DECOMPOSITION THEOREMS FOR HARDY SPACES ON CONVEX DOMAINS OF FINITE TYPE

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DECOMPOSITION THEOREMS FOR HARDY SPACES ON CONVEX DOMAINS OF FINITE TYPE

Sandrine Grellier, Marco Peloso

To cite this version:

Sandrine Grellier, Marco Peloso. DECOMPOSITION THEOREMS FOR HARDY SPACES ON CONVEX DOMAINS OF FINITE TYPE. Illinois Journal of Mathematics, 2002, 46, pp.207-232.

�hal-00076918�

CONVEX DOMAINS OF FINITE TYPE

SANDRINE GRELLIER AND MARCO M. PELOSO

Abstract. In this paper we study the holomorphic Hardy spacesH^p(Ω), where Ω is a convex domain of finite type in Cⁿ. We show that for 0 < p ≤ 1, the spaceH^p(Ω) admits an atomic decomposition. More precisely, we prove that each f ∈ H^p(Ω) can be written asf =PS(P∞

j=0νjaj) =P∞

j=0νjPS(aj), wherePS is the Szeg¨o projection and thea_j’s are real variablep-atoms on the boundary∂Ω andP∞

j=0|νj|^p <

∼ kfk^p_Hp(Ω).

Moreover, we prove the following factorization theorem. Eachf ∈ H^p(Ω) can be written as f =P∞

j=0fjgj, wherefj ∈ H^2p, gj ∈ H^2p, andP∞

j=0kfjk_H2pkgjk_H2p

<∼ kfk_Hp(Ω).

Finally, we extend these theorems to a class of domains of finite type that includes the strongly pseudoconvex domains and the convex domains of finite type.

Introduction

Let Ω be a smoothly bounded domain in Cⁿ. For 0 < p≤ ∞, let L^p(Ω) denote the Lebesgue space with respect to the volume form,L^p(∂Ω) be the Lebesgue spaces on∂Ω with respect to the induced surface measure dσ and, for 0< p ≤1, H^p(∂Ω) be the real-variable Hardy spaces on ∂Ω.

We let H^p(Ω) denote the Hardy space of holomorphic functions on Ω, with norm given by

kfk^p_Hp(Ω) := sup

0<ε<ε0

Z

δ(w)=ε

|f(w)|^pdσε(w),

whereδ(w) is the distance fromwto∂Ω anddσ_εdenotes the surface measure on the manifold {δ(w) = ε}. To any f ∈ H^p(Ω) corresponds a unique boundary function in L^p(∂Ω), that we still denote by f, obtained as normal almost everywhere limit, [St1]. Thus, we may identifyH^p(Ω) with a closed subspace of L^p(∂Ω).

1991 Mathematics Subject Classification. 32A37 47B35 47B10 46E22.

Key words and phrases. Hardy spaces, atomic decomposition, finite type domains, convex domains.

This work was done within the project TMR Network “Harmonic Analysis”. We thank the European Commission and the mentioned Network for the support provided.

1

The Hilbert space orthogonal projection P_S of L²(∂Ω) ontoH²(Ω) is given by the Szeg¨o projection

P_Sf(z) = Z

∂Ω

f(ζ)S_Ω(z, ζ)dσ(ζ), whereSΩ(z, ζ) is the Szeg¨o kernel.

When Ω is a smoothly bounded convex domain of finite type, there exists a natural pseudo-distance d_b on ∂Ω (see [Mc]) which makes ∂Ω into a space of homogenous type. The real-variable Hardy space H^p(∂Ω), 0 < p ≤ 1, is defined as a space of distributions on ∂Ω, in terms of atoms, in the following sense. Each distribution f ∈H^p(∂Ω) can be written asf =P∞

j=0ν_ja_j, where {ν_j} ∈`^p, thea_j’s arep-atoms and the series is assumed to converge in the sense of distributions (see Section 1 for the precise definition).

In this paper we prove that, for 0 < p≤ 1, the Hardy space H^p(Ω) continuously embeds into H^p(∂Ω). In other words, every f ∈ H^p(Ω) has boundary values that belong toH^p(∂Ω), so that it admits an atomic decomposition.

To be more precise, for g a distribution on ∂Ω,PS(g) is the holomorphic function in Ω defined for z ∈Ω by

PS(g)(z) = hg, SΩ(z,·)i

where S_Ω(z, ζ) denotes the Szeg¨o kernel, which is C^∞(∂Ω) in the second variable whenever z ∈Ω.

We prove that any f ∈ H^p(Ω) can be written as P_S(P

ν_ja_j) where P

ν_ja_j is a distribution that belongs to H^p(∂Ω). Moreover, it holds that P_S(P

ν_ja_j) = Pν_jP_S(a_j), with equality in the H^p(Ω)-sense, that is, the series converges to f in theH^p(Ω)-norm.

On the other hand, we prove that the Szeg¨o projection PS maps continuously H^p(∂Ω) into H^p(Ω), for 0< p≤1.

Moreover, we prove that on H^p(Ω) a (weak) factorization theorem holds true.

More precisely, we prove that, given anyf ∈ H^p(Ω) there existf_j, g_j ∈ H^2p(Ω) such thatf =P∞

j=0f_jg_j andP∞

j=0kf_jk_H^2pkg_jk_H^2p ≤ckfk_H^p_(Ω), withcindependent of f.

Finally, in Section 8 we extend the above results to the H-domains, a class of smooth, bounded domains of finite type that includes the strongly pseudoconvex domains and the convex domains. Such domains are a natural extension of the mentioned domains, and were studied in [BPS3].

These results extend classical results to the case of convex domains of finite type, and more generally to the case of H-domains. The atomic decomposition of the (holomorphic) Hardy spaces was first proved in the case of the unit ball by Garnett and Latter [GL], and later extended to strongly pseudoconvex domains and domains of finite type inC² by Krantz and Li [KL2], and independently by Dafni [D]. Related results about duality between H¹ and BMO appear in [KL1] for strongly pseudo- convex domains and domains of finite type in C² and in [KL3] for convex domains of finite type. The factorization theorem is classical in dimension 1, while in several

variables it was first proved by Coifman, Rochberg and Weiss [CRW], in the case of the unit ball, for the space H¹(Ω). This theorem was extended to the case of strongly pseudoconvex domains in [KL2] for all H^p(Ω), 0 < p ≤ 1 and to convex domains of finite type in [BPS2] forH¹(Ω).

Applications of these results to the regularity properties of small Hankel operators appear, to name a few, in [CRW], [KL2] and [BPS2]. Another classical characteriza- tion of Hardy spaces, the area integral one, is proved to hold true in [KL4] in convex domains of finite type in Cⁿ. Finally, we mentioned that the optimal approach re- gions for a Fatou-type theorem for H^p functions are described in [DFi], in the case of a convex domain of finite type.

The geometry of convex domains of finite type was first described by McNeal [Mc]. Those results were later applied to the analysis of the mapping properties of the Bergman projection [McS1] and Szeg¨o projection [McS2] by McNeal and Stein.

In the case of strongly pseudoconvex domains and finite type domains in C² the geometry was determined by canonical vector fields. The natural quasi-distance on ∂Ω was the control distance determined by these vector fields, i.e. the Carnot metric. The situation of convex domains of finite type is essentially more general.

The “weight” of each vector field may vary from point to point, and one needs to take into consideration the different order of contact of complex lines with∂Ω. For these reasons it is natural to consider adiameter function τ(ζ, λ, r), which gives the diameter of the largest one-dimensional disc in the direction of λ, that fits inside the region{z⁰ : %(z⁰)< r}. Here % denotes a fixed smooth defining function for Ω.

As a consequence, in order to define the cancellation property for p-atoms for small values of p, we need to consider the pairing between f ∈ H^p(Ω) and smooth bump functions whose derivatives in all tangential directions can be controled in terms of the diameter function.

We remark that, althought the main lines of the proof of the atomic decomposition are standard, some of arguments have been greatly simplified with respect to the classical ones. On the other hand in order to prove the factorization theorem we adapt an idea from [BPS2], where the result was proved in the case p = 1. The proof does neither rely on the explicit expression of the Szeg¨o kernel, as in [CRW], nor its asyptotic expansion, [KL4]. Instead, it based on a recent result by Diederich and Fornæss on the existence of support functions on convex domains of finite type.

We use the notation A <∼ B to indicate thatA≤c·B where the constantc does not depend on the important parameters on which the functionsA and B depend.

(Tipically, the constant c will only depend on the geometry of the domain Ω.) We use the symbols >

∼ and ≈ analogously.

1. Basic facts and notation

Let Ω be a smoothly bounded convex domain in Cⁿ. A point ζ ∈∂Ω is said to be of finite type if the order of contact of complex lines with∂Ω at this point is finite, see [BS] and references therein. The type of the point is the least upper bound of the various orders of contact. We say that Ω is of finite type MΩ if every point on

∂Ω is of finite type≤M_Ω.

Let Ω = {z ∈Cⁿ : ρ(z)<0}. There exists ε₀ >0 such that for |ε| ≤ ε₀ the sets Ω_ε ={z ∈Cⁿ : ρ(z)< ε} are all convex, and the normal projection π : U →∂Ω is well defined and smooth, whereU ={z ∈Cⁿ: δ(z)< ε₀}.

The basic geometric facts about convex domains of finite type were first proved by McNeal [Mc], see also [McS1], [McS2] and [DFo]. By recalling the results that are involved in the present work we take the opportunity to review the main elements of the construction and set some notation.

For z ∈U and λ∈ Cⁿ a unit vector, we denote by τ(z, λ, r) the distance from z to the surface{z⁰ : %(z⁰) = %(z) +r} along the complex line determined by λ.

For eachz ∈U andr < ε₀there exists a special set of coordinates{w₁^z,r, . . . , w_n^z,r}, that we callr-extremal. The first vector v⁽¹⁾ is given by the direction transversal to the boundary, in the sense that the shortest distance fromz to the set{z⁰ : %(z⁰) =

%(z) +r} is realized in the complex line determined by v⁽¹⁾.

The vector v⁽²⁾ is chosen among the vectors orthogonal to v⁽¹⁾ in such a way that τ(z, v⁽²⁾, r) is maximal. We repeat the same process until we determine an orthonormal basis {v⁽¹⁾, . . . , v⁽ⁿ⁾}. We denote by (w1, . . . , wn) the coordinates with respect to this basis. Notice that these coordinates (w₁, . . . , w_n) = (w₁^z,r, . . . , w_n^z,r) depend on z and r. However, the transversal direction w₁ does not depend on r.

For k = 1, . . . , n we set

(1) τ_k(z, r) =τ(z, v^(k), r), and define the polydiscQ(z, r)

(2) Q(z, r) ={w: |wk|< τk(z, r), k = 1, . . . , n}.

Basic relations among these quantities are the following, see [McS2] Prop. 1.1 and also [BPS2] Lemma 2.1.

Proposition 1.1. There exists a constantC > 0depending only on Ωsuch that for any unit vector λ∈Cⁿ, 0< r≤ε₀, z ∈U, and 0< η <1 we have:

(i) η^1/2τ(z, λ, r) <

∼ τ(z, λ, ηr) <

∼ η^1/MΩτ(z, λ, r);

(ii) η^1/2Q(z, C⁻¹r)⊂Q(z, ηr)⊂η^1/MΩQ(z, Cr);

(iii) if w∈Q(z, δ) then τ(z, λ, r)≈τ(w, λ, r).

We define the quasi-distance d_b :U×U by setting (3) d_b(z, w) = inf{δ: w∈Q(z, δ)},

and the function d

(4) d(z, w) =d_b(z, w) +δ(z) +δ(w).

Notice that d is initially defined onU ×U and we extend it to Cⁿ×Cⁿ by setting d(z, w) = ψ(%(z))ψ(%(w))d(z, w) + (1−ψ(%(z)))(1−ψ(%(w)))|z−w|,

where ψ is a smooth cut-off function on R such that ψ(t) = 1 for |t| ≤ ε₀/2 and ψ(t) = 0 for |t| ≥ε₀.

On the boundary we will use a family of “balls” centered at ζ ∈ ∂Ω of radius δ defined as

B(ζ, δ) = Q(ζ, δ)∩∂Ω.

For any unit vector λ we introduce the differential operator (5) L_λ = (∂_λ%)∂_x1 −(∂_x1%)∂_λ,

wherew₁ =x₁+iy₁ is the transversal direction fixed earlier. Here,∂_λ is the standard vector field defined byλas∂_λf =hλ, dfi, for a smooth functionf, its real differential df, and where h,i denotes the usual pairing between a one-form and a vector.

Notice that Lλ is always a tangential vector field. Ifλ ∈S²ⁿ⁻¹ is itself tangent to

∂Ω, thenLλ is the directional derivative in the direction λ.

For Λ = (λ₁, . . . , λ_q) aq-list of vectors in S²ⁿ⁻¹ and µ= (µ₁, . . . , µ_q) aq-index we set|µ|=µ₁+· · ·+µ_n,

(6) L^µ_Λ =L^µ_λ¹

1· · ·L^µ_λ^q

q, and

(7) τ^µ(z,Λ, δ) = τ(z, λ₁, δ)^µ1· · ·τ(z, λ_q, δ)^µq.

Finally we recall the fundamental estimates for the Szeg¨o kernel and its derivatives [McS2] (called interior estimates ofS-type, see [McS2] Def. 4 and Thm. 3.6)

(8)

L^µ_Λ,zL^µ_Λ⁰0,z⁰S_Ω(z, z⁰) <

∼

τ^−µ(z,Λ, δ)τ^−µ0(z⁰,Λ⁰, δ) σ B(π(z), δ) ,

whereδ =d(z, z⁰), z, z⁰ ∈Ω×Ω\∆_∂Ω, ∆_∂Ω denotes the diagonal on ∂Ω.

We conclude this section with the definition of the real-variable Hardy spaces.

We need first to introduce the notion ofp-atoms. Let ζ0 ∈∂Ω,r0 < ε0 and N be a positive integer. On C^∞(B(ζ0, r0)) we introduce the norm

(9) kφkS_N(B(ζ0,r0))= sup

Λ

X

|µ|=N

kL^µ_Λφk_L^∞_(B(ζ0,r0))τ^µ(ζ₀,Λ, r₀).

Definition 1.2. We set

`₀ =

(1−1/p)(M_Ω+ 2n−2)

, and N_p =`₀+ 1,

where [x] denotes the integral part of x. A measurable function a on ∂Ω is called a p-atom if either it is the constant function on ∂Ω equal to σ(∂Ω)^−1/p, or if:

(i) suppa⊆B(ζ₀, r₀);

(ii) |a(ζ)| ≤σ B(ζ₀, r₀)^−1/p

; (iii) R

∂Ωa(ζ)dσ(ζ) = 0;

(iv) for allφ ∈ C^∞ B(ζ₀, r₀)

we have

| Z

∂Ω

a(ζ)φ(ζ)dσ(ζ)| ≤ kφk_{SNp(B(ζ0,r0))}σ B(ζ₀, r₀)^1−1/p .

Notice that the above condition (iv) replaces the classical higher moment condi- tion. It is in the same spirit as the analog condition in [KL2]. The difference here is given by the choice of the normk · kS_Np.

Real-variable Hardy spaces. Let 0 < p ≤ 1. The real Hardy space H^p(∂Ω) is the space of distributionsf on∂Ω which can be written as

(10) f =

∞

X

j=0

ν_ja_j,

where P

|ν_j|^p < ∞, the a_j’s are p-atoms and the series is assumed to converge in the sense of distributions.

With a standard abuse of notation, the “norm” onH^p(∂Ω) is defined as kfk^p_Hp = inf{P

j|ν_j|^p : f =P

jν_ja_j}. Setting d(f, g) = kf −gk^p_Hp we see that H^p(∂Ω) is a complete metric space. This implies that the series in (10) converges innorm. This is in fact obvious sincekPm2

j=m1ν_ja_jk^p_Hp ≤Pm2

j=m1|ν_j|^p which tends to 0 asm₁ → ∞.

This implies that P∞

j=1ν_ja_j converges in norm, hence in the sense of distributions, necessarly to f¹.

We point out that this definition ofH^p(∂Ω) is consistent with the definition given in [CW] in the case of a space of homogenous type for values ofpclose to 1, see also [KL5].

2. Statement of the main results The main results of the present work are the following.

Theorem 2.1. Let Ω be a smoothly bounded domain of finite type. Let 0< p ≤1.

Then there exists a constant c depending only on p and Ω such that the following holds. Given any f ∈ H^p(Ω) there exist constants ν_j and p-atoms a_j such that P∞

j=0ν_ja_j ∈H^p(∂Ω) and f =P_S

^∞ X

j=0

ν_ja_j

=

∞

X

j=0

ν_jP_S(a_j),

1In order to clarify one point on which there is a little bit of confusion in the litterature, we remark that, in [MTW] p. 513 it is shown that a generic H^p-function f infinite sum of atoms cannot be written as finite sum of atomsPN

1 νjbj, withPN

j=1|νj|^p ≈ kfk^p_Hp. This fact does not of course contradicts the norm convergence of the series in (10).

and moreover

∞

X

j=0

|ν_j|^p ≤ckfk^p_Hp(Ω).

Theorem 2.2. Let Ω be a smoothly bounded domain of finite type. Let 0< p ≤ 1 and H^p(∂Ω) be the real-variable Hardy space on∂Ω. Then

P_S :H^p(∂Ω)→ H^p(Ω) is bounded.

Finally, we have the factorization theorem.

Theorem 2.3. Let Ω be a smoothly bounded domain of finite type. Let 0< p ≤1, 1 < q < ∞ and q⁰ be its conjugate exponent. There exists a constant c depending only on p, q and Ω such that the following holds. Given any f ∈ H^p(Ω) there exist fj ∈ H^pq, gj ∈ H^pq0, j = 1,2, . . . such that

f =

∞

X

j=0

f_jg_j

and ∞

X

j=0

kf_jkH^pqkg_jk_Hpq0 ≤ckfkH^p(Ω).

We remark that this theorem was proved in [BPS2] for p = 1, by a different, more indirect, method. Our proof relies on the atomic decomposition obtained in Theorem 2.1.

The above theorems are valid on a class of smoothly bounded finite type domains which includes the convex domains as well as the strongly pseudoconvex domains.

These domains are introduced in [BPS3] and are calledH-domains. We will discuss the extension of these theorems to theH-domains in the last section.

3. Proof of Theorem 2.2 Let f =P∞

j=0νjaj ∈H^p(∂Ω). By definition, P_S

^∞ X

j=0

ν_ja_j

(z) =h

∞

X

j=0

ν_ja_j, S_Ω(z,·)i

=

∞

X

j=0

hν_ja_j, S_Ω(z,·)i

=

∞

X

j=0

ν_jP_S(a_j)(z),

since the serie converges in the sense of distributions. It remains to prove that this last term belongs toH^p(Ω), with norm controlled bykfk_H^p_(∂Ω).

We claim that there exists C > 0 such that kP_S(a)k_H^p_(Ω) ≤ C for any p-atom a.

From this it then follows that, for anym₁, m₂ ∈N,m₁ ≤m₂, k

m2

X

m1

ν_jP_S(a_j)k^p_Hp ≤C^p

m2

X

m1

|ν_j|^p

so that, by the assumption on {ν_j} and the completeness of H^p(Ω), one gets that P_S(f) = P∞

j=0ν_jP_S(a_j) belongs to H^p(Ω). Moreover, kP_S(f)k^p_Hp <

∼ P

j|ν_j|^p when- ever f =P

jνjaj, which gives the desired estimate.

Thus, we only need to estimate kP_S(a)kH^p for any p-atom a. Let ε >0. Then, Z

∂Ωε

|P_S(a)(ζ)|^pdσ_ε(ζ) = Z

∂Ω

|P_S(a)(ζ−εν(ζ))|^pdσ(ζ)

= Z

B(ζ0,2δ)

+ Z

cB(ζ0,2δ)

|P_S(a)(ζ_ε)|^pdσ(ζ)

=I+II,

whereν(ζ) denotes the outward unit normal atζ ∈∂Ω and we writeζ_ε =ζ−εν(ζ).

Since p < 2, I ≤

Z

B(ζ0,2δ)

|P_S(a)(ζ_ε)|²dσ(ζ) p/2

·σ B(ζ₀,2δ)1−p/2

≤ckak^p_L2(∂Ω)σ B(ζ₀,2δ)^1−p/2

≤c,

since kak_L²_(∂Ω) ≤σ B(ζ₀,2δ)1/2−1/p

and P_S maps L²(∂Ω_ε) into L²(∂Ω) with norm independent ofε, as a consequence of theT(1)-theorem of David and Journ´e and of the results in [McS2].

Next, set E_k={ζ ∈∂Ω : 2^kδ≤d(ζ, ζ₀)≤2^k+1δ}. Then, II =

∞

X

k=0

Z

E_k

|P_S(a)(ζ−εν(ζ))|^pdσ_ε(ζ), and

|P_S(a)(ζ_ε)|=| Z

∂Ω

S_Ω(ζ_ε, w)a(w)dσ(w)|

≤ kS_Ω(ζ_ε,·)k_{SNp(B(ζ0,δ))}·σ B(ζ₀,2δ)^1−1/p . Recall that the Szeg¨o kernel satisfies the estimate (8), so that

|L^µ_ΛSΩ(ζε, w)| <

∼

τ^−µ(w,Λ, d(ζ_ε, w)) σ(B(w, d(ζ_ε, w))) ,

and notice that forζ ∈E_k and w∈B(ζ₀, δ),

d(ζ_ε, w) =ε+d_b(ζ, w)≥ε+d_b(ζ, ζ₀)−d_b(ζ₀, w)

≥ε+ 2^kδ−δ≥δ2^k−1. Therefore, for ζ ∈E_k,

kS_Ω(ζ_ε,·)kS_Np(B(ζ0,δ)) = sup

Λ

X

|µ|=N_p

kL^µ_ΛS_Ω(ζ_ε,·)k_L^∞_(B(ζ0,δ))·τ^µ(ζ₀,Λ, δ)

= sup

Λ

X

|µ|=Np

sup

w∈B(ζ₀,δ)

τ^−µ(w,Λ, d(ζ_ε, w))

σ(B(w, d(ζ_ε, w))) ·τ^µ(ζ0,Λ, δ)

∼< sup

Λ

X

|µ|=Np

sup

w∈B(ζ0,δ)

τ^µ(ζ0,Λ, δ)

τ^µ(w,Λ, δ2^k−1) · 1

σ(B(w, δ2^k−1))

∼< sup

Λ

X

|µ|=Np

τ^µ(ζ₀,Λ, δ)

τ^µ(ζ₀,Λ, δ2^k−1) · 1

σ(B(w, δ2^k−1))

∼<

X

|µ|=N_p

2^{−k|µ|/MΩ} 1

σ(B(w, δ2^k−1)). Going back to the estimation of II,

II =

∞

X

k=1

Z

Ek

|P_S(a)(ζ)|^pdσ_ε(ζ)

∼<

X

k

kS_Ω(ζ_ε,·)k^p_S

Np(B(ζ0,δ))σ B(ζ₀, δ)p−1

σ(E_k)

∼<

X

k

X

|µ|=Np

2^{−kp|µ|/MΩ} σ(B(ζ₀, δ))^p−1 σ(B(w, δ2^k))^p−1

∼<

X

k

X

|µ|=Np

2^−k

(|µ|/M_Ω)p+(1+(2n−2)/M_Ω)(p−1)

≤c,

since the term in brackets in the exponent is positive, due to our choice ofN_p. 2 4. Maximal functions and a partition of unity

In the proof of the atomic decomposition of H^p(Ω) we are going to use some maximal operators, that now we introduce. These operators are standard variants of classical ones, see [FS], [St2], and also [KL2].

Given ζ ∈∂Ω we define the approach region A_γ(ζ) as the subset of Ω given by A_γ(ζ) = {z ∈Ω : d(ζ, π(z))< γδ(z)}.

We define the non-tangential maximal function f_γ^∗

(11) f_γ^∗(ζ) = sup

z∈A(ζ)

|f(z)|, and the tangential variant

(12) f_N^∗∗(ζ) = sup

w∈Ω

δ(w) δ(w) +d(ζ, π(w))

N

|f(w)|.

We consider a space of smooth bump functions at ζ:

K^M_γ (ζ) ={g ∈ C^∞(∂Ω) : suppg ⊆B(ζ0, t0), with ζ0 ∈ Aγ(ζ) and kgkM,ζ0,t0 ≤1}, where

(13) kgk_M,ζ0,t0 = sup

Λ,|µ|≤M

τ^µ(ζ₀,Λ, t₀)σ(B(ζ₀, t₀))kL^µ_Λgk_L^∞_(B(ζ0,t0)).

Following [McS2] Def. 2, we say that a function ψ is a smooth bump function of orderN on B(ζ₀, t₀)⊆∂Ω ifψ ∈ C^∞(B(ζ₀, t₀)) and

sup

Λ

|L^µ_Λψ(z)|τ^µ(ζ₀,Λ, t₀)≤C_ψ,

for all z ∈B(ζ0, t0) and|µ| ≤N. If Cψ = 1, ψ is called a normalized smooth bump function of orderN.

The grand maximal function is defined as

(14) Kγ,M(f)(ζ) = sup

g∈K_γ^M(ζ)

Z

∂Ω

f(w)g(w)dσ(w) .

Lemma 4.1. With the definitions above, there exist c = c(Ω) and N = N(γ, M) such that

K_γ,Mf(ζ) <

∼ f_cγ^∗(ζ) +f_N^∗∗(ζ).

Proof. We wish to estimate |R

∂Ωf(w)g(w)dσ(w)| for g ∈ K^M_γ (ζ). Given such a g, there existζ0andt0such that suppg ⊆B(ζ0, t0) andd(ζ, ζ0)< γt0, i.e. ζ0−t0ν(ζ0)∈ Aγ(ζ). Using the holomorphicity of f by integration by parts, see Lemma 6.5 in [BPS1], we see that there exists a differential operatorY_k+1 with smooth coefficients, of orderk+ 1 such that

| Z

∂Ω

f(w)g(w)dσ(w)|=| Z

Ω

f(w)Y_k+1g˜(w)δ^k(w)dV(w)|, where ˜g is a smooth extension of g, say

˜ g(w) =

(g(π(w))g₁(%(w)) if |%(w)|<2t₀

0 otherwise,

and g₁ ∈ C₀^∞ [−2t₀,2t₀]

, g₁(t) = 1 if |t| ≤ t₀/2. Here k is a positive integer to be selected later.

Notice that |g₁^(j)(t)| ≤cjt^−j₀ and hence

|Y_k+1g(w)|˜ <

∼

k+1

X

j=0

1

t^j₀ sup

Λ,|µ|=k+1−j

kL^µ_Λgk_L^∞_(B(ζ0,t0))

<∼

k+1

X

j=0

1

t^j₀ sup

Λ,|µ|=k+1−j

1

τ^µ(ζ₀,Λ, t₀) · 1 σ(B(ζ₀, t₀))

<∼ 1

t^k+1₀ · 1 σ(B(ζ₀, t₀)). Now we write,

| Z

∂Ω

f(w)g(w)dσ(w)| ≤ Z

Ωt0

|f(w)Y_k+1g(w)|δ˜ ^k(w)dV(w) +

Z

Ω\Ωt0

|f(w)Y_k+1g(w)|δ˜ ^k(w)dV(w)

=I+II.

Notice that w ∈ supp ˜g implies that d(π(w), ζ) ≤ cγt₀. So, if w ∈ Ω_t0 ∩supp ˜g thenw ∈ A_cγ(ζ). Hence,

I <∼ f_cγ^∗(ζ) Z

B(ζ0,t0)

Z 2t0

t0

t^k

t^k+1₀ σ(B(ζ0, t))dtdσ(w)

<∼ f_cγ^∗(ζ).

Moreover, since

|f(w)| ≤f_N^∗∗(ζ)

1 + d(π(w), ζ) δ(w)

N

<∼ f_N^∗∗(ζ)

1 + t₀ δ(w)

N

, we have

II <∼ Z

B(ζ0,t0)

Z ∞

t0

|f(w)| t^k

t^k+1₀ σ(B(ζ₀, t₀))dtdσ(w)

<∼ Z ∞

t0

f_N^∗∗(ζ)

1 + t₀ t

N

t^k t^k+1₀ dt

<∼ f_N^∗∗(ζ), if k−N <−1. 2

The next two lemmas are classical, and they hold on any smoothly bounded domain. The first one is due to Stein, see [St1], Sec. 9. The second one is a version of a result of Fefferman and Stein, [FS] Lemma VI.1.

Lemma 4.2. Let Ω be any smoothly bounded domain, 0< p≤ ∞. Then kf_γ^∗k_L^p_(∂Ω) <

∼ kfkH^p(Ω).

Lemma 4.3. Let Ω be any smoothly bounded domain, 0 < p ≤ ∞. Then, for N large enough,

kf_N^∗∗k_L^p_(∂Ω) <

∼ kfkH^p(Ω).

We now introduce a smooth partition of unity on any open set O ⊆∂Ω.

Lemma 4.4. Let O ⊆ ∂Ω be an open set. Then there exist a collection of balls B_i =B(ζ_i, r_i), functionsφ_i ∈ C^∞(∂Ω), i= 0,1,2, . . ., and constantsα >1> β >0, depending only onΩ, such that the following conditions hold:

(i) 0≤φ_i ≤1;

(ii) suppφ_i ⊆B_i; (iii) φ_i = 1 on _α¹B_i; (iv) P∞

i=0φi =χO;

(v) for eachithere existsζi ∈ ^cO such that, for any integerN we havecNφi/kφik_L¹

∈ K^N_α(ζ_i), for some c_N =c(N,Ω).

Proof. By [McS2] Prop. 1.9, given any ζ₀, δ > 0 and C > 1, there exist normalized smooth bump functions of orderN on B(ζ₀, Cδ), that are identically equal to 1 on B(ζ₀, δ).

Given this, the proof now proceeds as the proof of Lemma 4.3 in [KL2]. We give the details for sake of completeness.

Let {B(ζ_i, r_i)} be a sequence of balls satisfying:

(a) ∪iβB(ζi, ri) =O;

(b) B(ζi, ri)⊆ O, αB(ζi, ri)∩ ^cO 6=∅;

(c) _α¹B(ζ_i, r_i) are pairwise disjoint;

(d) no point in O lies in more than NΩ of the balls B(ζi, ri),

for some 0< β <1< α. Such a sequence exists, being a Whitney covering for O.

Given B_i =B(ζ_i, r_i) let ψ_i be a normalized smooth bump function supported in B_i, that equals 1 on _α¹B_i. By (d) above,

1≤X

i

ψ_i(ζ)≤N_Ω for all ζ ∈ O.

Now set

φ_i(ζ) = ψ_i(ζ)/X

j

ψ_j(ζ).

Then, (i)-(iv) are clearly satisfied. We only need to check (v).

By (b) above, let ξ_i ∈αB(ζ_i, r_i)∩ ^cO. Then ζ_i −r_iν(ζ_i) ∈ A_γ(ξ_i), suppφ_i ⊆ B_i, withγ >1, and

sup

Λ,|µ|≤N

τ^µ(ξ_i,Λ, r_i)σ B(ξ_i, r_i)

kL^µ_Λφ_ik_L^∞_(∂Ω) <

∼ c_Nσ B(ξ_i, r_i)

∼<cNkφik_L¹_(∂Ω).

5. Beginning of the proof of Theorem 2.1 We let k₀ to be the least integer such that

(15) kK_γ,M(f) +f_γ^∗k_L^p_(∂Ω) ≤2^k0. For a positive integer k define

(16) O_k={z ∈∂Ω : K_γ,Mf(z) +f_γ^∗(z)>2^k0+k}.

For each k we fix a Whitney covering and a partition of χO_k, {φ^k_i}, as in Lemma 4.4.

Letζ₀ ∈∂Ω and B(ζ₀, r₀) be fixed. The ball B(ζ₀, r₀) is contained in the polydisc Q(ζ₀, r₀). By the results in [Mc] there exists an r₀-extremal (orthonormal) basis {v⁽¹⁾, . . . , v⁽ⁿ⁾} on Q(ζ₀, r₀), where v⁽¹⁾ is transversal to the boundary. We denote by (w₁, . . . , w_n) the coordinates with respect to this basis and we defineV<sub>`</sub>(ζ<sub>0</sub>, r<sub>0</sub>) to be the space of polynomials of degree ≤` in Imw₁, w₂, w₂, . . . , w_n, w_n.

We remark that V_{`</sub>(ζ<sub>0</sub>, r<sub>0</sub>) does not depend on r<sub>0</sub> since v<sup>(1)</sup> does not depend onr<sub>0</sub>. Hence, we simply writeV<sub>`}(ζ₀).

Letφ₀ be a smooth bump function supported onB(ζ₀, r₀). We denote byL²_φ0(dσ) theL²-space with respect to the probability measure (φ0/kφ₀k_L¹)dσ. We letP_φ0 de- note the orthogonal projection ofL²_φ

0(dσ) ontoV_`(ζ₀) (which is obviously contained inL²_φ

0(dσ)).

In what follows, we will write w = s +it, s = (s₁, s⁰), t = (t₁, t⁰), s₁, t₁ ∈ R, s⁰, t⁰ ∈Rⁿ⁻¹, so that V<sub>`</sub>(ζ<sub>0</sub>) is the set of polynomials in s<sup>0</sup>, t<sub>1</sub>, t<sup>0</sup>, of degree ≤`.

Let {π_J} be an orthonormal basis for V<sub>`0</sub>, where `₀ = [(1−1/p)(M_Ω−2n−2)]

and J = (j⁰, j) = (0, j₂⁰, . . . , j_n⁰, j₁, . . . , j_n,),|J|=j₂⁰ +· · ·+j_n⁰ +j₁+· · ·+j_n. Lemma 5.1. Let π_J be as above. Then,

(i) |π_J(z)| ≤c_J on the support of φ₀;

(ii) |L^µ_Λπ_J(z)| ≤c_J,µτ^−µ(ζ₀,Λ, r₀) on the support ofφ₀.

Proof. We begin with (i). Recall that ζ₀ and r₀ are fixed. Then B := B(ζ₀, r₀) = Q(ζ₀, r₀)∩∂Ω, whereQ(ζ₀, r₀) is the polydisc centered atζ₀and polyradiusτ₁(ζ₀, r₀)

=r₀, τ₂(ζ₀, r₀), . . . , τ_n(ζ₀, r₀).

Let

π_J(w) = X

|γ⁰|+|γ|≤`0

a_γ⁰_,γs^γ0t^γ,

whereγ⁰ = (0, γ₂⁰, . . . , γ_n⁰),γ = (γ₁, γ₂, . . . , γ_n). Then, kπJk_L^∞_(B) ≤ sup

(s⁰,t)∈B

X

|γ⁰|+|γ|≤`0

aγ⁰,γs^γ0t^γ

= sup

(˜s⁰,˜t)∈B˜

X

|γ⁰|+|γ|≤`0

a_γ⁰_,γs˜^γ0t˜^γτ^γ0+γ(ζ₀, r₀) ,

where ˜s = (0, s⁰₂/τ2, . . . , s⁰_n/τn), ˜t = (t1/τ1, t2/τ2, . . . , tn/τn), τj = τj(ζ0, r0), and B˜ = (−1,1)²ⁿ⁻¹.

Hence, using the equivalence of all norms on finite dimensional vector spaces, we have

kπ_Jk_L^∞_(B)≤ sup

(s⁰,t)∈B

X

|γ⁰|+|γ|≤`₀

a_γ⁰_,γs^γ0t^γ

<∼

X

|γ⁰|+|γ|≤`₀

a_γ⁰_,γτ^γ0+γ(ζ₀, r₀)˜s^γ0˜t^γ L²( ˜B)

=σ(B)^−1/2

X

|γ⁰|+|γ|≤`0

a_γ⁰_,γs^γ0t^γ L²(B)

≈

X

|γ⁰|+|γ|≤`0

a_γ⁰_,γs^γ0t^γ L²_φ

0(B)

= 1.

Here we use the fact that, since φ₀ = 1 on B(ζ₀, r₀) and suppφ₀ ⊆ αB, kφ₀k_L¹ ≈ σ(B(ζ₀, r₀)). This proves (i).

It remains to prove the estimates on the derivatives of π_J. We are going to show that, for any (γ⁰, γ) such that |γ⁰|+|γ| ≤`0, there exists c^J_γ0,γ so that

(17) |aγ⁰,γ| ≤ c^J_γ0,γ

τ^γ0+γ(ζ₀, r₀).

Assuming this estimate for the moment, it follows that, for any β⁰, β with |β⁰|+

|β| ≤`₀,

∂_s^β0∂_t^βπ_J(w) =

X

γ⁰≥β⁰,γ≥β,|γ⁰|+|γ|≤`0

c_β⁰_,βa_γ⁰_,γs^γ0−β0t^γ−β

≤ cJ,γ⁰,γ

τ^γ0+γ(ζ₀, r₀) ·τ^{γ0−β0+γ−β}(ζ₀, r₀)

≤ c_J,γ⁰_,γ τ^β0+β(ζ₀, r₀).

The estimate for any derivative L^µ_Λ follows from the fact that any Lλ is a linear combination fo the ∂sj’s and ∂t_k’s, say Lλ =Pn

j=2aj∂sj +Pn

k=1bk∂t_k, and that 1

τ(ζ₀, λ, r₀) ≈

n

X

j=1

|aj|+|bj| τ_j(ζ₀, r₀), (herea₁ = 0), see [McS2] Prop. 1.1.

It remains to prove (17). Since the π_J’s are real-analytic they satisfy the mean- value property. In particular,

τ^γ0+γ(ζ₀, r₀)

∂_s^γ0∂_t^γπ_J(ζ₀)

≤ c σ(B(ζ₀, r₀))

Z

B(ζ0,r0)

|π_J(w)|dσ(w)≤c˜_J,

so that

|a_γ⁰_,γ|=

∂_s^γ0∂_t^γπ_J(ζ₀)

≤ c˜_J τ^γ0+γ(ζ0, r0), which proves (17) and finishes the proof. 2

Lemma 5.2. With the notation fixed above, there exists c > 0 such that for f ∈ H^p(Ω)

|P_φ^k

i(f)(z)φ^k_i(z)| ≤c2^k. Furthermore,

P_φ^k+1

j

f− P_φ^k+1

j (f) φ^k_i

(z)φ^k+1_j (z)

≤c2^k+1.

Proof. We use Lemma 5.1 with ζ_i^k=ζ0, φ^k_i =φ0, and B_i^k(ζ_i^k, r^k_i) =B0. Then P_φ^k

i(f)(z) = X

|J|≤`₀

c_J(f)π_J(z), where

cJ(f) = Z

∂Ω

f(w)πJ(w)φ^k_i(w)dσ(w).

The estimate on the π_J’s and φ^k_i imply that π_J ·φ^k_i ∈ K^M_γ (ζ_i^k) so that

|cJ(f)| ≤Kγ,M(f)(ζ_i^k)≤c2^k.

This estimate, together with Lemma 5.1 again, imply the bound on P_φ^k

i(f)φ^k_i. Next we prove the second estimate in the statement. Set h = f − P_φ^k+1

j (f) φ^k_i. Then,

c^(k+1)_J (h) = Z

∂Ω

f − P_φk+1 j (f)

(w)φ^k_i(w)π_J^(k+1)(w)dσ_φk+1

j (w)

= Z

∂Ω

f(w)φ^k_i(w)π^(k+1)_J (w)φ^k+1_j (w) dσ(w) kφ^k+1_j k_L¹

− Z

∂Ω

P_φ^k+1

j (f)(w)φ^k_i(w)φ^k+1_j (w)π_J^(k+1)(w) dσ(w) kφ^k+1_j k_L¹

=I+II.

By condition (v) in Lemma 4.4, for some ξ_j^k+1 ∈ ^cO_k+1,

|I| ≤K_γ,M(f)(ξ_j^k+1)≤2^k+1,

by construction. On the other hand, by the previous majorization,

|P_φk+1

j (f)(w)φ^k+1_j (w)| ≤c2^k+1,

while|π_J^(k+1)(w)| ≤c, so that the desired conclusion follows. 2

6. Atomic decomposition

In this section we define the atomic decomposition of a given functionf ∈ H^p(Ω), and finish the proof of Theorem 2.1. As noticed before, f admits boundary values defined a.e. on ∂Ω, that we still denote byf.

We write,

f = f −

∞

X

i=0

f φ^k_i +

∞

X

i=0

f φ^k_i

=f_k+

∞

X

i=0

f φ^k_i

=f_k+

∞

X

i=0

P_φ^k

i(f)φ^k_i +

∞

X

i=0

f − P_φ^k

i(f) φ^k_i

=h_k+

∞

X

i=0

f − P_φk i(f)

φ^k_i,

whereh_k =f_k+P∞ i=0P_φ^k

i(f)φ^k_i. Notice that,

(18)

∞

X

i=0

P_φ^k

i(f)(z)φ^k_i(z)

≤c₀2^k,

since no point in^cO_k lies in more thanN_Ω of theB(ζ_i^k, r^k_i)’s. Moreover, notice that supp

^∞ X

i=0

f − P_φk i(f)

φ^k_i

⊆ O_k,

so that the function on the left hand-side above tends to 0 pointwise as k → ∞.

This implies thath_k →f pointwise a.e., so that the following equality holds a.e.

(19) f =h₀+

∞

X

k=0

(h_k+1−h_k).

Now notice that

(20) f −h_k=

∞

X

i=0

f − P_φ^k

i(f) φ^k_i

and that (21)

∞

X

i=0

P_φ^k+1

j

f− P_φ^k+1

j (f) φ^k_i

=P_φ^k+1

j

f − P_φ^k+1

j (f)

= 0.