Demonstratio Mathematica 2021; 54: 140–150

Research Article

Eddy Kwessi* On the equivalence between weak BMO and the space of derivatives of the Zygmund class

https://doi.org/10.1515/dema-2021-0013 received January 1, 2021; accepted April 30, 2021

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 H 1 strictly contains the special atom space.

Keywords: BMO, derivative, distributions, Zygmund class

MSC 2020: 42B05, 42B30, 30B50, 30E20

1 Introduction

The space of functions of bounded means oscillation has taken central stage in the mathematical literature after the work of Charles Fefferman [1], where he showed that it is the dual space of the real Hardy space 1, a long sought-after result. Right after, Ronald Coifman [2] showed this result using a different method. Essentially, he showed that 1 has an atomic decomposition. De Souza [3] showed there is a subset B1 of 1, formed by special atoms that is contained in 1. This space B1 has the particularity that it contains some functions whose Fourier series diverge. The question of whether B1 is equivalent to 1 was never truly answered explicitly, but it was always suspected that the inclusion B1 ⊂ 1 was strict, that is, there must be at least one in 1 that is not in B1. However, such a function had neither been constructed nor given. Since the dual space (1)∗ of 1 is BMO and B1 ⊂ 1, 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 in essence, that BMOw is the dual of B1 would also prove that B1 ⊂ 1 with a strict inclusion. Moreover, it was 1 ( ) already proved that (B )* ≅ Λ∗′, where Λ∗′ is the space of derivative in the sense of distributions of functions 1 ∗ w in the Zygmund class Λ∗, see, for example, [3] and [4]. Therefore, if(B )≅BMO , then by transition, we would w have ΛBMO∗′ ≅ . Henceforth, we will adopt the following notations:  ={zz ∈ :1 ∣∣< } is the open unit disk and let  ={zz ∈ :1 ∣∣= } is the unit sphere. For an integrable function f on a measurable set A, and the Lebesgue 1 measure λ on A, we will write ⨍A fξdλξ() ()≔∫ fξdλξ () (). We will start by recalling the necessary λA()A definitions and important results. The interested reader can see, for example, [5] for more information.

 * Corresponding author: Eddy Kwessi, Department of , Trinity University, San Antonio, TX, United States of America, e-mail: [email protected]

Open Access. © 2021 Eddy Kwessi, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. Equivalence between weak BMO and derivatives of Zygmund space  141

Definition 1. Let 0 <<∞p be a real number. The Hardy Space  pp≔() is the space of holomorphic functions f defined on  and satisfying

1  2π  p 1 iξ p ∥∥f  p ≔sup  ∫ ∣(fre )∣ dλξ () <∞. 01≤

1 n n n  Let f ∈()Lloc , Q be a hypercube in , and λ be the on for some n ∈ . Put

# fQ =⨍⨍fξdλξ () (), fQ = fξdλξ () (). Q Q

1 n n fi For f ∈(Lloc ) and x ∈ ,wede ne

# # M ()()=fxsup⨍ ∣()− fξ fdλξQ ∣ () , (1.1) Qx∋ Q

( ) M()()=fxsup⨍ [()− fξ fdλξQ ] () , 1.2 Qx∋ Q

mf()= xsup fQ , (1.3) Qx∋ 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 integr- able functions f for which the operator M# is bounded, that is,

nn1 #∞  n BMO()={∈fLloc () : Mf ()∈()} L .

We can endowed BMO(n) with the

# # ∥∥fMfMfxBMO()n =∥ ()∥≔∞ sup ()(). x∈n The space of functions of weak bounded mean oscillation is defined as the space of locally integrable functions f for which the operator M is bounded, that is,

wn1 n∞ n BMO()={∈fLloc () : MfL ∈ ()}.

Remark 1.1. It follows from the above definitions that BMO()⊆nwn BMO ().

Let us recall the definition of the space of functions of vanishing mean oscillation VMO(n) and introduce the space of functions of weak vanishing mean oscillations VMO wn( ).

Definition 3. 1 n Let f ∈(Lloc ).

n fR∈VMO ( ) if lim ∣()− fξfdλξQ ∣ () = 0. λQ()→0⨍ Q

wn ffξfdλξ∈VMO ( ) if lim [()−Q ] ()= 0. λQ()→0 ⨍ Q 142  Eddy Kwessi

Remark 1.2. It follows from the above definition that VMO(n) is a subspace of VMO wn( ) which is itself a subspace of BMOwn( ). We will show (see Theorem 2.8 below) that VMO w is in fact closed subspace of BMOw.

Henceforth, BMO()nwn , BMO (), and VMO wn( ) will simply be referred to as BMO, BMOw, and VMOw. Now, we consider A() as the space of analytic functions defined on the unit disk . 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 is defined as

 1 + reiθ  Pr()=θ Re  .  1 − reiθ 

Definition 4. The space of analytic functions of bounded mean oscillation is defined as

   iθ 1 iξ  BMOA()=FA ∈ ()∃∈ ; f BMO () : Fre ( )=∫Pθr (−)( ξfe ) dλξ ().  2π     We endow BMO() with the norm

   1 iξ iθ  ∥FF ∥BMOA() ≔ ∣ (0sup )∣ +∫Pθr ( − ξ )∣ fe ( ) − Fre ( )∣ dλξ ( ) < ∞. 01≤

We can now define the space BMOAw of analytic function of weak bounded mean oscillation.

Definition 5. An analytic function F on  is said to be of weak bounded mean oscillation if there exists f ∈(BMOw ) such that

iθ 1 iξ F()=re Pθr (−)()() ξfe dλξ. 2π ∫ 

We endow BMOAw() with the norm

   1 iξ iθ  ∥∥FFBMOAw() ≔∣()∣+0sup ∫Pθr ( − ξ )[( fe )− Fre ( )]() dλξ <∞. 01≤

Definition 6. n n For n ≥ 1, we consider the hypercube of given as Jahah= ∏j=1[−jjjj, +] where aj, hj are 1 real numbers with hj > 0. Let ϕLJ∈() with ϕJ()=∫ ϕξ ( ) dλξ ( ). J The special atom (of type 1) is a function b : IJ⊆→ such that bξ()=1on\ J I or 1 bξ()= [()−()]χξ χξ, ϕI() R L where 2n−1 • n−1 n n−1 RI=⋃ij for some ii12,,…∈{… , i2 1,2,,2} with ii12<<⋯< i2 and L = I\R. j=1 • {II12,,… , I 2n} is the collection of sub-cubes of I, cut by the hyperplanes x1122==…=ax,,, a xn an. • χA represents the characteristic function of set A. Equivalence between weak BMO and derivatives of Zygmund space  143

y

a 2 + h 2

L 2 R 2 L

1 a 2 I L 1 R 1 1 − I a 2 h 2 R

x

a − h a a + h (a) (b) 1 1 1 1 1

a3 + h3

z

+ a3 a2 h2

a1 − h1 a2 a3 − h3

a1 a2 − h2

y x a1 + h1 (c)

Figure 1: Illustration of the special atom, for n = 1 in (a), n = 2 in (b), and n = 3 in (c).

Definition 7. The special atom space is defined as

∞ ∞   1  B =→()=()∣∣<∞fJ:; fξ∑∑ αbξnn ; αn  ,    n=00n=  where the bn’s are special atoms of type 1. ∞ 1 1 fi B is endowed with the norm ∥∥fαB =inf ∑n=0 ∣n∣,wherethein mum is taken over all representations of f (Figure 1).

Now, we define the Zygmund class of functions.

Definition 8.  k n Let k ∈ . A function f is said to be in the Zygmund class Λ∗( ) of functions of order k if f ∈(Ckn−1  ) and ∣∂ααfx ( + h ) + ∂ fx ( − h ) −2 ∂ α fx ( )∣ ∥∥f kn = sup <∞. Λ∗() ∑ ∣∣=αkxh, ∣∣h

1 n In particular, for k = 1, we have ΛΛ∗ ≔(∗ ), and hence   0 n ∣(+)+(−)−()∣fx h fx h2 fx Λ:sup∗ =∈()∥∥≔fC fΛ∗ <∞.  xh,0> 2h 

One important note about the space Λ∗ is that it contains the so-called Weierstrass functions that are ff k known to be continuous everywhere but nowhere di erentiable. Therefore, the space Λ∗ is the space of k−1 - derivatives of functions in Λ∗ , where the derivative is taken in the sense of distributions. Another equiva k k−1 ff lent way to see Λ∗ is to consider functions of Λ∗ that are either di erentiable or limits of convolutions with the Poisson kernel, that is, f ()=ξfPξlim ( ∗r )() where Pr(θ) is the Poisson kernel. r→1 144  Eddy Kwessi

2 Main results

w Our first result is about the constant fQ in the definition of BMO . If fact, the constant fQ can be replaced with any non-negative constant. The same can be said about BMO as well.

Proposition 2.1. 1 Let f ∈ Lloc. (1) For any non-negative real number α, we have

( ) sup⨍⨍[()−fξ fQ ] dλξ () ≤ 2 sup [()−] fξ αdλξ (). 2.1 Qx∋∋Qx Q Q

(2) f ∈ BMOw if and only if for any x ∈ n and any cube Q ∋ x, there exists a non-negative number αQ∈ such that

sup ⨍[()−]fξ αdλξ () <∞. (2.2) Qx∋ Q

Proof. To prove assertion (1), fix x ∈ n and a cube Q ⊆ n containing x. Observe that for every non- negative α we have,

⨍⨍[()−fξ fQ ] dλξ () ≤2 [()−] fξ αdλξ (). Q Q

Indeed, for any non-negative real number α, we have

⨍⨍⨍[fξ ( )− fQ ] dλξ ( ) = [ fξ ( )− αdλξ ] ( )+ [ α − fQ ] dλξ ( ) Q QQ

≤⨍⨍ [()−fξ αdλξ ] () + [ α − fQ ] dλξ () QQ

≤⨍⨍⨍ [()−f ξ αdλξ ] () + αdλξ () − f () ξ dλξ () QQQ

≤⨍⨍⨍ [()−f ξ αdλξ ] () + [ α − f ()]() ξ dλξ ≤2. [()− f ξ αdλξ ] () QQQ

To conclude, we take the supremum over of all cubes Q ∋ x.

Assertion (2) follows immediately from (1). □

We can define two equivalent norms on BMOw and prove that endowed with these norms, BMOw is in fact a Banach space.

Proposition 2.2. Consider the following: for every f ∈ BMOw, we put

w ∗ ∥∥fmfMfandfmffξfdλξBMO =∥ ∥∞∞ +∥ ∥ ∥∥BMOw =∥ ∥∞ +2supinf ⨍ [()−Q ] (). Qx∋ α>0 Q Equivalence between weak BMO and derivatives of Zygmund space  145

Then,

w ∗ w ∥∥ffBMO ≤∥∥BMOw ≤2 ∥∥ fBMO .

Proof. The proof is an immediate consequence of Proposition 2.1. We note that the proof can also be obtained from the closed-graph theorem, but that will require to first prove that endowed with the two norms, BMOw is a Banach space. □

w Theorem 2.3. The space ()BMO , ∥⋅∥BMOw is complete.

Proof.

(1) In the proof that ∥⋅∥BMOw is a norm, homogeneity and the triangle inequality are easy to prove. As for positivity, we note that ∥∥ffBMOw =0sup ⇔Q =0 and f ()=ξfQ on all cubes Q ∋ x. It follows immedi- Qx∋ ately that f = 0. w (2) Let {fnn} ∈ be a Cauchy sequence in BMO . Let ϵ > 0 and N ∈  such that ∀nm, ∈ N, we have ∥−ffnm ∥BMOw <ϵ. That is,

  sup ff ffξ ff dλξ ϵ. (2.3) (−)+[(−)()−(−)]()

In particular, from (2.3), we have that sup(−)

∣fnQ,,− f mQ ∣=⨍⨍ fξdλξn () () − fξdλξm () () ≤ ⨍ [ ffξdλξffnm − ]() () =( nmQ − ) ≤sup ( ffnmQ − ) < ϵ. Qx∋ Q Q Q

Hence, for fixed Q, {}fnQ, is Cauchy sequence in . Let fQ = lim fnQ, . We note from the above inequalities that n→∞

( ffnmQ−)=[−]()()≥∣−⨍ ffξdλξfn m nQ,, f mQ ∣≥ f nQ ,, − f mQ. Q

Therefore, given a cube Q ⊂ n containing x, we have

∣Mfxnm( )− Mfx ( )∣≤sup ⨍ [( fξnm ( )− fξ ( ))−( f nQmQ,, − f )] dλξ ( ) Qx∋ Q

≤sup⨍ [(fξnm ( )− f ( ξ ))−( f nmQ − f ) ] dλξ ( ) + sup ∣( fnmQnQmQ − f ) −( f,, − f )∣ Qx∋∋Qx Q

≤∥ffnm − ∥BMOw +sup ∣( fnQ , − f mQ , )∣< 2ϵ. Qx∋

∞ It follows that {Mfnn} ∈ is a Cauchy sequence in Lloc. Let hMf= lim n. ∞ 1 1 n→∞ Since Lloc⊂ L loc, we have hL∈ loc.

hx( )=lim Mfxn ( )= lim sup⨍⨍ [ fξfdλξnnQ ( )−, ] ( ) = sup [ lim fξfdλξnQ ( )− ] ( ) . n→∞n →∞ Qx∋∋Qx n→∞ Q Q

Since hx( ) is finite on any cube Q ∋ x, that follows that f ()=ξfξlim n () is finite a.e. on Q. Thus, hMf= , w n→∞ for some f ∈ BMO and lim∥−∥ffn BMOw →0. □ n→∞ 146  Eddy Kwessi

Theorem 2.4. (Hölder’s type inequality) Let g ∈ BMOw and a hyper-cube J ⊂ n. Consider the following

1  1 ∗ w operator TBg : → given by Tfg()=∫ fξgξdλξ ()() (). Then, TBg ∈( ) with ∥∥Tgg ()B1 ∗ ≤∥∥BMO . J w 1 ∗ Moreover, the operator HB:BMO →( ) defined as Hg()= Tg is onto.

Proof. By linearity of the , Tg is a linear. To start, we study the action of this operator on special atoms. Indeed, let ξI∈⊆J and suppose f (ξbξ )= ( )=1 [ χξχξ ( )− ( )], where RL, are sub-cubes of I ϕI() RL such that IRL=∪and RL∩=∅. Therefore, we have:

Tg()= b∫∫∫ bξ ()() gξ dλξ ()= bξ ()() gξ dλξ ()= bξ ()[()− gξ gI ] dλξ (),since ∫ bξ () gdλξI ()= 0. JII I

Taking the supremum over all I ∋ x and using the fact that b()≤ξ 1 , we have ϕI()

w ( ) ∣Tbg()∣≤sup inf ⨍ [()− gξ αdλξ ] () ≤∥∥ gBMO . 2.4 Ix∋ α>0 IJ⊆ I

∞ ∞ 1 Now suppose f ()=∑ξcbξnn () with ∑ ∣∣<∞cn is an element of B , where the bn’s are special atoms n=1 n=1 ∞ defined on sub-cubes In of J with IRLnn=∪n and RLnn∩=∅. Let II=⋃n. We have for α > 0 n=1  ∞   ∞  Tg( f )=∫∫∑∑ cbnn ( ξ ) gξdλξ ( ) ( )= cbnn ( ξ ) [ gξ ( )− αdλξ ] ( ) n 11n J  =  I  =  ∞ ∞ =∑∑cbξgξαdλξn∫∫n ()[()− ] ()= cbξgξαdλξn n ()[()− ] (). n=11n= I In

It follows from (2.4) that

 ∞  ∣Tfg()∣≤∑ ∣ cn ∣⋅∥∥ gBMOw . n=1 

Talking the infimum over all representations of f yields as follows:

∣Tfg()∣≤∥∥⋅∥∥ fB1 gBMOw .

1 Therefore, Tg is a bounded linear operator on B with

w ∥∥TTfgg ()B1 ∗ =sup ∣()∣≤∥∥g BMO . (2.5) ∥∥f B1 =1 Now suppose thatT is a bounded linear functional on B1, that is, TB∈(1 )∗. We want to show that there exists w 1 ∗ a function g ∈ BMO such that Tf()= Tg ()= f∫ fξgξdλξ ()() (). That TB∈( ) implies the existence of an J absolute constant C such that 1 ( ) ∣Tf()∣≤ Cf ∥∥B1 , ∀∈ f B. 2.6 Recall that a function G : J →  is said to be absolutely continuous on J if for every positive number ϵ, there fi - exists a positive number δ, such that for a nite sequence of pairwise disjoint sub cubes Ixynn=( , n) of J, ∞ ∞ ∑∑(−)<⇒yxn n δ ∣()−()∣< GyGxn n ϵ. (2.7) n=11n= - Suppose such a sequence of sub cubes exists. Now, consider G()=xTχ ([−xhxh, +)) for some real number h > 0. Then, given an cube In and a real number hn, we have by linearity of T that

  G yGxTχ χ  . ()−()=n n  [−yhyhn,, +)nnnnn − [− xhxh +)  n n  Equivalence between weak BMO and derivatives of Zygmund space  147

fi Now, if we de ne Lnnnnn=[xhxh −, + ) and Ryhyhn =[n −n, n + n), we see that RLnn∩=∅and their union forms a single cube Jn if we choose hyxn =(n −n )/2. Moreover, in that case, the length of the cube Jn is yn +−+=(−hxhnnn2 yxn n). Therefore, by linearity  1    G()−()=(yGxyxTn 2 −n ) χχ− . n n   [−yhyhn n,,n +)nnnnn [− xhxh +)   2(−)yxn n   

1   1 We observe that bn()=ξχξχξ ()− ()is a special atom with norm in B equal to 1. 2(−)yx [−yhyhn n,,n +)nnnnn [− xhxh +)  n n   It follows that using (2.6)

∣Gy()−()∣≤(−)∣()∣≤(−)∥∥=(−)n Gxn 22 yn xnn Tb Cyn xnn bB1 2 Cyn xn . Hence,

∞ ∞ ∑∑∣(Gyn )−( Gxn )∣≤2 C ( yn − xn ). n=11n= We conclude by noting that given ϵ 0, (2.7) is satisfied if we choose δ ϵ . We conclude that the function > = 2C G is absolutely continuous on J. Therefore, G is differentiable almost everywhere, that is, there exists x gL∈ 1 such that G()=xgξdλξ∫ () () for all cubes IabJ=[, ]⊆ . Let I ∋ x be a sub-cube of J. Therefore, a sup 1 ∫ gξ() dλξ ()<∞because an absolutely continuous function is a function with bounded variation. λI() I Ix∋ 1 1 Since gL∈ , we have that ggξdλξI =()()<∫ ∞. It follows that λI() I

1 1 Mgx()=sup ∫∫(()−gξ gI ) dλξ () ≤sup gξ() dλξ ()+sup gI <∞. Ix∋∋∋λI() IxλI() Ix I I w w 1 ∗ This proves that g ∈ BMO . That is, HB:BMO →( ) is onto with Hg()= T = Tg. Identifying T with the g, it follows from (2.5) that

w ∥gTT ∥()BB11∗∗ =∥ ∥ () =∥g ∥ ()B 1 ∗ ≤∥ g ∥BMO . (2.8) □

Remark 2.5. We observe that the above result can be obtained differently. In fact, we recall that it was [ ] 1 2 proved in 3 that the dual space of B is equivalent to Λ∗. Let x ∈>=[−+]⊆=Jh,0, I x hx , h JL , [xhx− , ), and Rxxh=[, + ]. Let b ξχξχξ1 . We have ()=2h [RL ()− ()]

∥∥ff1 ∗ ≅∥∥2 =sup bξfξdλξbξfξfdλξf ()∂() ()= sup ()(∂()−∂) ()≤∥∥ w . ()B Λ∗ ∫∫I BMO Ix∋∋Ix hh>>00I I

w 2 1 ∗ This shows that BMO⊆≅() Λ∗ B .

Theorem 2.6. 1 ∗ 1 w w 1 The dual space (B ) of B is BMO with ∥∥ggBMO ≅∥∥()B ∗.

1  1  [ ] To prove Theorem 2.6,werecallthatB ( ) has an analytic extension BA( ) to the unit disk. In fact, in 8 , 1  1  it was shown that BA( ) consists of functions F that are Poisson integrals of functions in B ( ),thatis, 1 π 1 + ez−iξ 1 π iξ 1  1 F()=zfedλξ∫ iξ ( ) () where f ∈(B ). Moreover, the norm ∥∥FFzdzB () =∫∫ ∣′()∣ is equivalent to 2π −π 1 − ez− A 0 −π 1 1  1  the norm ∥∥f B ( ).ThisallowstoidentifyBA( ) with B ( ) and thus the following:

Proposition 2.7. 1  fi 1  BA( ) can be identi ed with a closed subspace of L ( ). 148  Eddy Kwessi

1 1 Proof. Let {fnn} ∈ be a sequence in B () that converges to f in L (). We need to show that the Poisson π iξ 1  1  1 1 + ez− iξ 1  integral of f is in BA( ). Since fn ∈()∀∈Bn, , then Fn()=zfedλξ∫ n ( ) () belongs to BA( ). 2π −π 1 − ez−iξ 4 Let F z be the Poisson integral of f . We note that if zeiθ, then ξθ2 (−)ξθ 1 ez−iξ 22 ξθ. ( ) = ( −)−2 ≤∣− ∣≤(−) 1 π 2e−iξ 1 π 2 iξ iξ 1 Therefore, we have that F′(zfedλξ ) =∫ iξ 2 ( ) ( ) and ∣Fz′( )∣ ≤ ∫ iξ 2 ∣(fe )∣()≤ dλξ C ∥∥ fL ( ). 2π −π (−1 ez− ) 2π −π ∣−1 ez− ∣ It follows that,

1 π

∥−∥=FF11 ∣ FzFzdzCff′()− ′()∣ ≤ ∥ − ∥  . n BA ∫∫ n n L () 0 −π 1  1 1  1  Since fn → f in L ( ), it follows that F ∈ BA and Fn → F in BA( ). The result follows by identifying BA( ) and B1( ). □

We note that there is an extension of this result to the polydisk  n and polysphere  n, see [9].

Proof of Theorem 2.6. ffi 1 It is su cient to show that there exists a constant M > 0 such that ∥∥T (())B  ∗ ≥ n M ∥∥g BMOw( ). The extension to  is natural, using the results in [7,9]. With Proposition 2.7, the proof follows along the lines of the proof of Proposition 7.3 in [6]. Let TB∈(1 ( ))∗. Since B1() is a closed subspace of L1() by Proposition 2.7, then by Hahn-Banach

1  ∗ 11 1  ∗ Theorem,T can be extended to a bounded linear operatorTL′∈( ( )) with∥∥TT(BL ( ))∗∗ =∥′∥ ( ( )) .Since(L ())= ∞ ∞ 1 iξ iξ L (), then there exists gL∈() such that ∥gTT ∥∞  =∥ ′∥11∗∗ =∥ ∥ and Tf()= g ( e )( fe ) dλξ (), 0 L () ()LB () 2π∫ 0 1  1  inξ for all f ∈(BA ). Note that here, we identify f with its correspondent in BA( ). Now, let ∑n∈ Aen be ∞ 2 2 the Fourier series of g0. Since gL0 ∈()⊂( L), we have that ∑n∈∣∣

π π 1 ge()iξ 1 ge()iξ gz()= 00de()=iξ dξ. 2πi ∫∫eziξ − 2π 1 − ez−iξ −−π π π 1 −inξ n 1 −inξ iξ Since iξ =∑  ezand Aegedλξn =()(∫ 0 ), we have 1 − ez n∈ 2π −π π 1 ge()iξ gz()= 0 dξ= A zn. ∫ −iξ ∑ n 2π 1 − ez n∈ −π π iξ 2   iθ 1 ge0() 1 1 This implies that gH∈(). For θ ∈ , ge()=∫ dλξ (). Moreover, given f ∈()⊆(BHA ), and 2π −π 1 − ee−iξ iθ using the Cauchy integral formula, we have:

π π  π  π 1 1 1 fe()iξ 1 ge()()=iθ fe iθ dθ ge ()iθ  dλ() ξ dθ =ge (iθ )( fedθTf iθ ) = (). 2π ∫∫∫∫2π 002π 1 − ee−iξ iθ  2π −−−π π  π  −π We also observe that

π π 1 1 1 ez−iξ  1 1 ez−iξ + iξ + iξ ( ) gz()= ∫∫ +()=1ge0 dξ Ge0()() dλξ, 2.9 2π 2 1 − ez−iξ  2π 1 − ez−iξ −π −π π iξ1 iξ 1 iξ1 iξ where G0()=egegedλξgeA0 ()+∫ 0 ()()=(()+0 0). 2 ()2π −π 2

Put G1020=()=(ReGG , Im G), and π π −iξ −iξ 1 1 + ez iξ 1 1 + ez iξ U()=z Ge12()() dλξ; Vz ()= Ge()() dλξ. 2π ∫∫1 − ez−iξ 2π 1 − ez−iξ −π −π Equivalence between weak BMO and derivatives of Zygmund space  149

Then gUi=+V. Moreover, U and V are analytic in  since they represent the Poisson integral of G12, G ∈ ∞ 1 w L ()⊆B (). Observe that BMO⊆ BMO . Moreover, Theorem 3.2 in [6] shows that ∥∥ggBMO ≅∥∥ BMOA. It therefore follows that there exists a constant C > 0 such that

w Cg∥∥BMO ≤ Cg ∥∥ BMO ≤∥∥ g BMOA ≤∥′∥ T()LB11∗∗ =∥∥ T (). □

2.1 Discussion

We note that BMOw Λ′ with fCfw, where gfwith gfin the sense of distributions. ⊆ ∗ ∥∥ΛBMO∗′ ≤ ∥∥ ∥∥ΛΛ∗′ =∥∥∗ ′= 1 ∗ 1 ∗ w w Since (B )≅Λ∗′, and from Theorem 2.6 above (B )≅BMO , it follows that BMO≅ Λ∗′. The consequence is that there exists c 0 such that c ffw , that is, those two norms are equivalent. We finish by > ∥∥BMO ≤∥∥ Λ∗′ w noting Λ∗′ has an analytic characterization, so we would expect the analytic characterization of BMO to also be equivalent to that of Λ∗′. Another takeaway from Theorem 2.6 is that it provides another proof that BMO is strictly contained in BMOw otherwise we would have had B1 ≅ 1, which is not true. In other words, there exists f ∈ 1 such that f ∉ B1. Our next result is about the closeness of VMO w in BMOw.

Theorem 2.8. VMO w is a closed subspace of BMOw.

w w w Proof. Let {fnn} ∈ be a sequence in VMO that converges to f ∈ BMO . Let us prove that f ∈ VMO . That w fn → f as n →∞in BMO is equivalent to lim∥−∥ffn BMOw =0. The latter is also equivalent, by definition of n→∞ the norm in BMOw to

lim sup(−)=ffnQ 0 and lim Mffx (−)()=n 0. n→∞ Qx∋ n→∞

w Since f ∈ BMO , then sup ⨍[()−fξ fQ ] dλξ ()<∞. Therefore, Qx∋ Q

⨍⨍[fξ ( )− fQ ] dλξ ( ) = [ fξ ( )− fnnnQnQQ ( ξ )+ f ( ξ )− f,, + f − f ] dλξ ( ) Q Q

≤⨍⨍⨍ [()−f ξ fn ()] ξ dλ () ξ + [ fnnQ ()− ξ f,, ] dλ () ξ + [ fnQ − f Q ] dλ () ξ Q Q Q

≤⨍⨍ [()−f ξ fn ()] ξ dλ () ξ + [ fnnQnQQ ()− ξ f,, ] dλ () ξ +∣ f − f ∣ Q Q

≤ [fξ ( )− f ( ξ )] dλξ ( ) +∥ f − f ∥ww ≤∥ f − f ∥1 +∥ f − f ∥ . ⨍ nnnBMOLloc n BMO Q

w w 1 Since {fnn} ∈ is a sequence of function is BMO , we have that lim∥−∥ffn BMOw =0. Since BMO ⊂ Lloc,we n→∞ w w have that lim∥−∥ffn L 1 =0. Hence, VMO is a closed subspace of BMO . □ n→∞ loc

Remark 2.9. We note that it was proved in [6] that VMO is a closed subspace of BMO. A stronger result even showed that if we restrict ourselves to  , then the space of complex-valued and continuous functions C()⊆VMO () and the BMO()-closure of C() is precisely VMO(). It turns out this result is also true for VMOw. 150  Eddy Kwessi

Theorem 2.10. The BMOw-closure of C() is precisely VMOw(), that is

w C()BMO =VMOw ().

Proof. We observe thatC()⊆VMO ()⊆ VMOww ()⊆  BMO ( ).Therefore,sinceVMOw() is closed inBMOw(), we have that

ww C()BMO() ⊆VMOww () BMO () = VMO ().

w For the other direction, let f ∈(VMO ), and consider the sequence {Rfϵϵ()}>0 of rotations of f by angle ϵ, iθ i(−) θ ϵ defined on  as Rfeϵ()()=( fe ); θ ∈ . Then, from Theorem 2.1 in [6], we have that for all ϵ0,>()∈Rfϵ w C() and lim∥()−∥RfϵBMO f () =0.SinceBMO()⊆ BMO (), we also have that lim∥()−∥RfϵBMO f w() =0. ϵ0→ ϵ0→ w  That is, f ∈()C  BMO ( ). □

Acknowledgments: The author would like to extend his thanks to the two anonymous referees for their comments and suggestions, which helped greatly improve the manuscript. In particular, the author wishes to thank Professor Tepper Gill (Howard University) and Corine Michelle Kwessi Nyandjou for their insightful comments and discussions, and Professor Geraldo de Souza (Auburn University) for his suggestions.

Funding information: This material is based upon the work supported by the National Science Foundation under Grant No. DMS-1440140, the National Security Agency under Grant No. H98230-20-1-0015, and the Sloan Grant under Grant No. G-2020-12602, while the author participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the summer of 2020.

Conflict of interest: The author states no conflict of interest.

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