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Sergei Treil

Publications and source records attributed to Sergei Treil.

At least 55 records · Page 3Linked to original sources
arXiv2010

Sharp weighted estimates for dyadic shifts and the $A_2$ conjecture

We give a self-contained proof of the $A_2$ conjecture, which claims that the norm of any Calderon-Zygmund operator is bounded by the first degree of the $A_2$ norm of the weight. The original proof of this result by the first author relied on a subtle and rather difficult reduction to a testing condition by the last three authors. Here we replace this reduction by a new weighted norm bound for dyadic shifts - linear in the $A_2$ norm of the weight and quadratic in the complexity of the shift -, which is based on a new quantitative two-weight inequality for the shifts. These sharp one- and two-weight bounds for dyadic shifts are the main new results of this paper. They are obtained by rethinking the corresponding previous results of Lacey-Petermichl-Reguera and Nazarov-Treil-Volberg. To complete the proof of the $A_2$ conjecture, we also provide a simple variant of the representation, already in the original proof, of an arbitrary Calderon-Zygmund operator as an average of random dyadic shifts and random dyadic paraproducts. This method of the representation amounts to the refinement of the techniques from nonhomogeneous Harmonic Analysis.

math.CA↗
arXiv2010

$H^1$ and dyadic $H^1$

In this paper we give a simple proof of the fact that the average over all dyadic lattices of the dyadic $H^1$-norm of a function gives an equivalent $H^1$-norm. The proof we present works for both one-parameter and multi-parameter Hardy spaces. The results of such type are known. The first result (for one-parameter Hardy spces) belongs to Burgess Davis (1980). Also, by duality, such results are equivalent to the "BMO from dyadic BMO" statements proved by Garnett-Jones(1982} for one parameter case, and by Pipher-Ward (2008) for two-parameter case. While the paper generalizes these results to the multi-parameter setting, this is not its main goal. The purpose of the paper is to present an approach leading to a simple proof, which works in both one-parameter and multi-parameter cases. The main idea of treating square function as a Calderon--Zygmind operator is a commonplace in harmonic analysis; the main observation, on which the paper is based, is that one can treat the random dyadic square function this way. After that, all is proved by using the standard and well-known results about Calderon--Zygmind operators in the Hilbert-space-valued setting. As an added bonus, we get a simple proof of the (equivalent by duality) inclusion $\text{BMO}\subset \text{BMO}_d$, $H^1_d \subset H^1$ in the multi-parameter case. Note, that unlike the one-parameter case, the inclusions in the general situation are far from trivial.

math.CA↗
arXiv2010

On $A_2$ conjecture and corona decomposition of weights

We consider here a problem of finding the sharp estimate for the boundedness of an arbitrary Calderón-Zygmund operator in $L^2(w)$, $w\in A_2$. We first prove that for $A_2$ weight $w$ one has that the norm a Calderon--Zygmund operator $T$ in $L^2(w)$ is bounded by the sum of its weak norm, the weak norm of its adjoint, and the $A_2$ norm of the weight. From this result we derive that $\|T\|_{L^2(w)\rightarrow L^2(w)} \le C\,[w]_{A_2}\log (1+[w]_{A_2})$. We believe that the logarithmic factor is superflous. The approach is based on $2$-weight estimates technique and, hence, on non-homogeneous harmonic analysis.

math.AP↗
arXiv2010

Two weight estimate for the Hilbert transform and corona decomposition for non-doubling measures

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".

math.AP↗
arXiv2008

Rank one perturbations and singular integral operators

We consider rank one perturbations $A_α=A+α(\cdot,φ)φ$ of a self-adjoint operator $A$ with cyclic vector $φ\in\mathcal H_{-1}(A)$ on a Hilbert space $\mathcal H$. The spectral representation of the perturbed operator $A_α$ is given by a singular integral operator of special form. Such operators exhibit what we call 'rigidity' and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms $T_\varepsilon$ are uniformly (in $\varepsilon$) bounded operators from $L^2(μ)$ to $L^2(μ_α)$, where $μ$ and $μ_α$ are the spectral measures of $A$ and $A_α$, respectively. As an application, a sufficient condition for $A_α$ to have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of $A$ with respect to $φ$. Some examples, like Jacobi matrices and Schrödinger operators with $L^2$ potentials are considered.

math.FA↗
arXiv2007

Analytic projections, Corona Problem and geometry of holomorphic vector bundles

The main result of the paper is the theorem giving a sufficient condition for the existence of a bounded analytic projection onto a holomorphic family of (generally infinite-dimensional) subspaces (a holomorphic sub-bundle of a trivial bundle). This sufficient condition is also necessary in the case of finite dimension or codimension of the bundle. A simple lemma of N. Nikolski connects the existence of a bounded analytic projection with the Operator Corona Problem (existence of a bounded analytic left inverse for an operator-valued function), so as corollaries of the main result we obtain new results about the Operator Corona Problem. In particular, we find a new sufficient condition, a complete solution in the case of finite codimension, and, a solution of the generalized Corona Problem.

math.CA↗
arXiv2007

A theorem about three quadratic forms

We prove a self-improvement property regarding quadratic forms on arbitrary vector spaces. We discuss several consequences of this result, in particular those concerning dimension-free L^p estimates of certain singular integral operators (Riesz transforms).

math.FA↗
arXiv2007

The problem of ideals of $H^\infty$: beyond the exponent 3/2

The paper deals with the problem of ideals of $H^\infty$: describe increasing functions $ϕ\ge 0$ such that for all bounded analytic functions $f_1,f_2,...,f_n, τ$ in the unit disc $D$ the condition $|τ(z) | \le ϕ(\sum_k |f_k(z)|)$ for all $z\in D$, implies that $τ$ belong to the ideal generated by $f_1,f_2,...,f_n$. It was proved earlier by the author that the function $ϕ(s) =s^2$ does not work. The main result of the paper is that one can take for $ϕ$ any function of form $ϕ(s) =s^2 ψ(\ln s^{-2})$, where $ψ$ is a bounded non-increasing function on $[0, \infty)$ satisfying $\int_0^\infty ψ(x) dx <\infty$.

math.CV↗
arXiv2007

Estimates in corona theorems for some subalgebras of $H^\infty$

We obtain estimates in the corona theorem for the algebra of analytic functions in the unit disc whose nth derivative is bounded, and its subalgebras defined by the boundary continuity of the nth derivative. The corona theorem for such algebras is trivial and known, but corona theorem with estimates is new and much harder to prove. As an auxiliary result of independent interest we get the continuity of the best estimate in such algebras (including the classical case of the algebra $H^\infty$ of bounded analytic functions). We also show that for a fixed $n$ the best estimate is the same for all algebras we consider.

math.CA↗
arXiv2007

Carleson Potentials and the Reproducing Kernel Thesis for Embedding Theorems

In this note we present a new proof of the Carleson Embedding Theorem on the unit disc and unit ball. The only technical tool used in the proof of this fact is Green's formula. The starting point is that every Carleson measure gives rise to a bounded subharmonic function. Using this function we construct a new related Carleson measure that allows for a simple embedding. In the case of the disc this gives the best known constant, with the previous best given by N. Nikolskii.

math.CA↗
arXiv2004

The Matrix-Valued $H^{p}$ Corona Problem in the Disk and Polydisk

In this paper we consider the matrix-valued $H^{p}$ corona problem in the disk and polydisk. The result for the disk is rather well known, and is usually obtained from the classical Carleson Corona Theorem by linear algebra. Our proof provides a streamlined way of obtaining this result and allows one to get a better estimate on the norm of the solution. In particular, we were able to improve the estimate found in the recent work of T. Trent. Note that, the solution of the $H^{\infty}$ matrix corona problem in the disk can be easily obtained from the $H^{2}$ corona problem either by factorization, or by the Commutant Lifting Theorem. The $H^{p}$ corona problem in the polydisk was originally solved by K.C. Lin. The solution used Koszul complexes and was rather complicated because one had to consider higher order $\bar{\partial}$-equations. Our proof is more transparent and it improves upon Lin's result in several ways. First, we were able to show that the norm of the solution is independent of the number of generators. Additionally, we illustrate that the norm of the solution grows at most proportionally to the dimension of the polydisk. Our approach is based on one that was originated by M. Andersson. In the disk it essentially depends on Green's Theorem and duality to obtain the estimate. In the polydisk we use Riesz projections to reduce the problem to the disk case.

math.CV↗
arXiv1997

Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators in nonhomogeneous spaces

In the paper we consider Calderón-Zygmund operators in nonhomogeneous spaces. We are going to prove the analogs of classical results for homogeneous spaces. Namely, we prove that a Calderón-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 not available. We obtain also the weak type of maximal singular operator from these inequalities.

math.FA↗