A Characterization of Hardy Space with Non-doubling Measurescfa/notes/eisenach_yang_3.pdf · X. Tolsa, The space H1 for nondoubling measures in terms of a grand maximal operator,

A Characterization of Hardy Spacewith Non-doubling Measures

D. Yang (jointed with Guoen Hu, Yan Meng & Wengu Chen)

[email protected]

School of Mathematical Sciences

Beijing Normal University

A Characterization of Hardy Space with Non-doubling Measures – p. 1/23

1. Introduction

• X. Tolsa, Painleve’s problem and the semiadditivity of analyticcapacity, Acta Math. 190(2003), 105-149.

From FEATURED REVIEW of AMS by John B. Garnett:

1) Let E ⊂ C be compact and Ω = C \ E connected. ThePainleve problem, posed in 1947 by L. V. Ahlfors [Duke Math.J. 14 (1947), 1-11] is to characterize geometrically thosesets E such that every bounded analytic function on Ωis constant.

2) The above paper and its companion [X. Tolsa, Bilipschitzmaps, analytic capacity, and the Cauchy integral, Ann. of Math.(2), to appear] give complete and surprising answers to thePainlevé problem.


References

3) Tolsa’s solution has three noteworthy ingredients. Oneof them is that it depends on sophisticated ideas from theCalderon-Zygmund theory of singular integrals, such as theT (b) theorem for non-doubling measures.

• X. Tolsa, BMO, H1 and Calderon-Zygmund operators for nondoubling measures, Math. Ann. 319(2001), 89-149.

• X. Tolsa, The space H1 for nondoubling measures in terms of agrand maximal operator, Trans. AMS. 355(2003), 315-348.

• G. Hu, Y. Meng and D. Yang, New atomic characterization ofH1 space with non-doubling measures and its applications, Math.Proc. Cambridge Philos. Soc. 138 (2005), 151-171.


References (2)

• W. Chen, Y. Meng & D. Yang, Calderon-Zygmund operatorson Hardy spaces without the doubling condition, Proc. AMS.133(2005), 2671-2680.

• Y. Han & D. Yang, Triebel-Lizorkin spaces with non-doublingmeasures, Studia Math. 162 (2004), 105-140.

• D. Deng, Y. Han & D. Yang, Besov spaces with non-doublingmeasures, Trans. AMS. (to appear).


Non-doubling measure

• (Rd, | · |, µ): Rd ⇐= d-dimensional Euclidean space;

| · | ⇐= Euclidean distance;

µ is a Radon measure on Rd satisfying that

there are constants C > 0 and n ∈ (0, d] such that for allx ∈ R

d, r > 0 ,

(1.1) µ(B(x, r)) ≤ Crn,

where B(x, r) is the open ball centered at x and havingradius r.

• µ is not necessary to be doubling. Thus, (Rd, | · |, µ) is alsocalled a non-homogeneous space.


2. New atomic characterization of H1(µ)

• X. TolsaDefn 1 (maximal function) Given f ∈ L1

loc (µ), we set

MΦf(x) = supϕ∼x

∣∣∣∣∫

Rd

fϕ dµ

∣∣∣∣ ,

where the notation ϕ ∼ x means that ϕ ∈ L1(µ) ∩ C1(Rd)and satisfies

i) ‖ϕ‖L1(µ) ≤ 1,

ii) 0 ≤ ϕ(y) ≤ 1|y−x|n for all y ∈ R

d, and

iii) |∇ϕ(y)| ≤ 1|y−x|n+1 for all y ∈ R

d.


Hardy space H1(µ)

Defn 2 The Hardy space H1(µ) is the set of all functionsf ∈ L1(µ) satisfying that

∫Rd f dµ = 0 and MΦf ∈ L1(µ).

Moreover, the norm of f ∈ H1(µ) is defined by

‖f‖H1(µ) = ‖f‖L1(µ) + ‖MΦf‖L1(µ).

• Let α > 1 and β > αn. A cube Q is said to be a(α, β)-doubling cube if µ(αQ) ≤ βµ(Q), where αQ denotesthe closed cube with the same center as Q and

l(αQ) = αl(Q).

(l(Q) is the side length of Q.)


Hardy space H1(µ)

• Given two cubes Q ⊂ R in Rd, set

KQ,R = 1 +

NQ,R∑

k=1

µ(2kQ)

l(2kQ)n,

where NQ,R is the smallest positive integer k such that

l(2kQ) ≥ l(R)(> l(2k−1Q)

).

• For example, if µ is the d-dimensional Lebesgue measure,then KQ,R ≈ 1 + log2

l(R)l(Q) . Moreover, if Q and R are dyadic

cubes, l(Q) = 2k and l(R) = 2k+s with s ∈ N, thenKQ,R = 1 + s.


Atomic Hardy space H1, patb,γ(µ)

Defn 3 Let p ∈ (1,∞], ρ > 1 and γ ∈ N. A function b ∈ L1loc(µ)

is called a (p, γ)-atomic block if

(1) there exists some cube R such that supp (b) ⊂ R,

(2)∫

Rd b dµ = 0,

(3) for j = 1, 2, there are functions aj supported on cubeQj ⊂ R and numbers λj ∈ R such that b = λ1a1 + λ2a2,and

‖aj‖Lp(µ) ≤ [µ(ρQj)]1/p−1

K−γQj , R

.

Then we denote

|b|H1, patb,γ(µ) = |λ1| + |λ2|.


Atomic Hardy space H1, patb,γ(µ) (2)

Defn 4 We say that f ∈ H1, patb,γ(µ) if there are (p, γ)-atomic

blocks bii∈N such that

f =∞∑

i=1

bi

with∑∞

i=1 |bi|H1, patb,γ(µ) < ∞. The H

1, patb,γ(µ) norm of f is

defined by

‖f‖H1, patb,γ(µ) = inf

∑

i

|bi|H1, patb,γ(µ)

.


New characterization of H1(µ)

• Let H1, patb (µ) = H

1, patb, 1(µ). Tolsa proved that H

1, patb (µ) = H1(µ)

for all p ∈ (1,∞] with equivalence of norms, namely, thecase γ = 1 of the following theorem.

Thm 1 For any integer γ ∈ N and any p ∈ (1,∞],

H1, patb,γ(µ) = H

1, patb (µ) = H1(µ)

with an equivalent norm.

• Denote by Q the smallest doubling cube which contains Q

and has the same center as Q.


The space RγBMO (µ)

Defn 5 Let γ ∈ N and ρ > 1. A function f ∈ L1loc(µ) is in

RγBMO (µ) if and only if there exists some constant C > 0

such that for any cube Q centered at some point of supp (µ),

1

µ(ρQ)

∫

Q|f(y) − m

Q(f)| dµ(y) ≤ C

and|mQ(f) − mR(f)| ≤ CK

γQ,R

for any two doubling cubes Q ⊂ R, where

mQ(f) =1

µ(Q)

∫

Qf dµ.


Dual space

Thm 2 For any integer γ ∈ N,

H1(µ)∗ = RγBMO (µ).

• When γ = 1, denote RγBMO (µ) simply by RBMO(µ). Thm2 with γ = 1 was obtained by Tolsa.


3. Calderón-Zygmund operator

Let K be a function on Rd × R

d \ (x, y) : x = y satisfyingthat for x 6= y,

(3.1) |K(x, y)| ≤ C|x − y|−n,

and for |x − y| ≥ 2|x − x′|,

(3.2) |K(x, y)−K(x′, y)|+|K(y, x)−K(y, x′)| ≤ C|x − x′|δ

|x − y|n+δ,

where δ ∈ (0, 1] and C > 0 is a constant.


C-Z operator (2)

• The C-Z operator associated to the above kernel K and themeasure µ is formally defined by

(3.3) Tf(x) =

∫

Rd

K(x, y)f(y)dµ(y).

But, this integral may not be convergent for many functions.

• The truncated operators Tε for ε > 0 is defined by

Tεf(x) =

∫

|x−y|>εK(x, y)f(y)dµ(y).

• T is bounded on L2(µ) if the operators Tεε>0 arebounded on L2(µ) uniformly on ε > 0.


C-Z operator (3)

• Suppose T is bounded on L2(µ). Let T be the weak limit asε → 0 of some subsequence of operators Tεε>0. T is stillbounded on L2(µ); moreover, for f ∈ L2(µ) with compactsupport and a. e. x ∈ R

d \ supp (f),

(3.4) T f(x) =

∫

Rd

K(x, y)f(y) dµ(y)

with the same K as before. T is also bounded from L1(µ)

into weak-L1(µ) and from H1(µ) into L1(µ).


The operator T

• T ∗1 = 0 means that for any bounded function b withcompact support and

∫Rd b dµ = 0,

(3.5)

∫

Rd

T b(x) dµ(x) = 0.

(For a such function b, b ∈ H1(µ) and therefore, T b ∈ L1(µ).Also, if T b ∈ H1(µ), then (3.5) is also necessary.)

Thm 3 Let K be the function on Rd × R

d \ (x, y) : x = y

satisfying (3.1) and (3.2). Suppose that the operator T in (3.4)is bounded on L2(µ) and T ∗1 = 0 as in (3.5), then T isbounded on H1(µ).(The key ingredient of the proof is Thm 1 with γ > 1.)


4. Commutator Tb

• Let b ∈ RBMO(µ). The commutator Tb is defined by

Tb(f) = bT f − T (bf).

Thm 4 Let b ∈ RBMO(µ). Then Tb is bounded from H1(µ) toL1,∞(µ), that is, there exists a constant C > 0 such that forall λ > 0 and all functions f ∈ H1(µ),

µ(

x ∈ Rd :

∣∣∣Tbf(x)∣∣∣ > λ

)≤ C‖b‖RBMO(µ)λ

−1‖f‖H1(µ).

(Again, the key ingredient of the proof is Thm 1 with γ > 1.)


Hörmander condition

• If we replace (3.2) of the kernel K by the followingHormander condition that

supR>0

|x−x′|<R

∞∑

l=1

l

∫

2lR<|x−y|≤2l+1R

(|K(x, y) − K(x′, y)|

+|K(y, x) − K(y, x′)|)

dµ(y) < ∞,

Thm 4 is still true.

——————————Recall

(3.2) |K(x, y)−K(x′, y)|+|K(y, x)−K(y, x′)| ≤ C|x − x′|δ

|x − y|n+δ


5. An interpolation theorem

• an interpolation theorem and its application to theboundedness of the commutator Tb.

• Sharp maximal operator (Tolsa):

M ]f(x) = supQ3x

1

µ(32Q)

∫

Q

∣∣∣f(y) − m

Q(f)

∣∣∣ dµ(y)

+ supx∈Q⊂R

Q, R doubling

|mQ(f) − mR(f)|

KQ,R.

—————Recall:

M ]f(x) = supQ3x

1

µ(Q)

∫

Q

∣∣f(y) − mQ(f)∣∣ dµ(y).


An interpolation theorem (2)

Thm 5 Let 1 < p0 < ∞, T1 and T2 be two sublinear operators.Suppose that

(i) T1 is bounded from H1(µ) to L1,∞(µ), that is, there is apositive constant C such that for any λ > 0,

µ(

x ∈ Rd : |T1f(x)| > λ

)≤

C

λ‖f‖H1(µ);

(ii) T2 is bounded on Lp0(µ);

(iii) there is a positive constant B such that for any functionf ∈ L∞(µ) ∩ Lp0(µ),

M ](T1f)(x) ≤ T2f(x) + B‖f‖L∞(µ).

Then T1 is bounded on Lp(µ) for any 1 < p < p0.A Characterization of Hardy Space with Non-doubling Measures – p. 21/23

Commutator Tb

Thm 6 Let b ∈ RBMO(µ), and Tb be the same as in Thm 4.Suppose that for some fixed p0 with 1 < p0 < ∞, themaximal operator T ∗ defined by

T ∗f(x) = supε>0

∣∣∣∫

|x−y|>εK(x, y)f(y)dµ(y)

∣∣∣

= supε>0

∣∣∣Tε(f)(x)∣∣∣

is bounded on Lp0(µ). Then the commutator Tb is alsobounded on Lp(µ) provided that 1 < p < p0.


Some remarks

• Is it possible to give a Littlewood-Paley characterization forH1(µ)?

• Nothing is known so far on Hardy spaces Hp(µ) whenp < 1?

• The same questions are for Besov and Triebel-Lizorkinspaces when p ≤ 1?


Documents

A Characterization of Hardy Space with Non-doubling Measurescfa/notes/eisenach_yang_3.pdf · X. Tolsa, The space H1 for nondoubling measures in terms of a grand maximal operator,