My research interests are in areas related to analysis. Mostly, I am interested in problems in analysis of several complex variables, harmonic analysis and operator theory, and specifically the interaction between these subjects. Many of my research projects deal with extending results in complex and harmonic analysis to more variables. Preprints of my work can be found below as well as slides from Talks that I have given. Some recent projects have included questions related to the Corona problem (joint with Sergei Treil), extension of some results in Harmonic Analysis to several complex variables (joint with Sergei Treil and Stefanie Petermichl), and Multi-Parameter Riesz Commutators (joint with Michael Lacey, Stefanie Petermichl, and Jill Pipher). More recently, I have considered the question of when paraproducts belong to various Schatten classes (joint with Michael Lacey), and the characterization of when the product of two little Hankel operators on the polydisc are compact (joint with Dechao Zheng). I have also recently done some work that relates to the Corona problem and to the stable rank of certain sub-algebras of bounded analytic functions (some of this is joint work with Amol Sasane and Raymond Mortini). Finally, I have also spent some time thinking about questions that are motived by the Generalized-Beurling operator. In particular I am thinking about weighted estimates for these operators, asymptotics of the operator for large values of p, and iterates of the operator (some of this is joint work with Leonid Slavin and Stefanie Petermichl).

Currently, I am thinking about questions that arise in the theory of Dirichlet spaces of analytic functions. In joint work with Nicola Arcozzi, Eric Sawyer and Richard Rochberg, we obtained a characterization of the bilinear forms on the Dirichlet space in terms of Carleson measures for the space. This answered a long-standing question of Rochberg's on this space. We are in the process of writing this up and determining all the implications of the result. Finally, this semester I plan to spend some time thinking about a question due to Sergei Treil, namely give a real variable characterization of the annhilator of the space of bounded analytic functions on the polydisc. The solution to this problem would have applications to the Corona problem in several complex variables.

Here you can find copies of my papers.

1. New Estimates for the Beurling-Ahlfors Operator on Differential Forms, with S. Petermichl and L. Slavin, (submitted to ).


2. Stabilization in Real H^∞, (submitted to Pub. Mat.).


3. Simultaneuos Stabilization in the Real Disc Algebra, with R. Mortini, (submitted to Studia Math.).


4. Composition of Haar Paraproducts: The Random Case, with D. Bilyk, X. Li, and M. Lacey, (submitted to Math. Anal.).


5. Toeplitz Corona Theorems for the Polydisk and the Unit Ball, with T. Trent, Complex Anal. Oper. Theory, to appear.


6. The Bass and Topological Stable Ranks for Real H^∞ and the Real Disc Algebra, with R. Mortini, J. Reine Angew. Math., to appear.


7. Multi-Parameter Riesz Commutators, with M. Lacey, S. Petermichl, and J. Pipher, Amer. J. Math., to appear.


8. The Corona Theorem and Stable Rank for C+BH^∞, with R. Mortini and A. Sasane, Houston J. Math., to appear.


9. Analytic Projections, the Corona Problem and Geometry of Holomorphic Vector Bundles, with S. Treil, J. Amer. Math. Soc., to appear.


10. Carleson Potentials and the Reproducing Kernel Thesis for Embedding Theorems, with S. Petermichl and S. Treil, Illinois J. of Math. 51 (2007), no. 4, 1249-1263.


11. A Weighted Estimate for the Square Function on the Unit Ball, with S. Petermichl, Ark. Mat. 45 (2007), no. 2, 337-350.


12. A Note About Stabilization in the Real Disc Algebra, Math. Nachr., to appear.


13. Remarks on Product VMO, with M. Lacey and E. Terwilleger, Proc. Amer. Math. Soc. 134 (2006), no. 2, pp. 465-474.


14. The Matrix-Valued H^p Corona Problem in the Disk and Polydisk, with S. Treil, J. Funct. Anal. 226 (2005), no. 1, pp. 138-172.

Here are the transparencies for talks I have given.

1. Remarks on Product VMO, IWOTA 2004


2. Multi-Parameter Riesz Commutators, SEAM XXII


3. Multi-Parameter Riesz Commutators, Sixth Prairie Analysis Seminar


4. Bounded Analytic Projections and the Corona Problem, SEAM XXIII


5. Multi-Parameter Riesz Commutators, AMS 2008 Spring Central Sectional Meeting


6. Bilinear forms on the Dirichlet Space, St. Petersburg Meeting in Mathematical Analysis

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 Foundation or the University of South Carolina.