DOCSLIB.ORG

Fourier Multipliers on Weighted $ L^ P $ Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fourier Multipliers on Weighted $ L^ P $ Spaces FOURIER MULTIPLIERS ON WEIGHTED Lp SPACES SEBASTIAN KROL´ Abstract. The paper provides a complement to the classical results on Fourier multipliers on Lp spaces. In particular, we prove that if q ∈ (1, 2) and a func- tion m : R → C is of bounded q-variation uniformly on the dyadic intervals p in R, i.e. m ∈ Vq (D), then m is a Fourier multiplier on L (R,wdx) for every p ≥ q and every weight w satisfying Muckenhoupt’s Ap/q-condition. We also obtain a higher dimensional counterpart of this result as well as of a result by E. Berkson and T.A. Gillespie including the case of the Vq(D) spaces with q > 2. New weighted estimates for modified Littlewood-Paley functions are also provided. 1. Introduction and Statement of Results For an interval [a,b] in R and a number q ∈ [1, ∞) denote by Vq ([a,b]) the space of all functions m :[a,b] → C of bounded q-variation over [a,b], i.e., kmkVq([a,b]) := sup |m(x)| + kmkVarq ([a,b]) < ∞, x∈[a,b] n−1 q 1/q where kmkVarq ([a,b]) := sup{( i=0 |m(ti+1) − m(ti)| ) } and the supremum is taken over all finite sequences a =: t0 <t1 < ... < tn := b (n ∈ N). We write D for the dyadic decomposition of RP, i.e., D := ±(2k, 2k+1]: k ∈ Z , and set R C Vq(D) := m : → : sup km|I kVq (I) < ∞ (q ∈ [1, ∞)). I∈D Moreover, let Ap(R) (p ∈ [1, ∞)) be the class of weights on R which satisfy the R Muckenhoupt Ap condition. Denote by [w]Ap the Ap-constant of w ∈ Ap( ). If w ∈ A∞(R) := ∪p≥1Ap(R) we write Mp(R, w) for the class of all multipliers on Lp(R, w) (p> 1), i.e., ∞ p Mp(R, w) := {m ∈ L (R): Tm extends to a bounded operator on L (R, w)} . Here Tm stands for the Fourier multiplier with the symbol m, i.e., (Tmf) = mf (f ∈ arXiv:1403.4477v2 [math.CA] 13 May 2014 R R S( )). Note that Mp( , w) becomes a Banach space under the norm kmkMp(R,w) := R kTmkL(Lp(R,w)) (m ∈ Mp( , w)). b b The main result of the paper is the following complement to results due to D. Kurtz [18], R. Coifman, J.-L. Rubio de Francia, S. Semmes [8], and E. Berkson, T. Gillespie [4]. Theorem A. (i) Let q ∈ (1, 2]. Then, Vq(D) ⊂ Mp(R, w) for every p ≥ q and every Muckenhoupt weight w ∈ Ap/q (R). R 1 1 −1 (ii) Let q > 2. Then, Vq(D) ⊂ Mp( , w) for every 2 ≤ p < ( 2 − q ) and every 1 1 −1 Muckenhoupt weight w ∈ Ap/2 with sw > (1 − p( 2 − q )) . 2010 Mathematics Subject Classification. 42B25 (42B15). Key words and phrases. weighted Fourier multipliers, weighted inequalities, Littlewood-Paley square functions, Muckenhoupt weights. The author was supported by the Alexander von Humboldt Foundation. 1 2 SEBASTIAN KROL´ Here, for every w ∈ A∞(R), we set sw := sup{s ≥ 1 : w ∈ RHs(R)} and we write w ∈ RHs(R) if 1/s −1 1 b 1 b sup w(x)sdx w(x)dx < ∞. b − a b − a a<b Za ! Za ! Recall that, by the reverse H¨older inequality, sw ∈ (1, ∞] for every Muckenhoupt weight w ∈ A∞(R). For the convenience of the reader we repeat the relevant material from the liter- ature, which we also use in the sequel. Recall first that in [18] D. Kurtz proved the following weighted variant of the classical Marcinkiewicz multiplier theorem. Theorem 1 ([18, Theorem 2]). V1(D) ⊂ Mp(R, w) for every p ∈ (1, ∞) and every Muckenhoupt weight w ∈ Ap(R). As in the unweighted case, Theorem 1 is equivalent to a weighted variant of the Littlewood-Paley decomposition theorem, which asserts that for the square D D function S corresponding to the dyadic decomposition D of R, kS fkp,w h kfkp,w p (f ∈ L (R, w)) for every p ∈ (1, ∞) and w ∈ Ap(R); see [18, Theorem 1], and also [18, Theorem 3.3]. Here and subsequently, if I is a family of disjoint intervals in R, we write SI for the Littlewood-Paley square function corresponding to I, i.e., I 2 1/2 2 R S f := I∈I |SI f| (f ∈ L ( )). Recall also that in [26] J.-L. Rubio de Francia proved the following extension of the classicalP Littlewood-Paley decomposition theorem. Theorem 2 ([26, Theorem 6.1]). Let 2 <p< ∞ and w ∈ Ap/2(R). Then for an arbitrary family I of disjoint intervals in R the square function SI is bounded on Lp(R, wdx). Applying Rubio de Francia’s inequalities, i.e. Theorem 2, R. Coifman, J.-L. Ru- bio de Francia, and S. Semmes [8] proved the following extension and improvement of the classical Marcinkiewicz multiplier theorem. (See Section 2 for the definition of R2(D).) Theorem 3 ([8, Th´eor`eme 1 and Lemme 5]). Let 2 ≤ q < ∞. Then, Vq(D) ⊂ R 1 1 1 Mp( ) for every p ∈ (1, ∞) such that | p − 2 | < q . Furthermore, R2(D) ⊂ M2(R, w) for every w ∈ A1(R). Subsequently, a weighted variant of Theorem 3 was given by E. Berkson and T. Gillespie in [4]. According to our notation their result can be formulated as follows. Theorem 4 ([4, Theorem 1.2]). Suppose that 2 ≤ p< ∞ and w ∈ Ap/2(R). Then, there is a real number , depending only on and , such that 1 1 1 s> 2 p [w]Ap/2 s > 2 − p and Vq(D) ⊂ Mp(R, w) for all 1 ≤ q<s. Note that the part (i) of Theorem A fills a gap which occurs in Theorem 1 and the weighted part of Theorem 3. The part (ii) identifies the constant s in Berkson- 1 1 −1 ′ sw Gillespie’s result, i.e., Theorem 4, as ( − ′ ) , where s := , and in general, 2 swp w sw−1 this constant is best possible. Except for some details, the proofs given below reproduce well-known argu- ments from the Littlewood-Paley theory; in particular, ideas which have been FOURIER MULTIPLIERS ON WEIGHTED Lp SPACES 3 presented in [18], [8], [26], and [29]. A new point of our approach is the follow- I ing result on weighted estimates for modified Littlewood-Paley functions Sq (·) := ′ ′ q 1/q ( I∈I |SI (·)| ) (q ∈ (1, 2]), which may be of independent interest. TheoremP B. (i) Let q ∈ (1, 2), p > q, and w ∈ Ap/q (R). Then, there exists a constant C > 0 such that for any family I of disjoint intervals in R I p R kSq fkp,w ≤ Ckfkp,w (f ∈ L ( , wdx)). R Moreover, for every q ∈ (1, 2), p > q and V ⊂ Ap/q ( ) with supw∈V [w]Ap/q < ∞ I R sup kSq fkp,w : w ∈V, I a family of disjoint intervals in , kfkp,w =1 < ∞. (ii) For any family I of disjoint intervals in R and every Muckenhoupt weight R I 2 R 2 R w ∈ A1( ), the operator S2 maps L ( , wdx) into weak-L ( , wdx), and I sup kS fk 2,∞ : w ∈V, I a family of disjoint intervals in R, kfk 2 =1 < ∞ 2 Lw Lw n R o for every V ⊂ A1( ) with supw∈V [w]A1 < ∞. Moreover, if q ∈ (1, 2), then for any well-distributed family I of disjoint intervals R R I q R in and every Muckenhoupt weight w ∈ A1( ), the operator Sq maps L ( , wdx) into weak-Lq(R, wdx). Recall that a family I of disjoint intervals in R is well-distributed if there exists λ> 1 such that supx∈R I∈I χλI (x) < ∞, where λI denotes the interval with the same center as I and length λ times that of I. P 2 I Note that the validity of the A1-weighted L -estimates for square function S = I R S2 corresponding to an arbitrary family I of disjoint intervals in , i.e., I 2 R R kS2 fk2,w ≤ Cwkfk2,w (f ∈ L ( , wdx), w ∈ A1( )), is conjectured by J.-L. Rubio de Francia in [26, Section 6, p.10]; see also [12, Section 8.2, p. 187]. Theorem B(ii), in particular, provides the validity of the weak variant of Rubio de Francia’s conjecture. Notice that in contrast to the square function I I operators S2 , in general, operators Sq (q ∈ [1, 2)) are not bounded on (unweighted) Lq(R); see [9]. Moreover, in [24] T.S. Quek proved that if I is a well-distributed ′ R I q R q,q R family of disjoint intervals in , then the operator Sq maps L ( ) into L ( ) for ′ every q ∈ (1, 2). Note that this result is in a sense sharp, i.e., Lq,q (R) cannot be replaced by Lq,s(R) for any s < q′; see [24, Remark 3.2]. Therefore, Theorem B provides also a weighted variant of this line of researches. Cf. also relevant results given by S.V. Kisliakov in [17]. Furthermore, as a consequence of our approach we also get a higher dimensional analogue of Theorem A, see Theorem C in Section 4, which extends earlier results by Q. Xu [29]; see also M. Lacey [19, Chapter 4]. Since the formulation of Theorem C is more involved and its proof is essentially the iteration of one-dimensional arguments we refer the reader to Section 4 for more information. The part (ii) of Theorem A is a quantitative improvement of [4, Theorem 1.2] due to E. Berkson and T.
Load more

Recommended publications
  • The Heisenberg Group Fourier Transform
    THE HEISENBERG GROUP FOURIER TRANSFORM NEIL LYALL Contents 1. Fourier transform on Rn 1 2. Fourier analysis on the Heisenberg group 2 2.1. Representations of the Heisenberg group 2 2.2. Group Fourier transform 3 2.3. Convolution and twisted convolution 5 3. Hermite and Laguerre functions 6 3.1. Hermite polynomials 6 3.2. Laguerre polynomials 9 3.3. Special Hermite functions 9 4. Group Fourier transform of radial functions on the Heisenberg group 12 References 13 1. Fourier transform on Rn We start by presenting some standard properties of the Euclidean Fourier transform; see for example [6] and [4]. Given f ∈ L1(Rn), we define its Fourier transform by setting Z fb(ξ) = e−ix·ξf(x)dx. Rn n ih·ξ If for h ∈ R we let (τhf)(x) = f(x + h), then it follows that τdhf(ξ) = e fb(ξ). Now for suitable f the inversion formula Z f(x) = (2π)−n eix·ξfb(ξ)dξ, Rn holds and we see that the Fourier transform decomposes a function into a continuous sum of characters (eigenfunctions for translations). If A is an orthogonal matrix and ξ is a column vector then f[◦ A(ξ) = fb(Aξ) and from this it follows that the Fourier transform of a radial function is again radial. In particular the Fourier transform −|x|2/2 n of Gaussians take a particularly nice form; if G(x) = e , then Gb(ξ) = (2π) 2 G(ξ). In general the Fourier transform of a radial function can always be explicitly expressed in terms of a Bessel 1 2 NEIL LYALL transform; if g(x) = g0(|x|) for some function g0, then Z ∞ n 2−n n−1 gb(ξ) = (2π) 2 g0(r)(r|ξ|) 2 J n−2 (r|ξ|)r dr, 0 2 where J n−2 is a Bessel function.
    [Show full text]
  • Lecture Notes: Harmonic Analysis
    Lecture notes: harmonic analysis Russell Brown Department of mathematics University of Kentucky Lexington, KY 40506-0027 August 14, 2009 ii Contents Preface vii 1 The Fourier transform on L1 1 1.1 Definition and symmetry properties . 1 1.2 The Fourier inversion theorem . 9 2 Tempered distributions 11 2.1 Test functions . 11 2.2 Tempered distributions . 15 2.3 Operations on tempered distributions . 17 2.4 The Fourier transform . 20 2.5 More distributions . 22 3 The Fourier transform on L2. 25 3.1 Plancherel's theorem . 25 3.2 Multiplier operators . 27 3.3 Sobolev spaces . 28 4 Interpolation of operators 31 4.1 The Riesz-Thorin theorem . 31 4.2 Interpolation for analytic families of operators . 36 4.3 Real methods . 37 5 The Hardy-Littlewood maximal function 41 5.1 The Lp-inequalities . 41 5.2 Differentiation theorems . 45 iii iv CONTENTS 6 Singular integrals 49 6.1 Calder´on-Zygmund kernels . 49 6.2 Some multiplier operators . 55 7 Littlewood-Paley theory 61 7.1 A square function that characterizes Lp ................... 61 7.2 Variations . 63 8 Fractional integration 65 8.1 The Hardy-Littlewood-Sobolev theorem . 66 8.2 A Sobolev inequality . 72 9 Singular multipliers 77 9.1 Estimates for an operator with a singular symbol . 77 9.2 A trace theorem. 87 10 The Dirichlet problem for elliptic equations. 91 10.1 Domains in Rn ................................ 91 10.2 The weak Dirichlet problem . 99 11 Inverse Problems: Boundary identifiability 103 11.1 The Dirichlet to Neumann map . 103 11.2 Identifiability . 107 12 Inverse problem: Global uniqueness 117 12.1 A Schr¨odingerequation .
    [Show full text]
  • CONVERGENCE of FOURIER SERIES in Lp SPACE Contents 1. Fourier Series, Partial Sums, and Dirichlet Kernel 1 2. Convolution 4 3. F
    CONVERGENCE OF FOURIER SERIES IN Lp SPACE JING MIAO Abstract. The convergence of Fourier series of trigonometric functions is easy to see, but the same question for general functions is not simple to answer. We study the convergence of Fourier series in Lp spaces. This result gives us a criterion that determines whether certain partial differential equations have solutions or not.We will follow closely the ideas from Schlag and Muscalu's Classical and Multilinear Harmonic Analysis. Contents 1. Fourier Series, Partial Sums, and Dirichlet Kernel 1 2. Convolution 4 3. Fej´erkernel and Approximate identity 6 4. Lp convergence of partial sums 9 Appendix A. Proofs of Theorems and Lemma 16 Acknowledgments 18 References 18 1. Fourier Series, Partial Sums, and Dirichlet Kernel Let T = R=Z be the one-dimensional torus (in other words, the circle). We consider various function spaces on the torus T, namely the space of continuous functions C(T), the space of H¨oldercontinuous functions Cα(T) where 0 < α ≤ 1, and the Lebesgue spaces Lp(T) where 1 ≤ p ≤ 1. Let f be an L1 function. Then the associated Fourier series of f is 1 X (1.1) f(x) ∼ f^(n)e2πinx n=−∞ and Z 1 (1.2) f^(n) = f(x)e−2πinx dx: 0 The symbol ∼ in (1.1) is formal, and simply means that the series on the right-hand side is associated with f. One interesting and important question is when f equals the right-hand side in (1.1). Note that if we start from a trigonometric polynomial 1 X 2πinx (1.3) f(x) = ane ; n=−∞ Date: DEADLINES: Draft AUGUST 18 and Final version AUGUST 29, 2013.
    [Show full text]
  • Inner Product on Matrices), 41 a (Closure of A), 308 a ≼ B (Matrix Ordering
    Index A : B (inner product on matrices), 41 K∞ (asymptotic cone), 19 A • B (inner product on matrices), 41 K ∗ (dual cone), 19 A (closure of A), 308 L(X,Y ) (linear maps X → Y ), 312 A . B (matrix ordering), 184 L∞(), 311 A∗(adjoint operator), 313 L p(), 311 BX (open unit ball), 17 λ (Lebesgue measure), 317 C(; X) (continuous maps → X), λmin(B) (minimum eigenvalue of B), 315 142, 184 ∞ Rd →∞ C0 ( ) (space of test functions), 316 liminfn xn (limit inferior), 309 co(A) (convex hull), 314 limsupn→∞ xn (limit superior), 309 cone(A) (cone generated by A), 337 M(A) (space of measures), 318 d D(R ) (space of distribution), 316 NK (x) (normal cone), 19 Dα (derivative with multi-index), 322 O(g(s)) (asymptotic “big Oh”), 204 dH (A, B) (Hausdorff metric), 21 ∂ f (subdifferential of f ), 340 δ (Dirac-δ function), 3, 80, 317 P(X) (power set; set of subsets of X), 20 + δH (A, B) (one-sided Hausdorff (U) (strong inverse), 22 − semimetric), 21 (U) (weak inverse), 22 diam F (diameter of set F), 319 K (projection map), 19, 329 n divσ (divergence of σ), 234 R+ (nonnegative orthant), 19 dµ/dν (Radon–Nikodym derivative), S(Rd ) (space of tempered test 320 functions), 317 domφ (domain of convex function), 54, σ (stress tensor), 233 327 σK (support function), 328 epi f (epigraph), 19, 327 sup(A) (supremum of A ⊆ R), 309 ε (strain tensor), 232 TK (x) (tangent cone), 19 −1 f (E) (inverse set), 308 (u, v)H (inner product), 17 f g (inf-convolution), 348 u, v (duality pairing), 17 graph (graph of ), 21 W m,p() (Sobolev space), 322 Hess f (Hessian matrix), 81
    [Show full text]
  • The Plancherel Formula for Parabolic Subgroups
    ISRAEL JOURNAL OF MATHEMATICS, Vol. 28. Nos, 1-2, 1977 THE PLANCHEREL FORMULA FOR PARABOLIC SUBGROUPS BY FREDERICK W. KEENE, RONALD L. LIPSMAN* AND JOSEPH A. WOLF* ABSTRACT We prove an explicit Plancherel Formula for the parabolic subgroups of the simple Lie groups of real rank one. The key point of the formula is that the operator which compensates lack of unimodularity is given, not as a family of implicitly defined operators on the representation spaces, but rather as an explicit pseudo-differential operator on the group itself. That operator is a fractional power of the Laplacian of the center of the unipotent radical, and the proof of our formula is based on the study of its analytic properties and its interaction with the group operations. I. Introduction The Plancherel Theorem for non-unimodular groups has been developed and studied rather intensively during the past five years (see [7], [11], [8], [4]). Furthermore there has been significant progress in the computation of the ingredients of the theorem (see [5], [2])--at least in the case of solvable groups. In this paper we shall give a completely explicit description of these ingredients for an interesting family of non-solvable groups. The groups we consider are the parabolic subgroups MAN of the real rank 1 simple Lie groups. As an intermediate step we also obtain the Plancherel formula for the exponential solvable groups AN (see also [6]). In that case our results are more extensive than those of [5] since in addition to the "infinitesimal" unbounded operators we also obtain explicitly the "global" unbounded operator on L2(AN).
    [Show full text]
  • Functional Analysis an Elementary Introduction
    Functional Analysis An Elementary Introduction Markus Haase Graduate Studies in Mathematics Volume 156 American Mathematical Society Functional Analysis An Elementary Introduction https://doi.org/10.1090//gsm/156 Functional Analysis An Elementary Introduction Markus Haase Graduate Studies in Mathematics Volume 156 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 46-01, 46Cxx, 46N20, 35Jxx, 35Pxx. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-156 Library of Congress Cataloging-in-Publication Data Haase, Markus, 1970– Functional analysis : an elementary introduction / Markus Haase. pages cm. — (Graduate studies in mathematics ; volume 156) Includes bibliographical references and indexes. ISBN 978-0-8218-9171-1 (alk. paper) 1. Functional analysis—Textbooks. 2. Differential equations, Partial—Textbooks. I. Title. QA320.H23 2014 515.7—dc23 2014015166 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by the American Mathematical Society.
    [Show full text]
  • Methods of Applied Mathematics
    Methods of Applied Mathematics Todd Arbogast and Jerry L. Bona Department of Mathematics, and Institute for Computational Engineering and Sciences The University of Texas at Austin Copyright 1999{2001, 2004{2005, 2007{2008 (corrected version) by T. Arbogast and J. Bona. Contents Chapter 1. Preliminaries 5 1.1. Elementary Topology 5 1.2. Lebesgue Measure and Integration 13 1.3. Exercises 23 Chapter 2. Normed Linear Spaces and Banach Spaces 27 2.1. Basic Concepts and Definitions. 27 2.2. Some Important Examples 34 2.3. Hahn-Banach Theorems 43 2.4. Applications of Hahn-Banach 48 2.5. The Embedding of X into its Double Dual X∗∗ 52 2.6. The Open Mapping Theorem 53 2.7. Uniform Boundedness Principle 57 2.8. Compactness and Weak Convergence in a NLS 58 2.9. The Dual of an Operator 63 2.10. Exercises 66 Chapter 3. Hilbert Spaces 73 3.1. Basic Properties of Inner-Products 73 3.2. Best Approximation and Orthogonal Projections 75 3.3. The Dual Space 78 3.4. Orthonormal Subsets 79 3.5. Weak Convergence in a Hilbert Space 86 3.6. Exercises 87 Chapter 4. Spectral Theory and Compact Operators 89 4.1. Definitions of the Resolvent and Spectrum 90 4.2. Basic Spectral Theory in Banach Spaces 91 4.3. Compact Operators on a Banach Space 93 4.4. Bounded Self-Adjoint Linear Operators on a Hilbert Space 99 4.5. Compact Self-Adjoint Operators on a Hilbert Space 104 4.6. The Ascoli-Arzel`aTheorem 107 4.7. Sturm Liouville Theory 109 4.8.
    [Show full text]
  • THE PLANCHEREL FORMULA for COUNTABLE GROUPS Bachir Bekka
    THE PLANCHEREL FORMULA FOR COUNTABLE GROUPS Bachir Bekka To cite this version: Bachir Bekka. THE PLANCHEREL FORMULA FOR COUNTABLE GROUPS. Indagationes Math- ematicae, Elsevier, 2021, 32 (3), pp.619-638. 10.1016/j.indag.2021.01.002. hal-02928504v2 HAL Id: hal-02928504 https://hal.archives-ouvertes.fr/hal-02928504v2 Submitted on 26 Oct 2020 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. THE PLANCHEREL FORMULA FOR COUNTABLE GROUPS BACHIR BEKKA Abstract. We discuss a Plancherel formula for countable groups, which provides a canonical decomposition of the regular represen- tation of such a group Γ into a direct integral of factor represen- tations. Our main result gives a precise description of this decom- position in terms of the Plancherel formula of the FC-center Γfc of Γ (that is, the normal sugbroup of Γ consisting of elements with a finite conjugacy class); this description involves the action of an appropriate totally disconnected compact group of automorphisms of Γfc. As an application, we determine the Plancherel formula for linear groups. In an appendix, we use the Plancherel formula to provide a unified proof for Thoma's and Kaniuth's theorems which respectively characterize countable groups which are of type I and those whose regular representation is of type II.
    [Show full text]
  • Real Analysis a Comprehensive Course in Analysis, Part 1
    Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Real Analysis A Comprehensive Course in Analysis, Part 1 http://dx.doi.org/10.1090/simon/001 Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Providence, Rhode Island 2010 Mathematics Subject Classification. Primary 26-01, 28-01, 42-01, 46-01; Secondary 33-01, 35-01, 41-01, 52-01, 54-01, 60-01. For additional information and updates on this book, visit www.ams.org/bookpages/simon Library of Congress Cataloging-in-Publication Data Simon, Barry, 1946– Real analysis / Barry Simon. pages cm. — (A comprehensive course in analysis ; part 1) Includes bibliographical references and indexes. ISBN 978-1-4704-1099-5 (alk. paper) 1. Mathematical analysis—Textbooks. I. Title. QA300.S53 2015 515.8—dc23 2014047381 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright.
    [Show full text]
  • Paley-Wiener Theorems with Respect to the Spectral Parameter Susanna Dann Louisiana State University and Agricultural and Mechanical College, [email protected]
    Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2011 Paley-Wiener theorems with respect to the spectral parameter Susanna Dann Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Applied Mathematics Commons Recommended Citation Dann, Susanna, "Paley-Wiener theorems with respect to the spectral parameter" (2011). LSU Doctoral Dissertations. 3986. https://digitalcommons.lsu.edu/gradschool_dissertations/3986 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. PALEY-WIENER THEOREMS WITH RESPECT TO THE SPECTRAL PARAMETER A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Susanna Dann B.S., Fachhochschule Stuttgart, Germany, 2004 M.S., Louisiana State University, 2006 August 2011 To my family ii Acknowledgments It is a genuine pleasure to thank my advisor Professor Gestur Olafsson´ for his time, patience and guidance during my years of graduate study at Louisiana State University. I am also thankful to all my teachers, colleagues and students from and through whom I learned a lot. Special thanks go to professors Gestur Olafsson, Am- bar Sengupta, Mark Davidson, Raymond Fabec, Lawrence Smolinsky and Jacek Cygan and to my former and present fellow graduate students Natalia Ptitsyna, Amber Russell, Maria Vega, Maiia Bakhova, Jens Christensen, Silvia Jim´enezand Keng Wiboonton.
    [Show full text]
  • Reduced C*-Algebra for Reductive P-Adic Hx~Ps
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Reduced C*-Algebra for Reductive p-adic hx~ps R. J. PLYMEN Receivtxi June 17. 1988 1.1I Let G be a reductive group over H. For any field F we can consider the group G(F) of F-points on G. For example, let G be the general linear group GIL(n). By taking F to be the real field BB,the complex field @, the p-adic field Q,,, or the finite ikid F,, we obtain successively the groups GL(n, [w), GL(n, @I),GL(n, Q$,,),and the finite group GL(n, q). 1.2. At first glance, the groups G(F) for different fields F appear to have little in common with each other. I. Ceifand has conjectured that the struc- ture of the repre~~~~at~onsof G(F) has fundamental features which do not depend on a choice of F. This conjecture was a theme in Kazhdan’s talk at the ICM 86 18, p. 8491. We shah restrict ourselves to the real field R, the complex field c, and focal fields. We recall that the local fields are the finite extensions of the field Q, of p-adic numbers, and the finite extensions of the power series field F,,((x)). 1.3. We shall obtain, in conformity with the Gelfand conjecture, a struc- ture theorem for the reduced C*-algebra of G(F) which is independent of F. The reduced C*-algebra C:(G(F)) is a version of the group algebra of G(F) which is well-adapted to the Plancherel theory of G(F).
    [Show full text]
  • Lecture 7: Inversion, Plancherel and Convolution
    Lecture 7: Inversion, Plancherel and Convolution Johan Thim ([email protected]) April 2, 2021 \You should not drink and bake" |Mark Kaminski 1 Inversion of the Fourier Transform So suppose that we have u 2 G(R) and have calculated the Fourier transform F u(!). Can we from F u(!) recover the function we started with? Considering that the Fourier transform is constructed by the multiplication with e−i!x and then integration, what would happen if we multiplied with ei!x and integrate again? Formally, 1 R 1 R 1 F u(!)ei!x d! = lim u(t)e−i!tei!x dt d! = lim u(t)e−i!(x−t) dt d! ˆ−∞ R!1 ˆ−R ˆ−∞ R!1 ˆ−R ˆ−∞ 1 R = lim u(t) e−i!(x−t) d! dt; R!1 ˆ−∞ ˆ−R where we changed the order of integration (this can be motivated) but we're left with something kind of weird in the inner parenthesis and we would probably like to move the limit inside the outer integral. First, let's look at the expression in the inner parenthesis: R e−i!(x−t) R e−iR(x−t) eiR(x−t) 2 sin(R(x − t)) e−i!(x−t) d! = = − + = ; x 6= t: ˆ−R −i(x − t) !=−R i(x − t) i(x − t) x − t The Dirichlet kernel (on the real line) Definition. We define the Dirichlet kernel for the Fourier transform by sin(Rx) D (x) = ; x 6= 0; R > 0; R πx and DR(0) = R/π.
    [Show full text]