arXiv subjects

Fedor Nazarov

Publications and source records attributed to Fedor Nazarov.

At least 55 records · Page 3Linked to original sources

arXiv2014

Functions of perturbed noncommuting self-adjoint operators

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.

math.FA↗

arXiv2014

Reflectionless measures for Calderón-Zygmund Operators I: Basic Theory

We study the properties of reflectionless measures for an $s$-dimensional Calderón-Zygmund operator $T$ acting in $\mathbb{R}^d$, where $s\in (0,d)$. Roughly speaking, these are the measures $μ$ for which $T(μ)$ 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 theory.

math.AP↗

arXiv2014

A Bellman function proof of the $L^2$ bump conjecture

We approach the problem of finding the sharp sufficient condition of the boundedness of all two weight Calderon--Zygmund operators. We solve this problem in $L^2$ by writing a formula for a Bellman function of the problem.

math.CA↗

arXiv2014

Global estimates for kernels of Neumann series and Green's functions

We obtain global pointwise estimates for kernels of the resolvents $(I-T)^{-1}$ of integral operators \[Tf(x) = \int_Ω K(x, y) f(y) d ω(y)\] on $L^2(Ω, ω)$ under the assumptions that $||T||_{L^2(ω) \rightarrow L^2 (ω)} <1$ and $d(x,y)=1/K(x,y)$ is a quasi-metric. Let $K_1=K$ and $K_j(x,y) = \int_Ω K_{j-1} (x,z) K(z,y) \, d ω(z)$ for $j \geq 1$. Then $$ K(x,y) e^{c K_2 (x,y)/K(x,y)} \leq \sum_{j=1}^{\infty} K_j(x,y) \leq K(x,y) e^{C K_2 (x,y)/K(x,y)}, $$ for some constants $c,C>0$. Our estimates yield matching bilateral bounds for Green's functions of the fractional Schrödinger operators $(-\triangle)^{α/2}-q$ with arbitrary nonnegative potentials $q$ on $\mathbb{R}^n$ for $0<α<n$, or on a bounded non-tangentially accessible domain $Ω$ for $0<α\le 2$. In probabilistic language, these results can be reformulated as explicit bilateral bounds for the conditional gauge associated with Brownian motion or $α$-stable Lévy processes.

math.AP↗

arXiv2014

Bellman approach to the one-sided bumping for weighted estimates of Calderón--Zygmund operators

We give again (see also arXiv:1112.0676) a proof of weighted estimate of any Calderón-Zygmund operator. This is under a universal sharp sufficient condition that is weaker than the so-called bump condition. Bump conjecture was recently solved independently and simultaneously by A. Lerner (arXiv:1202.1860) and Nazarov--Reznikov-Treil-Volberg (arXiv:1202.2406). The latter paper uses the Bellman approach. Immediately a very natural and seemingly simple question arises how to strengthen the bump conjecture result by weakening its assumptions in a natural symmetric way. This is what we are dealing with here. However we meet an unexpected and, in our opinion, deep obstacle, that allows us to achieve only a partial result. The result of the present article is slightly stronger than the one in arXiv:1112.0676. Our proof consists of two main parts: reduction to a simple model operator, construction of Bellman function for estimating this simple operator. The newer feature is that the domain of definition of our Bellman function is infinitely dimensional.

math.CA↗

arXiv2013

The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions

We show that, given a set $E\subset \mathbb R^{n+1}$ 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 $E\subset\mathbb R^{n+1}$ with $H^n(E)<\infty$ is removable for Lipschitz harmonic functions if and only if it is purely $n$-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions.

math.CA↗

arXiv2013

Reflectionless measures for Calderón-Zygmund operators

We study the properties of reflectionless measures for a Calderón-Zygmund operator T. Roughly speaking, these are measures $μ$ for which T(μ) vanishes (in a weak sense) on the support of the measure. We describe the relationship between certain well-known problems in harmonic analysis and geometric measure theory and the classification of reflectionless measures. As an application of our theory, we give a new proof of a recent theorem of Eiderman, Nazarov, and Volberg, which states that in $\mathbb{R}^d$, the s-dimensional Riesz transform of a non-trivial $s$-dimensional measure is unbounded if $s\in (d-1,d)$.

math.AP↗

arXiv2013

Three revolutions in the kernel are worse than one

An example is constructed of a purely unrectifiable measure $μ$ for which the singular integral operator whose kernel triples and reverses the argument of a complex number is bounded $L^2(μ)$. This is in sharp contrast with the results known for the Cauchy transform, whose kernel reverses the argument of a complex number.

math.CA↗

arXiv2013

Reflectionless measures and the Mattila-Melnikov-Verdera uniform rectifiability theorem

A new proof is given of the Mattila-Melnikov-Verdera theorem on the uniform rectifiability of an Ahlfors-David regular measure whose associated Cauchy transform operator is bounded.

math.AP↗

arXiv2013

The proof of $A_2$ conjecture in a geometrically doubling metric space

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 for these dyadic shifts, made in [16] and later in [19].

math.CA↗

arXiv2013

Log-integrability of Rademacher Fourier series, with applications to random analytic functions

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.

math.CV↗

arXiv2012

On the uniform rectifiability of AD regular measures with bounded Riesz transform operator: the case of codimension 1

We prove that if $μ$ is a d-dimensional Ahlfors-David regular measure in $\R^{d+1}$, then the boundedness of the $d$-dimensional Riesz transform in $L^2(μ)$ 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 $μ$.

math.AP↗

arXiv2012

The fractional Riesz transform and an exponential potential

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

math.AP↗

arXiv2012

Carleson--Buckley measures beyond the scope of $A_\infty$ and their applications

Carleson measures are ubiquitous in Harmonic Analysis. In the paper of Fefferman--Kenig--Pipher in 1991 an interesting class of Carleson measures was introduced for the need of regularity problems of elliptic PDE. These Carleson measures were associated with $A_\infty$ weights. In discrete setting (we need exactly discrete setting here) they were studied by Buckley's, where they were associated with dyadic $A\infty^d$. Our goal here is to show that such Carleson--Buckley measures (in discrete setting) exists for virtually any positive function (weight). Of course some modification is needed, because it is known that Carleson property of Buckley's measure are equivalent to the weight to be in $A_\infty^d$. However a very natural generalization of those facts exist for weights more general $A_\infty$, and of course, in a special case of $A_\infty^d$ it gives Buckley's results. Our generalization of Buckley's inequality beyond the scope of $A_\infty$ allows us to prove the so-called bump conjecture.

math.CA↗

arXiv2012

Functions of perturbed tuples of self-adjoint operators

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 Hölder 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 self-adjoint operators $(A_1,...,A_n)$ and $(B_1,...,B_n)$. We also consider the case of arbitrary moduli of continuity and the case when the operators $A_j-B_j$ belong to the Schatten--von Neumann class $\bS_p$.

math.FA↗

arXiv2012

The $s$-Riesz transform of an $s$-dimensional measure in $\R^2$ is unbounded for $1<s<2$

In this paper, we prove that for $s\in(1,2)$ there exists no totally lower irregular finite positive Borel measure $μ$ in $\R^2$ with\break $\mathcal H^s(\suppμ)<+\infty$ such that $\|Rμ\|\ci{L^\infty(m_2)}<+\infty$, where $Rμ=μ\ast\frac{x}{|x|^{s+1}}$ and $m_2$ is the Lebesgue measure in $\R^2$. Combined with known results of Prat and Vihtilä, this shows that for any non-integer $s\in(0,2)$ and any finite positive Borel measure in $\R^2$ with $\mathcal H^s(\suppμ)<+\infty$, we have $\|Rμ\|\ci{L^\infty(m_2)}=\infty$.

math.AP↗

arXiv2012

An asymmetric convex body with maximal sections of constant volume

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.

math.MG↗

arXiv2011

Non-uniqueness of convex bodies with prescribed volumes of sections and projections

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.

math.CA↗