Bergman-type Singular Operators and the Characterization of Carleson Measures for Besov--Sobolev Spaces on the Complex Ball

The purposes of this paper are two fold. First, we extend the method of non-homogeneous harmonic analysis of Nazarov, Treil and Volberg to handle "Bergman--type" singular integral operators. The canonical example of such an operator is the Beurling transform on the unit disc. Second, we use the methods developed in this paper to settle the important open question about characterizing the Carleson measures for the Besov--Sobolev space of analytic functions $B^\sigma_2$ on the complex ball of $\mathbb{C}^d$. In particular, we demonstrate that for any $\sigma> 0$, the Carleson measures for the space are characterized by a "T1 Condition". The method of proof of these results is an extension and another application of the work originated by Nazarov, Treil and the first author.Comment: v1: 31 pgs; v2: 31 pgs, title changed, typos corrected, references added; v3: 33 pages, typos corrected, references added, presentation improved based on referee comments

A Study of the Matrix Carleson Embedding Theorem with Applications to Sparse Operators

In this paper, we study the dyadic Carleson Embedding Theorem in the matrix weighted setting. We provide two new proofs of this theorem, which highlight connections between the matrix Carleson Embedding Theorem and both maximal functions and $H^1$-BMO duality. Along the way, we establish boundedness results about new maximal functions associated to matrix $A_2$ weights and duality results concerning $H^1$ and BMO sequence spaces in the matrix setting. As an application, we then use this Carleson Embedding Theorem to show that if $S$ is a sparse operator, then the operator norm of $S$ on $L^2(W)$ satisfies: $$\| S\|_{L^2(W) \rightarrow L^2(W)} \lesssim [W]_{A_2}^{\frac{3}{2}},$$ for every matrix $A_2$ weight $W$.Comment: 14 page

Spectral Characteristics and Stable Ranks for the Sarason Algebra H∞+CH^\infty+C

We prove a Corona type theorem with bounds for the Sarason algebra $H^\infty+C$ and determine its spectral characteristics. We also determine the Bass, the dense, and the topological stable ranks of $H^\infty+C$.Comment: v1: 16 page

Weighted LpL^p Estimates for the Bergman and Szeg\H{o} Projections on Strongly Pseudoconvex Domains with Near Minimal Smoothness

We prove the weighted $L^p$ regularity of the ordinary Bergman and Cauchy-Szeg\H{o} projections on strongly pseudoconvex domains $D$ in $\mathbb{C}^n$ with near minimal smoothness for appropriate generalizations of the $B_p/A_p$ classes. In particular, the $B_p/A_p$ Muckenhoupt type condition is expressed relative to balls in a quasi-metric that arises as a space of homogeneous type on either the interior or the boundary of the domain $D$.Comment: 40 pages, introduction reorganized and some typos correcte

Stabilization in HR∞(D)H^\infty_{\mathbb{R}}(\mathbb{D})

In this paper we prove the following theorem: Suppose that f_1,f_2\in H^\infty_\R(\D), with \norm{f_1}_\infty,\norm{f_2}_{\infty}\leq 1, with \inf_{z\in\D}(\abs{f_1(z)}+\abs{f_2(z)})=\delta>0. Assume for some $\epsilon>0$ and small, $f_1$ is positive on the set of $x\in(-1,1)$ where \abs{f_2(x)}0 sufficiently small. Then there exists g_1, g_1^{-1}, g_2\in H^\infty_\R(\D) with \norm{g_1}_\infty,\norm{g_2}_\infty,\norm{g_1^{-1}}_\infty\leq C(\delta,\epsilon) and f_1(z)g_1(z)+f_2(z)g_2(z)=1\quad\forall z\in\D. Comment: v1: 22 pages, 2 figures, to appear in Pub. Mat; v2: 32 pages, 5 figures. The earlier version incorrectly claimed a characterization, as was pointed out by R. Mortini. A key hypothesis was strengthened with the main result remaining the sam