DOCSLIB.ORG

Lecture Notes, Singular Integrals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes, Singular Integrals Lecture notes, Singular Integrals Joaquim Bruna, UAB May 18, 2016 Chapter 3 The Hilbert transform In this chapter we will study the Hilbert transform. This is a specially important operator for several reasons: • Because of its relationship with summability for Fourier integrals in Lp- norms. • Because it constitutes a link between real and complex analysis • Because it is a model case for the general theory of singular integral op- erators. A main keyword in the theory of singular integrals and in analysis in general is cancellation. We begin with some easy examples of what this means. 3.1 Some objects that exist due to cancellation One main example of something that exists due to cancellation is the Fourier 2 d 2 transform of functions in L (R ). In fact the whole L -theory of the Fourier transform exists thanks to cancellation properties. Let us review for example 1 d 2 d the well-known Parseval's theorem, stating that for f 2 L (R )\L (R ) one has kf^k2 = kfk2, a result that allows to extend the definition of the Fourier 2 d transform to L (R ). Formally, Z Z Z Z jf^(ξ)j2 dξ = f(x)f(y)( e2πiξ·(y−x)dξ)dxdy: Rd Rd Rd Rd This being equal to R jf(x)j2 dx means formally that Rd Z 2πiξ·x e dξ = δ0(x): Rd Along the same lines, let us look at the Fourier inversion theorem, stating 1 d 1 d that whenever f 2 L (R ); f^ 2 L (R ) one has 1 Z f(x) = f^(ξ)e2πiξ·x dξ: Rd Again, the right hand side, by a formal use of Fubini's theorem becomes Z Z Z Z ( f(y)e−2πiξ·y dy)e2πiξ·x dξ = f(y)( e−2πiξ·(y−x) dy) dξ: Rd Rd Rd Rd If this is to be equal to f(x) we arrive to the same formal conclusion, namely that superposition of all frequencies is zero outside zero. An intuitive way to understand this is by noting that (d = 1) Z R sin(2πRx) e2πiξx dξ = ; R πx 1 is zero for x an integer multiple of R , so when R ! +1 the zeros become more and more dense. 1 d The Fourier transform exists trivially if f 2 L (R ), but not so trivially 2 d 2 if f 2 L (R ). It exists, in the L -sense Z −2πixξ f^(ξ) = lim fn(x)e dx; n R 1 d 2 d 2 d with fn arbitrary in L (R ) \ L (R ) such that fn ! f in L (R ). The most trivial choice is to take fR(x) to be equal to f(x) if jxj < R and zero elsewhere, so that Z R f^(ξ) = lim f(x)e−2πixξ dx; R −R the limit being in the L2-sense. It is a general and very deep result due to L. Carleson that this limits exists a.e. To illustrate all this we will consider the Fourier transform of two partic- 1 ular functions, none of which in L (R), and study the pointwise behaviour of the limit above. For instance, R sin x dx is conditionally convergent, meaning that R x Z R sin x lim dx (3.1) R!1 −R x R sin x exists while j x j dx = +1. The later can be proved by estimating from R π below the contribution to the integral at points jx − ( 2 + kπ)j < δ; k 2 Z. Alternatively, noting that the Fourier transform of 1[−a;a] is Z a sin 2πaξ e−2πxξ dx = ; a πξ 2 we can argue that it cannot be integrable, because if it were, by Fourier in- version theorem, 1[−a;a] would be continuous. We note too that by Parseval's theorem Z Z sin 2πaξ 2 2 ( ) dξ = 1[−a;a] = 2a: R πξ R Regarding the value of the integral, we will see that the limit Z R sin 2πaξ 2πiξx lim e dξ = 1[−a;a](x); (3.2) R!+1 −R πξ that as we know exists in the L2 sense, exists too pointwise a.e., in fact for x 6= ±a. Indeed, by using residues we have that 1 Z R eiαξ lim dξ R!1 2πi −R ξ 1 eiαξ 1 1 equals 2 Res( ξ ; 0) = 2 if α > 0 and − 2 if α < 0, and this implies (3.2). For x = 0 we get that Z R sin 2πaξ lim dξ = 1; R!+1 −R πξ from which it follows that the integral in (3.1) equals π. Another example are the Fresnel integrals Z Z sin x2dx; cos x2dx: R R By the change of variable u = x2 one has 2 Z R 1 Z R sin u sin x2dx = p du; 0 2 0 u so the integral is not absolutely convergent. To compute the integral we will see more generally that Z R Z R −απx2 −2πixξ −απx2 1 − π ξ2 lim e e dx = 2 lim e cos 2πxξ dx = p e α : R!+1 −R R!+1 0 α (3.3) for Re α ≥ 0 and all ξ 2 C. For Re α > 0, the above is the Fourier transform −απx2 of fα(x) = e , a function in the Schwarz class. Since it is holomorphic π 2 p1 − α ξ in α and for α; ξ real equals α e , the same holds true by analytic continuation first in α and then in ξ, that is Z −απx2 −2πixξ 1 − π ξ2 F (α; ξ) = e e dx = p e α ; Re α > 0; ξ 2 C: R α 3 By making Re α ! 0 we can already conclude that the Fourier transform, in 2 π 2 −απx p1 − α ξ the sense of distributions, of e is α e also if Re α = 0. However, this does not imply (3.3). To compute the limit in (3.3) when Re α = 0 we will exploit the analyticity in x as follows. We consider the contour iπt 1 in the z-plane given by the segment [0;R], the arc t 7! Re ; 0 ≤ t ≤ 4 , i π and the segment from Re 4 to 0, and apply the Cauchy theorem to the entire function e−απz2 cos(2πzξ). The contribution of the arc is bounded by aRe−bR2+cR with some positive constants a; b; c depending on ξ, and so has limit zero when R ! 1. Hence, parametrizing with z = t 1+p i , we get 2 R R Z 2 1 + i Z 2 1 + i lim e−απx cos(2πxξ) dx = p lim e−απit cos(2π p tξ)dt: R!+1 0 2 R!1 0 2 If Re iα > 0, that is, Im α < 0 this shows that the limits exist and equals 1 + i 1 + i 1 − π ξ2 p F (iα; p ξ) = p e α ; 2 2 α p as claimed. For ξ = 0 and α = − i we get the value pπ for the Fresnel π 2 2 integrals. 3.2 The conjugate Poisson kernel and the Cauchy transform One method to study the summability of the Fourier integral is by the means method. The Abel method consists in using Z f^(ξ)e−2πtjξje2πix·ξ dξ: Rd The behaviour as t ! 0 is easily understood. Indeed, by Fubini's theorem the above equals Z Z −2πtjξj 2πiξ(x−y) f(y)( e e dξ)dy = (f ∗ Pt)(x); Rd Rd with Z −2πtjξj 2πix·ξ Pt(x) = e e dξ: Rd Let us compute explicitly Pt in d = 1: Z +1 Z 0 2πξ(ix−t) 2πξ(ix+t) Pt(x) = e dξ + e dξ = A + B; 0 −∞ with 1 1 A = ;B = A = : 2π(t − ix) 2π(t + ix) 4 This gives 1 t P (x) = : t π x2 + t2 1 −2πjξj Note that Pt(x) = t P (x=t), with P = P1, and that P1; e are a Fourier pair, P is positive with integral one. In dimension d > 1 the computation is −d harder, the result being Pt(x) = t P (x=t), with d+1 1 Γ( 2 ) P (x) = cd d+1 ; cd = d+1 ; (jxj2 + 1) 2 π 2 It is convenient to look as (f ∗ Pt)(x) as a function of (x; t) in the upper half space t > 0. Noticing that P is in all Lp-spaces we define p d Definition 1. For f 2 L (R ); 1 ≤ p ≤ +1, we define the Poisson trans- form u = P [f] as the function defined in the upper half-space t > 0 by Z Z 1 t ^ −2πtjξj 2πix·ξ P [f](x; t) = (f∗Pt)(x) = f(y) 2 2 dy = f(ξ)e e dξ π Rd (x − y) + t Rd Since we will need it again in a more general form we state again the main −d properties of convolutions with functions of the form ht(x) = t h(x=t), with 1 d h 2 L (R ). 1 d R −d Proposition 1. Let h 2 L (R ); m = h, and set ht(x) = t h(x=t).. For p d f 2 L (R ); 1 ≤ p ≤ +1, consider f ∗ ht. Then p d • kf ∗ htkp ≤ kfkpkhk1, and f ∗ ht ! mf as t ! 0 in L (R ); 1 ≤ p < +1. • If f is bounded and continuous at x0 then f ∗ ht(x0) ! mf(x0). • If f is bounded and uniformly continuous then f ∗ht ! mf uniformly.
Load more

Recommended publications
  • Arxiv:1507.07356V2 [Math.AP]
    TEN EQUIVALENT DEFINITIONS OF THE FRACTIONAL LAPLACE OPERATOR MATEUSZ KWAŚNICKI Abstract. This article discusses several definitions of the fractional Laplace operator ( ∆)α/2 (α (0, 2)) in Rd (d 1), also known as the Riesz fractional derivative − ∈ ≥ operator, as an operator on Lebesgue spaces L p (p [1, )), on the space C0 of ∈ ∞ continuous functions vanishing at infinity and on the space Cbu of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner’s subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain. 1. Introduction We consider the fractional Laplace operator L = ( ∆)α/2 in Rd, with α (0, 2) and d 1, 2, ... Numerous definitions of L can be found− in− literature: as a Fourier∈ multiplier with∈{ symbol} ξ α, as a fractional power in the sense of Bochner or Balakrishnan, as the inverse of the−| Riesz| potential operator, as a singular integral operator, as an operator associated to an appropriate Dirichlet form, as an infinitesimal generator of an appropriate semigroup of contractions, or as the Dirichlet-to-Neumann operator for an appropriate harmonic extension problem. Equivalence of these definitions for sufficiently smooth functions is well-known and easy. The purpose of this article is to prove that, whenever meaningful, all these definitions are equivalent in the Lebesgue space L p for p [1, ), ∈ ∞ in the space C0 of continuous functions vanishing at infinity, and in the space Cbu of bounded uniformly continuous functions.
    [Show full text]
  • Singular Integrals on Self-Similar Sets and Removability for Lipschitz Harmonic Functions in Heisenberg Groups
    SINGULAR INTEGRALS ON SELF-SIMILAR SETS AND REMOVABILITY FOR LIPSCHITZ HARMONIC FUNCTIONS IN HEISENBERG GROUPS VASILIS CHOUSIONIS AND PERTTI MATTILA Abstract. In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In addition to obtaining results in a very general setting, the purpose of this work is twofold; we shall extend some results in Euclidean spaces to more general kernels than previously considered, and we shall obtain in Heisenberg groups some applications to harmonic (in the Heisenberg sense) functions of some results known earlier in Euclidean spaces. 1. Introduction The Cauchy singular integral operator on one-dimensional subsets of the complex plane has been studied extensively for a long time with many applications to analytic functions, in particular to analytic capacity and removable sets of bounded analytic functions. There have also been many investigations of the same kind for the Riesz singular integral op- erators with the kernel x=jxj−m−1 on m-dimensional subsets of Rn. One of the general themes has been that boundedness properties of these singular integral operators imply some geometric regularity properties of the underlying sets, see, e.g., [DS], [M1], [M3], [Pa], [T2] and [M5]. Standard self-similar Cantor sets have often served as examples where such results were first established. This tradition was started by Garnett in [G1] and Ivanov in [I1] who used them as examples of removable sets for bounded analytic functions with positive length.
    [Show full text]
  • Estimates for the Maximal Singular Integral in Terms of the Singular Integral: the Case of Even Kernels
    Annals of Mathematics 174 (2011), 1429{1483 http://dx.doi.org/10.4007/annals.2011.174.3.2 Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels By Joan Mateu, Joan Orobitg, and Joan Verdera Abstract Let T be a smooth homogeneous Calder´on-Zygmund singular integral n operator in R . In this paper we study the problem of controlling the max- imal singular integral T ?f by the singular integral T f. The most basic 2 n form of control one may consider is the estimate of the L (R ) norm of ? 2 n T f by a constant times the L (R ) norm of T f. We show that if T is an even higher order Riesz transform, then one has the stronger pointwise inequality T ?f(x) ≤ CM(T f)(x), where C is a constant and M is the Hardy-Littlewood maximal operator. We prove that the L2 estimate of T ? by T is equivalent, for even smooth homogeneous Calder´on-Zygmund operators, to the pointwise inequality between T ? and M(T ). Our main result characterizes the L2 and pointwise inequalities in terms of an alge- Ω(x) braic condition expressed in terms of the kernel jxjn of T , where Ω is an even homogeneous function of degree 0, of class C1(Sn−1) and with zero n−1 integral on the unit sphere S . Let Ω = P Pj be the expansion of Ω in spherical harmonics Pj of degree j. Let A stand for the algebra generated by the identity and the smooth homogeneous Calder´on-Zygmund opera- tors.
    [Show full text]
  • A Technique for Solving the Singular Integral Equations of Potential Theory
    Old Dominion University ODU Digital Commons Mathematics & Statistics Theses & Dissertations Mathematics & Statistics Spring 2007 A Technique for Solving the Singular Integral Equations of Potential Theory Brian George Burns Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/mathstat_etds Part of the Mathematics Commons Recommended Citation Burns, Brian G.. "A Technique for Solving the Singular Integral Equations of Potential Theory" (2007). Doctor of Philosophy (PhD), Dissertation, Mathematics & Statistics, Old Dominion University, DOI: 10.25777/q9kv-0x34 https://digitalcommons.odu.edu/mathstat_etds/6 This Dissertation is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion in Mathematics & Statistics Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. A TECHNIQUE FOR SOLVING THE SINGULAR INTEGRAL EQUATIONS OF POTENTIAL THEORY by Brian George Burns M.Sci. 1998, University of Glasgow M.Sc. 1999, University of Stirling M.S. 2005, Old Dominion University A Dissertation Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the Requirement for the Degree of DOCTOR OF PHILOSOPHY COMPUTATIONAL AND APPLIED MATHEMATICS OLD DOMINION UNIVERSITY May 2007 John Tweed (Director) Ion Melrose John/A. Adam Richard D. Noren Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT A TECHNIQUE FOR SOLVING THE SINGULAR INTEGRAL EQUATIONS OF POTENTIAL THEORY Brian George Burns Old Dominion University, 2007 Director: Dr. John Tweed The singular integral equations of Potential Theory are investigated using ideas from both classical and contemporary mathematics. The goal of this semi-analytic ap­ proach is to produce numerical schemes that are both general and computationally simple.
    [Show full text]
  • Singular Integrals on Homogeneous Spaces and Some Problems Of
    ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze A. KORÁNYI S. VÁGI Singular integrals on homogeneous spaces and some problems of classical analysis Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 25, no 4 (1971), p. 575-648 <http://www.numdam.org/item?id=ASNSP_1971_3_25_4_575_0> © Scuola Normale Superiore, Pisa, 1971, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction 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/ SINGULAR INTEGRALS ON HOMOGENEOUS SPACES AND SOME PROBLEMS OF CLASSICAL ANALYSIS A. KORÁNYI and S. VÁGI CONTENTS Introduction. Part I. General theory. 1. Definitions and basic facts. 2. The main result on LP -continuity. 3. An L2-theorem. 4. Preservation of Lipschitz classes. 5. Homogeneous gauges and kernel Part 11. Applications. 6. The Canohy-Szego integral for the generalized halfplane D. 7. The Canchy-Szego integral for the complex unit ball. 8. The functions of Littlewood-Paley and Lusin on D. 9. The Riesz transform on Introduction. In this paper we generalize some classical results of Calderon and Zygmund to the context of homogeneous spaces of locally compact groups and use these results to solve certain problems of classical type which can not be dealt with by the presently existing versions of the theory of sin- gular integrals.
    [Show full text]
  • Singular Integral Operators with Operator-Valued Kernels, And
    Singular integral operators with operator-valued kernels, and extrapolation of maximal regularity into rearrangement invariant Banach function spaces Ralph Chill, Alberto Fiorenza To cite this version: Ralph Chill, Alberto Fiorenza. Singular integral operators with operator-valued kernels, and ex- trapolation of maximal regularity into rearrangement invariant Banach function spaces. Journal of Evolution Equations, Springer Verlag, 2014, 14 (4), pp.795-828. 10.1007/s00028-014-0239-1. hal- 01275671 HAL Id: hal-01275671 https://hal.archives-ouvertes.fr/hal-01275671 Submitted on 17 Feb 2016 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. SINGULAR INTEGRAL OPERATORS WITH OPERATOR-VALUED KERNELS, AND EXTRAPOLATION OF MAXIMAL REGULARITY INTO REARRANGEMENT INVARIANT BANACH FUNCTION SPACES RALPH CHILL AND ALBERTO FIORENZA Abstract. We prove two extrapolation results for singular integral operators with operator-valued kernels and we apply these results in order to obtain the following extrapolation of Lp-maximal regularity: if an autonomous Cauchy problem on a Banach space has Lp-maximal regularity for some p (1; ), 2 1 then it has Ew-maximal regularity for every rearrangement-invariant Banach function space E with Boyd indices 1 < pE qE < and every Muckenhoupt ≤ 1 weight w ApE .
    [Show full text]
  • Weighted Inequalities for Singular Integral Operators on the Half-Line Ralph Chill, Sebastian Król
    Weighted inequalities for singular integral operators on the half-line Ralph Chill, Sebastian Król To cite this version: Ralph Chill, Sebastian Król. Weighted inequalities for singular integral operators on the half-line. 2016. hal-01278755 HAL Id: hal-01278755 https://hal.archives-ouvertes.fr/hal-01278755 Preprint submitted on 24 Feb 2016 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. WEIGHTED INEQUALITIES FOR SINGULAR INTEGRAL OPERATORS ON THE HALF-LINE RALPH CHILL AND SEBASTIAN KRÓL ABSTRACT. We prove weighted estimates for singular integral operators which oper- ate on function spaces on a half-line. The class of admissible weights includes Muck- enhoupt weights and weights satisfying Sawyer’s one-sided conditions. The kernels of the operators satisfy relaxed Dini conditions. We apply the weighted estimates to extrapolation of maximal Lp regularity of first order, second order and fractional or- der Cauchy problems into weighted rearrangement invariant Banach function spaces. In particular, we provide extensions, as well as a unification of recent results due to Auscher and Axelsson, and Chill and Fiorenza. 1. MOTIVATION AND DESCRIPTION OF THE MAIN RESULT Weighted Lp-estimates for singular integral operators play an important role in har- monic analysis and in its applications to elliptic and parabolic partial differential equa- tions, as well as abstract evolution equations on Banach spaces, where singular inte- gral operators with operator valued kernels naturally arise via representation formu- las for solutions.
    [Show full text]
  • A Renormalized Riesz Potential and Applications
    A renormalized Riesz potential and applications Mihai Putinar Abstract. The convolution in Rn with |x|−n is a very singular operator. Endowed with a proper normalization, and regarded as a limit of Riesz potentials, it is equal to Dirac’s distribution δ. However, a different normalization turns the non-linear operator: −2 −n Ef = exp( |x| ∗ f), |Sn−1| into a remarkable transformation. Its long history (in one dimension) and some of its recent applications in higher dimensions make the subject of this exposition. A classical extremal problem studied by A. A. Markov is related to the operation E in one real variable. Later, the theory of the spectral shift of self-adjoint perturbations was also based on E. In the case of two real variables, the transform E has ap- peared in operator theory, as a determinantal-characteristic function of certain close to normal operators. The interpretation of E in com- plex coordinates reveals a rich structure, specific only to the plane setting. By exploiting an inverse spectral function problem for hy- ponormal operators, applications of this exponential transform to im- age reconstruction, potential theory and fluid mechanics have recently been discovered. In any number of dimensions, the transformation E, applied to characteristic functions of domains Ω, can be regarded as the geometric mean of the distance function to the boundary. This interpretation has a series of unexpected geometric and analytic con- sequences. For instance, for a convex algebraic Ω, it turns out that the operation E is instrumental in converting finite external data (such as field measurements or tomographic pictures) into an equation of the boundary.
    [Show full text]
  • The L(Log L) Endpoint Estimate for Maximal Singular Integral Operators
    The L(log L) endpoint estimate for maximal singular integral operators Tuomas Hytonen¨ and Carlos Perez´ J. Math. Anal. Appl. 428 (2015), no. 1, 605–626. Abstract We prove in this paper the following estimate for the maximal operator T ∗ associated to the singular integral operator T: Z ∗ 1 kT f k 1;1 j f x j M w x dx; w ≥ ; < ≤ : L (w) . ( ) L(log L) ( )( ) 0 0 1 Rn This follows from the sharp Lp estimate ∗ 0 1 1=p0 kT f k p p k f k ; < p < 1; w ≥ ; < δ ≤ : L (w) . ( ) Lp(M (w)) 1 0 0 1 δ L(log L)p−1+δ As as a consequence we deduce that Z ∗ 1;1 kT f kL (w) . [w]A1 log(e + [w]A1 ) j f j w dx; Rn extending the endpoint results obtained in [LOP] and [HP] to maximal singular integrals. Another consequence is a quantitative two weight bump estimate. 1 Introduction and main results Very recently, the so called Muckenhoupt-Wheeden conjecture has been disproved by Reguera-Thiele in [RT]. This conjecture claimed that there exists a constant c such that for any function f and any weight w (i.e., a nonnegative locally integrable function), there holds Z kH f kL1;1(w) ≤ c j f j Mwdx: (1) R where H is the Hilbert transform. The failure of the conjecture was previously obtained by M.C. Reguera in [Re] for a special model operator T instead of H. This conjecture was motivated by a similar inequality by C.
    [Show full text]
  • On Fourier Multiplier Transformations of Banach-Valued Functions by Terry R
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 285, Number 2, October 1984 ON FOURIER MULTIPLIER TRANSFORMATIONS OF BANACH-VALUED FUNCTIONS BY TERRY R. MCCONNELL1 ABSTRACT. We obtain analogues of the Mihlin multiplier theorem and Little- wood-Paley inequalities for functions with values in a suitable Banach space B. The requirement on B is that it have the unconditionality property for martingale difference sequences. 1. Introduction. A number of authors have considered the extension of the Calderón-Zygmund theory of singular integrals to vector-valued functions (see, e.g., [1, 2, and 6]). Recently Burkholder [5] has obtained definitive results on transforms of Banach space-valued martingales, which are the analogues of singular integrals in probability theory, and these results have been applied to the study of singu- lar integrals by Burkholder and McConnell [6] and by Bourgain [2 and 3]. In the present paper we use martingale methods to obtain extensions of the Mihlin- Hörmander multiplier theorem and the Littlewood-Paley inequalities to the case of Banach-valued functions. For suitable real-valued test functions / defined on R" let / denote the Fourier transform of /, (1.1) /(A) = (2tt)-" / e~iXxf(x)dx, and / the inverse Fourier transform. To each bounded Borel function m on Rn we may associate an operator Tm defined on the test functions / by (1.2) Tmf(x) = (m/)v(x). The Calderón-Zygmund singular integral transformations provide an interesting class of examples (see, e.g., [22, p. 39]) and the fact that such operators are bounded on the Lp spaces for 1 < p < co suggests the following question: under what condi- tions on m does Tm defined in (1.2) extend to a bounded linear transformation of the Lp spaces? A general though by no means complete answer is given by Theorem A, an improvement due to Hörmander [11] of the Mihlin multiplier theorem [17].
    [Show full text]
  • Appendix A. Special Cases of the Marcinkiewicz Interpolation Theorem
    Appendix A. Special Cases of the Marcinkiewicz Interpolation Theorem In a number of places in Chapters 2, 3 and 5, we employ simple forms of the Marcinkiewicz interpolation theorem. The purpose of this appendix is to present for the reader's convenience statements and proofs of the theorems involved and to explain the concepts associated with them. More general versions of the Mar­ cinkiewicz theorem can be found in [9], Section 13.8 and [33], Appendix B. Let us agre'e that the measure spaces appearing below are always (J-finite. A.I. The Concepts of Weak Type and Strong Type A.I.t. Strong type. Let (M, JIt, 11) and (N, .AI, v) be measure spaces and p an index in the range [1, ex)]. Suppose D is a subset of V(Il) and T is a mapping from D into the space of complex-valued measurable functions on N, or the space of nonnegative extended-real-valued measurable functions on N, or the space of equiv­ alence classes of one of these. We say T is of strong type (p, p), or simply of type (p, p), on D if there is a constant B such that (1) for allfin D. The smallest number B for which (1) holds is then termed the (p, p) norm of Ton D and is denoted II TIIp,p, when D is understood. Even if there may not exist a finite constant B for which (1) holds, it is customary to define (provided of course that D #- {On. So we may say that T is of type (p, p) on D if and only if IITllp,p < 00.
    [Show full text]
  • A “Local Mean Oscillation” Decomposition and Some Its Applications
    A \LOCAL MEAN OSCILLATION" DECOMPOSITION AND SOME ITS APPLICATIONS ANDREI K. LERNER 1. Introduction A recent result by the author [39] establishes a pointwise control of an arbitrary measurable function in terms of its local mean oscillations. Soon after that, in a surprising work [12], D. Cruz-Uribe, J. Martell and C. P´erezshowed that this result can be effectively applied in a variety of questions, including sharp weighted inequalities for classical singular integrals and the dyadic square function. In turn, based on [12] and on a recent concept of the intrinsic square function by M. Wilson [54], the author [40] obtained sharp weighted estimates for essentially any Littlewood-Paley operator. The aim of these notes is to present a unified, extended and almost self-contained exposition of the above-mentioned works [39, 12, 40]. 2. The space BMO 2.1. The classical approach to BMO. The mean oscillation of a locally integrable function f over a cube Q ⊂ Rn is defined by Z 1 j − j Ω(f; Q) = j j f(x) fQ dx; Q Q R 1 where fQ = jQj Q f. It is easy to see that Z Z 1 1 inf jf(x) − cjdx ≤ Ω(f; Q) ≤ 2 inf jf(x) − cjdx: 2R j j 2R j j c Q Q c Q Q The space of functions with bounded mean oscillation, BMO(Rn), 2 1 Rn consists of all f Lloc( ) such that kfkBMO ≡ sup Ω(f; Q) < 1: Q⊂Rn This space was introduced by F. John and L. Nirenberg in [30]. In the same work the following fundamental property of BMO-functions was 2000 Mathematics Subject Classification.
    [Show full text]