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 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 [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 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 = {z ∈ C : |z| < 1} is the open unit disk and let T = {z ∈ C : |z| = 1} is the unit sphere. For an integrable function f on a measurable set A, and Z 1 Z the 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 , Trinity University, San Antonio, TX, USA, [email protected]

1 p p Definition 1. Let 0 < p < ∞ 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) < ∞ . H 0≤r<1 2π 0

1 n n n Let f ∈ Lloc(R ),Q be a hypercube in R , and λ be the Lebesgue measure on R for some n ∈ N. Put Z Z # fQ = − f(ξ)dλ(ξ), fQ = − f(ξ)dλ(ξ) . Q Q 1 n n For f ∈ Lloc(R ) and x ∈ 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 # ∞ n o BMO(R ) = f ∈ Lloc(R ): M (f) ∈ L (R ) .

n We can endowed BMO(R ) with the

# # kfkBMO( n) = M (f) = M (f)(x) . R ∞ 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 ∞ n o BMO (R ) = f ∈ Lloc(R ): Mf ∈ 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 ∈ Lloc(R ). Z n f ∈ VMO(R ) if lim − |f(ξ) − fQ| dλ(ξ) = 0 . λ(Q)→0 Q Z w n f ∈ 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 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 ∈ A(D); ∃f ∈ 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 := |F (0)| + sup P (θ − ξ) f(eiξ) − F (reiθ) dλ(ξ) < ∞ . BMOA(D) r 0≤r<1 2π T θ∈T

In other words, BMOA(D) is the space of Poisson 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 ∈ 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 := |F (0)| + sup P (θ − ξ) f(e ) − F (re ) dλ(ξ) < ∞ . BMOA (D) r 0≤r<1 2π T θ∈T

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 ∈ 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 J\I or 1 b(ξ) = [χ (ξ) − χ (ξ)] , φ(I) R L

where

2n−1 [ n • R = Iij for some i1, i2, ··· , i2n−1 ∈ {1, 2, ··· , 2 } with i1 < i2 < ··· < i2n−1 and j=1 L = I\R .

3 • {I1,I2, ··· ,I2n } 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 X X B = f : J → R; f(ξ) = αnbn(ξ); |αn| < ∞ , n=0 n=0 where the bn’s are special atoms of type 1. ∞ 1 X B is endowed with the norm kfkB1 = inf |αn| , 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 ∈ N. A function f is said to be in the Zygmund class Λ∗(R ) of functions of order k−1 n k if f ∈ C (R ) and

X |∂αf(x + h) + ∂αf(x − h) − 2∂αf(x)| kfk k n = sup < ∞ . Λ∗(R ) x,h |h| |α|=k

1 n In particular, for k = 1, we have Λ∗ := Λ∗(R ), and hence ( ) |f(x + h) + f(x − h) − 2f(x)| Λ = f ∈ C0( n): kfk := sup < ∞ . ∗ 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. Therefore, the space Λ∗ is the space k−1 of derivatives of functions in Λ∗ , where the derivative is taken in the sense of distributions. Another k k−1 equivalent way to see Λ∗ is to consider functions of Λ∗ that are either differentiable or limits of convolutions with the Poisson kernel, that is, f(ξ) = lim(f ∗ Pr)(ξ) where Pr(θ) is the Poisson kernel. r→1

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.

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

sup − [f(ξ) − fQ]dλ(ξ) ≤ 2 sup − [f(ξ) − α]dλ(ξ) . (2.1) Q3x Q Q3x Q

w n (2) f ∈ BMO if and only if for any x ∈ R and any cube Q 3 x, there exists is a non-negative number α ∈ Q such that Z

sup − [f(ξ) − α]dλ(ξ) < ∞. (2.2) Q3x Q

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

− [f(ξ) − fQ]dλ(ξ) ≤ 2 − [f(ξ) − α]dλ(ξ) . Q Q Indeed, for any non-negative real number α, we have Z Z Z

− [f(ξ) − fQ]dλ(ξ) = − [f(ξ) − α]dλ(ξ) + − [α − fQ]dλ(ξ) Q Q Q Z Z

≤ − [f(ξ) − α]dλ(ξ) + − [α − fQ]dλ(ξ) Q Q Z Z Z

≤ − [f(ξ) − α]dλ(ξ) + − αdλ(ξ) − − f(ξ)dλ(ξ) Q Q Q

5 Z Z

≤ − [f(ξ) − α]dλ(ξ) + − [α − f(ξ)]dλ(ξ) Q Q Z

≤ 2 − [f(ξ) − α]dλ(ξ) . Q

To conclude, we taking the supremum over of all cubes Q 3 x.

(2) 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 a every f ∈ BMOw, we put Z ∗ kfkBMOw = kmfk∞ +kMfk∞ and kfkBMOw = kmfk∞ + 2 sup inf − [f(ξ) − fQ]dλ(ξ) . Q3x α>0 Q

Then ∗ kfkBMOw ≤ kfkBMOw ≤ 2 kfkBMOw . 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 , k · kBMOw ) is complete.

Proof. (1). In the proof that k · kBMOw is a norm, homogeneity and the triangle inequality are easy to prove. As for positivity, we note that kfkBMOw = 0 ⇐⇒ sup fQ = 0 and f(ξ) = fQ on all cubes Q3x Q 3 x. It follows immediately that f = 0 .

(2). Let {f } be a Cauchy sequence in BMOw. Let  > 0 and N ∈ such that ∀n, m ∈ N, we n n∈N N have kfn − fmkBMOw < . That is,  Z 

sup (fn − fm)Q + − [(fn − fm)(ξ) − (fn − fm)Q]dλ(ξ) <  . (2.3) Q3x Q

In particular, from (2.3), we have that sup (fn − fm)Q < , therefore Q3x Z Z

|fn,Q − fm,Q| = − fn(ξ)dλ(ξ) − − fm(ξ)dλ(ξ) Q Q Z

≤ − [fn − fm](ξ)dλ(ξ) = (fn − fm)Q Q ≤ sup (fn − fm)Q <  . Q3x

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

(fn − fm)Q = − [fn − fm](ξ)dλ(ξ) ≥ |fn,Q − fm,Q| ≥ fn,Q − fm,Q . Q

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

|Mfn(x) − Mfm(x)| ≤ sup − [(fn(ξ) − fm(ξ)) − (fn,Q − fm,Q)]dλ(ξ) Q3x Q Z

≤ sup − [(fn(ξ) − fm(ξ)) − (fn − fm)Q]dλ(ξ) + sup |(fn − fm)Q − (fn,Q − fm,Q)| Q3x Q Q3x

≤ kfn − fmkBMOw + sup |(fn,Q − fm,Q)| < 2 . Q3x

∞ It follows that {Mfn} is a Cauchy-sequence in L . Let h = lim Mfn. n∈N loc n→∞ ∞ 1 1 Since Lloc ⊂ Lloc, we have h ∈ Lloc .

h(x) = lim Mfn(x) n→∞ Z

= lim sup − [fn(ξ) − fn,Q]dλ(ξ) n→∞ Q3x Q Z

= sup − [ lim fn(ξ) − fQ]dλ(ξ) . Q3x Q n→∞

Since h(x) is finite on any cube Q 3 x, that follows that f(ξ) = lim fn(ξ) is finite a.e. on Q. Thus n→∞ w h = Mf, for some f ∈ BMO and lim kfn − fk w → 0. n→∞ BMO Theorem 2.4 (Holder’s¨ type inequality). Let g ∈ BMOw and a hyper-cube J ⊂ n. Consider the Z R 1 1 ∗ following operator Tg : B → R given by Tg(f) = f(ξ)g(ξ)dλ(ξ). Then, Tg ∈ (B ) with J kTgk(B1)∗ ≤ kgkBMOw . w 1 ∗ Moreover, the operator H : BMO → (B ) defined as H(g) = Tg is onto.

Proof. By linearity of the , Tg is a linear. To start, we study the action of this operator on special 1 atoms. Indeed, let ξ ∈ I ⊆ J and suppose f(ξ) = b(ξ) = φ(I) [χR (ξ) − χL (ξ)], where R,L are sub-cubes of I such that I = R ∪ L and R ∩ L = ∅. Therefore we have: Z Z Tg(b) = b(ξ)g(ξ)dλ(ξ) = b(ξ)g(ξ)dλ(ξ) J I Z Z = b(ξ)[g(ξ) − gI ]dλ(ξ), since b(ξ)gI dλ(ξ) = 0 . I I

1 Taking the supremum over all I 3 x and using the fact that b(ξ) ≤ φ(I) , we have Z

|Tg(b)| ≤ sup inf −[g(ξ) − α]dλ(ξ) ≤ kgkBMOw . (2.4) I3x α>0 I I⊆J

∞ ∞ X X 1 Now suppose f(ξ) = cnbn(ξ) with |cn| < ∞ is an element of B , where the bn’s are special n=1 n=1 ∞ [ atoms defined on sub-cubes In of J with In = Rn ∪ Ln and Rn ∩ Ln = ∅. Let I = In. We have n=1 for α > 0 ∞ ! Z X Tg(f) = cnbn(ξ) g(ξ)dλ(ξ) J n=1

7 ∞ ! Z X = cnbn(ξ) [g(ξ) − α] dλ(ξ) I n=1 ∞ X Z = cn bn(ξ)[g(ξ) − α] dλ(ξ) n=1 I ∞ X Z = cn bn(ξ)[g(ξ) − α] dλ(ξ) . n=1 In It follows from (2.4) that ∞ ! X |Tg(f)| ≤ |cn| · kgkBMOw . n=1 Talking the infimum over all representations of f yields:

|Tg(f)| ≤ kfkB1 · kgkBMOw .

1 Therefore Tg is a bounded linear operator on B with

kTgk(B1)∗ = sup |Tg(f)| ≤ kgkBMOw . (2.5) kfkB1 =1

Now suppose that T is a bounded linear functional on B1, that is, T ∈ (B1)∗. We want to show that Z w 1 ∗ there exists a function g ∈ BMO such that T (f) = Tg(f) = f(ξ)g(ξ)dλ(ξ). That T ∈ (B ) J implies the existence of an absolute constant C such that

1 |T (f)| ≤ C kfkB1 , ∀f ∈ B . (2.6)

Recall that a function G : J → R is said to be absolutely continuous on J if for every positive number , there exists a positive number δ, such that for a finite sequence of pairwise disjoint sub-cubes In = (xn, yn) of J, ∞ ∞ X X (yn − xn) < δ =⇒ |G(yn) − G(xn)| <  . (2.7) n=1 n=1   Suppose such a sequence of sub-cubes exists. Now consider G(x) = T χ[x−h,x+h) for some real number h > 0. Then, given an cube In and a real number hn, we have by linearity of T that h i G(y ) − G(x ) = T χ − χ . n n [yn−hn,yn+hn) [xn−hn,xn+hn)

Now if we define Ln = [xn − hn, xn + hn) and Rn = [yn − hn, yn + hn), we see that Rn ∩ Ln = ∅ their union forms a single cube Jn if we choose hn = (yn − xn)/2. Moreover, in that case, the length of the cube Jn is yn + hn − xn + hn = 2(yn − xn). Therefore by linearity

 1 h i G(y ) − G(x ) = 2(y − x )T χ − χ . n n n n [yn−hn,yn+hn) [xn−hn,xn+hn) 2(yn − xn) 1 h i We observe that b (ξ) = χ (ξ) − χ (ξ) is a special atom with n [yn−hn,yn+hn) [xn−hn,xn+hn) 2(yn − xn) norm in B1 equal to 1. It follows that using (2.6)

|G(yn) − G(xn)| ≤ 2(yn − xn) |T (bn)| ≤ 2C(yn − xn) kbnkB1 = 2C(yn − xn) .

8 Hence, ∞ ∞ X X |G(yn) − G(xn)| ≤ 2C (yn − xn) . n=1 n=1  We conclude by noting that given  > 0,(2.7) is satisfied if we choose δ = . We conclude that 2C the function G is absolutely continuous on J. Therefore, G is differentiable almost everywhere, that Z x is, there exists g ∈ L1 such that G(x) = g(ξ)dλ(ξ) for all cubes I = [a, b] ⊆ J. Let I 3 x be a Z 1 a sub-cube of J. Therefore, sup g(ξ)dλ(ξ) < ∞ because an absolutely continuous function I3x λ(I) I Z 1 1 is a function with bounded variation. Since g ∈ L , we have that gI = g(ξ)dλ(ξ) < ∞. It λ(I) I follows that

Z Z 1 1 Mg(x) = sup (g(ξ) − gI )dλ(ξ) ≤ sup g(ξ)dλ(ξ) + sup gI < ∞ . I3x λ(I) I I3x λ(I) I I3x

w w 1 ∗ This proves that g ∈ BMO . That is, H : BMO → (B ) is onto with H(g) = T = Tg. Identifying T with the g, it follows from (2.5) that

kgk(B1)∗ = kT k(B1)∗ = kTgk(B1)∗ ≤ kgkBMOw . (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 ∈ J, h > 0,I = [x − h, x + h] ⊆ 1 J, L = [x − h, x), and R = [x, x + h]. Let b(ξ) = [χ (ξ) − χ (ξ)]. We have 2h R L Z ∼ kfk 1 ∗ = kfk 2 = sup b(ξ)∂f(ξ)dλ(ξ) (B ) Λ∗ I3x I h>0 Z

= sup b(ξ)(∂f(ξ) − ∂fI )dλ(ξ) I3x I h>0

≤ kfkBMOw . w 2 ∼ 1 ∗ This shows that BMO ⊆ Λ∗ = (B ) . 1 ∗ 1 w ∼ Theorem 2.6. The dual space (B ) of B is BMO with kgkBMOw = kgk(B1)∗ .

1 1 To prove Theorem 2.6, we recall that B (T) has an analytic extension BA(D) to the unit disk. In 1 fact, in [1], it was shown that BA(D) consists of functions F that are Poisson integrals of functions Z π −iξ 1 1 1 + e z iξ 1 in B (T), that is, F (z) = −iξ f(e )dλ(ξ) where f ∈ B (T). Moreover, the norm 2π −π 1 − e z Z 1 Z π 0 1 kF k 1 = F (z) dz is equivalent to the norm kfk 1 . This allows to identify B ( ) B (D) B (T) A D A 0 −π 1 with B (T) and thus the following: 1 1 Proposition 2.7. BA(D) can be identified with a closed subspace of L (T).

9 Proof. Let {f } be a sequence in B1( ) that converges to f in L1( ). We need to show that the n n∈N T T Z π −iξ 1 1 1 1 + e z iξ Poisson integral of f is in BA(D). Since fn ∈ B (T), ∀ n ∈ N, then Fn(z) = −iξ fn(e )dλ(ξ) 2π −π 1 − e z 1 iθ 2 belongs to BA(D). Let F (z) be the Poisson integral of f. We note that if z = e , then (ξ − θ) − Z π −iξ (ξ−θ)4 −iξ 2 2 0 1 2e iξ 2 ≤ 1 − e z ≤ (ξ − θ) . Therefore, we have that F (z) = −iξ 2 f(e )dλ(ξ) 2π −π (1 − e z) Z π 0 1 2 iξ and F (z) ≤ f(e ) dλ(ξ) ≤ C kfk 1 . It follows that , 2 L (T) 2π −π |1 − e−iξz|

Z 1 Z π 0 0 kF − F k 1 = F (z) − F (z) dz ≤ C kf − fk 1 . n B n n L (T) A 0 −π

1 1 1 Since fn → f in L (T), it follows that F ∈ BA and Fn → F in BA(D). The result follows by 1 1 identifying BA(D) and B (T). n n We note that there is an extension of this result to the polydisk D and polysphere T , see [7].

Proof of theorem 2.6. It sufficient to show that there exists a constant M > 0 such that kT k 1 ∗ ≥ (B (T)) n M kgk w . The extension to is natural, using the results in [7,8] . With Proposition 2.7, the BMO (T) R proof follows along the lines of the proof of Proposition 7.3 in [6]. 1 ∗ 1 1 Let T ∈ (B (T)) . Since B (T) is a closed subspace of L (T) by Proposition 2.7, then by Hahn- 0 1 ∗ Banach Theorem, T can be extended to a bounded linear operator T ∈ (L ( )) with kT k 1 ∗ = T (B (T)) 0 1 ∗ ∞ ∞ kT k(L1( ))∗ . Since (L (T)) = L (T), then there exists g0 ∈ L (T) such that kgkL∞( ) = T Z T 0 1 iξ iξ 1 kT k(L1)∗ = kT k(B1)∗ and T (f) = g0(e )f(e )dλ(ξ), for all f ∈ BA(D). Note that here, 2π T 1 X inξ we identify f with its correspondent in BA(D). Now let Ane be the Fourier series of g0. Since n∈N ∞ 2 X 2 g0 ∈ L (T) ⊂ L (T), we have that |An| < ∞ . This means that g0 is holomorphic. Let n∈Z Z π iξ Z π iξ 1 g0(e ) iξ 1 g0(e ) g(z) = iξ d(e ) = −iξ dξ . 2πi −π e − z 2π −π 1 − e z Z π 1 X −inξ n 1 −inξ iξ Since iξ = e z and An = e g0(e )dλ(ξ), we have 1 − e z 2π −π n∈N Z π iξ 1 g0(e ) X n g(z) = −iξ dξ = Anz . 2π −π 1 − e z n∈N

Z π iξ 2 iθ 1 g0(e ) This implies that g ∈ H (D). For θ ∈ R, g(e ) = −iξ iθ dλ(ξ). Moreover, given 2π −π 1 − e e 1 1 f ∈ BA(D) ⊆ H (D), and using the Cauchy integral formula, we have:

Z π Z π  Z π iξ  1 iθ iθ 1 iθ 1 f(e ) g(e )f(e )dθ = g0(e ) −iξ iθ dλ(ξ) dθ 2π −π 2π −π 2π −π 1 − e e Z π 1 iθ iθ = g0(e )f(e )dθ 2π −π = T (f) .

10 We also observe that Z π  −iξ  Z π −iξ 1 1 1 + e z iξ 1 1 + e z iξ g(z) = −iξ + 1 g0(e )dξ = −iξ G0(e )dλ(ξ), (2.9) 2π −π 2 1 − e z 2π −π 1 − e z  Z π  iξ 1 iξ 1 iξ 1  iξ  where G0(e ) = g0(e ) + g0(e )dλ(ξ) = g0(e ) + A0 . 2 2π −π 2 Put G1 = Re(G0),G2 = Im(G0), and

Z π −iξ Z π −iξ 1 1 + e z iξ 1 1 + e z iξ U(z) = −iξ G1(e )dλ(ξ); V (z) = −iξ G2(e )dλ(ξ) . 2π −π 1 − e z 2π −π 1 − e z

Then g = U + iV . Moreover, U and V are analytic in D since they represent the Poisson integral of ∞ 1 w G1,G2 ∈ L (T) ⊆ B (T). Observe that BMO ⊆ BMO . Moreover, Theorem 3.2 in [6] shows that ∼ kgkBMO = kgkBMOA. It therefore follows that there exists a constant C > 0 such that 0 C kgkBMOw ≤ C kgkBMO ≤ kgkBMOA ≤ T (L1)∗ = kT k(B1)∗ .

2.1 Discussion

w 0 0 We note that BMO ⊆ Λ with kfk 0 ≤ C kfk w where kgk 0 = kfk with g = f in the ∗ Λ∗ BMO Λ∗ Λ∗ 1 ∗ ∼ 0 1 ∗ ∼ w sense of distributions. Since (B ) = Λ∗, and from Theorem 2.6 above (B ) = BMO , it follows w ∼ 0 that BMO = Λ . The consequence is that there exists c > 0 such that c kfk w ≤ kfk 0 , that is, ∗ BMO Λ∗ 0 those two norms are equivalent. We finish by noting Λ∗ has an analytic characterization, so we would w 0 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 w 1 ∼ 1 1 BMO otherwise we would have had B = H , which is not true. In other words, there exists f ∈ H such that f∈ / B1. Our next result is about the closeness of VMOw in BMOw. Theorem 2.8. VMOw is a closed subspace of BMOw. Proof. Let {f } be a sequence in VMOw that converges to f ∈ BMOw. Let us prove that f ∈ n n∈N w w VMO . That fn → f as n → ∞ in BMO is equivalent to lim kfn − fk w = 0. The latter is n→∞ BMO also equivalent, by definition of the norm in BMOw to

lim sup (fn − f)Q = 0 and lim M(fn − f)(x) = 0 . n→∞ Q3x n→∞ Z w Since f ∈ BMO , then sup − [f(ξ) − fQ]dλ(ξ) < ∞. Therefore Q3x Q Z Z

− [f(ξ) − fQ]dλ(ξ) = − [f(ξ) − fn(ξ) + fn(ξ) − fn,Q + fn,Q − fQ]dλ(ξ) Q Q Z Z Z

≤ − [f(ξ) − fn(ξ)]dλ(ξ) + − [fn(ξ) − fn,Q]dλ(ξ) + − [fn,Q − fQ]dλ(ξ) Q Q Q Z Z

≤ − [f(ξ) − fn(ξ)]dλ(ξ) + − [fn(ξ) − fn,Q]dλ(ξ) + |fn,Q − fQ| Q Q Z

≤ − [f(ξ) − fn(ξ)]dλ(ξ) + kfn − fkBMOw Q

11 ≤ kfn − fk 1 + kfn − fk w . Lloc BMO

w Since {fn} is a sequence of function is BMO , we have that lim kfn − fk w = 0. Since n∈N n→∞ BMO w 1 w w BMO ⊂ L , we have that lim kfn − fk 1 = 0. Hence VMO is a closed subspace of BMO . loc n→∞ Lloc

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 T, then the space of complex-valued and continuous functions C(T) ⊆ VMO(T) and the BMO(T)-closure of C(T) is precisely VMO(T). It turns out this result is also true for VMOw.

w w Theorem 2.10. The BMO -closure of C(T) is precisely VMO (T), that is

w BMO w C(T) = VMO (T) .

w w w Proof. We observe that C(T) ⊆ VMO(T) ⊆ VMO (T) ⊆ BMO (T). Therefore since VMO (T) w is closed in BMO (T), we have that

w w BMO (T) BMO (T) w C(T) ⊆ VMOw(T) = VMO (T) .

w For the other direction , let f ∈ VMO (T), and consider the sequence {R(f)}>0 of rotations of f iθ i(θ−) by angle , defined on T as R(f)(e ) = f(e ); θ ∈ R. Then from Theorem 2.1 in [6], we have w that for all  > 0,R(f) ∈ C( ) and lim kR(f) − fk = 0. Since BMO( ) ⊆ BMO ( ), T →0 BMO(T) T T w BMO (T) we also have that lim kR(f) − fk w = 0. That is, f ∈ C( ) . →0 BMO (T) T

Acknowledgment

This material is based upon work supported by the National Science Foundation under Grant No. DMS- 1440140, 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 Sci- ences Research Institute in Berkeley, California, during the summer of 2020.

References

[1] S. Bloom and G. De Souza. Atomic decomposition of generalized Lipschitz spaces. Illinois Journal of Mathematics, pages 682–686, 1989.

[2] R. Coifman. A real variable characterization of hp. Studia Mathematica, 51:269–274, 1974.

[3] G. De Souza. Spaces formed by special atoms. PhD thesis, SUNY at Albany, 1980.

[4] C. Fefferman. Characterization of bounded mean oscillation. Bulletin of the American Mathematical Society, 77(4):587–588, 1971.

[5] T. Gill. Banach spaces for the schwartz distributions. Real Analysis and Exchange, 43(1):1–36, 2017.

[6] D. Girela. Analytic functions of bounded mean oscillation. Complex Functions Spaces (Merkrijarvi¨ 1999), Univ. Joensuu Dept. Math. Rep. Ser.(4):61–170, 2001.

12 [7] E. Kwessi and G. De Souza. Analytic characterization of high weighted special atom spaces,. Complex Variables and Elliptic Equations, 2020.

[8] E. Kwessi, G. De Souza, D. Ngalla, and N. Mariama. The special atom space in higher . Demonstratio Mathematica, 33:131–151, 2020.

[9] A. Zygmund. Trigonometric Series, Vol. I, II. Cambridge Mathematical Library, 2002.

13