170 research outputs found

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

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    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 ϵ>0\epsilon>0 and small, f1f_1 is positive on the set of x(1,1)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

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    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 B2σB^\sigma_2 on the complex ball of Cd\mathbb{C}^d. In particular, we demonstrate that for any σ>0\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+CH^\infty+C

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