FUNCTIONS IN DOMAINS OF FINITE TYPE

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HARDY-SOBOLEV SPACES OF COMPLEX TANGENTIAL DERIVATIVES OF HOLOMORPHIC

FUNCTIONS IN DOMAINS OF FINITE TYPE

SANDRINE GRELLIER^∗

Abstract

In this paper, we prove Fefferman-Stein like characterizations of Hardy-Sobolev spaces of complex tangential derivatives of holomorphic functions in domains of finite type inCⁿ. We also study the relationship between these complex tangential Hardy-Sobolev spaces and the usual ones. We also obtain partial results on domains not necessarily of finite type.

0. Introduction

In this paper, we consider Hardy-Sobolev spaces of complex tangential derivatives of holomorphic functions in some domain inCⁿ. Let us precise the definition when n = 2. ForL a complex tangential derivative in, k ∈ ^N andua holomorphic function in, we denote by∇_T^kuthe(k+1)-tuple of functions given by(u, Lu, . . . , L^ku). Then, we consider, for 0< p <∞, the spaceH_k,T^p ()of holomorphic functionsuin for which the normal maximal function of|∇_T^ku|belongs toL^p(∂). We call the complex tangential Hardy-Sobolev space of orderkH_k,T^p (). One has to put in parallel the usual Hardy-Sobolev spaceH_k^p()which is defined in terms of the total gradient.

For this last one, Fefferman-Stein like characterizations hold in terms of the Littlewood-Paley function, the area integral or the maximal admissible function. These characterizations are proved when is strictly pseudoconvex or of finite type inC²where one can define geometrically adapted admissible approach regions. Since derivation preserves holomorphy, this follows from the corresponding characterizations of the Hardy space of holomorphic functions.

We prove here analogous characterizations of H_k,T^p () when is of finite type inCⁿwith the main difficulty that complex tangential derivation does not preserve holomorphy. Here, we say that is of finite typem when the Lie

∗The author would like to thank Aline Bonami for valuable suggestions about that work. The author is partially supported by theEuropean Commission(TMR 1998-2001 NetworkHarmonic Analysis).

Received May 5, 1999.

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brackets up to ordermof the complex tangential vector fields generate all the tangential space. Let us point out that part of the characterizations ofH_k,T^p () (as well as all the characterizations ofH_k^p()) hold without any assumptions of finite type on. In this case, we use a family of admissible approach re- gionsA_α^(m)(ζ ),ζ ∈ ∂, which are arbitrarily large, asm increases, around Levi flat points ζ but which coïncide with the hyperbolic approach regions around stricly pseudoconvex points and which fit the domain around points of finite typeminC².

Moreover, we study the relationship betweenH_k,T^p ()andH_k^p(). Note that in [11] and in [6], results were given in strictly pseudoconvex domains (or more generally in domains of finite type 2, the case of the unit ball inCⁿhave been done previously in [1]). In this case,H_k^p()identifies withH2^pk,T(). The situation cannot be as simple in the general case, since the inclusion H_k^p()⊂H₂^p_k,T()cannot be improved because of the strictly pseudoconvex points. To obtain converse inclusions, some finite type hypothesis is necessary.

One needs to recover all complex derivatives from complex tangential ones.

Whenis of finite typem, we prove that a holomorphic function inH_k,T^p () is in the usual Hardy-Sobolev space of orderk/m.

Let us now describe precisely the setting.

Let⊂^Cⁿbe a bounded, smooth domain, given by = {z∈^Cⁿ;r(z) <0}

with r a C^∞ function such that |∇r| = 1 on ∂ = {r = 0}. For δ > 0 and z ∈ , denote by τ(z, δ)the function (eventually infinite) constructed by Catlin which gives, when is of finite type m, the size in the complex tangential directions of the polydiscs that fits the domain aroundz(we will recall the precise definition ofτ(z, δ)in §1.1). Form≥2 an integer, denote by τm(z, δ):= min{τ(z, δ), δ¹^/m}and byQm(z, δ)the corresponding polydiscs.

It gives a non-isotropic pseudo-distancedmon∂. This is equivalent to Catlin’s pseudo-distance whenis of finite typeµ, for anym≥µand gives arbitrarily large balls in complex tangential directions around flat points asmgrows.

We identify a small neighborhood of∂in , denoted by ∩U, with

∂×[0, s0[ via a diffeomorphism:

:∂×[0, s0[→∩U (ζ,0)=ζ, ζ ∈∂.

Forz∈∩U, letπ(z)∈∂andδ(z)≥0 be such that(π(z), δ(z))=z; δ(z)is equivalent to the distance to∂. In the following, we will writeτm(z) forτm(z, δ(z))and we will forget the subscriptmwhen there is no ambiguity.

We define the following quantities for any smooth functionuand any aper- tureα >0:

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• The normal maximal function:

for any ζ ∈∂, Nu(ζ )=sup{|u((ζ, t))|; 0< t < s0}.

• The maximal admissible function:

for any ζ ∈∂, M_α^(m)u(ζ )=sup

|u(z)|; z∈A_α^(m)(ζ ) whereA_α^(m)(ζ )denotes the admissible approach region:

A_α^(m)(ζ )= {(z, t); z∈∂,0< t < s0, dm(z, ζ ) < αt}.

• The Littlewood-Paley function:

for any ζ ∈∂, g(u)(ζ )= _s₀

0 |u◦(ζ, t)|²dt t

1/2

.

• The admissible area function:

for any ζ ∈∂, S_α^(m)u(ζ )=

Aα^(m)(ζ )|u(z)|² dV (z) δ(z)²τ(z, δ(z))²ⁿ⁻²

1/2

. Define the complex Hardy spaceH^pas the space of holomorphic functions uwhose normal maximal functions are inL^p(∂). It follows from standard method (see [7] and [4] for harmonic functions and [3] and [11] in this context) that H^p, 0 < p < ∞, is characterized in terms of any of the preceding functionals. Namely, it is equivalent for a holomorphic function u to be in H^p(), to have M_α^(m)u ∈ L^p(∂), org(δ∇u) ∈ L^p(∂)or S_α^(m)(δ∇u) ∈ L^p(∂), independently on the apertureαand on the choice ofm.

One can then consider Hardy-Sobolev spacesH_k^p()of holomorphic functions, that is the spaces of holomorphic functions which have derivatives up to order k in H^p(). Since derivatives of holomorphic functions are still holomorphic, it is a corollary of the previous characterizations ofH^p()that similar characterizations hold forH_k^p().

Fork∈^N,r ∈^N^∗,m∈^N\ {0,1},0< p <∞, for a holomorphic function uin, the following are equivalent

u∈H_k^p() N (|∇^ku|)∈L^p(∂)

M_α^(m)(|∇^ku|)∈L^p(∂) for some α∈]0,1[

g(δ^r|∇^r^+ku|)∈L^p(∂)

S_α^(m)(δ^r|∇^r^+ku|)∈L^p(∂) for some α∈]0,1[.

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(the symbol∇^kdenotes the collection of all the derivatives of order less thank).

We want here to prove the analogs for spaces involving only complex tangential derivatives. We want also to link these spaces to the usual Hardy- Sobolev spaces. Namely, denote by∇_T^kuthe collection of all possible com- position of order less thankof theLij’s,i < j, given by

Lij = ∂r

∂zj

∂

∂zi − ∂r

∂zi

∂

∂zj.

As before, denote byH_k,T^p ()the set of holomorphic functionsuinsuch thatN(|∇_T^ku|)∈L^p(∂).

Our first result holds without any assumption on the type of.

Theorem0.1.Fork∈^N,m≥2an integer and0< p <∞, the following are equivalent for a holomorphic functionuin.

(i) u∈H_k,T^p (),

(ii) M_α^(m)(|∇_T^ku|)∈L^p(∂)for someα ∈]0,1[.

Furthermore, ifS_α^(m)(τ_m^−kδ^r|∇^ru|)∈L^p(∂)for somer ∈^Nso that2r−k ≥1 thenu∈H_k,T^p ().

Remark 0.2. The last statement implies in particular that H_k/^p₂() ⊂ H_k,T^p ()(sincecδ(z)¹^/²≤τm(z)).

Remark0.3. Whenis Levi flat around some point, part (ii) states that the supremum can be taken over arbitrarily large admissible regions around this point.

Theorem0.4.Letbe a bounded smooth domain of finite typeminCⁿ. For k∈^N, and1−_mn+¹ ₁ < p <∞, the following are equivalent for a holomorphic functionuin.

(i) u∈H_k,T^p (),

(ii) M_α^(m)(|∇_T^ku|)∈L^p(∂)for someα ∈]0,1[, (iii) g(δ|∇∇_T^ku|)∈L^p(∂),

(iv) S_α^(m)(δ|∇∇_T^ku|)∈L^p(∂)for someα∈]0,1[,

(v) S_α^(m)(τ_m^−kδ^r|∇^ru|)∈L^p(∂)for somer ∈^Nso that2r−k≥1.

Remark0.5. The last statement implies that, when is of finite typem, a function inH_k,T^p ()is also in the ordinary Hardy-Sobolev spaceH_k/m^p () (sinceτm(z) ≤ Cδ(z)¹^/m). We recover in this context the well known phe- nomenon of finite type domains: complex tangential derivatives of holomorphic functions behave at least as well as global derivatives of order 1/min domains

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of finite typem. It actually says something more subtle. A complex tangential gradient of orderkbehaves likeτ_m^−kδ^r|∇^ru|and conversely in domains of finite typem. In particular, this means that∇_T^kubehaves asan ordinary gradient whose order changes from point to point.

Remark0.6. In this paper, we only give the proof of Theorem 0.4 when 0< p <2. Whenp≥2, the result follows from singular integrals machinery and some commutation properties (see [12]).

The key point in the proofs of Theorem 0.1 and 0.4 is the use of mean-value properties for complex tangential derivatives. Forz∈, denote byQm(z)the set

Qm(z):= {w∈; δ(z)/2≤δ(w)≤2δ(z); dm(π(z), π(w))≤δ(z)/2}.

Denote by Mean^Q^m^(z)(|F|)the mean-value of|F|overQm(z). We prove the following.

Theorem 0.7 (Mean-value inequality). For k, l ∈ ^N, 0 < p < ∞ and m≥2an integer, there exists a constantC >0such that, foruholomorphic function inandzin∩U,

δ(z)^lp|∇^l∇_T^ku(z)|^p≤CMean^Q^m^(z)(|∇_T^ku|^p).

To get these mean-value properties, we improve the usual freezing coeffi- cient method which consists in taking the coefficients ofLto be constant up to a remaining term so that it preserves holomorphy. As this is not sufficient here, we “freeze” the coefficients to a higher order by using a Taylor expansion of the coefficients ofLup to a sufficiently large order.

To prove the link between complex tangential derivatives and ordinary derivatives, we use the pointwise estimates between complex tangential gradients and ordinary gradients proved in [10]. Namely, one has the following:

Pointwise estimates([10]).Fork ∈ ^N,ua holomorphic function in, andz∈,

(1) τ(z)²^k|∇_T^ku(z)|²≤CMean^Q(z)(|u|²).

Moreover ifis of finite typeminCⁿthen for) >0there existsC())so that (2) δ(z)²^k|∇^ku(z)|²≤Mean^Q(z)(C())τ²^k|∇_T^ku|²+)²|u|²).

The paper is organized as follows. In section 1, we recall some basic defin- itions and properties of the geometry and prove Theorem 0.7. Theorem 0.1

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follows at once. In section 2, we establish the relations between usual area integrals and area integrals of complex tangential derivatives. In section 3, we conclude by showing the links between area integrals and maximal functions of complex tangential derivatives.

As said before, we proved these results in the context of domains of finite type 2 in [11]. The main innovation in this paper is to develop a new technic which allows to overcome the technical difficulties which appear form >2.

In the following, we will use the symbolA <∼Bif there exists a universal constantC so thatA≤ CB. Similarly, we will write A BifA <∼ B and B <∼A.

1. Geometry and mean-value properties

In this paragraph, we will assume for simplicity thatn=2.

1.1. Geometry

Assume is a domain inC². Let us recall the following facts from [5] (see also [8]). Letz0 ∈ ∂, as|∇r|(z0) = 1, we may assume that _∂z^∂r

1 = 0 in a neighborhoodV (z⁰)ofz⁰. Then

Lemma1.1.Let M ∈ ^N,M ≥ 2. For anyz ∈ V (z0)∩, there exists a biholomorphic mappingz:C²→^C²such that-:=r◦zsatisfies:

-(ζ )=r(z)+Re(ζ1)+

j,k∈N j,k≥1;j+k≤M

aj,k(z)ζ2jζ2k+O

|ζ2|^M+¹+ |ζ1| |ζ| .

Moreover

z(ζ )=

z1+d0(z)ζ1+ M k=1

dk(z)ζ2k, z2+ζ2

where d⁰(.), dk(.);k = 1, . . . , M depend smoothly on z and d⁰(.) = 0 in V (z0).

It is easy to extend this result to arbitrary dimension (this is done for instance in [10]). It is important to note that this change of variables is independent on any assumption on the type of . Now fix m ≥ 2 an integer and take M ≥min the preceding lemma. DefineAl(z):= max

|aj,k(z)|;j+k=l . Forδ > 0, denote byτ(z, δ)= min _δ

Al(z)

1/l, l = 2, . . . , m

. This defines a function onV (z⁰)∩with values inR₊. Whenis of finite typem, there existsl ∈ {2, . . . , m}such thatAl(z) = 0 forz ∈ ∂and by continuity for z ∈ V (z0)sufficiently small so that τ(z, δ)takes finite values. Now define

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τm(z, δ):= min{τ(z, δ), δ¹^/m}. Remark that ifis of finite typem, then, for anyµ≥m,τmτµτ. Define the polydisc aroundzby

Qm(z, δ)=z(Rm(z, δ))=z

ζ ∈^C²; |ζ1|< δ, |ζ2|< τm(z, δ) . The following properties hold:

(1) there exists a constantC >0 such that, for anyz∈V (z0)and 0< δ <1, 1

Cδ¹^/²≤τm(z, δ)≤Cδ¹^/m. (2) ifδ < δthen_δ

δ

1/2

τm(z, δ)≤τm(z, δ)≤_δ

δ

1/mτm(z, δ). (3) for any 0< δ <1 andz∈Qm(z, δ),τm(z, δ)τm(z, δ).

(4) there exists a constantC >0 such that, ifz∈Qm(z, δ), thenQm(z, δ)⊂ Qm(z, δ)andQm(z, δ)⊂Qm(z, Cδ).

By definition, there exists a constantcsuch that, for anyz∈V (z0), Qm(z, cδ(z))⊂.

We will noteQm(z)=Qm(z, cδ(z))=z(Rm(z))andτm(z)=τm(z, cδ(z)). (5) In addition, for anyζ ∈Qm(z),τm(ζ)τm(z).

It follows from these properties that

dm(z, ζ )=inf{δ >0, z∈Qm(ζ, δ)∩∂}

defines a pseudo-distance on∂.

1.2. Mean-value property for complex tangential derivatives and applications

LetEbe a measurable subset of. Denote by Mean^E(F )the mean-value of

|F|overEwith respect to the Lebesgue measure.

We prove the following proposition.

Proposition1.2. Fork, l, r, m ∈ ^N,m ≥ 2,0 < p < ∞, there exists a constantC >0such that, for any holomorphic functionuinand anyzin ∩U,

δ(z)^lp|∇^l+r∇_T^ku(z)|^p ≤CMean^Q^m^(z)(|∇^r∇_T^ku|^p).

Once this proposition is proved it follows by standard methods (see [7] or [14] for instance) that

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Corollary 1.3. For k, m ∈ ^N, m ≥ 2, α > 0, 0 < p < ∞ and a holomorphic functionu,

M_α^(m)(|∇_T^ku|)

L^p(∂) <∼ N(|∇_T^ku|)

L^p(∂),

and for any ζ ∈∂, g(δ|∇∇_T^ku|)(ζ) <∼S_α^(m)(δ|∇∇_T^ku|)(ζ ).

This gives the equivalence between (i) and (ii) of Theorem 0.1 and that (iv) implies (iii) in Theorem 0.4. Note that, in fact, the implication(iv) ⇒ (iii) does not need any finite type hypothesis.

Let us now prove Proposition 1.2. First, remark that|∇^r∇_T^ku| |∇_T^k∇^ru|: for k = r = 1, the commutator of any first order derivative and ∇T is a differential operator of order 1 with smooth coefficients. As∇T contains the identity by definition, we can write|∇∇Tu| |∇T∇u|. For largerrandk, the result follows from induction.

So, as ordinary derivatives preserve holomorphy, it is enough to consider the caser =0. We are going to writeL^kuas a sum of a function satisfying mean- value properties and of a remaining term. For this we introduce the following class of functions.

Deﬁnition1.4. LetK=(k¹, k²)be a multi-index of positive integers. A functionF ∈C^∞()is called (AB)_K if ^∂^kj^F

∂ζjkj =0 forj =1,2 in.

To simplify notation, we will assume thatK is fixed in the following and we will write (AB) instead of (AB)_K.

For anyζ ∈^Candr >0, we denote byD(ζ, r)the disc{z∈^C; |z−ζ| ≤r}. The terminology (AB) comes from Ahern and Bruna who proved the following lemma (cf [1]):

Lemma1.5.For (l1, l2)and(m1, m2) ∈ ^N²,0 < p < ∞, there exists a constantCsuch that, for any (AB)-functionF in, anyζ =(ζ1, ζ2)∈and anyr =(r¹, r²)∈(]0,+∞[)²such thatD(ζ¹, r¹)×^D(ζ², r²)⊂,

r1^p(l¹^+m¹⁾r2^p(l²^+m²⁾

∂^l¹^+l²^+m¹^+m²F

∂ζ¹^l¹∂ζ²^l²∂ζ1^m¹∂ζ²^m²(ζ )

^p≤CMean^D^(ζ¹^,r¹^)×^D^(ζ²^,r²⁾(|F|^p).

Given z ∈ ∩U, let w = z(ζ )and- = r ◦z(ζ). Denote byL =

∂ζ∂-2

∂ζ∂1 − _∂ζ^∂-₁_∂ζ^∂₂ a holomorphic complex tangential vector field. Recall that Qm(z) = z(Rm(z)) where Rm(z) = Rm(z, cδ(z)) and Rm(z, δ) = {ζ ∈ C²; |ζ1|< δ,|ζ2|< τm(z, δ)}. We have the following lemma:

Lemma 1.6.For ζ ∈ Rm(z), k, l ∈ ^N and a holomorphic function f in z(),

L^kf (ζ )=Fklf (ζ)+Rklf (ζ)

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whereFklf is an (AB)-function and for0< p <∞,ζ ∈Rm(z),0≤j ≤l, δ(z)^jp|∇^jRklf (ζ )|^p ≤CMean^R^m^(z)

|f|^p .

Proof. It follows easily by induction onk ∈^Nthat there exist some con- stantscr,s, 1≤r+s ≤k, such that

L^k =

1≤r+s≤k

Ek,r,s

cr,s

k j=1

∂^m^j⁺ⁿ^j-

∂ζ1^m^j∂ζ2ⁿ^j

∂^r+s

∂ζ1^r∂ζ2^s

,

whereEk,r,s denotes the set of couples (mj, nj), j = 1, dots, k, in lexico- graphical order, which satisfy_k

j=1mj = k−r and_k

j=1nj = k−s with mj+nj ≥1.

For anyN ∈^N, we can write-=T0^N-+R0^N-whereT0^N-stands for the Taylor expansion of-up to orderN.

Assume for simplicity thatl =0. ChooseN =2k−1. Since-isC^∞, one has

∂^m^j⁺ⁿ^j-

= ∂^m^j⁺ⁿ^jT0^N-

+rN,mj,mj where rN,mj,mj =O

|ζ|^N⁺¹^−m^j⁻ⁿ^j .

Now, forζ ∈Rm(z),|ζ| ≤τm(z)so one obtains, sincemj+nj ≤2k−(r+s), L^kf (ζ )=

1≤r+s≤k

_k

j=1

∂^m^j⁺ⁿ^jT0^N-

+O(τm(z)^N+¹⁻²^k+(r+s))

∂^l+sf

∂ζ1^r∂ζ2^s

(ζ )

=Fk0f (ζ )+Rk0f (ζ ).

By definition,Fk0is an(AB)K-function forK=K(N)large enough.

But, by the mean-value properties satisfied byf, for anyζ ∈Rm(z)

1≤r+s≤k

O

τm(z)^N⁺¹⁻²^k+(r+s)_p ∂^r^+sf

∂ζ1^r∂ζ2^s

^p(ζ)≤CMean^R^m^(z)

τ_m^p(N+¹⁻²^k)|f|^p

≤CMean^R^m^(z)(|f|^p).

This gives the lemma.

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onRm(z). For the first term, we use Lemma . to get a bound as Mean^R^m^(z) (|Fklf|^p). This in turn is bounded by

Mean^R^m^(z)(|L^kf|^p+ |Rklf|^p)≤CMean^R^m^(z)(|L^kf|^p+ |f|^p).

Going back to, it gives the result forLsinceL corresponds to a smooth non-vanishing function timesL.

Remark 1.7. It is by an analogous method that the pointwise estimates quoted in the introduction are proved in [10].

In the following, we will forget the subscriptmto simplify the notations.

2. Area Integrals

2.1. Area integrals and area integrals of complex tangential derivatives First, recall that usual methods, involving Hardy inequality and mean-value properties, allow to prove that, for 0< p≤2 anduholomorphic in, (∗) Sα(δ^r+ητ^−k|∇^ru|)L^p(∂) <∼ Sα(δ^l+ητ^−k|∇^lu|)L^p(∂)+sup

K |u|

as long asr+η−k/2 andl+η−k/2 are positive, whereKdenotes a compact subset of(see [4] and [3]). The same kind of method using part(1)of the pointwise estimates of the introduction gives that

Sα(δ^η+¹|∇∇_T^ku|)L^p(∂<∼ Sα(δ^r+ητ^−k|∇^ru|)L^p(∂)+sup

K |u|

forr+η−k/2>0 andη >−1 (see [11] in the context of domains of type 2 and [9]).

We prove now a converse inequality. For 0< p≤2,Sα(δ^r^+ητ^−k|∇^ru|)p

can be estimated by theL^p-norm ofSα(δ^j+η|∇^j∇_T^ku|)whenr+η−k/2>0 andj+η >0. This estimate is proved in [9]. We give here a simplified proof.

By(∗), it is sufficient to prove the required estimate for somerbig enough.

Apply the converse pointwise estimates (2) to the component of∇^lu,lwill be chosen large enough, and integrate overAα(ζ)to get

Sα(δ^k+l+ητ^−k|∇^k+lu|)(ζ )

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equivalent toSα(δ^r+ητ^−k|∇^ru|)p for anyr+η−k/2> 0. So, as theL^p- norms of the area integralsSα are independent on the apertureα, it gives an a priori estimate for)small enough. We get rid of the a priori assumption as in [11] by applying this inequality in) = {z∈;δ(z) > )}and letting)goes to 0. Eventually we get the following result.

Proposition2.1. Assumeis of finite type inCⁿ. Fork, r, j ∈^N,0< p≤ 2,α, η∈^Rso thatr+η−k/2>0,j+η >0, foruholomorphic in

Sα(δ^r+ητ^−k|∇^ru|)p <∼ Sα(δ^η+j|∇^j∇_T^ku|)p+sup

K |u|.

2.2. An embedding result

In this section, we prove a key estimate to deal with the remaining terms.

Proposition2.2. Assumeis of finite typeminCⁿ.

Letµ∈]0,1/m[. For1− _n+(⁽¹^/m−µ)₁_/m−µ) < p ≤2, there existsq ≥p,q >1 so that

Sα(δ¹^−µ|∇∇_T^k−¹u|)q ≤ Mα(|∇_T^ku|)p.

Proof. By the preceding paragraph, Sα(δ¹^−µ|∇∇_T^k−¹u|)q is success- ively bounded by

Sα(δ^k−µτ^−k+¹|∇^ku|)q, Sα(δ^−µτ|∇_T^ku|)q, Sα(δ¹^/m−µ|∇_T^ku|)q

up to sup_K|u|. This in turn is bounded by

CMα(δ¹^/m−µ−)|∇_T^ku|)L^q(∂)

for any) >0 since Sα(δ¹^/m−µ|∇_T^ku|)(ζ )

≤Mα(δ¹^/m−µ−)|∇_T^ku|)(ζ)×

Aα(ζ)

δ(z)⁾dV (z) δ(z)²τ²(z, δ(z))

1/2

. Now, using the atomic decomposition of spaces of homogeneous type (see [2]), one can show (see [11]) that

Mα(δ¹^/m−µ−)|∇_T^ku|)L^q(∂) ≤ Mα(|∇_T^ku|)L^p(∂)

if 1/m−µ−)≥n/p−n/q.

It is possible to find such aqby assumption on the range ofp.

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3. Characterizations of complex tangential Hardy-Sobolev spaces

3.1. Estimate on the normal maximal function by the Littlewood-Paley function

In this paragraph, we prove that (iii) implies (i) of Theorem 0.4. More precisely, we prove, without finite type hypothesis the following result.

Proposition3.1. Fork∈^N,0< p <∞anduholomorphic in, N(|∇_T^ku|)L^p(∂)<∼ g(δ|∇∇_T^ku|)L^p(∂)+s0^θSα(δ¹^−θ|∇∇_T^k−¹u|)L^q(∂)

for anyq >1,q ≥p, anyθ ∈]0,1[.

Remark3.2. When is of finite type, it gives an a-priori estimate when 1− _mn+¹₁ < p≤2, since by Proposition 2.2, forθ sufficiently close to 0, one can chooseq >1,q≥p, so that

Sα(δ¹^−θ|∇∇_T^k−¹u|)L^q(∂)<∼ Mα(∇_T^ku)L^p(∂)<∼ N(|∇_T^ku|)L^p(∂). So, ifu∈C^∞()∩H(), fors⁰small enough, we have

N(|∇_T^ku|)L^p(∂)≤Cg(δ∇∇_T^ku)L^p(∂).

To obtain the general result, one has to apply this estimate in) = {(z, t), t >

)}(since a holomorphic function in is in particularC^∞())) and to let) goes to zero. On one hand

₎₀

) t²|f|²((ζ, t))dt

t ≤g(δ|f|)(ζ),

on the other hand, the monotone convergence theorem proves that

)→lim0 sup

)<t<)0

|∇_T^ku|L^p(∂) = N(|∇_T^ku|)L^p(∂).

Proof. The method is analogous to the one used in [11]. The trick is to write∇_T^kuas the sum of a harmonic function and of a remaining term.

Write∇_T^ku= (∇_T^ku)⁰+(∇_T^ku)h where(∇_T^ku)⁰is the (vector)-solution to the Dirichlet problem

<v=<(∇_T^ku)in v=0 on∂.

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Then,

(∇_T^ku)0◦(ζ, t)≤ _s0

0

|∇(∇_T^ku)0◦(ζ, s)|ds+sup

K |u|

≤s0^θ

_s₀

0

∇(∇_T^ku)⁰◦(ζ, s)²s²⁻²^θds s

1/2

+sup

K |u|

whereKis a compact subset of, 0< θ <1.

So, it gives

N(|∇_T^ku|)L^p(∂)≤ N(|(∇_T^ku)h|)L^p(∂)+ N(|(∇_T^ku)0|)L^p(∂)

<∼ g(δ∇(∇_T^ku)h)L^p(∂)

+s0^θ

_s₀

0

|∇(∇_T^ku)0◦(., s)|²s²⁻²^θds s

1/2

L^p(∂)+sup

K |u|

<∼ g(δ∇∇_T^ku)L^p(∂)+ g(δ∇∇_T^ku)⁰L^p(∂)

+s0^θ

_s₀

0

|∇(∇_T^ku)0◦(., s)|²s²⁻²^θds s

1/2

L^p(∂)+sup

K |u|

<∼ g(δ∇∇_T^ku)L^p(∂)

+2s0^θ

_s₀

0

|∇(∇_T^ku)0◦(., s)|²s²⁻²^θds s

1/2

L^p(∂)+sup

K |u|.

Now, by estimates on the Dirichlet problem (see [11] appendix for study in this context or [13]) we obtain that, for someq >1,q ≥p

(∗)= _s₀

0

∇(∇_T^ku)⁰◦(., s)²s²⁻²^θds s

1/2 L^p(∂)

≤ _s₀

0

∇(∇_T^ku)⁰◦(., s)²s²⁻²^θds s

1/2 L^q(∂)

≤ <

∇_T^ku _W−1,(q,2)

θ ()

whereW_θ⁻¹^,(q,²⁾denotes the usual Sobolev space. Now, sinceuis holomorphic,

|<∇_T^ku| = |[<,∇_T^k]u|. Note that|[<,∇_T^k]u| |∇²∇_T^k−¹u|: First, recall that

|∇_T^k∇^ru| |∇^r∇_T^ku|. The commutator [<,∇_T^k] is obtained by derivating at least one and at most two complex tangential vector fields of the∇_T^k. If only one is derivated, one gets a term |∇²∇_T^k−¹u|, if two are derivated, one obtains

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a term |∇²∇_T^k−²u|. So, <

∇_T^ku _W−1,(q,2)

θ ()

_s₀

0

|∇∇_T^k−¹u|²s²⁻²^θds s

1/2 L^q(∂). Now, by the mean-value properties, this is bounded by

Sα

δ¹^−θ∇∇_T^k−¹u

L^q(∂). This ends the proof of the proposition.

3.2. Estimate of the area integral by the admissible maximal function In this paragraph, we adapt the method of [7] to our setting. We are going to prove the following result.

Proposition3.3. Let) > 0. For0< µ < 1,0< p < 2,α > 0andu holomorphic in,

Sα

δ|∇∇_T^ku| _p <∼ 1

)² +1 M_α

∇_T^ku _p +()+s0^µ) Sα

δ|∇∇_T^ku| _p+ Sα

δ¹^−µ|∇∇_T^k−¹u| _p+sup

K |u|.

Remark 3.4. Implication (ii) ⇒ (iv) of Theorem 0.7 follows: By §2.1, this gives an a-priori estimate when is of finite type m and 0 < p <

2. Indeed, Sα(δ¹^−µ|∇∇_T^k−¹u|)p is estimated by Sα(δ¹^−µτ|∇∇_T^ku|)p <∼ s0¹^/m−µSα(δ|∇∇_T^ku|)pif 1/m−µ >0.

We conclude that, whenis of finite typem, foru∈C^∞()∩H(), we have Sα

δ∇∇_T^ku _L_p_(∂) ≤C Mα

∇_T^ku _L_p_(∂)+sup

K |u|

.

It remains to show that this inequality is still valid for generalu. We apply this inequality in) = {z∈^Cⁿ;δ(z) > )}. One can verify that the constant involved is independent of) >0. We want to let) →0 in the inequality. Let us observe that, forζ)=(ζ, c))∈∂),Rα(ζ))⊂Rβ(ζ), for someβ > α. This allows to show that

Mα

∇_T^ku

L^p(∂))≤ Mβ

∇_T^ku

L^p(∂). Then, we conclude by Fatou’s Lemma that

Sα

δ∇∇_T^ku _L_p_(∂)≤C M_β

∇_T^ku _L_p_(∂)+sup

K |u|

.

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In the following, it will be convenient to have a defining function for which is harmonic near∂. We choose a pointx⁰ ∈ Kand denote byδ the Green’s function forwith singularityx0. Thus,δis harmonic in\ {x0}and δ(z)is comparable with the distance to the boundary, forz ∈ ∩U. Letλ and)be any real positive numbers andEbe the set

E),λ =E:=

z∈∂;Mα

∇_T^ku

(z)≤λ, Sγ

δ¹^−µ∇∇_T^k−¹u(z)≤λ, Sγ

δ∇∇_T^ku(z)≤ λ )+s0^µ

; for someγ > α.

LetE0be those points ofEof relative density ¹₂,D0, Dtheir complements.

By the maximal Theorem, σ (D0) ≤ Cσ(D). Proposition 3.3. follows from the following lemma.

Lemma3.5.There exists a constantCandγ > αsuch that, for every) >0

E0

Sα

δ∇∇_T^ku²(z) dσ (z)

≤C 1

)² +1

λ²σ(D0)+sup

K |u|² +

_λ

0 tσ

M_α(∇_T^ku)≥t dt +

)+s0^µ

2

ESγ

δ∇∇_T^ku²(z)dσ(z) +

ESγ

δ¹^−µ∇∇_T^k−¹u²(z) dσ(z)

.

Assume this lemma proved and let us prove Proposition 3.1. Write (∗)= Sα

δ∇∇_T^ku ^p

L^p(∂)

=p _∞

0

λ^p−¹σ Sα

δ∇∇_T^ku≥λ dλ

≤p _∞

0

λ^p−¹σ (D0) dλ +p _∞

M λ^p−³

E0

Sα

δ∇∇_T^ku²(z) dσ(z)dλ+M^pσ(∂).