DOCSLIB.ORG

Appendix 5.A. the Schwartz Space and the Fourier Transform May the Schwartz Be with You!1 in This Section, We Summarize Some

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Appendix 5.A. the Schwartz Space and the Fourier Transform May the Schwartz Be with You!1 in This Section, We Summarize Some Appendix 5.A. The Schwartz space and the Fourier transform May the Schwartz be with you!1 In this section, we summarize some results about Schwartz functions, tempered distributions, and the Fourier transform. For complete proofs, see [13, 15]. 5.A.1. The Schwartz space. Since we will study the Fourier transform, we consider complex-valued functions here. Definition 5.15. The Schwartz space S(Rn) is the topological vector space of functions f : Rn ! C such that f 2 C1(Rn) and xα@βf(x) ! 0 as jxj ! 1 n n n for every pair of multi-indices α; β 2 N0 . For α; β 2 N0 and f 2 S(R ) let α β (5.10) kfkα,β = sup x @ f : n R n A sequence of functions ffk : k 2 Ng converges to a function f in S(R ) if kfn − fkα,β ! 0 as k ! 1 n for every α; β 2 N0 . That is, the Schwartz space consists of smooth functions whose derivatives (including the function itself) decay at infinity faster than any power; we say, for short, that Schwartz functions are rapidly decreasing. When there is no ambiguity, we will write S(Rn) as S. 2 Example 5.16. The function f(x) = e−|xj belongs to S(Rn). More generally, 2 if p is any polynomial, then g(x) = p(x) e−|xj belongs to S. Example 5.17. The function 1 f(x) = (1 + jxj2)k does not belongs to S for any k 2 N since jxj2kf(x) does not decay to zero as jxj ! 1. Example 5.18. The function f : R ! R defined by 2 2 f(x) = e−x sin ex does not belong to S(R) since f 0(x) does not decay to zero as jxj ! 1. The space D(Rn) of smooth complex-valued functions with compact support n is contained in the Schwartz space S(R ). If fk ! f in D (in the sense of Defini- tion 3.7), then fk ! f in S, so D is continuously imbedded in S. Furthermore, if 1 n f 2 S, and η 2 Cc (R ) is a cutoff function with ηk(x) = η(x=k), then ηkf ! f in S as k ! 1, so D is dense in S. 1Spaceballs 132 5.A. THE SCHWARTZ SPACE AND THE FOURIER TRANSFORM 133 The topology of S is defined by the countable family of semi-norms k · kα,β given in (5.10). This topology is not derived from a norm, but it is metrizable; for example, we can use as a metric X cα,βkf − gkα,β d(f; g) = n 1 + kf − gkα,β α,β2N0 P where the cα,β > 0 are any positive constants such that n cα,β converges. α,β2N0 Moreover, S is complete with respect to this metric. A complete, metrizable topo- logical vector space whose topology may be defined by a countable family of semi- norms is called a Fr´echet space. Thus, S is a Fr´echet space. If we want to make explicit that a limit exists with respect to the Schwartz topology, we write f = S-lim fk; k!1 and call f the S-limit of ffkg. α α n If fk ! f as k ! 1 in S, then @ fk ! @ f for any multi-index α 2 N0 . Thus, the differentiation operator @α : S!S is a continuous linear map on S. 5.A.2. Tempered distributions. Tempered distributions are distributions (c.f. Section 3.3) that act continuously on Schwartz functions. Roughly speaking, we can think of tempered distributions as distributions that grow no faster than a polynomial at infinity.2 Definition 5.19. A tempered distribution T on Rn is a continuous linear functional T : S(Rn) ! C. The topological vector space of tempered distributions is denoted by S0(Rn) or S0. If hT; fi denotes the value of T 2 S0 acting on f 2 S, 0 then a sequence fTkg converges to T in S , written Tk *T , if hTk; fi ! hT; fi for every f 2 S: Since D ⊂ S is densely and continuously imbedded, we have S0 ⊂ D0. Moreover, a distribution T 2 D0 extends uniquely to a tempered distribution T 2 S0 if and only if it is continuous on D with respect to the topology on S. 1 0 Every function f 2 Lloc defines a regular distribution Tf 2 D by Z hTf ; φi = fφ dx for all φ 2 D: If jfj ≤ p is bounded by some polynomial p, then Tf extends to a tempered dis- 0 tribution Tf 2 S , but this is not the case for functions f that grow too rapidly at infinity. 2 Example 5.20. The locally integrable function f(x) = ejxj defines a regular 0 distribution Tf 2 D but this distribution does not extend to a tempered distribu- tion. x x 0 Example 5.21. If f(x) = e cos (e ), then Tf 2 D (R) extends to a tempered distribution even though the values of f(x) grow exponentially as x ! 1. This 2The name `tempered distribution' is short for `distribution of temperate growth,' meaning growth that is at most polynomial. 134 0 tempered distribution is the distributional derivative Tf = (Tg) of the regular 0 x distribution Tg where f = g and g(x) = sin(e ): Z hf; φi = −hg; φ0i = − sin(ex)φ(x) dx for all φ 2 S: The distribution Tf is decreasing in a weak sense at infinity because of the rapid oscillations of f. Example 5.22. The series X δ(n)(x − n) n2N where δ(n) is the nth derivative of the δ-function converges to a distribution in D0(R), but it does not converge in S0(R) or define a tempered distribution. We define the derivative of tempered distributions in the same way as for dis- n tributions. If α 2 N0 is a multi-index, then h@αT; φi = (−1)jαjhT;@αφi: We say that a C1-function f is slowly growing if the function and all of its deriva- n tives are of polynomial growth, meaning that for every α 2 N0 there exists a constant Cα and an integer Nα such that α 2Nα j@ f(x)j ≤ Cα 1 + jxj : If f is C1 and slowly growing, then fφ 2 S whenever φ 2 S, and multiplication by f is a continuous map on S. Thus for T 2 S0, we may define the product fT 2 S0 by hfT; φi = hT; fφi: 5.A.3. The Fourier transform on S. The Schwartz space is a natural one to use for the Fourier transform. Differentiation and multiplication exchange r^oles under the Fourier transform and therefore so do the properties of smoothness and rapid decrease. As a result, the Fourier transform is an automorphism of the Schwartz space. By duality, the Fourier transform is also an automorphism of the space of tempered distributions. Definition 5.23. The Fourier transform of a function f 2 S(Rn) is the func- tion f^ : Rn ! C defined by 1 Z (5.11) f^(k) = f(x)e−ik·x dx: (2π)n The inverse Fourier transform of f is the function fˇ : Rn ! C defined by Z fˇ(x) = f(k)eik·x dk: We generally use x to denote the variable on which a function f depends and k to denote the variable on which its Fourier transform depends. Example 5.24. For σ > 0, the Fourier transform of the Gaussian 1 2 2 f(x) = e−|xj =2σ (2πσ2)n=2 5.A. THE SCHWARTZ SPACE AND THE FOURIER TRANSFORM 135 is the Gaussian 1 2 2 f^(k) = e−σ jkj =2 (2π)n The Fourier transform maps differentiation to multiplication by a monomial and multiplication by a monomial to differentiation. As a result, f 2 S if and only ^ ^ ^ if f 2 S, and fn ! f in S if and only if fn ! f in S. Theorem 5.25. The Fourier transform F : S!S defined by F : f 7! f^ is a continuous, one-to-one map of S onto itself. The inverse F −1 : S!S is given by F −1 : f 7! fˇ. If f 2 S, then F [@αf] = (ik)αf;^ F (−ix)βf = @βf:^ The Fourier transform maps the convolution product of two functions to the pointwise product of their transforms. Theorem 5.26. If f; g 2 S, then the convolution h = f ∗ g 2 S, and h^ = (2π)nf^g:^ If f; g 2 S, then Z Z fg dx = (2π)n f^g^ dk: In particular, Z Z jfj2 dx = (2π)n jf^j2 dk: 5.A.4. The Fourier transform on S0. The main reason to introduce tem- pered distributions is that their Fourier transform is also a tempered distribution. If φ, 2 S, then by Fubini's theorem Z Z 1 Z φ ^ dx = φ(x) (y)e−ix·y dy dx (2π)n Z 1 Z = φ(x)e−ix·y dx (y) dy (2π)n Z = φψ^ dx: This motivates the following definition for the Fourier transform of a tempered distribution which is compatible with the one for Schwartz functions. Definition 5.27. If T 2 S0, then the Fourier transform T^ 2 S0 is the distribu- tion defined by hT^ ; φi = hT; φ^i for all φ 2 S: The inverse Fourier transform Tˇ 2 S0 is the distribution defined by hTˇ ; φi = hT; φˇi for all φ 2 S: We also write T^ = FT and Tˇ = F −1T .
Load more

Recommended publications
  • Arxiv:0704.2891V3 [Math.AG] 5 Dec 2007 Schwartz Functions on Nash Manifolds
    Schwartz functions on Nash manifolds Avraham Aizenbud and Dmitry Gourevitch ∗ July 11, 2011 Abstract In this paper we extend the notions of Schwartz functions, tempered func- tions and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case classically known properties of Schwartz functions on Rn and build some additional tools which are important in rep- resentation theory. Contents 1 Introduction 2 1.1 Mainresults................................ 3 1.2 Schwartz sections of Nash bundles . 4 1.3 Restricted topologyand sheaf properties . .... 4 1.4 Possibleapplications ........................... 5 1.5 Summary ................................. 6 1.6 Remarks.................................. 6 2 Semi-algebraic geometry 8 2.1 Basicnotions ............................... 8 arXiv:0704.2891v3 [math.AG] 5 Dec 2007 2.2 Tarski-Seidenberg principle of quantifier elimination anditsapplications............................ 8 2.3 Additional preliminary results . .. 10 ∗Avraham Aizenbud and Dmitry Gourevitch, Faculty of Mathematics and Computer Science, The Weizmann Institute of Science POB 26, Rehovot 76100, ISRAEL. E-mails: [email protected], [email protected]. Keywords: Schwartz functions, tempered functions, generalized functions, distributions, Nash man- ifolds. 1 3 Nash manifolds 11 3.1 Nash submanifolds of Rn ......................... 11 3.2 Restricted topological spaces and sheaf theory over them........ 12 3.3 AbstractNashmanifolds . 14 3.3.1 ExamplesandRemarks. 14 3.4 Nashvectorbundles ........................... 15 3.5 Nashdifferentialoperators . 16 3.5.1 Algebraic differential operators on a Nash manifold . .... 17 3.6 Nashtubularneighborhood . 18 4 Schwartz and tempered functions on affine Nash manifolds 19 4.1 Schwartzfunctions ............................ 19 4.2 Temperedfunctions. .. .. .. 20 4.3 Extension by zero of Schwartz functions . .. 20 4.4 Partitionofunity ............................. 21 4.5 Restriction and sheaf property of tempered functions .
    [Show full text]
  • 171 Composition Operator on the Space of Functions
    Acta Math. Univ. Comenianae 171 Vol. LXXXI, 2 (2012), pp. 171{183 COMPOSITION OPERATOR ON THE SPACE OF FUNCTIONS TRIEBEL-LIZORKIN AND BOUNDED VARIATION TYPE M. MOUSSAI Abstract. For a Borel-measurable function f : R ! R satisfying f(0) = 0 and Z sup t−1 sup jf 0(x + h) − f 0(x)jp dx < +1; (0 < p < +1); t>0 R jh|≤t s n we study the composition operator Tf (g) := f◦g on Triebel-Lizorkin spaces Fp;q(R ) in the case 0 < s < 1 + (1=p). 1. Introduction and the main result The study of the composition operator Tf : g ! f ◦ g associated to a Borel- s n measurable function f : R ! R on Triebel-Lizorkin spaces Fp;q(R ), consists in finding a characterization of the functions f such that s n s n (1.1) Tf (Fp;q(R )) ⊆ Fp;q(R ): The investigation to establish (1.1) was improved by several works, for example the papers of Adams and Frazier [1,2 ], Brezis and Mironescu [6], Maz'ya and Shaposnikova [9], Runst and Sickel [12] and [10]. There were obtained some necessary conditions on f; from which we recall the following results. For s > 0, 1 < p < +1 and 1 ≤ q ≤ +1 n s n s n • if Tf takes L1(R ) \ Fp;q(R ) to Fp;q(R ), then f is locally Lipschitz con- tinuous. n s n • if Tf takes the Schwartz space S(R ) to Fp;q(R ), then f belongs locally to s Fp;q(R). The first assertion is proved in [3, Theorem 3.1].
    [Show full text]
  • Function Spaces Mikko Salo
    Function spaces Lecture notes, Fall 2008 Mikko Salo Department of Mathematics and Statistics University of Helsinki Contents Chapter 1. Introduction 1 Chapter 2. Interpolation theory 5 2.1. Classical results 5 2.2. Abstract interpolation 13 2.3. Real interpolation 16 2.4. Interpolation of Lp spaces 20 Chapter 3. Fractional Sobolev spaces, Besov and Triebel spaces 27 3.1. Fourier analysis 28 3.2. Fractional Sobolev spaces 33 3.3. Littlewood-Paley theory 39 3.4. Besov and Triebel spaces 44 3.5. H¨olderand Zygmund spaces 54 3.6. Embedding theorems 60 Bibliography 63 v CHAPTER 1 Introduction In mathematical analysis one deals with functions which are dif- ferentiable (such as continuously differentiable) or integrable (such as square integrable or Lp). It is often natural to combine the smoothness and integrability requirements, which leads one to introduce various spaces of functions. This course will give a brief introduction to certain function spaces which are commonly encountered in analysis. This will include H¨older, Lipschitz, Zygmund, Sobolev, Besov, and Triebel-Lizorkin type spaces. We will try to highlight typical uses of these spaces, and will also give an account of interpolation theory which is an important tool in their study. The first part of the course covered integer order Sobolev spaces in domains in Rn, following Evans [4, Chapter 5]. These lecture notes contain the second part of the course. Here the emphasis is on Sobolev type spaces where the smoothness index may be any real number. This second part of the course is more or less self-contained, in that we will use the first part mainly as motivation.
    [Show full text]
  • Fourier Analysis in Function Space Theory
    Course No. 401-4463-62L Fourier Analysis in Function Space Theory Dozent: Tristan Rivi`ere Assistant: Alessandro Pigati Contents 1 The Fourier transform of tempered distributions 1 1.1 The Fourier transforms of L1 functions . 1 1.2 The Schwartz Space S(Rn)........................... 4 1.3 Frechet Spaces . 6 1.4 The space of tempered distributions S0(Rn).................. 12 1.5 Convolutions in S0(Rn)............................. 21 2 The Hardy-Littlewood Maximal Function 26 2.1 Definition and elementary properties. 26 2.2 Hardy-Littlewood Lp−theorem for the Maximal Function. 27 2.3 The limiting case p =1. ............................ 30 3 Quasi-normed vector spaces 32 3.1 The Metrizability of quasi-normed vector spaces . 32 3.2 The Lorentz spaces Lp;1 ............................ 36 3.3 Decreasing rearrangement . 37 3.4 The Lorentz spaces Lp;q ............................ 39 3.5 Functional inequalities for Lorentz spaces . 44 3.6 Dyadic characterization of some Lorentz spaces and another proof of Lorentz{ Sobolev embedding (optional) . 49 4 The Lp−theory of Calder´on-Zygmund convolution operators. 52 4.1 Calder´on-Zygmund decompositions. 52 4.2 An application of Calder´on-Zygmund decomposition . 54 4.3 The Marcinkiewicz Interpolation Theorem - The Lp case . 56 4.4 Calderon Zygmund Convolution Operators over Lp ............. 58 4.4.1 A \primitive" formulation . 60 4.4.2 A singular integral type formulation . 64 4.4.3 The case of homogeneous kernels . 69 4.4.4 A multiplier type formulation . 71 4.4.5 Applications: The Lp theory of the Riesz Transform and the Laplace and Bessel Operators . 74 4.4.6 The limiting case p =1........................
    [Show full text]
  • A Simpler Description of the $\Kappa $-Topologies on the Spaces
    A simpler description of the κ-topologies on p 1 the spaces DLp, L , M Christian Bargetz,∗ Eduard A. Nigsch,† Norbert Ortner‡ November 2017 Abstract p 1 The κ-topologies on the spaces DLp , L and M are defined by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre-)dual spaces. In many cases it is more convenient to work with a description of the topology by means of a family of semi-norms defined by multiplication and/or convolution with functions and by classical norms. We give such families of semi- norms generating the κ-topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence-space representation of the spaces DLp equipped with the κ-topology, which complements a result of J. Bonet and M. Maestre. As a byproduct, we give a characterisation of the compact subsets of 1 p 1 the spaces DLp , L and M . Keywords: locally convex distribution spaces, topology of uniform conver- gence on compact sets, p-integrable smooth functions, compact sets arXiv:1711.06577v2 [math.FA] 27 Nov 2018 MSC2010 Classification: 46F05, 46E10, 46A13, 46E35, 46A50, 46B50 ∗Institut f¨ur Mathematik, Universit¨at Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria. e-mail: [email protected]. †Wolfgang Pauli Institut, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. e-mail: [email protected]. ‡Institut f¨ur Mathematik, Universit¨at Innsbruck, Technikerstraße 13, 6020 Innsbruck, Austria. e-mail: [email protected]. 1 1 Introduction In the context of the convolution of distributions, L.
    [Show full text]
  • Intersections of Fréchet Spaces and (LB)–Spaces
    Intersections of Fr¶echet spaces and (LB){spaces Angela A. Albanese and Jos¶eBonet Abstract This article presents results about the class of locally convex spaces which are de¯ned as the intersection E \F of a Fr¶echet space F and a countable inductive limit of Banach spaces E. This class appears naturally in analytic applications to linear partial di®erential operators. The intersection has two natural topologies, the intersection topology and an inductive limit topology. The ¯rst one is easier to describe and the second one has good locally convex properties. The coincidence of these topologies and its consequences for the spaces E \ F and E + F are investigated. 2000 Mathematics Subject Classi¯cation. Primary: 46A04, secondary: 46A08, 46A13, 46A45. The aim of this paper is to investigate spaces E \ F which are the intersection of a Fr¶echet space F and an (LB)-space E. They appear in several parts of analysis whenever the space F is determined by countably many necessary (e.g. di®erentiability of integrabil- ity) conditions and E is the dual of such a space, in particular E is de¯ned by a countable sequence of bounded sets which may also be determined by concrete estimates. Two nat- ural topologies can be de¯ned on E \ F : the intersection topology, which has seminorms easy to describe and which permits direct estimates, and a ¯ner inductive limit topology which is de¯ned in a natural way and which has good locally convex properties, e.g. E \ F with this topology is a barrelled space.
    [Show full text]
  • On Schwartz Groups
    On Schwartz groups L. Au¼enhofer, M. J. Chasco, X. Dom¶³nguez,V. Tarieladze Abstract In this paper we introduce a notion of a Schwartz group, which turns out to be coherent with the well known concept of a Schwartz topo- logical vector space. We establish several basic properties of Schwartz groups and show that free topological Abelian groups, as well as free locally convex spaces, over a hemicompact k{space are Schwartz groups. We also prove that every hemicompact k{space topological group, in particular the Pontryagin dual of a metrizable topological group, is a Schwartz group. 1 Introduction The notions of Schwartz and nuclear locally convex spaces were introduced by A. Grothendieck in [14] and [13], respectively. An intensive study of these spaces was made in [14, 15, 17, 24, 18, 27], and many other papers. Some relevant problems in the theory of Schwartz spaces have been solved only recently, see in this connection [6, 7]. Many important spaces in Analysis and its applications are either nu- clear or Schwartz. Spaces in both classes satisfy some properties of ¯nite- dimensional spaces (e. g. their bounded subsets are precompact) which general Banach spaces do not have. A group version of the concept of a nuclear space, introduced by W. Banaszczyk in [5], has been proved to be useful in Harmonic Analysis and topological group theory. In this paper we de¯ne a group counterpart of the Schwartz notion. Our de¯nition uses only group-theoretic tools. For the underlying additive group of a topological vector space, our notion appears to be the usual notion of a Schwartz space.
    [Show full text]
  • Functional Properties of Hörmander's Space of Distributions Having A
    Functional properties of Hörmander’s space of distributions having a specified wavefront set Yoann Dabrowski, Christian Brouder To cite this version: Yoann Dabrowski, Christian Brouder. Functional properties of Hörmander’s space of distributions having a specified wavefront set. 2014. hal-00850192v2 HAL Id: hal-00850192 https://hal.archives-ouvertes.fr/hal-00850192v2 Preprint submitted on 3 May 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Communications in Mathematical Physics manuscript No. (will be inserted by the editor) Functional properties of H¨ormander’s space of distributions having a specified wavefront set Yoann Dabrowski1, Christian Brouder2 1 Institut Camille Jordan UMR 5208, Universit´ede Lyon, Universit´eLyon 1, 43 bd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France 2 Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie, Sorbonne Univer- sit´es, UMR CNRS 7590, UPMC Univ. Paris 06, Mus´eum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, F-75005 Paris, France. Received: date / Accepted: date ′ Abstract: The space Γ of distributions having their wavefront sets in a closed cone Γ has become importantD in physics because of its role in the formulation of quantum field theory in curved spacetime.
    [Show full text]
  • A Theorem on the Schwartz Space of a Reductive Lie Group (Associated Parabolic Subgroups/Plancherel Measure/Fourier Transform on a Reductive Lie Group)
    Proc. Nat. Acad. Sci. USA Vol. 72, No. 12, pp. 4718-4719, December 1975 Mathematics A theorem on the Schwartz space of a reductive Lie group (associated parabolic subgroups/Plancherel measure/Fourier transform on a reductive Lie group) JAMES ARTHUR Department of Mathematics, Yale University, New Haven, Connecticut 06520 Communicated by C.D. Mostow, September 22,1975 ABSTRACT The purpose of this paper is to define the and Fourier transform of an arbitrary tempered distribution on a reductive Lie group.- To this end we define a topologi- cal vector space, e(G),in terms of the classes of irreducible unitary representations of G,which plays the role of a dual Schwartz space. Our main theorem then asserts that the Then L^(Af,K) is canonically isomorphic to the space of usual £ Fourier transform, when restricted to functions Hilbert-Schmidt operators on X(U).For any A â ac we have in the Schwartz space, e(G) defined by Harish-Chandra, A the usual induced representation n-a,h of the group G, as well provides a toooloeical isomorphism from e(G)onto e(G). as the convolution algebra Ccm(G),on X(u). By means of the Let G be a reductive Lie group with Lie-algebra g. We make above isomorphism and the map f -> ruk(f),we obtain a map the three assumptions on G stated in (ref. 1, 5 3), and adopt the conventions and terminology of (ref. 1, 2 and 5 3). In particular K is a maximal compact subgroup of G. Further- from Ccm(G)to LmK).
    [Show full text]
  • On the Characterization of Trace Class Representations and Schwartz Operators
    On the characterization of trace class representations and Schwartz operators Gerrit van Dijk, ∗ Karl-Hermann Neeb, y Hadi Salmasian,z Christoph Zellner x January 3, 2016 Abstract In this note we collect several characterizations of unitary representa- tions (π; H) of a finite dimensional Lie group G which are trace class, i.e., for each compactly supported smooth function f on G, the operator π(f) is trace class. In particular we derive the new result that, for some m 2 N, m all operators π(f), f 2 Cc (G), are trace class. As a consequence the cor- responding distribution character θπ is of finite order. We further show π is trace class if and only if every operator A, which is smoothing in the sense that AH ⊆ H1, is trace class and that this in turn is equivalent to the Fr´echet space H1 being nuclear, which in turn is equivalent to the re- alizability of the Gaussian measure of H on the space H−∞ of distribution vectors. Finally we show that, even for infinite dimensional Fr´echet{Lie groups, A and A∗ are smoothing if and only if A is a Schwartz operator, i.e., all products of A with operators from the derived representation are bounded. Introduction Let (π; H) be a (strongly continuous) unitary representation of the (possibly infinite dimensional) Lie group G (with an exponential function). Let H1 be its subspace of smooth vectors. On this space we obtain by d dπ(x)v = π(exp tx)v dt t=0 the derived representation of g which we extend naturally to a representation of the enveloping algebra U(g), also denoted dπ.
    [Show full text]
  • Lecture Notes on Function Spaces
    Lecture Notes on Function Spaces space invaders Karoline Götze TU Darmstadt, WS 2010/2011 Contents 1 Basic Notions in Interpolation Theory 5 2 The K-Method 15 3 The Trace Method 19 3.1 Weighted Lp spaces . 19 3.2 The spaces V (p, θ, X0,X1) ........................... 20 3.3 Real interpolation by the trace method and equivalence . 21 4 The Reiteration Theorem 26 5 Complex Interpolation 31 5.1 X-valued holomorphic functions . 32 5.2 The spaces [X, Y ]θ and basic properties . 33 5.3 The complex interpolation functor . 36 5.4 The space [X, Y ]θ is of class Jθ and of class Kθ................ 37 6 Examples 41 6.1 Complex interpolation of Lp -spaces . 41 6.2 Real interpolation of Lp-spaces . 43 6.2.1 Lorentz spaces . 43 6.2.2 Lorentz spaces and the K-functional . 45 6.2.3 The Marcinkiewicz Theorem . 48 6.3 Hölder spaces . 49 6.4 Slobodeckii spaces . 51 6.5 Functions on domains . 54 7 Function Spaces from Fourier Analysis 56 8 A Trace Theorem 68 9 More Spaces 76 9.1 Quasi-norms . 76 9.2 Semi-norms and Homogeneous Spaces . 76 9.3 Orlicz Spaces LA ................................ 78 9.4 Hardy Spaces . 80 9.5 The Space BMO ................................ 81 2 Motivation Examples of function spaces: n 1 α p k,p s,p s C(R ; R),C ([−1, 1]),C (Ω; C),L (X, ν),W (Ω),W ,Bp,q, n where Ω domain in R , α ∈ R+, 1 ≤ p, q ≤ ∞, (X, ν) measure space, k ∈ N, s ∈ R+.
    [Show full text]
  • Topological Vector Spaces and Continuous Linear Functionals
    CHAPTER III TOPOLOGICAL VECTOR SPACES AND CONTINUOUS LINEAR FUNCTIONALS The marvelous interaction between linearity and topology is introduced in this chapter. Although the most familiar examples of this interaction may be normed linear spaces, we have in mind here the more subtle, and perhaps more important, topological vector spaces whose topologies are defined as the weakest topologies making certain collections of functions continuous. See the examples in Exercises 3.8 and 3.9, and particularly the Schwartz space S discussed in Exercise 3.10. DEFINITION. A topological vector space is a real (or complex) vector space X on which there is a Hausdorff topology such that: (1) The map (x, y) → x+y is continuous from X×X into X. (Addition is continuous.) and (2) The map (t, x) → tx is continuous from R × X into X (or C × X into X). (Scalar multiplication is continuous.) We say that a topological vector space X is a real or complex topolog- ical vector space according to which field of scalars we are considering. A complex topological vector space is obviously also a real topological vector space. A metric d on a vector space X is called translation-invariant if d(x+ z, y + z) = d(x, y) for all x, y, z ∈ X. If the topology on a topological vector space X is determined by a translation-invariant metric d, we call X (or (X, d)) a metrizable vector space. If x is an element of a metrizable vector space (X, d), we denote by B(x) the ball of radius 43 44 CHAPTER III around x; i.e., B(x) = {y : d(x, y) < }.
    [Show full text]