Commutators, paraproducts and BMO in non-homogeneous martingale settings

In this paper we investigate the relations between (martingale) BMO spaces, paraproducts and commutators in non-homogeneous martingale settings. Some new, and one might add unexpected, results are obtained. Some alternative proof of known results are also presented.Comment: 39 pages, 1 figure This material is based on the work supported by the National Science Foundation under the grant DMS-0800876. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundatio

H1H^1 and dyadic H1H^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.Comment: 14 pages. In the new version a minor error was corrected, see p. 8, reasoning before Section 1.3. The correct smoothness exponent in the averaged square function is 1/2, not N/2 as it was stated in the earlier versio

A remark on the reproducing kernel thesis for Hankel operators

We give a simple proof of the so called reproducing kernel thesis for Hankel operatorsComment: 8 pages. The proof now covers the case of vectorial Hankel operators. Added the discussion about the connection with the generalized embedding theorems and with the Carleson embedding theore

Sharp A2A_2 estimates of Haar shifts via Bellman function

We use the Bellman function method to give an elementary proof of a sharp weighted estimate for the Haar shifts, which is linear in the $A_2$ norm of the weight and in the complexity of the shift. Together with the representation of a general Calder\'{o}n--Zygmund operator as a weighted average (over all dyadic lattices) of Haar shifts, (cf. arXiv:1010.0755v2[math.CA], arXiv:1007.4330v1[math.CA]) it gives a significantly simpler proof of the so-called the $A_2$ conjecture. The main estimate is a very general fact about concave functions, which can be very useful in other problems of martingale Harmonic Analysis. Concave functions of such type appear as the Bellman functions for bounds on the bilinear form of martingale multipliers, thus the main estimate allows for the transference of the results for simplest possible martingale multipliers to more general martingale transforms. Note that (although this is not important for the $A_2$ conjecture for general Calder\'{o}n--Zygmund operators) this elementary proof gives the best known (linear) growth in the complexity of the shift.Comment: 23p

A remark on two weight estimates for positive dyadic operators

We give a simple proof of the Sawyer type characterization of the two weigh estimate for positive dyadic operators (also known as the bilinear embedding theorem).Comment: 9 pages. New version streamlined the "easy" part of the proof: the Carleson embedding theorem is used instead of the maximal function estimat

Regularizations of general singular integral operators

In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal, so the integral formally defining the operator or its bilinear form is not well defined (the integrand is not in L^1) even for nice functions. However, since the kernel only has singularities on the diagonal, the bilinear form is well defined say for bounded compactly supported functions with separated supports. One of the standard ways to interpret the boundedness of a singular integral operators is to consider regularized kernels, where the cut-off function is zero in a neighborhood of the origin, so the corresponding regularized operators with kernel are well defined (at least on a dense set). Then one can ask about uniform boundedness of the regularized operators. For the standard regularizations one usually considers truncated operators. The main result of the paper is that for a wide class of singular integral operators (including the classical Calderon-Zygmund operators in non-homogeneous two weight settings), the L^p boundedness of the bilinear form on the compactly supported functions with separated supports (the so-called restricted L^p boundedness) implies the uniform L^p-boundedness of regularized operators for any reasonable choice of a smooth cut-off of the kernel. If the kernel satisfies some additional assumptions (which are satisfied for classical singular integral operators like Hilbert Transform, Cauchy Transform, Ahlfors--Beurling Transform, Generalized Riesz Transforms), then the restricted L^p boundedness also implies the uniform L^p boundedness of the classical truncated operators.Comment: Introduced factor 1/2 in argument section 3.1, results unchange

Superexponential estimates and weighted lower bounds for the square function

We prove the following superexponential distribution inequality: for any integrable $g$ on $[0,1)^{d}$ with zero average, and any $\lambda>0$ $$|\{ x \in [0,1)^{d} \; :\; g \geq\lambda \}| \leq e^{- \lambda^{2}/(2^{d}\|S(g)\|_{\infty}^{2})},$$ where $S(g)$ denotes the classical dyadic square function in $[0,1)^{d}$. The estimate is sharp when dimension $d$ tends to infinity in the sense that the constant $2^{d}$ in the denominator cannot be replaced by $C2^{d}$ with $0<C<1$ independent of $d$ when $d \to \infty$. For $d=1$ this is a classical result of Chang--Wilson--Wolff [4]; however, in the case $d>1$ they work with a special square function $S_\infty$, and their result does not imply the estimates for the classical square function. Using good $\lambda$ inequalities technique we then obtain unweighted and weighted $L^p$ lower bounds for $S$; to get the corresponding good $\lambda$ inequalities we need to modify the classical construction. We also show how to obtain our superexponential distribution inequality (although with worse constants) from the weighted $L^2$ lower bounds for $S$, obtained in [5]

Clark model in general situation

For a unitary operator the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter $\gamma, |\gamma|=1$. Namely all such unitary perturbations are $U_\gamma:=U+(\gamma-1) (., b_1)_{\mathcal H} b$, where $b\in\mathcal H, \|b\|=1, b_1=U^{-1} b, |\gamma|=1$. For $|\gamma|<1$ operators $U_\gamma$ are contractions with one-dimensional defects. Restricting our attention on the non-trivial part of perturbation we assume that $b$ is cyclic for $U$. Then the operator $U_\gamma$, $|\gamma|<1$ is a completely non-unitary contraction, and thus unitarily equivalent to its functional model $\mathcal M_\gamma$, which is the compression of the multiplication by the independent variable $z$ onto the model space $\mathcal K_{\theta_\gamma}$, where $\theta_\gamma$ is the characteristic function of the contraction $U_\gamma$. The Clark operator $\Phi_\gamma$ is a unitary operator intertwining $U_\gamma, |\gamma|<1$ and its model $\mathcal M_\gamma$, $\mathcal M_\gamma \Phi_\gamma = \Phi_\gamma U_\gamma$. If spectral measure of $U$ is purely singular (equivalently, $\theta_\gamma$ is inner), operator $\Phi_\gamma$ was described from a slightly different point of view by D. Clark. When $\theta_\gamma$ is extreme point of the unit ball in $H^\infty$ was treated by D. Sarason using the sub-Hardy spaces introduced by L. de Branges. We treat the general case and give a systematic presentation of the subject. We find a formula for the adjoint operator $\Phi^*_\gamma$ which is represented by a singular integral operator, generalizing the normalized Cauchy transform studied by A. Poltoratskii. We present a "universal" representation that works for any transcription of the functional model. We then give the formulas adapted for the Sz.-Nagy--Foias and de Branges--Rovnyak transcriptions, and finally obtain the representation of $\Phi_\gamma$.Comment: 34 pages. 8/17/2013: changed the arXiv abstract, so the symbols display correctly; no changes in the tex

Approximation by analytic matrix functions. The four block problem

We study the problem of finding a superoptimal solution to the four block problem. Given a bounded block matrix function $\left(\begin{array}{cc}\Phi_{11} &\Phi_{12}\\\Phi_{21}&\Phi_{22}\end{array}\right)$ on the unit circle the four block problem is to minimize the $L^\infty$ norm of $\left(\begin{array}{cc} \Phi_{11}-F&\Phi_{12}\\\Phi_{21}&\Phi_{22}\end{array}\right)$ over $F\in H^\infty$. Such a minimizing $F$ (an optimal solution) is almost never unique. We consider the problem to find a superoptimal solution which minimizes not only the supremum of the matrix norms but also the suprema of all further singular values. We give a natural condition under which the superoptimal solution is unique

General Clark model for finite rank perturbations

All unitary perturbations of a given unitary operator $U$ by finite rank $d$ operators with fixed range can be parametrized by $(d\times d)$ unitary matrices $\Gamma$; this generalizes unitary rank one ($d=1$) perturbations, where the Aleksandrov--Clark family of unitary perturbations is parametrized by the scalars on the unit circle $\mathbb{T}\subset\mathbb{C}$. For a purely contractive $\Gamma$ the resulting perturbed operator $T_\Gamma$ is a contraction (a completely non-unitary contraction under the natural assumption about cyclicity of the range), so they admit the functional model. In this paper we investigate the Clark operator, i.e. a unitary operator that intertwines $T_\Gamma$ (presented in the spectral representation of the non-perturbed operator $U$) and its model. We make no assumptions on the spectral type of the unitary operator $U$; absolutely continuous spectrum may be present. We find a representation of the adjoint Clark operator in the coordinate free Nikolski--Vasyunin functional model. This representation features a special version of the vector-valued Cauchy integral operator. Regularization of this singular integral operator yield representations of the adjoint Clark operator in the Sz.-Nagy--Foias transcription. In the special case of inner characteristic functions (purely singular spectral measure of $U$) this representation gives what can be considered as a natural generalization of the normalized Cauchy transform (which is a prominent object in the Clark theory for rank one case) to the vector-valued settings.Comment: 46 pages. Added Section 9 on the Clark operator, re-worded abstract and introduction, included heuristic explanation in Section 6, fixed a few minor error