arXiv subjects
Publications and source records attributed to Brett D. Wick.
In this note we show that if the Corona data depends continuously (smoothly) on a parameter, the solutions of the corresponding Bezout equations can be chosen to have the same smoothness in the parameter.
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 $$ \inf_{z\in\D}(\abs{f_1(z)}+\abs{f_2(z)})=δ>0. $$ Assume for some $ε>0$ and small, $f_1$ is positive on the set of $x\in(-1,1)$ where $\abs{f_2(x)}<ε$ for some $ε>0$ sufficiently small. Then there exists $g_1, g_1^{-1}, g_2\in H^\infty_\R(\D)$ with $$ \norm{g_1}_\infty,\norm{g_2}_\infty,\norm{g_1^{-1}}_\infty\leq C(δ,ε) $$ and $$ f_1(z)g_1(z)+f_2(z)g_2(z)=1\quad\forall z\in\D. $$
Suppose H is a space of functions on X. If H is a Hilbert space with reproducing kernel then that structure of H can be used to build distance functions on X. We describe some of those and their interpretations and interrelations. We also present some computational properties and examples.
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 $B^σ_2$ on the complex ball of $\mathbb{C}^d$. In particular, we demonstrate that for any $σ> 0$, 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.
In this paper we extend a method of Arveson and McCullough to prove a tangential interpolation theorem for subalgebras of $H^\infty$. This tangential interpolation result implies a Toelitz corona theorem. In particular, it is shown that the set of matrix positivity conditions is indexed by cyclic subspaces, which is analogous to the results obtained for the ball and the polydisk algebra by Trent-Wick and Douglas-Sarkar.
We study the possible analogous of the Div-Curl Lemma in classical harmonic analysis and partial differential equations, but from the point of view of the multi-parameter setting. In this context we see two possible Div-Curl lemmas that arise. Extensions to differential forms are also given.
In this paper we study ``Bergman-type'' singular integral operators on Ahlfors regular metric spaces. The main result of the paper demonstrates that if a singular integral operator on a Ahlfors regular metric space satisfies an additional estimate, then knowing the ``T(1)'' conditions for the operator imply that the operator is bounded on $L^2$. The method of proof of the main result is an extension and another application of the work originated by Nazarov, Treil and the first author on non-homogeneous harmonic analysis.
Let $Ω$ be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by $H^\infty(Ω)$ the Banach algebra of all bounded holomorphic functions on $Ω$, with pointwise operations and the supremum norm. We show that the topological stable rank of $H^\infty(Ω)$ is equal to 2. The proof is based on Suarez's theorem that the topological stable rank of $H^\infty(\D)$ is equal to 2, where $\D$ is the unit disk. We also show that for domains symmetric to the real axis, the Bass and topological stable ranks of the real symmetric algebra $H^\infty_\R(Ω)$ are 2.
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 $H^\infty+C$.
We study the $H^{\infty}(\mathbb{B}_{n})$ Corona problem $\sum_{j=1}^{N}f_{j}g_{j}=h$ and show it is always possible to find solutions $f$ that belong to $BMOA(\mathbb{B}_{n})$ for any $n>1$, including infinitely many generators $N$. This theorem improves upon both a 2000 result of Andersson and Carlsson and the classical 1977 result of Varopoulos. The former result obtains solutions for strictly pseudoconvex domains in the larger space $H^{\infty}\cdot BMOA$ with $N=\infty $, while the latter result obtains $BMOA(\mathbb{B}_{n})$ solutions for just N=2 generators with $h=1$. Our method of proof is to solve $\overline{\partial}$-problems and to exploit the connection between $BMO$ functions and Carleson measures for $H^{2}(\mathbb{B}_{n})$. Key to this is the exact structure of the kernels that solve the $\overline{\partial}$ equation for $(0,q)$ forms, as well as new estimates for iterates of these operators. A generalization to multiplier algebras of Besov-Sobolev spaces is also given.
We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\mathbb{C}^{n}$ has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^σ$ has the "baby corona property" for all $σ\geq 0$ and $1<p<\infty $. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.
We establish new $p$-estimates for the norm of the generalized Beurling--Ahlfors transform $\mathcal{S}$ acting on form-valued functions. Namely, we prove that $\norm{\mathcal{S}}_{L^p(\R^n;Λ)\to L^p(\R^n;Λ)}\leq n(p^{*}-1)$ where $p^*=\max\{p, p/(p-1)\},$ thus extending the recent Nazarov--Volberg estimates to higher dimensions. The even-dimensional case has important implications for quasiconformal mappings. Some promising prospects for further improvement are discussed at the end.
We present results about spaces of holomorphic functions associated to the classical Dirichlet space. The spaces we consider have roles similar to the roles of $H^{1}$ and $BMO$ in the Hardy space theory and we emphasize those analogies.
Let $\mathcal{D}$ be the classical Dirichlet space, the Hilbert space of holomorphic functions on the disk. Given a holomorphic symbol function $b$ we define the associated Hankel type bilinear form, initially for polynomials f and g, by $T_{b}(f,g):= < fg,b >_{\mathcal{D}} $, where we are looking at the inner product in the space $\mathcal{D}$. We let the norm of $T_{b}$ denotes its norm as a bilinear map from $\mathcal{D}\times\mathcal{D}$ to the complex numbers. We say a function $b$ is in the space $\mathcal{X}$ if the measure $dμ_{b}:=| b^{\prime}(z)| ^{2}dA$ is a Carleson measure for $\mathcal{D}$ and norm $\mathcal{X}$ by $$ \Vert b\Vert_{\mathcal{X}}:=| b(0)| +\Vert | b^{\prime}(z)| ^{2}dA\Vert_{CM(\mathcal{D})}^{1/2}. $$ Our main result is $T_{b}$ is bounded if and only if $b\in\mathcal{X}$ and $$ \Vert T_{b}\Vert_{\mathcal{D\times D}}\approx\Vert b\Vert_{\mathcal{X}}. $$
In this note we study the problem of simultaneous stabilization for the algebra $A_\R(\D)$. Invertible pairs $(f_j,g_j)$, $j=1,..., n$, in a commutative unital algebra are called \textit{simultaneously stabilizable} if there exists a pair $(α,β)$ of elements such that $αf_j+βg_j$ is invertible in this algebra for $j=1,..., n$. For $n=2$, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_\R(\D)$ has stable rank two, we are faced here with a different situation. When $n=2$, necessary and sufficient conditions are given so that we have simultaneous stability in $A_\R(\D)$. For $n\geq 3$ we show that under these conditions simultaneous stabilization is not possible and further connect this result to the question of which pairs $(f,g)$ in $A_\R(\D)^2$ are totally reducible; that is, for which pairs do there exist two units $u$ and $v$ in $A_\R(\D)$ such that $uf+vg=1$.
We give a simple proof of L^p boundedness of iterated commutators of Riesz transforms and a product BMO function. We use a representation of the Riesz transforms by means of simple dyadic operators - dyadic shifts - which in turn reduces the estimate quickly to paraproduct estimates.
The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in $\mathbb{C}^n$.
In this note we prove that the Bass stable rank of $H^\infty_\R(\D)$ is two. This establishes the validity of a conjecture by S. Treil. We accomplish this in two different ways, one by giving a direct proof, and the other, by first showing that the topological stable rank of $H^\infty_\R(\D)$ is two. We apply these results to give new proofs of results by R. Rupp and A. Sasane stating that the Bass stable rank of $A_\R(\D)$ is two and the topological stable rank of $A_\R(\D)$ is two, settling a conjecture by the second author. We also present a $\bar\partial$-free proof of the second author's characterization of the reducible pairs in $A_\R(\D)$.