DOCSLIB.ORG

On the Equivalence Between Weak BMO and the Space of Derivatives of the Zygmund Class

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Equivalence Between Weak BMO and the Space of Derivatives of the Zygmund Class On the equivalence between Weak BMO and the space of derivatives of the Zygmund class Eddy Kwessi* Abstract In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show in particular that this proves that the Hardy space H1 strictly contains the space special atom space. Keywords: BMO, Derivative, Distributions, Zygmund Class AMS Subject Classification: 42B05, 42B30, 30B50, 30E20 1 Introduction The space of functions of bounded means oscillation has taken central stage in the mathematical litera- ture after the work of Charles Fefferman [4] where he showed that it is the dual space of the real Hardy 1 space H , a long sought after result. Right after, Ronald Coifman [2] showed this result using a different 1 method. Essentially, he showed that H has an atomic decomposition. De Souza [3] showed there is a 1 1 1 1 subset B of H , formed by special atoms that is contained in H . This space B has the particularity that it contains some functions whose Fourier series diverge. The question of whether B1 is equivalent 1 1 1 to H was never truly answered explicitly but it was always suspected that the inclusion B ⊂ H was 1 1 strict, that is, there must be at least one function in H that is not in B . However, such a function had 1 ∗ 1 1 1 never being constructed nor given. Since the dual space (H ) of H is BMO and B ⊂ H , it follows that the dual space (B1)∗ of B1 must be a superset of BMO. A natural superset candidate of BMO is therefore the space BMOw since BMO ⊂ BMOw. So essence, that BMOw is the dual of B1 would 1 1 1 ∗ ∼ 0 also prove that B ⊂ H with a strict inclusion. Moreover, it was already proved that (B ) = Λ∗, 0 where Λ∗ is the space of derivative (in the sense of distributions) of functions in the Zygmund class 1 ∗ ∼ w Λ∗, see for example [3] and [9]. Therefore, if (B ) = BMO , then by transition we would have arXiv:2104.08411v1 [math.FA] 17 Apr 2021 0 ∼ w Λ∗ = BMO . Henceforth, we will adopt the following notations: D = fz 2 C : jzj < 1g is the open unit disk and let T = fz 2 C : jzj = 1g is the unit sphere. For an integrable function f on a measurable set A, and Z 1 Z the Lebesgue measure λ on A, we will write − f(ξ)dλ(ξ) := f(ξ)dλ(ξ). We will start by A λ(A) A recalling the necessary definitions and important results. The interested reader can see for example [5] for more information. *Department of Mathematics, Trinity University, San Antonio, TX, USA, [email protected] 1 p p Definition 1. Let 0 < p < 1 be a real number. The Hardy Space H := H (D) is the space of holomorphic functions f defined on D and satisfying 1 Z 2π p p 1 iξ kfk p := sup f(re ) dλ(ξ) < 1 : H 0≤r<1 2π 0 1 n n n Let f 2 Lloc(R );Q be a hypercube in R , and λ be the Lebesgue measure on R for some n 2 N. Put Z Z # fQ = − f(ξ)dλ(ξ); fQ = − f(ξ)dλ(ξ) : Q Q 1 n n For f 2 Lloc(R ) and x 2 R , we define Z # # M (f)(x) = sup − f(ξ) − fQ dλ(ξ) (1.1) Q3x Q Z M(f)(x) = sup − [f(ξ) − fQ]dλ(ξ) (1.2) Q3x Q mf(x) = sup fQ : (1.3) Q3x where the supremum is taken over all hypercubes Q containing x. Now we can define the space of functions of bounded mean oscillation and its weak counterpart. Definition 2. The space of functions of bounded mean oscillation is defined as the space of locally integrable functions f for which the operator M # is bounded, that is, n n 1 n # 1 n o BMO(R ) = f 2 Lloc(R ): M (f) 2 L (R ) : n We can endowed BMO(R ) with the norm # # kfkBMO( n) = M (f) = M (f)(x) : R 1 The space of functions of weak bounded mean oscillation is defined as the space of locally inte- grable functions f for which the operator M is bounded, that is, w n n 1 n 1 n o BMO (R ) = f 2 Lloc(R ): Mf 2 L (R ) : n w n Remark 1.1. It follows from the above definitions that BMO(R ) ⊆ BMO (R ). n Let us recall the definition of the space of functions of vanishing mean oscillation VMO(R ) and w n introduce the space of functions of weak vanishing mean oscillations VMO (R ). 1 n Definition 3. Let f 2 Lloc(R ). Z n f 2 VMO(R ) if lim − jf(ξ) − fQj dλ(ξ) = 0 : λ(Q)!0 Q Z w n f 2 VMO (R ) if lim − [f(ξ) − fQ]dλ(ξ) = 0 : λ(Q)!0 Q n w n Remark 1.2. It follows from the above definition that VMO(R ) is a subspace of VMO (R ) which w n w is itself a subspace of BMO (R ). We will show (see Theorem 2.8 below) that VMO is in fact closed subspace of BMOw. 2 n w n w n w Henceforth, BMO(R ); BMO (R ), and VMO (R ) will simply be referred to as BMO; BMO , and VMOw. Now we consider A(D) as the space of analytic functions defined on the unit disk D. Following the work of Girela in [6] on the space of analytic functions of bounded means oscillations, we introduce their weak counterpart. Before, we recall that the Poisson Kernel is defined as 1 + reiθ P (θ) = Re : r 1 − reiθ Definition 4. The space of analytic functions of bounded mean oscillation is defined as Z iθ 1 iξ BMOA(D) = F 2 A(D); 9f 2 BMO(T): F (re ) = Pr(θ − ξ)f(e )dλ(ξ) : 2π T We can endowed BMO(D) with the norm 1 Z kF k := jF (0)j + sup P (θ − ξ) f(eiξ) − F (reiθ) dλ(ξ) < 1 : BMOA(D) r 0≤r<1 2π T θ2T In other words, BMOA(D) is the space of Poisson integrals of functions in BMO(T). We can now define the space BMOAw of analytic function of weak bounded mean oscillation. Definition 5. An analytic function F on D is said to be of weak bounded mean oscillation if there exists w f 2 BMO (T) such that Z iθ 1 iξ F (re ) = Pr(θ − ξ)f(e )dλ(ξ) : 2π T w We endow BMOA (D) with the norm Z 1 h iξ iθ i kF k w := jF (0)j + sup P (θ − ξ) f(e ) − F (re ) dλ(ξ) < 1 : BMOA (D) r 0≤r<1 2π T θ2T We recall the definition of special atom space B1, see [8]. n n Y Definition 6. For n ≥ 1, we consider the hypercube of R given as J = [aj − hj; aj + hj] where j=1 Z 1 aj; hj are real numbers with hj > 0. Let φ 2 L (J) with φ(J) = φ(ξ)dλ(ξ) . J The special atom (of type 1) is a function b : I ⊆ J ! R such that b(ξ) = 1 on JnI or 1 b(ξ) = [χ (ξ) − χ (ξ)] ; φ(I) R L where 2n−1 [ n • R = Iij for some i1; i2; ··· ; i2n−1 2 f1; 2; ··· ; 2 g with i1 < i2 < ··· < i2n−1 and j=1 L = InR . 3 • fI1;I2; ··· ;I2n g is the collection of sub-cubes of I, cut by the hyperplanes x1 = a1; x2 = a2; ··· ; xn = an. • χA represents the characteristic function of set A . Definition 7. The special atom space is defined as ( 1 1 ) 1 X X B = f : J ! R; f(ξ) = αnbn(ξ); jαnj < 1 ; n=0 n=0 where the bn’s are special atoms of type 1. 1 1 X B is endowed with the norm kfkB1 = inf jαnj ; where the infimum is taken over all representations of f : n=0 (a) (b) y a2 + h 2 L 2 R 2 L a2 1 I L 1 R 1 1 a2 − h 2 I R x a1 − h 1 a1 a1 + h 1 (c) a3 + h 3 z + a3 a2 h 2 a1 − h 1 a2 a3 − h 3 a1 a2 − h 2 y x a1 + h 1 Figure 1: Illustration of the the special atom, for n = 1 in (a) , n = 2 in (b), and n = 3 in (c). 4 Now we define the Zygmund class of functions. k n Definition 8. Let k 2 N. A function f is said to be in the Zygmund class Λ∗(R ) of functions of order k−1 n k if f 2 C (R ) and X j@αf(x + h) + @αf(x − h) − 2@αf(x)j kfk k n = sup < 1 : Λ∗(R ) x;h jhj jαj=k 1 n In particular, for k = 1, we have Λ∗ := Λ∗(R ), and hence ( ) jf(x + h) + f(x − h) − 2f(x)j Λ = f 2 C0( n): kfk := sup < 1 : ∗ R Λ∗ x;h>0 2h One important note about the space Λ∗ is that it contains the so-called Weierstrass functions that are k known to be continuous everywhere but nowhere differentiable.
Load more

Recommended publications
  • Mean Growth of the Derivative of Analytic Functions, Bounded Mean Oscillation and Normal Functions
    MEAN GROWTH OF THE DERIVATIVE OF ANALYTIC FUNCTIONS, BOUNDED MEAN OSCILLATION AND NORMAL FUNCTIONS Oscar Blasco, Daniel Girela and M. Auxiliadora Marquez´ Abstract. For a given positive function φ defined in [0, 1) and 1 ≤ p<∞, we consider L the space (p, φ) which consists of all functions f analytic in the unit disc ∆ for which 1/p 1 π iθ p → 2π −π f (re ) dθ =O(φ(r)), as r 1.A result of Bourdon, Shapiro and Sledd 1 −1 implies that such a space is contained in BMOA for φ(r)=(1− r) p .Among other results, in this paper we prove that this result is sharp in a very strong sense, showing that, 1 −1 for a large class of weight functions φ, the function φ(r)=(1− r) p is the best one to 1− 1 get L(p, φ) ⊂ BMOA.Actually, if φ(r)(1 − r) p ↑∞,asr ↑ 1, we construct a function f ∈L(p, φ) which is not a normal function.These results improve other obtained recently by the second author.We also characterize the functions φ, among a certain class of weight functions, to be able to embedd L(p, φ)intoHq for q>por into the space B of Bloch functions. 1. Introduction and statement of results. Let ∆ denote the unit disc {z ∈ C : |z| < 1} and T the unit circle {ξ ∈ C : |ξ| =1}.If 0 <r<1 and g is a function which is analytic in ∆, we set π 1/p 1 iθ p Mp(r, g)= g(re ) dθ , 0 <p<∞, 2π −π M∞(r, g) = max |g(z)|.
    [Show full text]
  • Research.Pdf (1.262Mb)
    Lossless Data Compression of Monitored Power Signals Using the PLHaar Transform A Thesis Presented to the Faculty of the Graduate School at the University of Missouri{Columbia In Partial Fulfillment of the Requirements for the Degree Master of Science By Nicholas J. Hudson Dr. Michael J. Devaney, Thesis Supervisor July 2015 c Nicholas Hudson 2015, All rights reserved. The undersigned have examined the thesis entitled Lossless Data Compression of Monitored Power Signals Using the PLHaar Transform presented by Nicholas Hudson and determined it worthy of acceptance. Professor Michael J. Devaney Professor Zhihai "Henry" He Professor Jeffrey Uhlmann for the old man Acknowledgements I've unmeasurable gratitude to my advisor Dr. Michael Devaney. Without his boundless patience, breadth of knowledge, excellent advice, and boundless patience not only would this work have been not just impossible but unimaginable to me. I'm both deeply honored and fantastically fortunate to be his student. I'd also like to thank Dr. James Keller, who not only introduced me to the non-Euclidean spaces fundamental to this work, but also extended my education with a philosophical ele- ment of value nigh-impossible to overstate. And I'm deeply indebted to my mother and brother, whose faith in me served as bedrock to this research. ii Contents Acknowledgements ii Contents iii List of Figures v List of Tables vi 1 Introduction 1 1.1 Motivation . .1 1.2 Relevance . .2 1.3 Proposed Approach . .4 1.4 Organization . .4 2 Background 5 2.1 History . .5 2.1.1 Fourier . .5 2.1.2 Windowed Transforms .
    [Show full text]
  • The Schur Lemma
    Appendix A The Schur Lemma Schur’s lemma provides sufficient conditions for linear operators to be bounded on Lp. Moreover, for positive operators it provides necessary and sufficient such condi- tions. We discuss these situations. A.1 The Classical Schur Lemma We begin with an easy situation. Suppose that K(x,y) is a locally integrable function on a product of two σ-finite measure spaces (X, μ) × (Y,ν), and let T be a linear operator given by T( f )(x)= K(x,y) f (y)dν(y) Y when f is bounded and compactly supported. It is a simple consequence of Fubini’s theorem that for almost all x ∈ X the integral defining T converges absolutely. The following lemma provides a sufficient criterion for the Lp boundedness of T. Lemma. Suppose that a locally integrable function K(x,y) satisfies sup |K(x,y)|dν(y)=A < ∞, x∈X Y sup |K(x,y)|dμ(x)=B < ∞. y∈Y X Then the operator T extends to a bounded operator from Lp(Y) to Lp(X) with norm − 1 1 A1 p B p for 1 ≤ p ≤ ∞. Proof. The second condition gives that T maps L1 to L1 with bound B, while the first condition gives that T maps L∞ to L∞ with bound A. It follows by the Riesz– − 1 1 Thorin interpolation theorem that T maps Lp to Lp with bound A1 p B p . This lemma can be improved significantly when the operators are assumed to be positive. A.2 Schur’s Lemma for Positive Operators We have the following necessary and sufficient condition for the Lp boundedness of positive operators.
    [Show full text]
  • Bounded Mean Oscillation and the Uniqueness of Active Scalar Equations
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 5, May 2015, Pages 3095–3118 S 0002-9947(2014)06040-6 Article electronically published on December 18, 2014 BOUNDED MEAN OSCILLATION AND THE UNIQUENESS OF ACTIVE SCALAR EQUATIONS JONAS AZZAM AND JACOB BEDROSSIAN Abstract. We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors. As special cases of our results, we provide a significantly simplified proof to the known uniqueness result for the 2D Euler equations in L1 ∩ BMO and provide a mild improvement to the recent results of Rusin for the 2D inviscid surface quasi-geostrophic (SQG) equations, which are now to our knowledge the best results known for this model. We also obtain what are (to our knowledge) the strongest known uniqueness results for the Patlak- Keller-Segel models with nonlinear diffusion. We obtain these results via tech- nical refinements of energy methods which are well-known in the L2 setting but are less well-known in the H˙ −1 setting. The H˙ −1 method can be consid- ered a generalization of Yudovich’s classical method and is naturally applied to equations such as the Patlak-Keller-Segel models with nonlinear diffusion and other variants. Important points of our analysis are an Lp-BMO interpo- lation lemma and a Sobolev embedding lemma which shows that velocity fields v with ∇v ∈ BMO are locally log-Lipschitz; the latter is known in harmonic analysis but does not seem to have been connected to this setting. 1. Introduction The term active scalar refers to a wide variety of nonlinear advection-diffusion equations for a scalar density which, through a nonlocal operator such as an elliptic PDE, gives rise to the advective velocity field (to the authors’ knowledge, the terminology is originally due to P.
    [Show full text]
  • Lecture Notes on Harmonic Analysis Updated: April 28, 2020 © Chengchun HAO Institute of Mathematics AMSS CAS - 237 - - 238 - Chengchun HAO
    VII BMO and Carleson Measures 7.1. The sharp maximal functions and BMO spaces .......... 237 7.2. John-Nirenberg theorem ....................... 243 7.3. Non-tangential maximal functions and Carleson measures ... 248 7.4. BMO functions and Carleson measures ............... 253 References ................................... 255 § 7.1 The sharp maximal functions and BMO spaces Functions of bounded mean oscillation were introduced by F. John and L. Niren- berg [JN61], in connection with differential equations. Definition 7.1.1. The mean oscillation of a locally integrable function f (i.e. a function be- 1 Rn Rn longing to Lloc( )) over a cube QZ in is defined as the following integral: ~ 1 fQ = jf(x) − Avg fjdx; jQj Q Q where Avg f is the average value of f on the cube Q, i.e. Q Z 1 Avg f = f(x)dx: Q jQj Q Definition 7.1.2. 1 Rn A BMO function is any function f belonging to Lloc( ) whose mean oscil- lation has a finite supremum over the set of all cubesa Q contained in Rn. For 2 1 Rn f Lloc( ), we define the maximal BMO function or the sharp maximal function Z 1 M #f(x) = sup jf(t) − Avg fjdt; Q3x jQj Q Q where the supremum is take over all cubes Q in Rn that contains the given point x, and M # is called the sharp maximal operator. Then we denote the # norm of f in this space by kfkBMO = kM fk1. The set Rn f 2 1 Rn k k 1g BMO( ) = f Lloc( ): f BMO < is called the function space of bounded mean oscillation or the BMO space.
    [Show full text]
  • Bounded Mean Oscillation Treated by John and Nirenberg [9], a Martingale Analogue Was Proved by Fefferman and Stein, A
    TRANSACTIONSOF THE AMERICANMATHEMATICAL SOCIETY Volume 193, 1974 BOUNDEDMEAN OSCILLATION AND REGULATEDMARTINGALES(l) BY CARL HERZ ABSTRACT. In the martingale context, the dual Banach space to H. is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for H.-martingales which involve the notion ofL -regulated ¿.-martingales where 1 < p < oo. The strongest decomposition theorem is for p = oo, and this provides full information about BMO. The weaker p = 2 decomposition is fundamental in the theory of martingale transforms. Introduction. Shortly after Charles Fefferman [A] proved that the dual Banach space to the Hardy space H. (in this context we view Hl as the space of func- tions on the circle which together with their conjugate function belong to Lj) was equivalent to the space BMO of functions of bounded mean oscillation treated by John and Nirenberg [9], a martingale analogue was proved by Fefferman and Stein, A. Garsia, and me; see [5] and [6]. In this Richard Gundy played a role which, at least for me, was of the highest value. There are related matters in [7] and [ll]. The martingale spaces BMO are defined for 1 < p < oo by the John-Nirenberg conditions JN plus a supplementary condition which is needed to handle the case of general martingales; see §2 below for the definitions. In the classical case, that of martingales on a dyadic stochastic base, John and Nirenberg proved that, while the defining conditions for membership in BMO appear to be more stringent with increasing p, they are all equivalent.
    [Show full text]
  • Carleson Measures and Elliptic Boundary Value Problems 3
    1 Carleson measures and elliptic boundary value 2 problems 3 4 Jill Pipher 5 Abstract. In this article, we highlight the role of Carleson measures in elliptic boundary value prob- 6 lems, and discuss some recent results in this theory. The focus here is on the Dirichlet problem, with 7 measurable data, for second order elliptic operators in divergence form. We illustrate, through selected 8 examples, the various ways Carleson measures arise in characterizing those classes of operators for 9 which Dirichlet problems are solvable with classical non-tangential maximal function estimates. 10 Mathematics Subject Classification (2010). Primary 42B99, 42B25, 35J25, 42B20. 11 Keywords. Carleson measures, elliptic divergence form equations, A1, boundary value problems. 12 1. Introduction 13 Measures of Carleson type were introduced by L. Carleson in [9] and [10] to solve a problem 14 in analytic interpolation, via a formulation that exploited the duality between Carleson mea- 15 sures and non-tangential maximal functions (defined below). Carleson measures have since 16 become one of the most important tools in harmonic analysis, playing a fundamental role in 17 the study of singular integral operators in particular, through their connection with BMO, 18 the John-Nirenberg space of functions of bounded mean oscillation. We aim to describe, 19 through some specific examples, the ubiquitous role of measures of this type in the theory 20 of boundary value problems, especially with regard to sharp regularity of “elliptic” measure, 21 the probability measure arising in the Dirichlet problem for second order divergence form 22 elliptic operators. Perhaps the first connection between Carleson measures and boundary 23 value problems was observed by C.
    [Show full text]
  • An Embedding Relation for Bounded Mean Oscillation on Rectangles
    AN EMBEDDING RELATION FOR BOUNDED MEAN OSCILLATION ON RECTANGLES. BENOˆIT F. SEHBA Abstract. In the two-parameter setting, we say a function belongs to the mean little BMO, if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by S. Pott and the author in relation with the multiplier algebra of the product BMO of Chang-Fefferman. We prove that the Cotlar-Sadosky space of functions of bounded mean oscillation bmo(TN ) is a strict subspace of the mean little BMO. 1. Introduction and results 1.1. Introduction. In the two-parameter case, the mean little BMO space consists of those functions such that their mean over any interval with re- spect to any of the two variables is uniformly in BMO(T). This space was introduced recently in the literature by S. Pott and the author in their way to the characterization of the multiplier algebra of the product BMO of Chang-Fefferman ([1, 5, 6]). Its definition is very close in spirit to the one of the little BMO of Cotlar and Sadosky ([2]) and this is somehow misleading. It is pretty clear that the little BMO embeds continuously into the mean little BMO and it was natural to ask if both spaces are the same. To find out, we use an indirect method; we characterize the multiplier algebra of the Cotlar-Sadosky space and the set of multipliers from the little BMO to the mean little BMO. arXiv:1703.00851v5 [math.CA] 15 Nov 2017 1.2.
    [Show full text]
  • Mean Oscillation
    Proyecciones Vol. 27, No 2, pp. 163—177, August 2008. Universidad Cat´olica del Norte Antofagasta - Chile FUNCTIONS OF BOUNDED (ϕ, p) MEAN OSCILLATION RENE´ ERLIN´ CASTILLO OHIO UNIVERSITY, U. S. A. JULIO CESAR´ RAMOS FERNANDEZ´ and EDUARD TROUSSELOT UNIVERSIDAD DE ORIENTE, VENEZUELA Received : November 2006. Accepted : July 2008 Abstract In this paper we extend a result of Garnett and Jones to the case of spaces of homogeneous type. 2000 Mathematics Subject Classification : Primary 32A37. Secondary 43A85. 164 Ren´eErl´ın Castillo, Julio C. Ramos and Eduard Trousselot 1. Introduction The space of functions of bounded mean oscillation, or BMO, naturally arises as the class of functions whose deviation from their means over cubes is bounded. L functions have this property, but there exist unbounded ∞ functions with bounded mean oscillation, for instance the function log x is | | in BMO but it is not bounded. The space BMO shares similar properties with the space L and it often serve as a substitute for it. The space of the ∞ functions with bounded mean oscillation BMO,iswellknownforitssev- eral applications in real analysis, harmonic analysis and partial differential equations. 1 The definition of BMO is that f BMO if supQ Q Q f(x) fQ dx = ∈ | | | − | 1 f BMO < where fQ = Q Q f(y)dy, Q is the LebesgueR measure of Q k k ∞ | | | | and Q is a cube in Rn, with sidesR parallel to the coordinate axes. In [1] Garnet and Jones gave comparable upper and lower bounds for the distance (1.1) dist (f,L )= inf f g BMO . ∞ g L k − k ∈ ∞ The bounds were expressed in terms of one constant in Jhon-Nirenberg inequality.
    [Show full text]
  • The Space of Functions of Bounded Mean Oscillation (BMO) Plays an Impor- Tant Role in Harmonic Analysis
    8BMO The space of functions of Bounded Mean Oscillation (BMO) plays an impor- tant role in Harmonic Analysis. d A function f,inL1(loc)inR is said to be a BMO function if Z 1 sup inf |f(y) − a|dy = kukBMO < ∞ (8.1) a | | x;r Bx;r y∈Bx;r where Bx;r is the ball of radius r centered at x,and|Bx;r| is its volume. Remark. The infimum over a can be replaced by the choice of Z 1 a =¯a = | | f(y)dy Bx;r y∈Bx;r giving us an equivalent definition. We note that for any a, Z | − |≤ 1 | − | a a¯ | | f(y) a dy Bx;r y∈Bx;r and therefore if a∗ is the optimal a, ∗ |a¯ − a |≤kfkBMO Remark. Any bounded function is in the class BMO and kfkBMO ≤kfk∞. Theorem 8.1 (John-Nirenberg).R Let f beaBMOfunctiononacubeQ | | k k ≤ of volume Q =1satisfying Q f(x)dx =0and f BMO 1. Then there are finite positive constants c1;c2, independent of f, such that, for any `>0 ` |{x : |f(x)|≥`}| ≤ c1 exp[− ] (8.2) c2 Proof. Let us define F (`)=sup|{x : |f(x)|≥`} f k k ≤ Rwhere the supremum isR taken over all functions with f BMO 1and k k1 ≤kk ≤ Q f(x)dx =0.Since Q f(x)dx = 0 implies that f f BMO 1, ≤ 1 d F (`) ` . Let us subdivide the cube into 2 subcubes with sides one half theR original cube. We pick a number a>1 and keep theR cubes Qi with 1 |f(x)|dx ≥ a.
    [Show full text]
  • BMO on Shapes and Sharp Constants
    BMO ON SHAPES AND SHARP CONSTANTS GALIA DAFNI AND RYAN GIBARA Abstract. We consider a very general definition of BMO on a domain in Rn, where the mean oscillation is taken with respect to a basis of shapes, i.e. a collection of open sets covering the domain. We examine the basic properties and various inequalities that can be proved for such functions, with special emphasis on sharp constants. For the standard bases of shapes consisting of balls or cubes (classic BMO), or rectangles (strong BMO), we review known results, such as the boundedness of rearrangements and its consequences. Finally, we prove a product decomposition for BMO when the shapes exhibit some product structure, as in the case of strong BMO. 1. Introduction First defined by John and Nirenberg in [37], the space BMO of functions of bounded mean oscillation has served as the replacement for L∞ in situations where considering bounded functions is too restrictive. BMO has proven to be important in areas such as harmonic analysis, partly due to the duality with the Hardy space established by Fefferman in [23], and partial differential equations, where its connection to elasticity motivated John to first consider the mean oscillation of functions in [36]. Additionally, one may regard BMO as a function space that is interesting to study in its own right. As such, there exist many complete references to the classical theory and its connection to various areas; for instance, see [27, 31, 39, 58]. 1 n The mean oscillation of a function f Lloc(R ) was initially defined over a cube Q with sides parallel to the axes as ∈ (1.1) f f , − | − Q| ZQ 1 where fQ = −Q f and −Q = Q Q.
    [Show full text]
  • HARMONIC MAPS Introduction 1 Harmonic Functions on Euclidean
    Article for the ‘Handbook of Global Analysis’, Elsevier (2007, to appear) HARMONIC MAPS Fred´ eric´ Helein´ and John C. Wood Introduction The subject of harmonic maps is vast and has found many applications, and it would require a very long book to cover all aspects, even superficially. Hence, we have made a choice; in particular, highlighting the key questions of existence, uniqueness and regularity of harmonic maps between given manifolds. Thus we shall survey some of the main methods of global analysis for answering these questions. We first consider relevant aspects of harmonic functions on Euclidean space; then we give a general introduc- tion to harmonic maps. The core of our work is in Chapters 3–6 where we present the analytical methods. We round of the article by describing how twistor theory and integrable systems can be used to construct many more harmonic maps. On the way, we mention harmonic morphisms: maps between Riemannian manifolds which pre- serve Laplace’s equation; these turn out to be a particular class of harmonic maps and exhibit some properties dual to those of harmonic maps. More information on harmonic maps can be found in the following articles and books; for generalities: [61, 62, 63, 219], analytical aspects: [21, 88, 103, 118, 131, 133, 135, 189, 204, 194], integrable systems methods: [73, 94, 117], applications to complex and Kahler¨ geometry: [63, 135], harmonic morphisms: [7], and other topics: [64, 231]. 1 Harmonic functions on Euclidean spaces Harmonic functions on an open domain Ω of Rm are solutions of the Laplace equation ∂2 ∂2 ∆f = 0, where ∆ := + ··· + (x1, ··· , xm) ∈ Ω.
    [Show full text]