170 research outputs found

### Stabilization in $H^\infty_{\mathbb{R}}(\mathbb{D})$

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

### 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

### Spectral Characteristics and Stable Ranks for the Sarason Algebra $H^\infty+C$

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

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