geostrophic (SQG) equation.
We consider surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions involving conservation of a certain family of moduli of continuity.
Denote by $μ_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty ω_j a^j$, where $P(ω_j=0)=P(ω_j=1)=1/2$ and all the choices are independent. For $0 1$ and $\log (1/3) /\log (1/4)$ is irrational.
Given a fixed $p\neq 2$, we prove a simple and effective characterization of all radial multipliers of $\cF L^p(\Bbb R^d)$, provided that the dimension $d$ is sufficiently large. The method also yields new $L^q$ space-time regularity results for solutions of the wave equation in high dimensions.
We prove that the unit cube $B^n_{\infty}$ is a strict local minimizer for the Mahler volume product $vol_n(K)vol_n(K^*)$ in the class of origin symmetric convex bodies endowed with the Banach-Mazur distance.
Let p(z) be a monic polynomial of degree n. Consider the lemniscate L={z:|p(z)|=1}. It has been conjectured that L has the largest length when p(z)=z^n-1. We show that the length of L attains a local maximum at this polynomial and prove the asymptotically sharp bound |L|<2n+o(n).
The paper is a comprehensive study of the existence, uniqueness, blow up and regularity properties of solutions of the Burgers equation with fractional dissipation. We prove existence of the finite time blow up for the power of Laplacian $α< 1/2,$ and global existence as well as analyticity of solution for $α\geq 1/2.$ We also prove the existence of solutions with very rough initial data $u_0 \in L^p,$ $1 < p < \infty.$ Many of the results can be extended to a more general class of equations, including the surface quasi-geostrophic equation.