Spectral Theory of Composition Operators on Hardy Spaces of the Unit Disc and of the Upper Half-Plane
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Fourier Spectrum Characterizations of Clifford $ H^{P} $ Spaces on $\Mathbf {R}^{N+ 1} + $ for $1\Leq P\Leq\Infty$
p n+1 Fourier Spectrum of Clifford H Spaces on R+ for 1 ≤ p ≤ ∞ Pei Dang, Weixiong Mai,∗ Tao Qian Abstract This article studies the Fourier spectrum characterization of func- p n+1 tions in the Clifford algebra-valued Hardy spaces H (R+ ), 1 ≤ p ≤ ∞. Namely, for f ∈ Lp(Rn), Clifford algebra-valued, f is further the non-tangential boundary limit of some function in p n+1 ˆ ˆ H (R+ ), 1 ≤ p ≤ ∞, if and only if f = χ+f, where χ+(ξ) = 1 ξ 2 (1 + i |ξ| ), the Fourier transformation and the above relation are suitably interpreted (for some cases in the distribution sense). These results further develop the relevant context of Alan McIn- tosh. As a particular case of our results, the vector-valued Clifford Hardy space functions are identical with the conjugate harmonic systems in the work of Stein and Weiss. The latter proved the cor- responding singular integral version of the vector-valued cases for 1 ≤ p < ∞. We also obtain the generalized conjugate harmonic systems for the whole Clifford algebra-valued Hardy spaces rather than only the vector-valued cases in the Stein-Weiss setting. Key words: Hardy space, Monogenic Function, Fourier Spectrum, Riesz Transform, Clifford Algebra, Conjugate Harmonic System arXiv:1711.02610v2 [math.CV] 14 Oct 2019 In memory of Alan McIntosh 1 Introduction The classical Paley-Wiener Theorem asserts that for a L2(R)-function f, scalar-valued, it is further the non-tangential boundary limit (NTBL) of a function in the Hardy H2 space in the upper half plane if and only if the Fourier transform of f, denoted by fˆ, satisfies the ˆ ˆ relation f = χ+f, where χ+ is the characteristic (indicator) function of the set (0, ∞), that This work was supported by the Science and Technology Development Fund, Macao SAR: 0006/2019/A1, 154/2017/A3; NSFC Grant No. -
Anisotropic Hardy Spaces and Wavelets
Anisotropic Hardy Spaces and Wavelets Marcin Bownik Author address: Department of Mathematics, University of Michigan, 525 East Uni- versity Ave., Ann Arbor, MI 48109 E-mail address: [email protected] 1 viii ANISOTROPIC HARDY SPACES AND WAVELETS Abstract In this paper, motivated in part by the role of discrete groups of dilations in wavelet theory, we introduce and investigate the anisotropic Hardy spaces associ- ated with very general discrete groups of dilations. This formulation includes the classical isotropic Hardy space theory of Fefferman and Stein and parabolic Hardy space theory of Calder´on and Torchinsky. Given a dilation A, that is an n n matrix all of whose eigenvalues λ satisfy λ > 1, define the radial maximal function× | | 0 −k −k Mϕf(x) := sup (f ϕk)(x) , where ϕk(x)= det A ϕ(A x). k∈Z | ∗ | | | Here ϕ is any test function in the Schwartz class with ϕ =0. For 0 <p< we p 6 ∞ introduce the corresponding anisotropic Hardy space HA as a space of tempered 0 p n R distributions f such that Mϕf belongs to L (R ). Anisotropic Hardy spaces enjoy the basic properties of the classical Hardy spaces. For example, it turns out that this definition does not depend on the choice of the test function ϕ as long as ϕ = 0. These spaces can be equivalently introduced in terms of grand, tangential, or6 nontangential maximal functions. We prove the Calder´on-Zygmund decompositionR which enables us to show the atomic p decomposition of HA. As a consequence of atomic decomposition we obtain the p description of the dual to HA in terms of Campanato spaces. -
Lower Bounds for the Ruelle Spectrum of Analytic Expanding Circle Maps
LOWER BOUNDS FOR THE RUELLE SPECTRUM OF ANALYTIC EXPANDING CIRCLE MAPS OSCAR F. BANDTLOW AND FRED´ ERIC´ NAUD Abstract. We prove that there exists a dense set of analytic expanding maps of the circle for which the Ruelle eigenvalues enjoy exponential lower bounds. The proof combines potential theoretic techniques and explicit calculations for the spectrum of expanding Blaschke products. 1. Introduction and statement One of the basic problems of smooth ergodic theory is to investigate the asymptotic behaviour of mixing systems. In particular, there is interest in precise quantitative results on the rate of decay of correlations for smooth observables. In his seminal paper [17], Ruelle showed that the long time asymptotic behaviour of analytic hyperbolic systems can be understood in terms of the Ruelle spectrum, that is, the spectrum of certain operators, known as transfer operators in this context, acting on suitable Banach spaces of holomorphic functions. Surprisingly, even for the simplest systems like analytic expanding maps of the circle, very few quantitative results on the Ruelle spectrum are known so far. Let us be more precise. With T denoting the unit circle in the complex plane C, a map τ : T ! T is said to be analytic expanding if τ has a holomorphic extension to a neighbourhood of T and we have inf jτ 0(z)j > 1 : z2T The Ruelle spectrum is commonly defined as the spectrum of the Perron-Frobenius or transfer operator (on a suitable space of holomorphic functions) given locally by d X 0 (Lτ f)(z) := !(τ) φk(z)f(φk(z)) ; (1) k=1 where φk denotes the k-th local inverse branch of the covering map τ : T ! T, and ( +1 if τ is orientation preserving; arXiv:1605.06247v1 [math.DS] 20 May 2016 !(τ) = (2) −1 if τ is orientation reversing. -
Additive Invariants on the Hardy Space Over the Polydisc
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Functional Analysis 253 (2007) 359–372 www.elsevier.com/locate/jfa Additive invariants on the Hardy space over the polydisc Xiang Fang 1 Department of Mathematics, Kansas State University, Manhattan, KS 64502, USA Received 21 March 2007; accepted 29 August 2007 Available online 10 October 2007 Communicated by G. Pisier Abstract In recent years various advances have been made with respect to the Nevanlinna–Pick kernels, especially on the symmetric Fock space, while the development on the Hardy space over the polydisc is relatively slow. In this paper, several results known on the symmetric Fock space are proved for the Hardy space over the polydisc. The known proofs on the symmetric Fock space make essential use of the Nevanlinna–Pick properties. Specifically, we study several integer-valued numerical invariants which are defined on an arbitrary in- variant subspace of the vector-valued Hardy spaces over the polydisc. These invariants include the Samuel multiplicity, curvature, fiber dimension, and a few others. A tool used to overcome the difficulty associated with non-Nevanlinna–Pick kernels is Tauberian theory. Published by Elsevier Inc. Keywords: Hardy space, polydisc; Samuel multiplicity; Curvature; Fiber dimension; Defect operator 0. Introduction and the main results The purpose of this paper is to prove several theorems on the Hardy space H 2(Dn) over the polydisc, whose symmetric Fock space versions are known, but the proofs rely on the properties of Nevanlinna–Pick kernels. In particular, our results allow one to formulate a theory of curvature invariant on H 2(Dn) in parallel to that on the symmetric Fock space [4]. -
Hardy Spaces, Bmo, and Boundary Value Problems for the Laplacian on a Smooth Domain in Rn
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 4, April 1999, Pages 1605{1661 S 0002-9947(99)02111-X HARDY SPACES, BMO, AND BOUNDARY VALUE PROBLEMS FOR THE LAPLACIAN ON A SMOOTH DOMAIN IN RN DER-CHEN CHANG, GALIA DAFNI, AND ELIAS M. STEIN Abstract. We study two different local Hp spaces, 0 <p 1, on a smooth ≤ domain in Rn, by means of maximal functions and atomic decomposition. We prove the regularity in these spaces, as well as in the corresponding dual BMO spaces, of the Dirichlet and Neumann problems for the Laplacian. 0. Introduction Let Ω be a bounded domain in Rn, with smooth boundary. The Lp regularity of elliptic boundary value problems on Ω, for 1 <p< , is a classical result in the theory of partial differential equations (see e.g. [ADN]).∞ In the situation of the whole space without boundary, i.e. where Ω is replaced by Rn, the results for Lp, 1 <p< , extend to the Hardy spaces Hp when 0 <p 1 and to BMO. Thus it is a natural∞ question to ask whether the Lp regularity of≤ elliptic boundary value problems on a domain Ω has an Hp and BMO analogue, and what are the Hp and BMO spaces for which it holds. This question was previously studied in [CKS], where partial results were ob- p p tained and were framed in terms of a pair of spaces, hr(Ω) and hz(Ω). These spaces, variants of those defined in [M] and [JSW], are, roughly speaking, the “largest” and “smallest” hp spaces that can be associated to a domain Ω. -
In Hardy Spaces of Several Variables
proceedings of the american mathematical society Volume 123, Number 1, January 1995 ON COMPACTNESS OF COMPOSITION OPERATORS IN HARDYSPACES OF SEVERALVARIABLES SONG-YINGLI AND BERNARDRUSSO (Communicated by Palle E. T. Jorgensen) Abstract. Characterizations of compactness are given for holomorphic com- position operators on Hardy spaces of a strongly pseudoconvex domain. 1. Introduction Let Q be a bounded domain in C" with C1 boundary. Let cp be a holomor- phic mapping from Q to Q. The composition operator C9 is defined formally as follows: C9(u)(z) = u(tp(z)) for all z £ Cl and any function a on Q. The study of such holomorphic composition operators has been active since the early 1970s (see Cowen [5] for details in the case of one variable). In the case of several complex variables, counterexamples have been constructed by several au- thors showing that composition operators can be unbounded on ßf2(Bn), where B„ is the unit ball in C" (see, for example, Cima and Wogen [1], Wogen's sur- vey paper [24], and the references therein). In this paper, we are concerned with compactness of composition operators. It was proved by Shapiro and Taylor [22] that Cf:ßfP(Bi) -» M7*(B\) is compact for one p £ (0, oo) if and only if it is compact on %7p(Bx) for all p £ (0, oo). There is a characterization of compactness for C9 : ß77v(Bx)-» %7P(BX)in terms of the Nevanlinna counting function, given by Shapiro [19]. Another characterization of compactness can be formulated in terms of a Carleson measure condition for the pullback mea- sure dv9 (see [16] for the case of the unit ball in C" ). -
Reproducing Kernel Hilbert Spaces and Hardy Spaces
Reproducing Kernel Hilbert Spaces and Hardy Spaces Reproducing Kernel Hilbert Spaces and Hardy Spaces Evan Camrud Iowa State University June 2, 2018 1 R: 1 ⇥ ⇤1 0 0 2 R3: 0 , 1 , 0 2 3 2 3 2 3 0 0 1 4 15 4 05 4 5 0 0 0 1 0 0 2 3 2 3 2 3 2 3 n 0 0 . 3 R : , ,..., . , . 6.7 6.7 6 7 6 7 6.7 6.7 617 607 6 7 6 7 6 7 6 7 607 607 607 617 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5 Question: How could we construct an ONB for an infinite-dimensional case? Reproducing Kernel Hilbert Spaces and Hardy Spaces Infinite Dimensional Hilbert Spaces Let’s look at some orthonormal bases (ONBs): 1 0 0 2 R3: 0 , 1 , 0 2 3 2 3 2 3 0 0 1 4 15 4 05 4 5 0 0 0 1 0 0 2 3 2 3 2 3 2 3 n 0 0 . 3 R : , ,..., . , . 6.7 6.7 6 7 6 7 6.7 6.7 617 607 6 7 6 7 6 7 6 7 607 607 607 617 6 7 6 7 6 7 6 7 4 5 4 5 4 5 4 5 Question: How could we construct an ONB for an infinite-dimensional case? Reproducing Kernel Hilbert Spaces and Hardy Spaces Infinite Dimensional Hilbert Spaces Let’s look at some orthonormal bases (ONBs): 1 R: 1 ⇥ ⇤ 1 0 0 0 0 1 0 0 2 3 2 3 2 3 2 3 n 0 0 . -
ON COMPLEMENTATION of VECTOR-VALUED HARDY SPACES WOLFGANG HENSGEN (Communicated by Paul S
proceedings of the american mathematical society Volume 104, Number 4, December 1988 ON COMPLEMENTATION OF VECTOR-VALUED HARDY SPACES WOLFGANG HENSGEN (Communicated by Paul S. Muhly) Abstract. Let X be a complex Banach space and 1 < p < oo. HP(X) resp. hp{X) denote the Hardy spaces of X-valued analytic resp. harmonic functions on the disc. LP(X) is the Lebesgue-Bochner space of X-valued integrable functions on the circle and HP(X) its Hardy-type subspace {/ € LP(X): ¡(n) = 0 Vn < 0}. It is proved that the following four conditions are equivalent: HP(X) is complemented in hp(X); the canonical analytic (or Riesz) projection is a bounded operator hp{X) -► HP{X); HP(X) is complemented in LP(X); analytic projection is a bounded operator LP(X) —»HP(X). It is well known that the last condition, in turn, is equivalent to the UMD property of X. 1. Introduction and statement of result. For 1 < p < oo, let LP(T, X) be the Lebesgue space on the unit circle T with normalized Lebesgue measure A, and let Hp C Lp be the usual Hardy space (so Hp = {f eLp: f(n) = 0 Vn < 0}, where fin) — fo*e~intfielt)Û 1S tne nin Fourier coefficient of /; n G Z). The famous M. Riesz theorem [15, p. 151ff.] states that for 1 < p < oo, Hp is complemented in Lp by the canonical "analytic (or Riesz) projection" / h-> fa, where fa is that Lp function with Fourier series Yn°=o fin)eint- F°r P = 1 or p = oo, / i—>fa is not a bounded operator in || • ||p (on the trigonometric polynomials, say). -
Arxiv:1703.05527V1 [Math.CA]
p( ) n n Interpolation between H · (R ) and L∞(R ): Real Method Ciqiang Zhuo, Dachun Yang ∗ and Wen Yuan Abstract Let p( ): Rn (0, ) be a variable exponent function satisfying the globally log-H¨older· continuous→ condition.∞ In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into “good” and “bad” parts and then prove the following real interpolation theorem between the variable Hardy space Hp(·)(Rn) and the space L∞(Rn): p(·) n ∞ n p(·)/(1−θ) n (H (R ),L (R )) ∞ = WH (R ), θ (0, 1), θ, ∈ where WHp(·)/(1−θ)(Rn) denotes the variable weak Hardy space. As an application, p(·) n the variable weak Hardy space WH (R ) with p− := ess inf ∈Rn p(x) (1, ) is x ∈ ∞ proved to coincide with the variable Lebesgue space WLp(·)(Rn). 1 Introduction In recent years, theories of several variable function spaces, based on the variable Lebesgue space, have been rapidly developed (see, for example, [3, 4, 10, 13, 24, 30, 31, 34, 35]). Recall that the variable Lebesgue space Lp(·)(Rn), with a variable exponent function p( ) : Rn (0, ), is a generalization of the classical Lebesgue space Lp(Rn). The study · → ∞ of variable Lebesgue spaces can be traced back to Orlicz [25], moreover, they have been the subject of more intensive study since the early work [22] of Kov´aˇcik and R´akosn´ık and [14] of Fan and Zhao as well as [7] of Cruz-Uribe and [11] of Diening, because of their intrinsic interest for applications into harmonic analysis, partial differential equations and variational integrals with nonstandard growth conditions (see also [1, 2, 20, 32] and their references). -
Gradient Systems
Ralph Chill, Eva Fasangovˇ a´ Gradient Systems – 13th International Internet Seminar – February 10, 2010 c by R. Chill, E. Faˇsangov´a Contents 0 Introduction ................................................... 1 1 Gradient systems in euclidean space .............................. 5 1.1 Thespace Rd .............................................. 5 1.2 Fr´echetderivative .............................. ............ 7 1.3 Euclideangradient............................... ........... 8 1.4 Euclideangradientsystems ........................ .......... 9 1.5 Algebraicequationsand steepest descent ............ ........... 12 1.6 Exercises ....................................... .......... 13 2 Gradient systems in finite dimensional space ...................... 15 2.1 Definitionofgradient ............................. .......... 17 2.2 Definitionofgradientsystem....................... .......... 18 2.3 Local existence for ordinary differential equations . ............. 18 2.4 Local and global existence for gradientsystems .. ..... .......... 21 2.5 Newton’smethod.................................. ......... 23 2.6 Exercises ....................................... .......... 25 3 Gradients in infinite dimensional space: the Laplace operator on a bounded interval ....................... 27 3.1 Definitionofgradient ............................. .......... 28 3.2 Gradientsofquadraticforms ....................... .......... 30 3.3 Sobolevspacesinonespacedimension ................ ........ 31 3.4 The Dirichlet-Laplace operator on a bounded interval . ......... -
Hardy Spaces and Unbounded Quasidisks
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 36, 2011, 291–300 HARDY SPACES AND UNBOUNDED QUASIDISKS Yong Chan Kim and Toshiyuki Sugawa Yeungnam University, Department of Mathematics Education 214-1 Daedong Gyongsan 712-749, Korea; [email protected] Tohoku University, Division of Mathematics, Graduate School of Information Sciences Aoba-ku, Sendai 980-8579, Japan; [email protected] Abstract. We study the maximal number 0 · h · +1 for a given plane domain such that f 2 Hp whenever p < h and f is analytic in the unit disk with values in : One of our main contributions is an estimate of h for unbounded K-quasidisks. 1. Introduction In his 1970 paper [10] Hansen introduced a number, denoted by h(); for a domain in the complex plane. The number h = h() is defined as the maximal one in [0; +1] so that every holomorphic function on any plane domain D with values in belongs to the Hardy class Hp(D) whenever 0 < p < h: The number was called by him the Hardy number of : If is bounded, then clearly h() = +1: Therefore, the consideration of h() is meaningful only when is unbounded. Hansen [10] studied the number by using Ahlfors’ distortion theorem. Also, in the same paper, he described it in terms of geometric quantities for starlike domains. Indeed, let 6= C be an unbounded starlike domain with respect to the origin. Let ®(t) be the length of maximal subarc of fz 2 T: tz 2 g for t > 0; where T stands for the unit circle fz 2 C: jzj = 1g: Observe that ®(t) is non-increasing in t by starlikeness. -
An Overview of Some Recent Developments on the Invariant Subspace Problem
Concrete Operators Review Article • DOI: 10.2478/conop-2012-0001 • CO • 2013 • 1-10 An overview of some recent developments on the Invariant Subspace Problem Abstract This paper presents an account of some recent approaches to Isabelle Chalendar1∗, Jonathan R. Partington2† the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions 1 Université de Lyon; CNRS; Université Lyon 1; INSA de Lyon; Ecole of specific topics, namely, universal operators, the Bishop op- Centrale de Lyon, CNRS, UMR 5208, Institut Camille Jordan 43 bld. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France erators, and Read’s Banach space counter-example involving a finitely strictly singular operator. 2 School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K. Keywords Received 24 July 2012 Invariant subspace • Universal operator • Weighted shift • Composition operator • Bishop operator • Strictly singular • Accepted 31 August 2012 Finitely strictly singular MSC: 47A15, 47B37, 47B33, 47B07 © Versita sp. z o.o. 1. History of the problem There is an important problem in operator theory called the invariant subspace problem. This problem is open for more than half a century, and there are many significant contributions with a huge variety of techniques, making this challenging problem so interesting; however the solution seems to be nowhere in sight. In this paper we first review the history of the problem, and then present an account of some recent developments with which we have been involved. Other recent approaches are discussed in [21]. The invariant subspace problem is the following: let X be a complex Banach space of dimension at least 2 and let T ∈ L(X), i.e., T : X → X, linear and bounded.