Research interests

Research interests and publications

General Research Area: Harmonic Analysis, Complex Analysis, and Operator Theory

Brief Research Summary: My research interests lie in the intersection of Harmonic Analysis, Complex Analysis and Operator Theory. Many problems I am working on have their origin in applications, such as Control Theory (H-infinity control, etc.), Stationary Random Processes.

My earlier research dealt with Hankel and Toeplitz operators, functional models of operators, spectral decompositions of operators, spectral theory of matrix- and operator-valued functions.

Another direction of research concerned the Corona Problem and related topics, in particular the relations between the Corona Problem and geometry of Hermitian bundles.

More recent research projects involve classical and non-classical harmonic analysis, such as: Bellman function, weighted norm inequalities for singular integral operators, matrix A_p weights, wavelet and frame decompositions, Calderon--Zygmund operators on non-homogeneous spaces. Some of the projects in this directions deal multi-parameter harmonic analysis.

Perturbation theory of linear operators and connections with singular integral operators is another direction of my research. Among recent projects in this direction are matrix-valued Clark Theory and Aronszajn--Donoghue theory for the finite rank perturbations.

The papers below give you general idea of my recent research projects.

My PhD Thesis

If you are interested, the link is here. Is is in Russian, no English translation, sorry.

Recent papers

All my recent papers are posted on the ArXive, so I am no longer posting the papers on my page.

You can follow this link to get to the ArXive page with my papers

Older papers

Below are the older, previously posted papers, I am not removing any links, since not all of them are available in the ArXive. Clicking on a link brings you either to an ArXive page with the paper or to a page with short abstract (sometimes there is no abstract) and links to .dvi , .ps and .PDF files with the text.

• Curvature Condition for Noncontractions does not imply Similarity to the Backward Shift, with Hyun Kwon, arXiv:0903.4423v1 [math.CA]

• Rank one perturbations and singular integral operators, with Constanze Liaw, Journal of Functional Analysis, Volume 257(2009), Issue 6, pp 1947-1975, see also arXiv:0810.2750v1 [math.FA]

• $H^1$ and dyadic $H^1$, Linear and Complex Analysis: Dedicated to V. P. Havin on the Occasion of His 75th Birthday (S. Kislyakov A. Alexandrov, A. Baranov, ed.), Advances in the Mathematical Sciences, vol. 226, AMS, 2009, pp.~179--194. see also arXiv:0809.3288v1 [math.CA]

• Similarity of operators and geometry of eigenvector bundles, with Hyun Kwon, Publicacions Matematiques, 53 (2009), 417-438, see also arXiv:0712.0114v1 [math.FA]

• A theorem about three quadratic forms, with Oliver Dragičević and Alexander Volberg, Int. Math. Res. Not. IMRN 2008, Art. ID rnn 072, 9 pp., see also arXiv:0710.3249v1 [math.FA]

• The problem of ideals of $H^\infty$: beyond the exponent 3/2, Journal of Functional Analysis, 253 (2007), 220-240, see also

• arXiv:math/0702806v1 [math.CV]

• Two weight inequalities for individual Haar multipliers and other well localized operators, with F. Nazarov, and A. Volberg, accepted by Math. Research Letters, see also arXiv:math/0702758v1 [math.CA]

• Carleson Potentials and the Reproducing Kernel Thesis for Embedding Theorems, with Stefanie Petermichl and Brett Wick, Illinois Journal of Mathematics, 51 (2007), no. 4, 1249--1263, see also arXiv:math/0701851v1 [math.CA].

• Estimates in corona theorems for some subalgebras of $H^{\infty}$, with A. Sasane, Arkiv fur Matematik, v 35, No 2, 351-380, see also arXiv:math/0702754v1 [math.CA]

• Analytic projections, Corona Problem and geometry of holomorphic vector bundles, with B. Wick, J. Amer. Math. Soc. 22 (2009), no. 1, 55--76., see also arXiv:math/0702756v1 [math.CA]

• Scalar and vector Muckenhoupt weights, with M. Lauzon, Indiana Univ. Math. J. 56 (2007), no. 4, 1989--2015..

• Approximation by analytic operator functions. Factorizations and very badly approximable functions, with V. Peller, Algebra i Analiz 17 (2005), no. 3, 160--183, see also arXiv:math/0407458v1 [math.FA]

• Very badly approximable matrix functions, with V. Peller, Selecta Math. (N.S.), 11 (2005), no. 1, 127--154, see also arXiv:math/0303186v1 [math.FA]

• The Matrix-Valued H^p Corona Problem in the Disk and Polydisk, with B. Wick

• An Operator Corona Theorem

• Lower bounds in the matrix corona theorem and the codimension one conjecture

• Common Complements of Two Subspaces of a Hilbert Space, with M. Lauzon

• Linear resolvent growth of rank one perturbation of a unitary operator does not imply its similarity to a normal operator, with N. Nikolski

• Bellman function in stochastic control and harmonic analysis, with F. Nazarov and A. Volberg

• Estimates in the Corona Theorem and ideals of $H^\infty$; a problem of T. Wolff

• Tb Theorem on non-homogeneous spaces, with F. Nazarov and A. Volberg

• Linear resolvent growth of a weak contraction does not imply its similarity to a normal operator (with S. Kupin)

• The gap between complex structured singular value and its upper bound is infinite

• Weak type estimates and Cotlar inequalities for Calderon–Zygmund operators on nonhomogeneous spaces (with F. Nazarov and A. Volberg)

• Cauchy Integral and Calderon-Zygmund operators on nonhomogeneous spaces (with F. Nazarov and A. Volberg)

• Completely regular multivariate stationary processes and the Muckenhoupt condition (with A. Volberg)

• Weak type estimates and Cotlar inequalities for Calderon-Zygmund operators on nonhomogeneous spaces (with F. Nazarov and A. Volberg)

• Unconditional bases of invariant subspaces of a contraction with finite defects

• A conterexample to infinitedimensional Carleson imbedding theorem (with F. Nazarov and A. Volberg)

• The hunt for a Bellman function: applications to estimates of singular integral operators and to other classical problems in harmonic analysis (with F. Nazarov)

• The weighted norm inequalities for Hilbert transform are now trivial (with F. Nazarov)

• Hilbert transform, Toeplitz operators and Hankel operators, and invariant weights , (with A. Volberg and D. Zheng)

• Continuous frame decomposition and a vector Hunt -- Muckenhoupt -- Wheeden Theorem (with A. Volberg)

• Wavelets and the angle between past and future (with A. Volberg)

• A simple proof of the Hunt -- Muckenhoupt -- Wheeden Theorem (With A. Volberg)

• Approximation by analytic matrix functions. The four block problem (with V. V. Peller)