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Sep 22, 2013
09/13

by
Alexander Kiselev; Fedor Nazarov

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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 H\"older 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 $C^\beta$ norm of the solution for some $\beta >0.$ We use elementary tools involving control of H\"older...

Source: http://arxiv.org/abs/0908.0923v2

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Sep 21, 2013
09/13

by
Fëdor Nazarov; Ekaterina Shulman

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\noindent In [1] L. Polterovich and Z. Rudnick considered the behavior of a one-parameter subgroup of a Lie group under the influence of a sequence of kicks. Among others they raise the following problem: {\it is the horocycle flow stably quasi-mixing on $SL(2,\mathbb{R})/\Gamma$?} Equivalently it can be reformulated in terms of boundedness of the sequences of products $ P_n(t) = \Phi_n H(t)\Phi_{n-1} H(t) \, ... \, \Phi_1 H(t) $ where $H(t) = \begin{pmatrix} 1 & t 0 & 1 \end{pmatrix}$...

Source: http://arxiv.org/abs/1008.3077v1

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Sep 20, 2013
09/13

by
Fedor Nazarov; Mikhail Sodin

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We present several results that show somewhat surprising equidistribution patterns in the asymptotic behaviour of the argument of entire functions of finite order.

Source: http://arxiv.org/abs/math/0410341v1

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Sep 17, 2013
09/13

by
Fedor Nazarov; Mikhail Sodin

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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.

Source: http://arxiv.org/abs/1003.4237v1

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Jun 29, 2018
06/18

by
Benjamin Jaye; Fedor Nazarov

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We characterize the non-atomic measures $\mu$ for which all Calder\'{o}n-Zygmund operators with antisymmetric kernels of a fixed non-integer dimension $s$ are bounded in $L^2(\mu)$ in terms of a positive quantity, the Wolff energy.

Topics: Classical Analysis and ODEs, Mathematics

Source: http://arxiv.org/abs/1604.02014

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Sep 21, 2013
09/13

by
Fedor Nazarov; Mikhail Sodin

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Let N(f) be a number of nodal domains of a random Gaussian spherical harmonic f of degree n. We prove that as n grows to infinity, the mean of N(f)/n^2 tends to a positive constant, and that N(f)/n^2 exponentially concentrates around that constant. This result is consistent with predictions made by Bogomolny and Schmit using a percolation-like model for nodal domains of random Gaussian plane waves.

Source: http://arxiv.org/abs/0706.2409v1

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Sep 21, 2013
09/13

by
Fedor Nazarov; Vladimir Peller

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We generalize earlier results of Peller, Aleksandrov - Peller, Aleksandrov - Peller - Potapov - Sukochev to the case of functions of $n$-tuples of commuting self-adjoint operators. In particular, we prove that if a function $f$ belongs to the Besov space $B_{\be,1}^1(\R^n)$, then $f$ is operator Lipschitz and we show that if $f$ satisfies a H\"older condition of order $\a$, then $\|f(A_1...,A_n)-f(B_1,...,B_n)\|\le\const\max_{1\le j\le n}\|A_j-B_j\|^\a$ for all $n$-tuples of commuting...

Source: http://arxiv.org/abs/1204.5134v1

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Sep 22, 2013
09/13

by
Alexander Kiselev; Fedor Nazarov

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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.

Source: http://arxiv.org/abs/0908.0925v1

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Jun 30, 2018
06/18

by
Vladimir Eiderman; Fedor Nazarov

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Let $\mu$ be a measure in $\mathbb R^d$ with compact support and continuous density, and let $$ R^s\mu(x)=\int\frac{y-x}{|y-x|^{s+1}}\,d\mu(y),\ \ x,y\in\mathbb R^d,\ \ 0

Topics: Classical Analysis and ODEs, Mathematics

Source: http://arxiv.org/abs/1701.04500

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Jun 30, 2018
06/18

by
Fedor Nazarov; Alexander Olevskii

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We construct a function on the real line supported on a set of finite measure whose spectrum has density zero.

Topics: Classical Analysis and ODEs, Mathematics

Source: http://arxiv.org/abs/1701.09114

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Jun 29, 2018
06/18

by
Fedor Nazarov; Yuval Peres

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In the trace reconstruction problem, an unknown bit string $x \in \{0,1\}^n$ is observed through the deletion channel, which deletes each bit of $x$ with some constant probability $q$, yielding a contracted string $\widetilde{x}$. How many independent copies of $\widetilde{x}$ are needed to reconstruct $x$ with high probability? Prior to this work, the best upper bound, due to Holenstein, Mitzenmacher, Panigrahy, and Wieder (2008), was $\exp(\widetilde{O}(n^{1/2}))$. We improve this bound to...

Topics: Mathematics, Information Theory, Statistics Theory, Statistics, Probability, Computing Research...

Source: http://arxiv.org/abs/1612.03599

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Sep 17, 2013
09/13

by
Fedor Nazarov; Mikhail Sodin

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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...

Source: http://arxiv.org/abs/1003.4251v1

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Sep 21, 2013
09/13

by
Fedor Nazarov; Alexander Volberg

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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 Calder\'on-Zygmund operators to dyadic shifts from Hyt\"onen's...

Source: http://arxiv.org/abs/1104.4893v2

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Jun 30, 2018
06/18

by
Benjamin Jaye; Fedor Nazarov

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We study the properties of reflectionless measures for an $s$-dimensional Calder\'on-Zygmund operator $T$ acting in $\mathbb{R}^d$, where $s\in (0,d)$. Roughly speaking, these are the measures $\mu$ for which $T(\mu)$ is constant on the support of the measure. In this series of papers, we develop the basic theory of reflectionless measures, and describe the relationship between the description of reflectionless measures and certain well-known problems in harmonic analysis and geometric measure...

Topics: Mathematics, Analysis of PDEs, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1409.8556

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Jul 19, 2013
07/13

by
Fedor Nazarov; Mikhail Sodin

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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...

Source: http://arxiv.org/abs/1005.4113v1

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Sep 21, 2013
09/13

by
Alexander Kiselev; Fedor Nazarov

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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...

Source: http://arxiv.org/abs/1106.4591v1

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Sep 21, 2013
09/13

by
Alexander Fryntov; Fedor Nazarov

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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|

Source: http://arxiv.org/abs/0808.0717v1

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Jun 28, 2018
06/18

by
Fedor Nazarov; Mikhail Sodin

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We study the asymptotic laws for the spatial distribution and the number of connected components of zero sets of smooth Gaussian random functions of several real variables. The primary examples are various Gaussian ensembles of real-valued polynomials (algebraic or trigonometric) of large degree on the sphere or torus, and translation-invariant smooth Gaussian functions on the Euclidean space restricted to large domains.

Topics: Mathematical Physics, Mathematics, Classical Analysis and ODEs, Probability

Source: http://arxiv.org/abs/1507.02017

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Jun 28, 2018
06/18

by
Benjamin Jaye; Fedor Nazarov

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We continue our study of the reflectionless measures associated to an $s$-dimensional Calder\'{o}n-Zygmund operator (CZO) acting in $\mathbb{R}^d$ with $s\in (0,d)$. Here, our focus will be the study of CZOs that are rigid, in the sense that they have few reflectionless measures associated to them. Our goal is to prove that the rigidity properties of a CZO $T$ impose strong geometric conditions upon the support of any measure $\mu$ for which $T$ is a bounded operator in $L^2(\mu)$. In this way,...

Topics: Analysis of PDEs, Mathematics, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1507.08329

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Jun 28, 2018
06/18

by
Andrei K. Lerner; Fedor Nazarov

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This book is a short introduction into dyadic analysis with applications to classical weighted norm inequalities.

Topics: Mathematics, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1508.05639

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Jun 30, 2018
06/18

by
Aleksei Aleksandrov; Fedor Nazarov; Vladimir Peller

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We consider functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the trace norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_1}\le\const(\|A_1-A_2\|_{\bS_1}+\|B_1-B_2\|_{\bS_1})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz type estimate in the operator norm.

Topics: Complex Variables, Functional Analysis, Mathematics, Spectral Theory, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1411.1815

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Jun 27, 2018
06/18

by
Aleksei Aleksandrov; Fedor Nazarov; Vladimir Peller

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We study perturbations of functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and $B$ that can be defined in terms of double operator integrals. We prove that if $f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following Lipschitz type estimate in the Schatten--von Neumann norm $\bS_p$, $1\le p\le2$ norm: $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_p}\le\const(\|A_1-A_2\|_{\bS_p}+\|B_1-B_2\|_{\bS_p})$. However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the...

Topics: Spectral Theory, Functional Analysis, Complex Variables, Mathematics, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1504.01189

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Sep 24, 2013
09/13

by
Fedor Nazarov; Dmitry Ryabogin; Artem Zvavitch

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We show that if $d\ge 4$ is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.

Source: http://arxiv.org/abs/1112.3976v1

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Sep 23, 2013
09/13

by
Fedor Nazarov; Xavier Tolsa; Alexander Volberg

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We prove that if $\mu$ is a d-dimensional Ahlfors-David regular measure in $\R^{d+1}$, then the boundedness of the $d$-dimensional Riesz transform in $L^2(\mu)$ implies that the non-BAUP David-Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of $\mu$.

Source: http://arxiv.org/abs/1212.5229v2

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Sep 23, 2013
09/13

by
Fedor Nazarov; Yuval Peres; Alexander Volberg

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Let $C_n$ be the $n$-th generation in the construction of the middle-half Cantor set. The Cartesian square $K_n$ of $C_n$ consists of $4^n$ squares of side-length $4^{-n}$. The chance that a long needle thrown at random in the unit square will meet $K_n$ is essentially the average length of the projections of $K_n$, also known as the Favard length of $K_n$. A classical theorem of Besicovitch implies that the Favard length of $K_n$ tends to zero. It is still an open problem to determine its...

Source: http://arxiv.org/abs/0801.2942v1

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Sep 21, 2013
09/13

by
Fedor Nazarov; Sergei Treil; Alexander Volberg

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We give necessary and sufficient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give sufficient conditions for two weight norm inequalities for the Hilbert transform.

Source: http://arxiv.org/abs/math/9711209v1

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Sep 21, 2013
09/13

by
Fedor Nazarov; Leonid Polterovich; Mikhail Sodin

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The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line {f=0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay...

Source: http://arxiv.org/abs/math/0402412v2

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Sep 21, 2013
09/13

by
Fedor Nazarov; Alon Nishry; Mikhail Sodin

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We prove that any power of the logarithm of Fourier series with random signs is integrable. This result has applications to the distribution of values of random Taylor series, one of which answers a long-standing question by J.-P. Kahane.

Source: http://arxiv.org/abs/1301.0529v1

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Sep 21, 2013
09/13

by
Fedor Nazarov; Sergei Treil; Alexander Volberg

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In the paper we consider Calder\'{o}n-Zygmund operators in nonhomogeneous spaces. We are going to prove the analogs of classical results for homogeneous spaces. Namely, we prove that a Calder\'{o}n-Zygmund operator is of weak type if it is bounded in $L^2$. We also prove several versions of Cotlar's inequality for maximal singular operator. One version of Cotlar's inequality (a simpler one) is proved in Euclidean setting, another one in a more abstract setting when Besicovich covering lemma is...

Source: http://arxiv.org/abs/math/9711210v1

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Sep 22, 2013
09/13

by
Fedor Nazarov; Sergei Treil; Alexander Volberg

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In this paper we are proving that Sawyer type condition for boundedness work for the two weight estimates of individual Haar multipliers, as well as for the Haar shift and other "well localized" operators.

Source: http://arxiv.org/abs/math/0702758v1

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Sep 20, 2013
09/13

by
Alexander Borichev; Fedor Nazarov; Mikhail Sodin

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Let F be a class of functions with the uniqueness property: if a function f in F 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| outside a small exceptional set. Such estimates are well-known and useful for polynomials, complex- and real-analytic functions, exponential polynomials. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes...

Source: http://arxiv.org/abs/math/0301217v2

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Sep 19, 2013
09/13

by
Gheorghe Craciun; Fedor Nazarov; Casian Pantea

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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...

Source: http://arxiv.org/abs/1010.3050v2

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Sep 21, 2013
09/13

by
Olga Holtz; Fedor Nazarov; Yuval Peres

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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 $\P_p(N n)$ decaying exponentially in $n$ for every $p \in S$. We prove that for $\alpha>0$ non-integer, $f$ is in the space $C^\alpha [0,1]$ if and only if...

Source: http://arxiv.org/abs/0808.1936v3

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Sep 17, 2013
09/13

by
Fedor Nazarov; Sergei Treil; Alexander Volberg

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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 Calder\'on-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...

Source: http://arxiv.org/abs/1003.1596v1

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Jun 29, 2018
06/18

by
Ronen Eldan; Fedor Nazarov; Yuval Peres

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We prove that any $\ell$ positive definite $d \times d$ matrices, $M_1,\ldots,M_\ell$, of full rank, can be simultaneously spectrally balanced in the following sense: for any $k < d$ such that $\ell \leq \lfloor \frac{d-1}{k-1} \rfloor$, there exists a matrix $A$ satisfying $\frac{\lambda_1(A^T M_i A) }{ \mathrm{Tr}( A^T M_i A ) } < \frac{1}{k}$ for all $i$, where $\lambda_1(M)$ denotes the largest eigenvalue of a matrix $M$. This answers a question posed by Peres, Popov and Sousi and...

Topics: Probability, Functional Analysis, Mathematics

Source: http://arxiv.org/abs/1606.01680

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Jul 20, 2013
07/13

by
Alexander Kiselev; Fedor Nazarov; Roman Shterenberg

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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 $\alpha < 1/2,$ and global existence as well as analyticity of solution for $\alpha \geq 1/2.$ 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...

Source: http://arxiv.org/abs/0804.3549v1

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Sep 17, 2013
09/13

by
Fedor Nazarov; Richard Oberlin; Christoph Thiele

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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.

Source: http://arxiv.org/abs/0912.3010v1

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Sep 21, 2013
09/13

by
Fedor Nazarov; Yuval Peres; Pablo Shmerkin

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Denote by $\mu_a$ the distribution of the random sum $(1-a) \sum_{j=0}^\infty \omega_j a^j$, where $P(\omega_j=0)=P(\omega_j=1)=1/2$ and all the choices are independent. For $0 1$ and $\log (1/3) /\log (1/4)$ is irrational.

Source: http://arxiv.org/abs/0905.3850v1

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Sep 23, 2013
09/13

by
Vladimir Eiderman; Fedor Nazarov; Alexander Volberg

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In this paper, we prove that for $s\in(1,2)$ there exists no totally lower irregular finite positive Borel measure $\mu$ in $\R^2$ with\break $\mathcal H^s(\supp\mu)

Source: http://arxiv.org/abs/1109.2260v3

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Sep 21, 2013
09/13

by
Yaryong Heo; Fedor Nazarov; Andreas Seeger

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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.

Source: http://arxiv.org/abs/0902.0943v2

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Jun 30, 2018
06/18

by
Michael Frazier; Fedor Nazarov; Igor Verbitsky

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We obtain global pointwise estimates for kernels of the resolvents $(I-T)^{-1}$ of integral operators \[Tf(x) = \int_{\Omega} K(x, y) f(y) d \omega(y)\] on $L^2(\Omega, \omega)$ under the assumptions that $||T||_{L^2(\omega) \rightarrow L^2 (\omega)} 0$. Our estimates yield matching bilateral bounds for Green's functions of the fractional Schr\"{o}dinger operators $(-\triangle)^{\alpha/2}-q$ with arbitrary nonnegative potentials $q$ on $\mathbb{R}^n$ for $0

Topics: Mathematics, Functional Analysis, Spectral Theory, Analysis of PDEs, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1403.3945

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Sep 23, 2013
09/13

by
Fedor Nazarov; Xavier Tolsa; Alexander Volberg

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We show that, given a set $E\subset\R^{n+1}$ with finite $n$-Hausdorff measure $H^n$, if the $n$-dimensional Riesz transform $$R_{H^n|E} f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)$$ is bounded in $L^2(H^n|E)$, then $E$ is $n$-rectifiable. From this result we deduce that a compact set $E\subset\R^{n+1}$ with $H^n(E)

Source: http://arxiv.org/abs/1212.5431v1

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Sep 22, 2013
09/13

by
Yaryong Heo; Fedor Nazarov; Andreas Seeger

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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 $L^p\to L^{p,\nu}$ inequalities for convolutions with radial kernels.

Source: http://arxiv.org/abs/1001.2789v1

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Sep 21, 2013
09/13

by
Benjamin Jaye; Fedor Nazarov; Alexander Volberg

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In this paper we study the $s$-dimensional Riesz transform of a finite measure $\mu$ in $\mathbf{R}^d$, with $s\in (d-1,d)$. We show that the boundedness of the Riesz transform of $\mu$ implies that a nonlinear potential of exponential type is finite $\mu$-almost everywhere. It appears to be the first result of this type for $s>1$.

Source: http://arxiv.org/abs/1204.2135v2

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Sep 23, 2013
09/13

by
Fedor Nazarov; Mikhail Sodin; Alexander Volberg

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We consider the zeroes of a random Gaussian Entire Function f and show that their basins under the gradient flow of the random potential U partition the complex plane into domains of equal area. We find three characteristic exponents 1, 8/5, and 4 of this random partition: the probability that the diameter of a particular basin is greater than R is exponentially small in R; the probability that a given point z lies at a distance larger than R from the zero it is attracted to decays as...

Source: http://arxiv.org/abs/math/0510654v3

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Jun 28, 2018
06/18

by
Fedor Nazarov; Alon Nishry; Mikhail Sodin

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We find the asymptotics of the counting function of zeroes of random entire functions represented by Rademacher Taylor series. We also give the asymptotics of the weighted counting function, which takes into account the arguments of zeroes. These results answer several questions left open after the pioneering work of Littlewood and Offord of 1948. The proofs are based on our recent result on the logarithmic integrability of Rademacher Fourier series.

Topics: Probability, Mathematics, Complex Variables

Source: http://arxiv.org/abs/1507.02340

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45

Sep 18, 2013
09/13

by
Fedor Nazarov; Dmitry Ryabogin; Artem Zvavitch

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We show that in all dimensions d>2, there exists an asymmetric convex body of revolution all of whose maximal hyperplane sections have the same volume. This gives the negative answer to the question posed by V. Klee in 1969.

Source: http://arxiv.org/abs/1201.0393v1

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35

Jun 29, 2018
06/18

by
Benjamin Jaye; Fedor Nazarov; Xavier Tolsa

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In this paper we provide an extension of a theorem of David and Semmes ('91) to general non-atomic measures. The result provides a geometric characterization of the non-atomic measures for which a certain class of square function operators, or singular integral operators, are bounded in $L^2(\mu)$. The description is given in terms of a modification of Jones' $\beta$-coefficients.

Topics: Metric Geometry, Classical Analysis and ODEs, Functional Analysis, Mathematics

Source: http://arxiv.org/abs/1612.04754

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Jun 29, 2018
06/18

by
Paata Ivanisvili; Benjamin Jaye; Fedor Nazarov

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In this paper we address the following question: given $ p\in (1,\infty)$, $n \geq 1$, does there exists a constant $A(p,n)>1$ such that $\| M f\|_{L^{p}}\geq A(n,p) \| f\|_{L^{p}}$ for any nonnegative $f \in L^{p}(\mathbb{R}^{n})$, where $Mf$ is a maximal function operator defined over the family of shifts and dilates of a centrally symmetric convex body. The inequality fails in general for the centered maximal function operator, but nevertheless we give an affirmative answer to the...

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1602.05895

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Sep 21, 2013
09/13

by
Fedor Nazarov; Alexander Reznikov; Alexander Volberg

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We give a proof of the $A_2$ conjecture in geometrically doubling metric spaces (GDMS), i.e. a metric space where one can fit not more than a fixed amount of disjoint balls of radius $r$ in a ball of radius $2r$. Our proof consists of three main parts: a construction of a random "dyadic" lattice in a metric space; a clever averaging trick from [3], which decomposes a "hard" part of a Calderon-Zygmund operator into dyadic shifts (adjusted to metric setting); and the estimates...

Source: http://arxiv.org/abs/1106.1342v2