A NOTE ON MARTINGALE HARDY SPACES
arXiv:1411.5407v2 [math.CA] 30 Nov 2014
JINGGUO LAI Abstract. We investigate the relation between Carleson sequence and balayage, and use this to give an easy proof of the equivalence of the L1 -norms of the maximal function and the square function in non-honogeneous martingale settings.
1. Introduction and the main theorem In this note, we attempt to give an easy proof of the celebrated theorem: The L1 -norms of the maximal function and square function are equivalent. Throughout the note, we will work on the real line R under Lebesgue measure, and assume our σ-algebras are generated by disjoint intervals. The notation ., & stand for one-sided estimates up to an absolute constant, and the notation ≈ stands for two-sided estimates up to an absolute constant. We will introduce the basic set-up following from [7]. Definition 1.1. A lattice L is a collection of non-trivial finite intervals of R with the following properties (i) L is a union of generations Lk , k ∈ Z, where each generation is a collection of disjoint intervals, covering R. (ii) For each k ∈ Z, the covering Lk+1 is a finite refinement of the covering Lk , i.e. each interval I ∈ Lk is a finite union of disjoint intervals J ∈ Lk+1 . We allow the situation where there is only one such interval J, i.e. J = I; this means that I ∈ Lk also belongs to the generation Lk+1 . We say a lattice L is homogenous if (i) Each interval I ∈ Lk is a union of at most r intervals J ∈ Lk+1 . |I| (ii) There exists a constant K < ∞ such that |J| ≤ K for every I ∈ Lk and every J ∈ Lk+1 , J ⊆ I. The standard dyadic lattice D = {([0, 1)+j)2k : j, k ∈ Z} is an example of homogenous lattice.
Remark 1.2. Let Ak be the σ-algebra generated by Lk , i.e. countable unions of T intervals in Lk . Let A−∞ be the largest σ-algebra contained in all Ak , k ∈ Z, i.e. A−∞ = k∈Z Ak , and let A∞ be the smallest σ-algebra containing all Ak , k ∈ Z. The structures of σ-algebras A−∞ and A∞ can be described as follows A−∞ is the σ-algebra generated by all the intervals I of form [ I= Ik , where Ik ∈ Lk , Ik ⊆ Ik−1 . k∈Z
Note that R is a disjoint union of such intervals I and at most countably many points (we might need to add left endpoints to the intervals I). Let us denote the collection of such intervals I by A0−∞ . Define 0 A0,fin −∞ = {I ∈ A−∞ : |I| < ∞};
Key words and phrases. Carleson sequence, balayage, maximal function, square function. 1
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JINGGUO LAI
’fin’ here is to remind that the set consists of intervals of finite measure. Instead of describing A∞ , let us describe the corresponding measurable functions. Namely, a function f (x) is A∞ -measurable, if it is Borel measurable and it is constant on intervals I. \ I= Ik , where Ik ∈ Lk , Ik ⊆ Ik−1 . k∈Z
Clearly, such intervals I do not intersect, so there can only be countably many of them. In this note, we always assume that all functions are A∞ -measurable. Definition 1.3. The averaging operator E is defined to be I Z 1 E f = hf i 1 = f 1 , I I I I |I| I
where 1 is the indicator function of the interval I. I Let Ek denote the conditional expectation X E f. Ek f = E [f |Ak ] = I
I∈Lk
For an interval I ∈ L, let rk(I) be the rank of the interval I, i.e. the largest number k such that I ∈ Lk . For an interval I ∈ L, rk(I) = k, a child of I is an interval J ∈ Lk+1 such that J ⊆ I (actually we can write J $ I). The colletion of all children of I is denoted by child(I). Correspondingly, I is called the parent of J. The difference operator ∆ is defined to be I X ∆ f = −E f + E f. I
I
J
J∈child(I)
Let ∆k denote the martingale difference ∆k f = Ek f − Ek−1 f =
X
∆ f. I
I∈L,rk(I)=k−1
To motivate the definition of the square function as well as for our later purpose, let us prove the following proposition: Proposition 1.4. For a function f (x) ∈ Lp , where p > 1, X X f= ∆ f+ (1.1) E f a.e. and in Lp . I
I∈A0,fin −∞
I∈L
Proof. One can easily see that X
I∈L,m≤rk(I) (1+ε)k }, and take Ek = {I ∈ L : I ∈ Ek }. Note that Ek is a finite disjoint union of maximal intervals in Ek , maximal is considered in the sense of inclusion. Claim 1 : |
P
I∈L hf iI aI |
≤ε
P
k∈Z (1
+ ε)k |Ek |Carl(|a |). I
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JINGGUO LAI
P P is a Carleson First note that I∈Ek |a | ≤ I⊆Ek |a | ≤ |Ek |Carl(|a |), because {|a |} I I I I∈L I squence. Using this fact, we can estimate ∞ X X X X X X k k k ε (1 + ε) |Ek |Carl(|a |) ≥ ε |a | = ε (1 + ε) |a | (1 + ε) I
k∈Z
=
∞ X l=0
≥ ≥ Claim 2 :
I∈Ek
k∈Z
R
Mf ≥ ε
P
k∈Z (1
∞ X l=0
∞ X l=0
k∈Z
X ε l+1 (1 + ε)
X
l=0 I∈Ek+l \Ek+l+1
X
I∈Ek+l \Ek+l+1
|a | I
h|f |i |a | (by definition of Ek ) I
k∈Z I∈Ek+l \Ek+l+1
I
X X ε |hf i a | = | hf i a |. l+1 I I I I (1 + ε) I∈L
I∈L
+ ε)k |Ek+1 |. (1+ε)k+1
k k∈Z (1+ε)
0
X
I
k∈Z
X ε (1 + ε)k+l+1 l+1 (1 + ε)
This is done by a commonly used trick Z Z ∞ XZ |{x : M f > t}|dt = Mf = ≥ε
I
|{x : M f > t}|dt
(1 + ε)k |{x : M f > (1 + ε)(k+1) }| = ε
k∈Z
X k∈Z
(1 + ε)k |Ek+1 |.
To finish, we combine the two claims and conclude that for any ε > 0, X | hf i a | ≤ (1 + ε)||f ||H∗1 Carl(|a |). I∈L
I
I
I
Finally, letting ε → 0, we prove the desired inequality.
Acknowledgement The author would like to thank his PhD thesis advisor, Serguei Treil, for suggesting this problem for his topics exam and for the invaluable guidance and support throughout the course of preparation. References [1] B. J. Davis, On the integrability of the martingale square function, Israel J. Math. 8, 187-190 (1970) [2] A. M. Garcia, Martingale Inequalities: Seminar Notes on Recent Progress, New York: Benjamin Inc. 1973, Mathematics Lecture Notes Series [3] A. M. Garcia, The Burgess Davis inequalities via Fefferman’s inequality, Arkiv F¨ or Matematik, Volume 11, Numbers 1-2 (1973), 229-237 [4] C. Fefferman, Characterizations of Bounded Mean Oscillation, Bull. Amer. Math. Soc., Volume 77, no. 4 (1971), 587-588 [5] J. B. Garnett, Bounded Analytic Functions, Graduate Texts in Mathematics 236. Springer 1981 [6] S. Pott and A. Volberg, Carleson measure and Balayage, Int. Math. Res. Not. IMRN 2010, no. 13, 24272436 [7] S. Treil, Commutators, Paraproducs and BMO in non-homogeneous martingale settings, ArXiv:1007.1210 [math.CA] (2010) Department of Mathematics, Brown University, Providence, RI 02912, USA E-mail address:
[email protected] 