On Petermichl's dyadic shift and the Hilbert transform - Helsinki.fi

ON PETERMICHL’S DYADIC SHIFT AND THE HILBERT

TRANSFORM

TUOMAS HYTÖNEN

Abstract. Petermichl’s representation for **the** **Hilbert** **transform** as an average

of **dyadic** **shift**s has important applications. Here it is shown that **the**

integrals involved in (a variant of) this representation converge both almost

everywhere **and** strongly in L p (R), p ∈ (1, ∞), which improves on **the** earlier

result of weak convergence in L 2 (R).

1. Introduction

Given a **fi**nite interval I = [a, b) ⊂ R, denote its left **and** right halves by I − :=

[a, (a + b)/2) **and** I + := [(a + b)/2, b). The associated st**and**ard Haar function

is de**fi**ned by h I := |I| −1/2( 1 I− − 1 I+

)

**and** a modi**fi**ed Haar function by HI :=

2 −1/2( h I− − h I+

)

, both normalized in L 2 (R).

Given a **dyadic** system of intervals D (what it means precisely, will be speci**fi**ed

below), **the** associated **dyadic** **shift**, introduced by Petermichl [3], is de**fi**ned by

Xf := ∑ I∈D

H I 〈h I , f〉,

where 〈g, f〉 := ∫ ∞

g(x)f(x) dx **and** **the** letter X (‘sha’) st**and**s for **shift**.

−∞ The

importance of this **shift** operator lies in **the** bridge which it provides between **the**

probabilistic–combinatorial **and** **the** analytic realms. **On** **the** one h**and**, it is closely

related to discrete martingale **transform**s **and** **the** simple structure of **the** operator

makes it tractable to combinatorial considerations involving induction on **the** scales;

on **the** o**the**r h**and**, **the** prototype singular integral operator, **the** **Hilbert** **transform**

(1) Hf(x) := lim H ε,R f(x), H ε,R f(x) := 1 ∫

f(x − y)

dy,

ε→0

π

R→∞

ε

2 TUOMAS HYTÖNEN

multiplication operator by a function b ∈ L ∞ (R), lies in **the** weak operator closure

in L (L 2 (R)) of **the** convex hull of **the** **dyadic** **shift**s. But it turns out that **the**

**Hilbert** **transform** is an average of **the** **dyadic** **shift**s in a much stronger sense:

Theorem 1.1. For r ∈ [1, 2) **and** β ∈ {0, 1} Z , let X β,r be **the** **dyadic** **shift** associated

to **the** **dyadic** system rD β , as de**fi**ned in Section 2. Let µ st**and** for **the**

canonical probability measure on {0, 1} Z which makes **the** coordinate functions β j

independent with µ(β j = 0) = µ(β j = 1) = 1/2. Then for all p ∈ (1, ∞) **and**

f ∈ L p (R),

Hf(x) = − 8 π 〈X〉f(x) := − 8 π

∫ 2

1

∫

{0,1} Z X β,r f(x) dµ(β) dr

r ,

where **the** integral converges both pointwise for a.e. x ∈ R **and** also in **the** sense of

an L p (R)-valued Bochner integral.

This is **the** present contribution, proven in **the** rest of **the** note.

2. Dyadic systems **and** **shift**s

Recall that **the** st**and**ard **dyadic** system is

D 0 := ⋃ j∈Z

D 0 j , D 0 j := { 2 j( [0, 1) + k ) : k ∈ Z } .

A general **dyadic** system may be de**fi**ned as a collection D = ⋃ j∈Z D j, where D j =

D j + x j for some x j ∈ R **and** **the** partition D j re**fi**nes D j+1 for each j ∈ Z. Since

only **the** value of x j mod 2 j is relevant for **the** de**fi**nition of D j , one may choose

x j ∈ [0, 2 j ), **and** **the** re**fi**nement property requires that x j+1 − x j ∈ {0, 2 j }. It

follows by recursion that

x j = ∑ i

ON PETERMICHL’S DYADIC SHIFT AND THE HILBERT TRANSFORM 3

It follows easily from Burkholder’s inequality for martingale **transform**s that partial

sums of **the** j-series above give uniformly bounded operators on L p (R), p ∈ (1, ∞);

cf. [5, Theorem 3.1]. Then **the** convergence of **the** double series both in L p (R) **and**

pointwise a.e. is deduced by st**and**ard considerations. Note that **the** partial sums

of **the** j-series are differences of two conditional expectations of X β,r f, **and** hence

dominated by M β,r (X β,r f), where M β,r is **the** **dyadic** maximal operator related to

**the** scaled **dyadic** system rD β . From **the** boundedness of M β,r **and** X β,r on L p (R),

with bounds independent of (β, r), one easily deduces by dominated convergence

that **the** interal de**fi**ning **the** average **dyadic** **shift** in Theorem 1.1 exists, pointwise

a.e. **and** in L p (R), **and** satis**fi**es

(3) 〈X〉f := lim

n→+∞

m→−∞

n∑

j=m

∫ 2

1

∫

{0,1} Z ∑

I∈rD β j

H I 〈h I , f〉 dµ(β) dr

r .

Observe that

rD β j

3. Evaluation of **the** integral

= r2j{ [0, 1) + k + ∑ 2 i−j β i ; k ∈ Z}.

i

4 TUOMAS HYTÖNEN

**and** **the**n

∫ R

k t ∗ f dt = φ ε ∗ f − φ R ∗ f − π

ε t

8 H ε,Rf,

where H ε,R is **the** truncated **Hilbert** **transform** as de**fi**ned in (1).

As k is odd, its primitive K is even, **and** φ is again an odd function. It is also

bounded **and** compactly supported. Thus it follows from st**and**ard molli**fi**cation

results that φ t ∗ f → 0, both in L p (R) **and** pointwise a.e., as ei**the**r t → 0 or t → ∞.

Hence

〈X〉f = − π lim H 2m ,2

8

nf =: −π n→+∞

8 Hf.

m→−∞

Note that **the** existence of **the** limit was a consequence of **the** proof, building at

**the** bottom on martingale convergence; none of **the** well-known results concerning

**the** L p (R) boundedness of **the** **Hilbert** **transform** nor **the** convergence of its truncated

versions was presupposed, but all this follows as a byproduct of **the** established

representation.

References

[1] T. Figiel. Singular integral operators: a martingale approach. In Geometry of Banach spaces

(Strobl, 1989), volume 158 of London Math. Soc. Lecture Note Ser., pages 95–110. Cambridge

Univ. Press, Cambridge, 1990.

[2] F. Nazarov, S. Treil, **and** A. Volberg. The T b-**the**orem on non-homogeneous spaces. Acta Math.,

190(2):151–239, 2003.

[3] S. Petermichl. Dyadic **shift**s **and** a logarithmic estimate for Hankel operators with matrix

symbol. C. R. Acad. Sci. Paris Sér. I Math., 330(6):455–460, 2000.

[4] S. Petermichl. The sharp bound for **the** **Hilbert** **transform** on weighted Lebesgue spaces in

terms of **the** classical A p characteristic. Amer. J. Math., 129(5):1355–1375, 2007.

[5] S. Petermichl **and** S. Pott. A version of Burkholder’s **the**orem for operator-weighted spaces.

Proc. Amer. Math. Soc., 131(11):3457–3461 (electronic), 2003.

Department of Ma**the**matics **and** Statistics, University of **Helsinki**, Gustaf Hällströmin

katu 2b, FI-00014 **Helsinki**, Finl**and**

E-mail address: tuomas.hytonen@helsinki.**fi**