DOCSLIB.ORG

1 Hardy Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
1 Hardy Space Math 752 Fall 2015 1 Hardy Space Denote by H(D) the space of all analytic functions F : D ! C. We define the Hardy classes by p H (D) = fF 2 H(D): kF kHp(D) < 1g where kF kHp(D) = sup kFrkp: 0<r<1 p (This is the same norm as for h (D), but the space on which it is defined is p p different.) Evidently, H (D) ⊂ h (D). We define Hp([−1=2; 1=2]) to be the vector space of f 2 Lp([−1=2; 1=2]) such that fb(n) = 0 for n < 0. This definition makes sense since θ 7! e−2πinθ is in Lq([−1=2; 1=2]) for every 1 ≤ q ≤ 1, so that the integral defining the Fourier coefficient converges absolutely by H¨older'sinequality. 2 For p = 2 we have the following identifications. If F 2 H (D), then 1 1 X n X n 2πinθ F (z) = anz = anr e : n=0 n=0 Evidently 1 X n 2 janr j ≤ kF kH2(D) < 1; n=0 and the dominated convergence theorem shows that 1 X 2 janj < 1: n=0 2 This can be used (in the same way as the statements for h (D) and 2 L (R)) to show that X 2πinθ 2 f(θ) = lim ane 2 H ([−=1; 2; 12]) N!1 jn|≤N 2 is the unique L -limit of Fr as r ! 1− with kF kH2 = kfk2. In the reverse direction, if f 2 H2([−1=2; 1=2]), then F defined by 2πiθ F (re ) = Pr ∗ f(θ) 1 is an analytic function with kF kH2 = kfk2. (This can be seen directly from the Fourier series representation for Fr.) We summarize these statements in the following theorem. Theorem 1. The map that sends f to F as above defines an isometry (i.e., 2 2 a norm preserving bijection) between H ([−1=2; 1=2]) and H (D). 2 The Hilbert transform 2 2 2πiθ Let U 2 h (D). Recall that there exists (an) 2 ` (Z) such that for z = re X jnj 2πinθ U(z) = anr e ; n2Z and recall that the harmonic conjugate of U has the series representation X jnj 2πinθ V (z) = −i sgn(n)anr e : n2Z This means of course that 1 X n 2πinθ U(z) + iV (z) = a0 + 2 anr e n=0 2 is in H (D). The representation of the harmonic conjugate suggests the following definition. 2 2 Definition 1. define the map H : ` (Z) ! ` (Z) defined by H((an)n2Z) = (−isgn(n)an)n2Z: We will usually abuse notation by writing H(an) = −i sgn(n)an: We note that if a0 = 0, then kH(an)k2 = kank2. We note that the composition F −1 ◦ H ◦ F maps L2([−1=2; 1=2]) to L2([−1=2; 1=2]). Problem. Is there a more explicit representation of F −1 ◦ H ◦ F? Intuitively, H should be a convolution operator in the sense that 1 X F −1 ◦ H ◦ F(f)(x) = (−isgn(n)fb(n)e2πinx n=−∞ 2 looks like the Fourier series of a convolution. However −isgn(n) is not in 2 2 ` (Z), i.e., the function that we convolve with cannot be in L (R). However, if r < 1, then the sequence (an(r))n2Z defined by jnj an = −i sgn(n)r 2 2 is an element of ` (Z), hence there exists Qr 2 L ([−1=2; 1=2]) such that jnj Qbr(n) = −i sgn(n)r for all n 2 Z. We will therefore first compute Qr and then investigate what happens if r ! 1−. We have 1 X re2πiθ rne2πinθ = 1 − re2πiθ n=1 and −1 1 X X re−2πiθ r−ne2πinθ = rne−2πinθ = 1 − re−2πiθ n=−∞ n=1 which gives pointwise X 2r sin(2πθ) −i sgn(n)rjnje2πinθ = : 1 − 2r cos(2πθ) + r2 n2Z For fixed r < 1 the series on the left is uniformly and absolutely con- vergent for all θ 2 [−1=2; 1=2], hence it defines a continuous function on [−1=2; 1=2]. Since C[−1=2; 1=2] ⊂ L2([−1=2; 1=2]), we obtain 2r sin(2πθ) Q (θ) = : r 1 − 2r cos(2πθ) + r2 Theorem 2. Let u 2 L2([−1=2; 1=2]) and 2πiθ U(re ) = u ∗ Pr(θ) 2 the corresponding element in h (D). Then the harmonic conjugate V of U with V (0) = 0 has the representation 2πiθ V (re ) = u ∗ Qr(θ) for r < 1 and θ 2 R. 3 2 Proof. Vr 2 L ([−1=2; 1=2]) with jnj Vbr(n) = −i sgn(n)ub(n)r ; hence Vr = Pr ∗ u. Recall that Pr ≥ 0 and Z 1=2 Pr(θ)dθ = 1 −1=2 2πiθ 1 for all r < 1, i.e., P defined by P (re ) = Pr(θ) is in h (D). This implied in particular the solution of the Dirichlet problem. Unfortunately, Qr does not have the same nice property. 1 Remark 1. Qr 2= h (D). Indeed, Z 1=2 2 1 + r jQr(t)jdt = log ; −1=2 π 1 − r and this diverges to infinity as r ! 1−. This means in particular that the boundary representation of the harmonic conjugate has additional technical difficulties. Summary of what we know so far: If u 2 L2([−1=2; 1=2]), then there 2 exists U 2 h (D) such that kUr − tk2 ! 0 as r ! 1−. The harmonic 2 conjugate V of U has the property that V 2 h (D) with kV kh2(D) ≤ kUkh2(D) 2 (by Parseval), hence there exists v 2 L ([−1=2; 1=2]) with kVr − vk2 ! 0 as r ! 1−. Moreover, X v(θ) = −i lim sgn(n)u(n)e2πinθ N!1 b jn|≤N for a.e. θ. Goal. Find a representation for v in terms of the starting function u without having to calculate Fourier coefficients. The first question that we need to consider is how such a representation should look like. We need a certain amount of complex analysis to answer this question. We start with a function F that is analytic on D(0;R), where R > 1. The strategy is as follows. We will use Cauchy's theorem to express F as an integral over a suitably chosen contour. We then will deform the contour to the circle jzj = 1. Finally, we will take real and imaginary parts to obtain a representation of V in terms of u. 4 3 Hilbert transform on the unit circle iθ Let F be analytic on D(0;R) with R > 1 and F (0) = 0. Fix z0 = e on the unit circle. Fix " > 0. Consider the contour Γ" that is the unit circle traced counter clockwise except about z0, where we change it outwards by a circle segment with center z0 and radius r" = 2 sin("=2): (Sketch this.) For " ≤ jθ − tj ≤ π the contour is parametrized by ζ = eit: On the small circle segment it is parametrized by iθ i(t+θ) ζ = e + r"e ; where jtj ≤ 2−1(π + "). (Calculate angle sums in the triangle with inner angle " and side lengths 1; 1; r".) Cauchy's theorem gives us Z iθ 1 F (ζ) F (e ) = iθ dζ: 2πi Γ" ζ − e Call the large circle segment Γ1;" and the small circle segment Γ2;". On the large segment we parametrize by ζ = eit and obtain Z Z it 1 F (ζ) 1 F (e ) it it dζ = it iθ ie dt: 2πi Γ1;" ζ − e 2πi "≤|t−θ|≤π e − e iθ i(t+θ) On the small circle segment we parametrize by ζ = e + r"e with −1 iθ i(t+θ) i(t+θ) jtj ≤ 2 (π + "), so that ζ − e = r"e . Since dζ = r"ie dt, we get Z 1 iθ i(t+θ) 1 iθ F (e + r"e )idt ! F (e ) 2πi jt|≤2−1(π+") 2 as " ! 0. Multiplying by 2, letting " ! 0 in the original formula and subtracting F (eiθ) from both sides we obtain Z F (eit) F (eiθ) = lim dt: i(θ−t) "!0 "≤|t−θ|≤π 1 − e 5 We would like to have a kernel that is purely imaginary on the unit circle, and we can simplify the integrand further. To do so, note that if we 0 use the contour Γ" that is obtained by tracing along the inner part of the small circle around eiθ rather than the outer part, and applying Cauchy's theorem to eiθF (ζ) ζ 7! ζ which is also analytic in D(0;R) due to the assumption that F (0) = 0, we obtain with a similar reasoning that 1 Z ei(θ−t)F (eit) F (eiθ) = lim dt: i(θ−t) "!0 π "≤|θ−t|≤π 1 − e Add both representations for F (eiθ) to obtain i Z F (eit) F (eiθ) = lim dt: "!0 π θ−t "≤|θ−t|≤π 2 tan( 2 ) Take imaginary parts on both sides. We obtain 1 Z <F (eit) =F (eiθ) = lim dt "!0 2π θ−t "≤|θ−t|≤π tan( 2 ) Change of variable by replacing θ by 2πθ and substituting t = 2πτ. Set also F (e2πiθ) = u(θ) + iv(θ). We then get Z u(t) v(θ) = lim dt: "!0 "≤|θ−t|≤1=2 tan π(θ − t) Based on this example we formally define the integral transform f 7! fe by Z f(θ − t) fe(θ) = lim dt: "!0 "≤|t|≤1=2 tan πt This is also called the Hilbert transform of f.
Load more

Recommended publications
  • 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.
    [Show full text]
  • THE S.V.D. of the POISSON KERNEL 1. Introduction This Paper
    THE S.V.D. OF THE POISSON KERNEL GILES AUCHMUTY Abstract. The solution of the Dirichlet problem for the Laplacian on bounded regions N Ω in R , N 2 is generally described via a boundary integral operator with the Poisson kernel. Under≥ mild regularity conditions on the boundary, this operator is a compact 2 2 2 linear transformation of L (∂Ω,dσ) to LH (Ω) - the Bergman space of L harmonic functions on Ω. This paper describes the singular value decomposition (SVD)− of this operator and related results. The singular functions and singular values are constructed using Steklov eigenvalues and eigenfunctions of the biharmonic operator on Ω. These allow a spectral representation of the Bergman harmonic projection and the identification 2 of an orthonormal basis of the real harmonic Bergman space LH (Ω). A reproducing 2 2 kernel for LH (Ω) and an orthonormal basis of the space L (∂Ω,dσ) also are found. This enables the description of optimal finite rank approximations of the Poisson kernel with error estimates. Explicit spectral formulae for the normal derivatives of eigenfunctions for the Dirichlet Laplacian on ∂Ω are obtained and used to identify a constant in an inequality of Hassell and Tao. 1. Introduction This paper describes representation results for harmonic functions on a bounded region Ω in RN ; N 2. In particular a spectral representation of the Poisson kernel for solutions of the Dirichlet≥ problem for Laplace’s equation with L2 boundary data is described. This leads to an explicit description of the Reproducing− Kernel (RK) for 2 the harmonic Bergman space LH (Ω) and the singular value decomposition (SVD) of the Poisson kernel for the Laplacian.
    [Show full text]
  • Genius Manual I
    Genius Manual i Genius Manual Genius Manual ii Copyright © 1997-2016 Jiríˇ (George) Lebl Copyright © 2004 Kai Willadsen Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License (GFDL), Version 1.1 or any later version published by the Free Software Foundation with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. You can find a copy of the GFDL at this link or in the file COPYING-DOCS distributed with this manual. This manual is part of a collection of GNOME manuals distributed under the GFDL. If you want to distribute this manual separately from the collection, you can do so by adding a copy of the license to the manual, as described in section 6 of the license. Many of the names used by companies to distinguish their products and services are claimed as trademarks. Where those names appear in any GNOME documentation, and the members of the GNOME Documentation Project are made aware of those trademarks, then the names are in capital letters or initial capital letters. DOCUMENT AND MODIFIED VERSIONS OF THE DOCUMENT ARE PROVIDED UNDER THE TERMS OF THE GNU FREE DOCUMENTATION LICENSE WITH THE FURTHER UNDERSTANDING THAT: 1. DOCUMENT IS PROVIDED ON AN "AS IS" BASIS, WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, WITHOUT LIMITATION, WARRANTIES THAT THE DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT IS FREE OF DEFECTS MERCHANTABLE, FIT FOR A PARTICULAR PURPOSE OR NON-INFRINGING. THE ENTIRE RISK AS TO THE QUALITY, ACCURACY, AND PERFORMANCE OF THE DOCUMENT OR MODIFIED VERSION OF THE DOCUMENT IS WITH YOU.
    [Show full text]
  • Convergence of Complete Ricci-Flat Manifolds Jiewon Park
    Convergence of Complete Ricci-flat Manifolds by Jiewon Park Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 © Massachusetts Institute of Technology 2020. All rights reserved. Author . Department of Mathematics April 17, 2020 Certified by. Tobias Holck Colding Cecil and Ida Green Distinguished Professor Thesis Supervisor Accepted by . Wei Zhang Chairman, Department Committee on Graduate Theses 2 Convergence of Complete Ricci-flat Manifolds by Jiewon Park Submitted to the Department of Mathematics on April 17, 2020, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Abstract This thesis is focused on the convergence at infinity of complete Ricci flat manifolds. In the first part of this thesis, we will give a natural way to identify between two scales, potentially arbitrarily far apart, in the case when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation. We use an estimate of Colding-Minicozzi of a functional that measures the distance to the tangent cone. In the second part of this thesis, we prove a matrix Harnack inequality for the Laplace equation on manifolds with suitable curvature and volume growth assumptions, which is a pointwise estimate for the integrand of the aforementioned functional. This result provides an elliptic analogue of matrix Harnack inequalities for the heat equation or geometric flows. Thesis Supervisor: Tobias Holck Colding Title: Cecil and Ida Green Distinguished Professor 3 4 Acknowledgments First and foremost I would like to thank my advisor Tobias Colding for his continuous guidance and encouragement, and for suggesting problems to work on.
    [Show full text]
  • 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.
    [Show full text]
  • Elementary Applications of Fourier Analysis
    ELEMENTARY APPLICATIONS OF FOURIER ANALYSIS COURTOIS Abstract. This paper is intended as a brief introduction to one of the very first applications of Fourier analysis: the study of heat conduction. We derive the steady state equation, which serves as our motivating PDE, and then iden- tify solutions to two fundamental scenarios: the first, a heat distribution on a circle; the second, one on a rod. In doing all of this there is much more room for depth that simply isn't explored for brevity's sake. A more comprehensive paper could explicitly define what constitutes a good kernel, the limitations of convergence of the Fourier transform, and different ways to circumvent these limitations. Theoretically, there's room to explore harmonic functions explic- itly, as well as Hilbert spaces and the L2 space. That would be the more analytic route to take; on the other hand there are many fascinating applica- tions that aren't broached either. In math, application is nearly synonymous with PDEs, and as far as partial differential equations are concerned, the most important feature that the Fourier transform has is the property of exchanging differentiation for multiplication. This property is really what makes bothering with the Fourier transforms of most functions worth it at all. Another equally important property of Fourier transforms is called scaling, which, in physics, directly leads to a concept called the Heisenberg Uncertainty Principle, which, like the whole of quantum physics, is very good at bothering people. Contents 1. Introduction 1 2. The heat equation 2 3. The Steady-state equation and the Laplacian 3 4.
    [Show full text]
  • Uniformization of Riemann Surfaces
    CHAPTER 3 Uniformization of Riemann surfaces 3.1 The Dirichlet Problem on Riemann surfaces 128 3.2 Uniformization of simply connected Riemann surfaces 141 3.3 Uniformization of Riemann surfaces and Kleinian groups 148 3.4 Hyperbolic Geometry, Fuchsian Groups and Hurwitz’s Theorem 162 3.5 Moduli of Riemann surfaces 178 127 128 3. UNIFORMIZATION OF RIEMANN SURFACES One of the most important results in the area of Riemann surfaces is the Uni- formization theorem, which classifies all simply connected surfaces up to biholomor- phisms. In this chapter, after a technical section on the Dirichlet problem (solutions of equations involving the Laplacian operator), we prove that theorem. It turns out that there are very few simply connected surfaces: the Riemann sphere, the complex plane and the unit disc. We use this result in 3.2 to give a general formulation of the Uniformization theorem and obtain some consequences, like the classification of all surfaces with abelian fundamental group. We will see that most surfaces have the unit disc as their universal covering space, these surfaces are the object of our study in 3.3 and 3.5; we cover some basic properties of the Riemaniann geometry, §§ automorphisms, Kleinian groups and the problem of moduli. 3.1. The Dirichlet Problem on Riemann surfaces In this section we recall some result from Complex Analysis that some readers might not be familiar with. More precisely, we solve the Dirichlet problem; that is, to find a harmonic function on a domain with given boundary values. This will be used in the next section when we classify all simply connected Riemann surfaces.
    [Show full text]
  • 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].
    [Show full text]
  • Lectures on Partial Differential Equations
    Lectures on Partial Differential Equations Govind Menon1 Dec. 2005 Abstract These are my incomplete lecture notes for the graduate introduction to PDE at Brown University in Fall 2005. The lectures on Laplace’s equation and the heat equation are included here. Typing took too much work after that. I hope what is here is still useful. Andreas Kl¨ockner’s transcript of the remaining lectures is also posted on my website. Those however have not been proofread. Comments are wel- come by email. Contents 1 Laplace’s equation 3 1.1 Introduction............................ 3 1.1.1 Minimal surfaces and Laplace’s equation . 3 1.1.2 Fields and Laplace’s equation . 5 1.1.3 Motivation v. Results . 5 1.2 Notation.............................. 5 1.3 Themeanvalueinequality. 6 1.4 Maximumprinciples ....................... 8 1.5 Thefundamentalsolution . 10 1.6 Green’s function and Poisson’s integral formula . 13 1.7 The mean value property revisited . 17 1.8 Harmonic functions are analytic . 17 1.9 Compactness and convergence . 21 1.10 Perron’s method for the Dirichlet problem . 22 1.11 Energy methods and Dirichlet’s principle . 26 1.12 Potentials of measures . 28 1.13 Lebesgue’sthorn .. .. .. .. .. .. .. 30 1.14 The potential of a compact set . 33 1.15 Capacity of compact sets . 37 1 2 1.16 Variational principles for capacity . 39 2 The heat equation 43 2.1 Motivation ............................ 43 2.2 Thefundamentalsolution . 43 2.3 Uniquenessofsolutions . 47 2.4 Theweakmaximumprinciple . 48 2.5 Themeanvalueproperty . 49 2.6 Thestrongmaximumprinciple . 53 2.7 Differenceschemes . .. .. .. .. .. .. 54 2.8 Randomwalks .......................... 56 2.9 Brownianmotion ........................
    [Show full text]
  • The Operator-Valued Poisson Kernel and Its Applications
    Irish Math. Soc. Bulletin 51 (2003), 21–44 21 The Operator-valued Poisson Kernel and its Applications ISABELLE CHALENDAR 1. Introduction Let H be a complex Hilbert space and let L(H) denote the algebra of all linear and bounded mappings from H to H. For T 2 L(H), its spectrum σ(T ) is the nonempty compact subset of C consisting of all ¸ 2 C such that T ¡ ¸Id is non-invertible in L(H). Its point spectrum σp(T ) is the (possibly empty) subset of σ(T ) consisting of those ¸ 2 C such that ker (T ¡ ¸Id) 6= f0g. We write D for the open unit disc in C, and T for the unit circle. The spaces Lp = Lp(T), 1 · p · 1 are the usual Lebesgue function spaces relative to nor- malized Lebesgue measure on T. 2 2 As usual, we define theP Hardy space H = H (D) as the space of all functions f : z 7! 1 a zn for which the norm kfk = ¡ ¢ n=0 n P1 2 1=2 2 n=0 janj is finite. It is well known that H (D) may be re- garded isometrically as a closed subspace H2(T) of L2 [31, 25], by identifying the Taylor coefficients of f with the Fourier coefficients of an L2(T) function. The Hardy space H1 is the set of bounded and analytic functions on D, which is isometrically isomorphic to H1(T) := ff 2 L1(T): fˆ(n) = 0; n < 0g via the mapping that 1 takes a function into its radial limit.
    [Show full text]
  • 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.
    [Show full text]
  • Notes on Distributions
    Notes on Distributions Peter Woit Department of Mathematics, Columbia University [email protected] April 26, 2020 1 Introduction This is a set of notes written for the 2020 spring semester Fourier Analysis class, covering material on distributions. This is an important topic not covered in Stein-Shakarchi. These notes will supplement two textbooks you can consult for more details: A Guide to Distribution Theory and Fourier Transforms [2], by Robert Strichartz. The discussion of distributions in this book is quite compre- hensive, and at roughly the same level of rigor as this course. Much of the motivating material comes from physics. Lectures on the Fourier Transform and its Applications [1], by Brad Os- good, chapters 4 and 5. This book is wordier than Strichartz, has a wealth of pictures and motivational examples from the field of signal processing. The first three chapters provide an excellent review of the material we have covered so far, in a form more accessible than Stein-Shakarchi. One should think of distributions as mathematical objects generalizing the notion of a function (and the term \generalized function" is often used inter- changeably with \distribution"). A function will give one a distribution, but many important examples of distributions do not come from a function in this way. Some of the properties of distributions that make them highly useful are: Distributions are always infinitely differentiable, one never needs to worry about whether derivatives exist. One can look for solutions of differential equations that are distributions, and for many examples of differential equations the simplest solutions are given by distributions.
    [Show full text]