DOCSLIB.ORG

Fourier Analysis in Function Space Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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......................... 77 5 The Lp−Theorem for Littlewood Paley Decompositions 83 5.1 Bernstein and Nikolsky inequalities . 83 5.2 Littlewood Paley projections . 85 p n 5.3 The spaces L (R ; `q).............................. 86 5.4 The Lp-theorem for Littlewood-Paley decompositions . 89 1 The Fourier transform of tempered distributions 1.1 The Fourier transforms of L1 functions Theorem-Definition 1.1. Let f 2 L1(Rn; C) define the Fourier transform of f as fol- lows: Z n − n −ix·ξ 8ξ 2 R fb(ξ) = (2π) 2 e f(x) dx: Rn We have that fb 2 L1(Rn) and n − 2 (1.1) kfbkL1(Rn) ≤ (2π) kfkL1(Rn): Moreover fb 2 C0(Rn) and (1.2) lim jfb(ξ)j = 0: jξj!+1 We shall also sometimes denote the Fourier transform of f by F(f). Remark 1.2. There are several possible normalizations for defining the Fourier transform of an L1 function such as for instance Z fb(ξ) := e−2iπx·ξf(x) dx: Rn None of them give a full satisfaction. The advantages of the one we chose are the following: i) f 7−! fb will define an isometry of L2 as we will see in Proposition 1.5. ii) With our normalization we have the convenient formula (see Lemma 1.11) 8k = 1 : : : n @ξk fb = −i ξk fb but the less convenient formula (see Proposition 1.32) −n g\∗ f = (2π) gbf:b Proof of Theorem 1.1. The first part of the theorem that is inequality (1.1) is straight- 0 n 1 n forward. We prove now that fb 2 C (R ). Let fk 2 C0 (R ) such that 1 n fk −! f strongly in L (R ): 1 n 1 It is clear that since fk 2 C0 (R ) the functions fbk are also C . Inequality (1.1) gives n − 2 kfb− fbkkL1(Rn) ≤ (2π) kf − fkkL1(Rn): Thus fb is the uniform limit of continuous functions and, as such, it is continuous. It remains to prove that jfj(ξ) uniformly converges to zero as jξj converge to infinity. In 1 1 n 1 n Proposition 1.9 we shall prove that (1.2) holds if f 2 C0 (R ). Let f 2 L (R ), let " > 0 1 and let ' 2 C0 such that " n (1.3) kf − 'k 1 n ≤ (2π) 2 : L (R ) 2 There exists R > 0 such that " (1.4) jξj > R =) j'(ξ)j ≤ : b 2 Combining (1.3) and (1.4) together with (1.1) applied to the difference f − ', we obtain jξj > R =) jfb(ξ)j ≤ kfb− 'bkL1 + j'b(ξ)j ≤ ": This implies (1.2) and Theorem 1.1 is proved. ∗ Exercise 1.3. Prove that for any a 2 R+ jξj2 2 1 −ajxj − 4a e\ = n e : (2a) 2 ∗ Prove that for any a 2 R+ n f[a(x) = a fb(aξ) x n where fa(x) := f( a ) for any x 2 R . It is then natural to ask among the functions which are continuous, bounded in L1 and converging uniformly to zero at infinity, which one is the Fourier transform of an L1 function. Unfortunately, there seems to be no satisfactory condition characterizing the space of Fourier transforms of L1(Rn). We have nevertheless the following theorem. Theorem 1.4. (Inverse of the Fourier transform) Let f 2 L1(Rn; C) such that fb 2 L1(Rn; C) then Z n − n ix·ξ 8x 2 R f(x) = (2π) 2 e fb(ξ) dξ: Rn Proof of Theorem 1.4. We can of course explicitly write Z Z Z − n ix·ξ − n ix·ξ ix·y (2π) 2 e fb(ξ) dξ = (2π) 2 e dξ e f(y) dy: Rn Rn Rn The problem at this stage is that we cannot a-priori reverse the order of integrations because the hypothesis for applying Fubini's theorem are not fullfilled: iξ(x−y) 1 n n (ξ; y) 7−! e f(y) 2= L (R × R ) unless f ≡ 0. 2 2 2 − " jξj The idea is to insert the Gaussian function e 4 where " is a positive number that we are going to take smaller and smaller. Introduce Z 2 2 Z −n ix·ξ − " jξj −iξ·y I"(x) := (2π) e e 4 dξ e f(y) dy: Rn Rn Now we have 2 2 − " jξj iξ(x−y) 1 n n (ξ; y) 7−! e 4 e f(y) 2 L (R × R ) and we can apply Fubini's theorem. We have in one hand Z 2 2 − n ix·ξ − " jξj I"(x) = (2π) 2 e e 4 fb(ξ) dξ: Rn We can bound the integrand uniformly as follows: 2 2 n ix·ξ − " jξj 4 8x; ξ 2 R e e fb(ξ) ≤ jfb∗ ξ)j: By assumption, the right-hand side of the inequality is integrable and we have moreover, for every x and ξ 2 2 ix·ξ − " jξj ix·ξ lim e e 4 fb(ξ) = e fb(ξ): ")0 Hence dominated convergence theorem implies that for any x 2 R Z − n ix·ξ (1.5) lim I"(x) = (2π) 2 fb(x) e dξ: ")0 Rn Applying Fubini gives also Z Z 2 2 −n −i(y−x)·ξ − " jξj I"(x) = (2π) f(y) dy e e 4 dξ n R Rn Z 2 − n − " jξj2 = (2π) 2 f(y) F e 4 (y − x) dy Rn using Exercise 1.3, we then obtain n Z 2 − n −|y−xj 2 2 2 "2 I"(x) = (2π) f(y) e n dy: Rn " One proves without much difficulties that for any Lebesgue point x 2 R for f the following holds n Z 2 − n − jy−xj 2 2 lim (2π) 2 f(y) e "2 dy = f(x): ")0 n Rn " Continuing this identity with (1.5) gives the theorem. The transformation Z 1 n − n ix·ξ f 2 L (R ) 7−! (2π) 2 e f(ξ) dξ Rn _ will be denoted f or also F −1(f). 3 Proposition 1.5. Let f and g 2 L1(Rn; C). When Z Z f(x) gb(x) dx = fb(x) g(x) dx: Rn Rn Let f 2 L1(Rn; C) such that fb 2 L1(Rn; C), then Z Z f(x) f(x) dx = fb(ξ) fb(x) dξ: Rn Rn This last identity is called Plancherel identity. Proof of Proposition 1.5. The proof of the first identity in Proposition 1.5 is a direct consequence of Fubini's theorem. The second identity can be deduced from the first one by taking g := F −1(f) and by observing that −1 F (f) = F(f) : The second identity is an invitation to extend the Fourier transform as an isometry of L2. The purpose of the present chapter is to extend the Fourier transform to an even larger class of distributions. To that aim we will first concentrate on looking at the Fourier transform in a \small" class of very smooth function with very fast decrease at infinity: the Schwartz space. 1.2 The Schwartz Space S(Rn) The Schwartz functions are C1 functions whose successive derivatives decrease faster than any polynominal at infinity. We shall use below the following notations: n α α1 αn 8α = (α; : : : ; αn) 2 N x := x1 ; : : : ; xn βn n β @ @ 8β(β; : : : ; βn) 2 @ f := ::: (f) N β1 βn @n1 @xn P and jαj := αi. Definition 1.6. The space of Schwartz functions is the following subspace of C1(Rn; C): 8 1 n 9 ' 2 C (R ; C) s.t. > > < α β = n 8p 2 N (') := sup kx @ 'k 1 n < +1 S(R ) := N p L (R ) : > jαj ≤ p > : jβj ≤ p ; The following obvious proposition holds Proposition 1.7. S(Rn) is stable under the action of derivatives and the multiplication by polynomials in C[x1; : : : ; xn]. We prove now the following proposition: 4 n Proposition 1.8.
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]
  • The Semi-M Property for Normed Riesz Spaces Compositio Mathematica, Tome 34, No 2 (1977), P
    COMPOSITIO MATHEMATICA EP DE JONGE The semi-M property for normed Riesz spaces Compositio Mathematica, tome 34, no 2 (1977), p. 147-172 <http://www.numdam.org/item?id=CM_1977__34_2_147_0> © Foundation Compositio Mathematica, 1977, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 34, Fasc. 2, 1977, pag. 147-172 Noordhoff International Publishing Printed in the Netherlands THE SEMI-M PROPERTY FOR NORMED RIESZ SPACES Ep de Jonge 1. Introduction It is well-known that if (0394, F, IL) is a u-finite measure space and if 1 ~ p 00, then the Banach dual L *p of the Banach space Lp = Lp(0394, IL) can be identified with Lq = Lq(L1, 03BC), where p-1 + q-1 = 1. For p =00 the situation is different; the space Li is a linear subspace of L*, and only in a very trivial situation (the finite-dimensional case) we have Li = Lfi. Restricting ourselves to the real case, the Banach dual L *~ is a (real) Riesz space, i.e., a vector lattice, and Li is now a band in L*. The disjoint complement (i.e., the set of all elements in L* disjoint to all elements in LI) is also a band in L*, called the band of singular linear functionals on Loo.
    [Show full text]
  • Burkholder's Inequalities in Noncommutative Lorentz
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 7, July 2010, Pages 2431–2441 S 0002-9939(10)10267-6 Article electronically published on March 24, 2010 BURKHOLDER’S INEQUALITIES IN NONCOMMUTATIVE LORENTZ SPACES YONG JIAO (Communicated by Marius Junge) Abstract. We prove Burkholder’s inequalities in noncommutative Lorentz spaces Lp,q(M), 1 <p<∞, 1 ≤ q<∞, associated with a von Neumann algebra M equipped with a faithful normal tracial state. These estimates generalize the classical inequalities in the commutative case. 1. Introduction Martingale inequalities and sums of independent random variables are important tools in classical harmonic analysis. A fundamental result due to Burkholder [1, 2] can be stated as follows. Given a probability space (Ω, F ,P), let {Fn}n≥1 be a nondecreasing sequence of σ-fields of F such that F = ∨Fn and let En be the conditional expectation operator relative to Fn. Given 2 ≤ p<∞ and an p L -bounded martingale f =(fn)n≥1, we have ∞ ∞ 1/2 1/p 2 p (1.1) fLp ≈ Ek−1(|df k| ) + |df k| . Lp Lp k=1 k=1 The first term on the right is called the conditioned square function of f, while the second is called the p-variation of f. Rosenthal’s inequalities [14] can be regarded as the particular case where the sequence df =(df 1,df2, ...) is a family of independent mean-zero random variables df k = ak. In this case it is easy to reduce Rosenthal’s inequalities to ∞ ∞ ∞ 1/2 1/p ≈ 2 p (1.2) ak Lp ak 2 + ak p .
    [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]
  • Arxiv:1712.01188V2 [Math.AP]
    GREEN’S FUNCTION FOR SECOND ORDER ELLIPTIC EQUATIONS WITH SINGULAR LOWER ORDER COEFFICIENTS SEICK KIM AND GEORGIOS SAKELLARIS Abstract. We construct Green’s function for second order elliptic operators of the form Lu = −∇ · (A∇u + bu) + c · ∇u + du in a domain and obtain pointwise bounds, as well as Lorentz space bounds. We assume that the matrix of principal coefficients A is uniformly elliptic and bounded and the lower order coefficients b, c, and d belong to certain Lebesgue classes and satisfy the condition d −∇· b ≥ 0. In particular, we allow the lower order coefficients to be singular. We also obtain the global pointwise bounds for the gradient of Green’s function in the case when the mean oscillations of the coefficients A and b satisfy the Dini conditions and the domain is C1,Dini. 1. Introduction Let Ω be a domain (i.e., an open connected set) in Rn with n ≥ 3. We consider second order elliptic operators in divergence form n n ij i i Lu = − Di(a (x)Dju + b (x)u) + c (x)Diu + d(x)u, iX,j=1 Xi=1 which hereafter shall be abbreviated as Lu = − div(A∇u + bu) + c · ∇u + du. We assume that the principal coefficients A = (aij) are measurable n × n matrices that are bounded and uniformly elliptic; i.e. there is a constant λ> 0 such that n λ|ξ|2 ≤ A(x)ξ · ξ = aij(x)ξiξj, ∀x ∈ Ω, ∀ξ ∈ Rn. (1.1) iX,j=1 We also assume that the lower order coefficients b = (b1,..., bn), c = (c1,..., cn), and d are such that b ∈ Lq(Ω), c ∈ Lr(Ω), d ∈ Ls(Ω) for some q, r ≥ n, s ≥ n/2.
    [Show full text]
  • Weak-Type Weights and Normable Lorentz Spaces 1
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 3, March 1996 WEAK-TYPE WEIGHTS AND NORMABLE LORENTZ SPACES MAR´IA J. CARRO, ALEJANDRO GARC´IA DEL AMO, AND JAVIER SORIA (Communicated by J. Marshall Ash) Abstract. We show that the Lorentz space Λ1(w) is a Banach space if and only if the Hardy-Littlewood maximal operator M satisfies a certain weak-type estimate. We also consider the case of general measures. Finally, we study some properties of several indices associated to these spaces. 1. Introduction We are going to study weighted Lorentz spaces of functions defined in Rn as follows (for standard notation we refer to [BS] and [GR]): If u is a weight in Rn, + w is a weight in R , fu∗ denotes the decreasing rearrangement of f with respect to the measure u(x) dx and 0 <p< , we define ∞ 1/p ∞ p p p Λu(w)= f; f Λu(w) = fu∗(t) w(t)dt < . k k 0 ∞ Z If u 1, we will only write Λp(w). Classical examples are obtained by choosing ≡ (p/q) 1 p q,p w(t)=t − .InthiscaseΛ(w)=L . A classical result of G.G. Lorentz (see [Lo]) shows that Λ1(w) is a norm, if and only if, w is a decreasing function. The problem of findingk·k conditions on w so that Λp(w) is a Banach space (that is, there exists a norm equivalent to Λp(w)) was solved, for p>1, by E. Sawyer ([Sa]). This condition is that the Hardy-Littlewoodk·k maximal operator is bounded on Λp(w).
    [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]
  • HARDY-TYPE OPERATORS in LORENTZ-TYPE SPACES DEFINED on MEASURE SPACES1 Qinxiu Sun∗, Xiao Yu∗∗ and Hongliang Li∗∗∗
    Indian J. Pure Appl. Math., 51(3): 1105-1132, September 2020 °c Indian National Science Academy DOI: 10.1007/s13226-020-0453-1 HARDY-TYPE OPERATORS IN LORENTZ-TYPE SPACES DEFINED ON MEASURE SPACES1 Qinxiu Sun¤, Xiao Yu¤¤ and Hongliang Li¤¤¤ ¤Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou 310023, China ¤¤Department of Mathematics, Shangrao Normal University, Shangrao 334001, China ¤¤¤Department of Mathematics, Zhejiang International Studies University, Hangzhou 310012, China e-mail: [email protected] (Received 2 June 2018; accepted 12 June 2019) Weight criteria for the boundedness and compactness of generalized Hardy-type operators Z T f(x) = u1(x) f(y)u2(y)v0(y) d¹(y); x 2 X; (0.1) fÁ(y)·Ã(x)g in Orlicz-Lorentz spaces defined on measure spaces is investigated where the functions Á; Ã; u1; u2; v0 are positive measurable functions. Some sufficient conditions of boundedness of G0 G1 G0 G1;1 T :Λv0 (w0) ! Λv1 (w1) and T :Λv0 (w0) ! Λv1 (w1) are obtained on Orlicz-Lorentz spaces. Furthermore, we achieve sufficient and necessary conditions for T to be bounded and p0 p1;q1 compact from a weighted Lorentz space Λv0 (w0) to another Λv1 (w1). It is notable that the function spaces concerned here are quasi-Banach spaces instead of Banach spaces. Key words : Hardy operator; Orlicz-Lorentz spaces; weighted Lorentz spaces; boundedness; compactness. 2010 Mathematics Subject Classification : 46E30, 46B42. 1. INTRODUCTION R x For the Hardy operator S defined by Sf(x) = 0 f(t)dt, the weighted Lebesgue-norm inequalities have been characterized by many authors (e.g.
    [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]