For the additive subgroup K+ of the 2-series field K, we may choose a Haar measuredx

Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 20 (2004), 225–231

www.emis.de/journals ISSN 1786-0091

SOME PROPERTIES FOR FUNCTIONS OF VMO(2^ω)

JUN TATEOKA

Dedicated to Professor W.R. Wade on his sixtieth birthday

Abstract. A function of bounded mean oscillation (BMO) is said to have vanishing mean oscillation or belong to VMO space if its mean oscillation is locally small in a uniform sense. Though there is an extensive literature on the BMO, very few mention is made on the properties for functions of VMO.

In this note, we discuss the connection between modulus of continuity and the approximation of functions by Walsh polynomials in VMO space on the dyadic group 2^ω, VMO(2^ω), the analogy between VMO(2^ω) and C(2^ω), the estimate for certain type of convolution operators on VMO(2^ω), the decom- position theorem for functions in VMO(2^ω) and the characterization of Walsh series which happen to be the Walsh-Fourier series of a function in VMO(2^ω).

1. Notation

Our results are stated in the situation that the dyadic group 2^ω is the additive subgroup of the ring of integers in the 2-series field K of formal Laurent series in one variable over the finite field GF(2). We need to set some basic notation. It is taken from Taibleson’s book [9] where the fundamentals are detailed. For the additive subgroup K⁺ of the 2-series field K, we may choose a Haar measuredx.

Letd(αx) =|α|dx, α6= 0 and call|α|the valuation ofα.

Let P⁰ = {x ∈ K : |x| ≤ 1} and P¹ = {x ∈ K : |x| < 1}. K is totally disconnected, hence the value is discrete valued. Thus there is an element ℘ofP¹ of maximum value. Then an element x∈K is represented as

(1) x=

X∞ k=j

ak℘^k, ak∈GF(2),

which can contain a finite number of terms with negative powers of℘. The ring of integers P⁰=©

x=P_∞

k=0ak℘^kª

coincides with the dyadic group 2^ωas an additive group. For E a measurable subset of K, let |E| =R

KΦE(x)dx, where ΦE is the characteristic function ofE anddx is Haar measure normalized so|2^ω|= 1. Then

|P¹| = |℘| = 2⁻¹. Let P^k = {x ∈ K : |x| ≤ 2^−k} and Φk be its characteristic function. For x=x0+P₋₁

k=jak℘^k, ak ∈GF(2), x0∈2^ω,set

(2) w(℘^k) =

(−1 k=−1,

1 k <−1, w(x0) = 1.

Then wis a character on K⁺. Forx, y∈K, letwy(x) =w(y·x). wis constant on cosets of 2^ω and ify∈P^k thenwy is constant on cosets ofP^−k.

2000Mathematics Subject Classification. 42C10.

Key words and phrases. Dyadic group, BMO space, VMO space.

225

We assume that all functions are complex valued and measurable. Iff ∈L¹(K) the Fourier transform of f is the function ˆf defined by

(3) fˆ(y) =

Z

K

f(u)wy(u)du.

Then we have (2^kΦk)ˆ= Φ−kand ((2^kΦk)∗(2^lΦl))ˆ= Φ−(k∧l), wherek∧l= min(k, l).

Let {u(n)}^∞_n=0 be a complete list of distinct coset representatives of 2^ω in K⁺. We defineu(0) = 0,u(1) =℘⁻¹and forn=b0+b1·2+b2·2²+· · ·+bs·2^s(bi= 0 or 1), u(n) =u(b0) +℘⁻¹u(b1) +· · ·+℘^−su(bs). Then©

w_u(n)|_P⁰ª_∞

n=0=©

w_u(n)ª_∞

n=0 is a complete set of characters on 2^ω. This is the Walsh-Paley system.

The Dirichlet kernels are the functions Dn(x) =

n−1X

k=0

w_u(k)(x), n≥1, D0(x)≡0.

If f ∈L¹(2^ω) the Walsh-Fourier coefficients {ck}^∞_k=0 ={fˆ(u(k))}^∞_k=0 are given by c_k=R

2^ωf(x)w_u(k)(x)dx. The Walsh-Fourier series is given by f(x)∼

X∞

k=0

ckwu(k)(x).

The n-th partial sum of the Walsh-Fourier series of f is denoted by Snf(x) and is defined as Snf(x) = P_n−1

k=0ckw_u(k)(x). If f ∈ L¹(2^ω), x ∈ 2^ω, n ≥ 0, then S2ⁿf(x) = 2ⁿR

x+Pⁿf(t)dt,as follows from the fact thatD2ⁿ = 2ⁿΦn.

S(2^ω) is the collection of the test functions on 2^ω. If φ ∈ S(2^ω) then φ is a

“polynomial”, that is, φ(x) =P₂ⁿ₋₁

k=0 φ(u(k))wˆ u(k)(x) for somen≥0. Cidenotes a constant.

2. Properties of VMO(2^ω) functions

Letf ∈L¹(2^ω). By a ball we mean a setB={y∈2^ω:|x−y| ≤2^−k}=x+P^k for somex∈2^ωandk∈N. Iff ∈L¹(2^ω), writef_B= _|B|¹ R

Bf(x)dxfor the average off overB. If

(4) sup

B

1

|B| Z

B

|f(x)−fB|dx=kfk∗<∞,

where the supremum is over all ballsB, then we sayfis of bounded mean oscillation, f ∈BMO(2^ω). It is clear that L^∞(2^ω)⊂BMO(2^ω) and for f ∈L^∞(2^ω), kfk∗ ≤ 2kfk∞. BMO(2^ω) is the dual space to H¹(2^ω). That is, each continuous linear functional `on H¹(2^ω) can be realized as a mapping

(5) `(g) =

Z

2^ω

f(x)g(x)dx, g∈H¹(2^ω),

when suitably defined, where f is a function in BMO(2^ω). This pairing allows to realize H¹(2^ω) as the dual of VMO(2^ω). (See [7], [8] and [12].)

For 0< δ <1, write

(6) Mδ(f) = sup

|B|≤δ

1

|B|

Z

B

|f(x)−fB|dx.

Then f ∈ BMO(2^ω) if and only if Mδ(f) is bounded and kfk∗ = limδ→1Mδ(f).

BMO(2^ω) is a Banach space with normM1(f) +|fˆ(0)|or M1(f) +kfk1. We say thatf has vanishing mean oscillation, f ∈VMO(2^ω), if (7) f ∈BMO(2^ω),andM0f = lim

δ→0Mδ(f) = 0.

SOME PROPERTIES FOR FUNCTIONS OF VMO(2 ) 227

VMO(2^ω) contains every continuous functions on 2^ω, C(2^ω). The unbounded func- tion log|x| belongs to BMO(2^ω). However, log|x|is not VMO(2^ω). The function log|log|x||is in VMO(2^ω), although that is not immediately evident. VMO(2^ω) is a closed subspace of BMO(2^ω), so it contains the BMO-closure of C(2^ω).

The following theorem shows several characterization of VMO(2^ω). (See [10] for the dyadic group case, [5] and [6] for the classical case).

The space VMO(2^ω) is translation invariant. Fory ∈ 2^ω, we let τy denote the operator of translation byy; that is, (τyf)(x) =f(x−y) for any functionf on 2^ω. Theorem 2.1. Forf a function in BMO(2^ω), the following conditions are equiv- alent:

(i) f is inVMO(2^ω);

(ii) lim|h|→0kτhf−fk∗= 0;

(iii) limn→∞k2ⁿΦn∗f−fk∗= 0;

(iv) f is in theBMO-closure ofC(2^ω).

Next lemma is a simple but useful fact. (See [6].)

Lemma 2.2 (Inequality of Young type). If f is a function inBMO(2^ω)andφ is an integrable function on 2^ω, thenφ∗f is inVMO(2^ω)andkf ∗φk∗≤ kφk1kfk∗. If, in addition, φis continuous function on2^ω, thenφ∗f is in continuous function on 2^ω.

Proof. Put f∗φ(t) =h(t). Then, we have 1

|B|

Z

B

|h(t)−hB|dt≤ kφk1 1

|B|

Z

B

|(τuf)(t)−(τuf)B|dt.

Hence,khk∗≤ kφk1kτufk∗=kφk1kfk∗.

For anyε >0, there exists a polynomialTsuch thatkφ−Tk1< ε. Then,f∗T∈ C(2^ω) and for small|B|,

1

|B|

Z

B

|f ∗(φ−T)(t)−(f∗(φ−T))_B|dt≤ kφ−Tk₁kfk_∗< εkfk_∗.

We obtain, by Theorem 2.1.,f∗φ∈VMO(2^ω). ¤

To study the analogy between VMO(2^ω) and C(2^ω), we introduce the analogue in VMO(2^ω) of the Lipschitz classes. Let ρ(δ) be a positive, continuous, non- decreasing function on (0, ∞) satisfying limδ→0ρ(δ) = 0, and ρ(2δ)≤C1ρ(δ).

A continuous functionf on 2^ωis said to belong to the class Lipρ(δ) if it satisfies ω(f, δ) =O(ρ(δ)), whereω(f, δ) = sup{kτhf−fk∞:|h| ≤δ}.

We shall say f in BMO(2^ω) belongs to BMO(ρ(δ)) providedMδ(f) =O(ρ(δ)).

We have VMO(2^ω)=∪ρ(δ)BMO(ρ(δ)).

Theorem 2.3. Ifρsatisfies the condition Z ₁

0

ρ(t)

t dt <∞, then BMO(ρ(δ))⊂Lip(σ(δ)), where

σ(δ) = Z _δ

0

ρ(t) t dt.

In particular, BMO(δ^α)=Lip(δ^α),0< α≤1.

The analogue of this theorem in the classical case was shown S. Spanne ([5]).

We omit the proof of this theorem.

We consider the translation invariant singular integrals on VMO(2^ω). G.I. Gaudry and I.R. Inglis proved the next theorem ([3] and [4]), which is obtained without the intervention of the space H¹(2^ω).

Theorem 2.4. SupposeK∈L¹(2^ω). If (i) |K(u(n))| ≤ˆ C2, for|u(n)| ≥2ⁿ⁻¹, (ii) R

2^ω\Pⁿ|K(x−y)−K(x)|dx≤C2 for|y| ≤2⁻ⁿ,

then, for all f ∈L^∞(2^ω),kK∗fk∗≤C3kfk∗, whereC3 depends onC2 only.

Corollary 2.5. If f ∈C(2^ω), thenK∗f ∈VMO(2^ω).

Proof. For a continuous function f and anyε >0, there exists a polynomial T ∈ S(2^ω) such thatkf−Tk∞< ε. ThenK∗T ∈S(2^ω) andkK∗f−K∗Tk∗< C3ε.

Hence, we have, by Theorem 2.1., K∗f ∈VMO(2^ω). ¤ J.B. Garnett and P.W. Jones ([2]) and J.-A. Chao ([1]) proved the following characterization of BMO regular martingales similar to the construction Carleson’s.

Theorem 2.6. Letf ∈BMO(2^ω). Then there exist a g∈L^∞(2^ω)with kgk∞≤C4kfk∗,

a sequence of balls {Bi} and a corresponding sequence of complex numbers {bi} such that P

Bi⊂B|bi| ≤C4kfk∗|B|for any given ball B, and f =g+X

i

biΦBi

|Bi| +C5

for a constant C5.

Theorem 2.7. (i) Letf ∈VMO(2^ω)andf(0) = 0. Then there exist ag∈C(2^ω) with kgk∞ ≤ C6kfk∗, a sequence of balls {Bi} and a corresponding sequence of complex numbers {bi} such that _|B|¹ P

Bi⊂B|bi| →0 as |B| →0 for any given ball B, and f =g+P

ibiΦ_Bi

|B_i|.

(ii) Let g ∈ C(2^ω) and {Bi} be a sequence of balls. Assume that to each Bi, there is a associated constant bi satisfying _|B|¹ P

Bi⊂B|bi| →0 as |B| →0. Then, if f =g+P

ibiΦ_Bi

|Bi|+C7 for a constantC7,f ∈VMO(2^ω).

Proof. (i) Since f ∈ BMO(2^ω), using Theorem 2.6., write f =g0+P

i0bi0

Φ_Bi

|Bi00|, where kg0k∞≤C4kfk∗, andP

Bi0⊂B|bi₀| ≤ C4kfk∗|B|for any ball B. By Theo- rem 2.1., there is a ball Bj0 such thatkf−f∗ ^Φ_|B^Bj0

j0|k∗<kfk∗/2, Let G0=g0∗ ΦBj0

|Bj0| and

B0=X

i0

|bi0|ΦBi0

|Bi₀| ∗ ΦBj0

|Bj₀| =X

i0

|bi0|ΦBi0∧j0

|Bi₀∧j₀|, then

G0∈C(2^ω), kG0k∞≤ kg0k∞≤C4kfk∗, kB0k∞≤ X

i0:Bi0⊂Bj0

|bi0|

|Bj0| ≤C4kfk∗, and so thatkf−G0−B0k∗<kfk∗/2.

Repeating the above argument with f−f∗ ^Φ_|B^Bj0

j0|, we obtain f−f∗ ΦBj0

|Bj0| =g1+X

i1

bi1

ΦBi1

|Bi1|,

SOME PROPERTIES FOR FUNCTIONS OF VMO(2 ) 229

withkg1k∞≤C4kf−f∗ ^Φ_|B^Bj0

j0|k∗< C4kfk∗/2, and X

Bi1⊂B

|bi1| ≤C4kf −f ∗ΦBj0

|Bj0|k∗|B|< C4kfk∗|B|/2.

There exists a ballB_j1 such that k(f−f∗ ΦBj0

|Bj0|)−(f−f ∗ΦBj0

|Bj0|)∗ΦBj1

|Bj1|k∗<kfk∗/2². LetG1=g1∗^Φ_|B^Bj1

j1| and B1=X

i1

|bi1|ΦBi1

|Bi1| ∗ ΦBj1

|Bj1| =X

i1

|bi1|ΦBi1∧j1

|Bi1∧j1|. ThenG1∈C(2^ω),kG1k∞+kB1k∞≤C4kfk∗ and

kf −G0−B0−G1−B1k∗<kfk∗/2².

Iterating we obtain sequences {Gn} ⊂C(2^ω) and{Bn}with the following prop- erties:

kGnk∞+kBnk∞≤C4kfk∗/2ⁿ⁻¹, (a)

kf − Xn

1

(Gk+Bk)k∗≤ kfk∗/2ⁿ⁺¹. (b)

By (a), the function g =P

nGn ∈ C(2^ω) and _|B|¹ P

Bi⊂B|bi| → 0 as|B| → 0 and from (b), f =g+P

ibiΦ_Bi

|Bi|. (ii) Letb(x) =P

ibiΦ_Bi

|B_i| andB=a+P^l. We shall show that I= 1

|B|

Z

B

|b(t)−b(a)|dt→0 as |B| →0.

I= 2^l Z

a+P^l

¯¯

¯¯

¯ X

i

bi

|Bi|(ΦBi(t)−ΦBi(a))

¯¯

¯¯

¯dt

= 2^l Z

a+P^l

¯¯

¯¯

¯¯ X

i:a+P^l⊂Bi

+ X

i:Bi⊂a+P^l

¯¯

¯¯

¯¯dt.

In fact, if Bi and Bj are two nondisjoint balls on 2^ω, then either Bi ⊂ Bj or Bj⊂Bi. Ifa+P^l⊂Bi, then ΦBi(x) = ΦBi(a) = 1 and I= 0.

IfBi⊂a+P^l, then I= 2^l

Z

a+P^l

| X

i:Bi⊂a+P^l

bi

|Bi|(ΦBi(t)−ΦBi(a))|dt

≤2^l X

i:Bi⊂a+P^l

|b_i|

|Bi| Z

Bi

|ΦBi(t)−ΦBi(a)|dt≤2^l+1 X

i:Bi⊂a+P^l

|bi| →0

as l→ ∞. This proves (ii). ¤

Let consider the Walsh series W(x) = P_∞

n=0anwu(n)(x), whose coefficients are arbitrary numbers. We can show those Walsh seriesW(x) which happen to be the Walsh-Fourier series of a function in VMO(2^ω).

Theorem 2.8. Let W(x) be a Walsh series. Then W(x) is the Walsh-Fourier series of a VMOfunction if and only ifkS2ⁿ(W)−S2^m(W)k∗→0 asn, m→ ∞.

Proof. If part. Since VMO(2^ω) is a Banach space, any Cauchy sequence is con- vergent to a limit function f. We shall show f ∈ VMO(2^ω). Since VMO(2^ω) is embedded continuously into L²(2^ω), we also have kS2ⁿ(W)−S2^m(W)k2 → 0 as n, m → ∞, so that W(x) is the Walsh-Fourier series of an L²(2^ω) function f and we have S2ⁿ(W)(x) → f(x) a.e. and kS2ⁿ(W)−fk2 → 0. Consequently, R

BS2ⁿ(W)(t)dt→R

Bf(t)dtasn→ ∞for any ballB, that is, (S2ⁿ(W))B →(f)B

as n→ ∞. An application of Fatou’s lemma to |S2ⁿ(W)−S2^m(W)| over the ball B shows

1

|B|

Z

|f−fB| ≤ lim

n→∞

1

|B|

Z

|S2ⁿ(W)(t)−(S2ⁿ(W))B|dt.

Since, for|B|small,S2^m(W)(t) = (S2^m(W))B, we have 1

|B| Z

|f−fB| ≤ lim

n→∞

1

|B|

Z

|S2ⁿ(W)(t)−S2^m(W)(t)−(S2ⁿ(W)−S2^m(W))B|dt

≤ lim

n→∞kS2ⁿ(W)−S2^m(W)k∗

and we obtainf ∈VMO(2^ω).

On the other hand, we can use the integral formula S2ⁿW(t) =

Z

(τuf)(t)D2ⁿ(u)du.

LetDn,mW(t) =S2ⁿW(t)−S2^mW(t). Then, for|B|small, 1

|B| Z

B

|(Dn,mW)(t)−(Dn,mW)B|dt

≤ 1

|B|

Z

B

Z

|((τuf)(t)−(τuf)B)(D2ⁿ(u)−D2^m(u))du|dt

≤ Z

(D2ⁿ(u) +D2^m(u)) 1

|B|

Z

B

|(τuf)(t)−(τuf)B|dtdu→0 as |B| →0.

This proves kS2ⁿ(W)−S2^m(W)k∗→0 asn, m→ ∞. ¤ References

[1] J. -A. Chao,H^pand BMO regular martingales, Lecture Notes in Math.908, Springer, 1982.

[2] J. B. Garnett and P. W. Jones,BMO from dyadic BMO, Pacific J. Math.,99(1982), 351–371.

[3] G. I. Gaudry and I. R. Inglis,Weak-strong convolution operators on certain disconnected groups, Studia Math.,64(1979), 1-12.

[4] I. R. Inglis,Addendum to the paper “Weak-strong convolution operators on certain discon- nected groups”, Studia Math.,70(1981), 161–163.

[5] D. Sarason,Function theory on the unit circle, Virginia Poly. Inst. and State Univ., Blacks- burg, 1979.

[6] D. Sarason,Functions of vanishing mean oscillation, Trans. Amer. Math. Soc.,207(1975), 391–405.

[7] F. Schipp,The dual space of martingale VMO space, Proc. Third Pannonian Symp. Math.

Stat. 1984, 305–315.

[8] F. Schipp, W. R. Wade, P. Simon and J. P´al, Walsh series: An introduction to dyadic harmonic analysis, Adam Hilger and Akad´emiai Kiad´o, 1990.

[9] M. H. Taibleson,“Fourier analysis on local fields”, Princeton Univ. Press, 1975.

[10] J. Tateoka:The modulus of continuity and the best approximation over the dyadic group, Acta Math. Hung.,59(1992), 115–120.

[11] C. Watari, personal, 1999.

[12] F. Weisz,Martingale Hardy spaces and their applications in Fourier- Analysis, Lecture Notes in Math.1568, Springer, 1994.

SOME PROPERTIES FOR FUNCTIONS OF VMO(2 ) 231

Department of Mathematics, Akita University,

Tegata, Akita 010-8502, Japan

E-mail address: [email protected]