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- Fedor Nazarov, Victor Petrovich Havin, Alexander Volberg
- 2002

Let Cn be the nth generation in the construction of the middle-half Cantor set. The Cartesian square Kn of Cn consists of 4n squares of side length 4−n. The chance that a long needle thrown at random in the unit square will meet Kn is essentially the average length of the projections of Kn, also known as the Favard length of Kn. A classical theorem of… (More)

hold with some constant C independent of f? (Unless otherwise specified, all integrals are taken with respect to the standard Lebesgue measure on R.) Denoting w := u−1, we can reformulate the above question as follows: When is the operator T := M√vT0M√w bounded in L ? (Here Mφ stands for the operator of multiplication by φ.) Such weighted norm inequalities… (More)

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 ≥ 1/2. We also prove the existence of solutions… (More)

- Russell Lyons, Fedor Nazarov
- Eur. J. Comb.
- 2011

We prove that in every bipartite Cayley graph of every nonamenable 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.

The classical theory of Calderón–Zygmund operators started with the study of convolution operators on the real line having singular kernels. (A typical example of such an operator is the so called Hilbert transform, defined by Hf(t) = ∫ R f(s) ds t−s .) Later it has developed into a large branch of analysis covering a quite wide class of singular integral… (More)

Recently, using DiGiorgi-type techniques, Caffarelli and Vasseur [1] showed that a certain class of weak solutions to the drift diffusion equation with initial data in L 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… (More)

- Fedor Nazarov, MIKHAIL SODIN
- 2008

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 tends to a positive constant a, and that N(f)/n exponentially concentrates around a. This result is consistent with predictions made by Bogomolny and Schmit using a percolation-like model for nodal domains of… (More)

- Fedor Nazarov, Serguei Treil, Alexander Volberg
- 1998

We are going to show that the classical Carleson embedding theorem fails for Hilbert space valued functions and operator measures. Contre-exemple à un théorème de Carleson sur le plongement ponderé à valeurs vectorielles. Résumé. Nous allons montrer que le théorème classique de Carleson sur le plongement ponderé est faux s’il s’agit des fonctions à valeurs… (More)

Given a fixed p 6= 2 we prove a simple and effective characterization of all radial multipliers of FL(R) provided that the dimension d is sufficiently large. The method also yields new L space-time regularity results for solutions of the wave equation in high dimensions.

A consequence of Littlewood’s Subordination Principle [10] is the (not at all obvious) fact that every composition operator restricts to a bounded operator on the Hardy space H2 (see also [5, Theorem 1.7, page 10] or [16, pp. 13–15]), and this in turn has inspired a lively enterprise connecting complex function theory with operator theory, the goal being to… (More)