Constructive description of Hardy-Sobolev spaces in C

arXiv:1601.03218v1 [math.CV] 13 Jan 2016

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Constructive description of Hardy-Sobolev spaces in C

Alexander Rotkevich

Received: date / Accepted: date

Abstract In this paper we study the polynomial approximations in Hardy- Sobolev spaces on for convex domains. We use the method of pseudoanalytical continuation to obtain the characterization of these spaces in terms of poly- nomial approximations.

Keywords Hardy-Sobolev Spaces·polynomial approximations, pseudoana- lytical continuation·Cauchy-Leray-Fantappi`e integral.

Mathematics Subject Classification (2010) 32E30·41A10

1 Introduction

The purpose of this paper is to give an alternative characterizations of Hardy- Sobolev (see. [1]) spaces

H_p^l(Ω) ={f ∈H(Ω) :kfkH^p(Ω)+ X

|α|≤l

k∂^αfkH^p(Ω)<∞} (1) on strongly convex domainΩ⊂Cⁿ.

We continue the research started in [14] and devoted to description of basic spaces of holomorphic functions of several variables in terms of polynomial approximations and pseudoanalytical continuation. In particular, we show that for 1 < p < ∞ and l ≥ 1 a holomorphic on a strongly convex domain Ω function f is in the Hardy-Sobolev spaceH_p^l(Ω) if and only if there exist a polynomial sequenceP₂^k such that

Z

∂Ω

dσ(z)

∞

X

k=1

|f(z)−P2^k(z)|²2^2lk

!p/2

<∞. (2)

Alexander Rotkevich

Mathematics and Mechanics Faculty, St.Petersburg State University, Universitetsky pr., 28, Peterhof, St. Petersburg, Russia.

E-mail: [email protected]

In the one variable case this condition follows from the characterization ob- tained by E.M. Dynkin [5] for Radon domains.

The paper is divided into five sections with one appendix. In section 2we give main definitions and preliminaries of this work. Section3is devoted to the Cauchy-Leray-Fantappi`e integral formula, the polynomial approximations and estimates of its kernel. We also define internal and external Kor´anyi regions, the multidimensional analog of Lusin regions. In section 4 we introduce the method of pseudoanalytical continuation and three constructions of the con- tinuation with different estimates. We use these constructions to obtain the characterization of Hardy-Sobolev spaces in terms of estimates of the pseudo- analytical continuation. To prove this result we use the special analog of the Krantz-Li area-integral inequality [7] for external Kor´anyi regions established in appendixA. Finally, section5 contains the proof of characteristics (2).

2 Main notations and definitions

Let Cⁿ be the space of n complex variables, n ≥ 2, z = (z1, . . . , zn), zj = xj+iyj;

∂jf = ∂f

∂zj

= 1 2

∂f

∂xj −i∂f

∂yj

, ∂¯jf = ∂f

∂z¯j

= 1 2

∂f

∂xj

+i∂f

∂yj

,

∂f =

n

X

k=1

∂f

∂zk

dzk, ∂f¯ =

n

X

k=1

∂f

∂¯zk

d¯zk, df=∂f+ ¯∂f.

The notation

h∂f(z), wi=

n

X

k=1

∂f(z)

∂zk

wk.

is used to indicate the action of∂f on the vectorw∈Cⁿ,and

|∂f¯ |=

∂f

∂z1

+. . .+

∂f

∂zn

.

The euclidian distance form the pointz∈Cⁿ to the setD⊂Cⁿ we denote as dist(z, D) = inf{|z−w| :w∈D}. Lebesgue measure inCⁿ we denote as dµ.

For a multiindexα= (α1, . . . , αn)∈Nⁿ we set|α| = α1 + . . . + αn

andα! =α1!. . . α2!,alsoz^α=z₁^α1. . . z^α_nⁿ and∂^αf = _∂z¯α1^∂|α|f 1 ...∂¯z^αnn .

Let Ω = {z∈Cⁿ:ρ(z)<0} be a strongly convex domain with a C³- smooth defining function. We need to consider a family of domains Ωt = {z∈Cⁿ:ρ(z)< t}that are also strongly convex for each|t|< ε,whereε >0 is small enough. The projection ofz∈Ωε\Ω−ε to the∂Ω we define asπ(z).

Forξ∈∂Ωtwe define the complex tangent space Tξ={z∈Cⁿ :h∂ρ(ξ), ξ−zi= 0}.

The space of holomorphic functions we denote asH(Ω) and consider the Hardy space (see [17], [6])

H^p(Ω) :=







f ∈H(Ω) : sup

−ε<t<0

Z

∂Ωt

|f(z)|^pdσt(z)<∞





 ,

wheredσtis induced Lebesgue measure on the boundary ofΩt.We also denote dσ=dσ0.Hardy-Sobolev spacesH_p^l(Ω) are defined by (1).

Throughout this paper we use notations., ≍.We letf .g iff ≤cgfor some constant c >0, that doesn’t depend on main arguments of functionsf andg and usually depend only on dimension nand domainΩ.Alsof ≍g if c⁻¹g≤f ≤cgfor somec >1.

3 Cauchy-Leray-Fantappi`e formula

In the context of theory of several complex variables there is no unique repro- ducing formula formula, however we could use the Leray theorem, that allows us to construct holomorphic reproducing kernels ([2], [11], [12]). For convex do- mainΩ={z∈Cⁿ:ρ(z)<0}this theorem brings us Cauchy-Leray-Fantappi`e formula, and forf ∈H¹(Ω) andz∈Ω we have

f(z) =KΩf(z) = 1 (2πi)ⁿ

Z

∂Ω

f(ξ)∂ρ(ξ)∧( ¯∂∂ρ(ξ))ⁿ⁻¹ h∂ρ(ξ), ξ−ziⁿ =

Z

∂Ω

f(ξ)K(ξ, z)ω(ξ), (3) whereω(ξ) = _(2πi)¹ n∂ρ(ξ)∧( ¯∂∂ρ(ξ))ⁿ⁻¹,andK(ξ, z) =h∂ρ(ξ), ξ−zi⁻ⁿ.

The (2n−1)-formωdefines on∂ΩtLeray-Levy measuredS,that is equiv- alent to Lebesgue surface measuredσt(for details see [2], [9], [10]). This allows us to identify Lebesgue, Hardy and Hardy-Sobolev spaces defined with respect to measuresdσt anddS. Also note, that measuredV defined by the 2n-form dω= (∂∂ρ)¯ ⁿ is equivalent to Lebesgue measure dµin Cⁿ.

By [13] the integral operatorKΩ defines a bounded mapping onL^p(∂Ω) toH^p(Ω) for 1< p <∞.

The function d(w, z) = |h∂ρ(w), w−zi| defines on ∂Ω quasimetric, and if B(z, δ) = {w ∈ ∂Ω : d(w, z) < δ} is a quasiball with respect to d then σ(B(z, δ))≍ δⁿ, see for example [13]. Therefore{∂Ω, d, σ} is a space of ho- mogeneous type.

Note also the crucial role in the forthcoming considerations of the following estimate that is proved in [14].

Lemma 3.1 Let Ω be strongly convex, then

d(w, z)≍ρ(w) +d(π(w), z), w∈Cⁿ\Ω, z∈∂Ω.

3.1 The polynomial approximation of Cauchy-Leray-Fantappi´e kernel

In lemma 3.3 here we construct a polynomial approximations of Cauchy- Leray-Fantappi´e kernel based on theorem by V.K. Dzyadyk about estimates of Cauchy kernel on domains on complex plane (theorem 1 in part 1 of sec- tion 7 in [3]). The approximation is choosed similarly to [16] and [15]. This construction allows us in theorem 5.1 to get polynomials that approximate holomorphic function with desired speed.

Lemma 3.2 Let Ω be a strongly convex domain with 0 ∈ Ω, then for every ξ∈Ωε\Ωthe value of λ= ^{h∂ρ(ξ), zi}_{h∂ρ(ξ), ξi} for z∈Ω lies in domain L(t), bounded by the bigger arc of the circle |λ|= R =R(Ω) and the chord {λ∈ C: λ = 1 +e^its, s∈R, |λ| ≤R}, wheret= ^π₂−arg(h∂ρ(ξ), ξi).

Proof Forξ∈∂Ω define Λ(ξ) =

λ∈C:λ=h∂ρ(ξ), zi

h∂ρ(ξ), ξi, z ∈Ω

. The convexity ofΩwith 0∈Ωimplies that

|h∂ρ(ξ), ξi|&|∂ρ(ξ)| |ξ|&1, (4) Reh∂ρ(ξ), z−ξi ≤0, z∈Ω, ξ¯ ∈Ωε\Ω. (5) The domainL(ξ)⊂Cis also convex and contains 0, thus the equality

h∂ρ(ξ), zi

h∂ρ(ξ), ξi = 1 +h∂ρ(ξ), z−ξi h∂ρ(ξ), ξi with estimates (4), (5) completes the proof of the lemma.

Lemma 3.3 Let Ω be a strongly convex domain and r > 0. Then for every k∈N there exist functionKk(ξ, z) defined for ξ∈Ωε\Ωand polynomial in z∈Ω withdegKk(ξ,·)≤kand following properties:

K(ξ, z)−K_k^glob(ξ, z) . 1

k^r 1

d(ξ, z)^d+r, d(ξ, z)≥ 1

k; (6)

K_k^glob(ξ, z)

.kⁿ, d(ξ, z)≤ 1

k. (7)

Proof Due to [3] and [16] for anyj∈Nthere exists function Tj(t, λ) polyno- mial inλwith degTj(t,·)≤j such that

1

1−λ−Tj(t, λ)

. 1 j^r

1

|1−λ|^1+r (8) forλ∈L(t)\n

λ:|1−λ|< ¹_jo

and coefficients of polynomialsTj(t, λ) contin- uously depend ont.Note also that by maximum principle

Tj(t, λ).j⁻¹, λ∈L(t)\

λ:|1−λ|<1 j

. (9)

Forj∈Nand (j−1)< k≤jndefine K_k^glob(ξ, z) =K_jn^glob(ξ, z) = 1

h∂ρ(ξ), ξiⁿT_jⁿ

t(ξ),h∂ρ(ξ), zi h∂ρ(ξ), ξi

. Due to definition ofTj polynomialsK_k^glob(ξ,·) satisfy relations (6) - (7).

3.2 Kor´anyi regions

Forξ∈∂Ω andε >0 we define the inner Kor´anyi regionas Dⁱ(ξ, η) ={τ∈Ω:π(τ)∈B(ξ,−ηρ(τ)), ρ(τ)>−ε}.

The strong convexity ofΩimplies that area-integral inequality by S. Krantz and S.Y. Li [7] forf ∈H^p(Ω), 0< p <∞,could be expressed as

Z

∂Ω

dσ(z)





 Z

Dⁱ(z,η)

|∂f(τ)|² dµ(τ) (−ρ(τ))ⁿ⁻¹







p/2

≤c(Ω, p) Z

∂Ω

|f|^pdσ. (10)

Consider the decomposition of vectorτ−ξ∈Cⁿ asτ =w+tn(ξ),where w∈Tξ, t∈C,and n(z) = _|∂ρ(ξ)|^∂ρ(ξ) is a complex normal vector atz. We define theexternal Kor´anyi region as

D^e(ξ, η) ={τ∈Cⁿ\Ω:τ =w+tn(z),

w∈Tz, t∈C, |w|< ηρ(τ), |Re(t)|< ηρ(τ), ρ(τ)< ε}. In appendix A we will proof the area-integral inequality similar to 10 for external regionsD^e.

We point out two rules for integration over regionsD^e(ξ, η).First, for every function F we have

Z

Ωε\Ω

|F(z)|dµ(z)≍ Z

∂Ω

dσ(ξ) Z

D^e(ξ,η)

|F(τ)|dµ(τ) ρ(τ)ⁿ. Second, if F(w) = ˜F(ρ(w)) then

Z

D^e(ξ,η)

|F(τ)|dµ(τ)≍

ε

Z

0

F(t)˜

tⁿdt.

Similar rules are valid for regionsDⁱ.

We could clarify the estimate ofd(τ, w) in lemma3.1forτ ∈D^e(z, η).

Lemma 3.4 Let Ω be a strongly convex domain andη >0, then

d(τ, w)≍ρ(τ) +d(z, w), z, w∈∂Ω, τ ∈D^e(z, η). (11)

Proof Forτ∈D^e(z, η) we haved(π(τ), z).ηρ(τ) and

d(τ, w).ρ(τ) +d(π(τ), w).ρ(τ) +d(π(τ), z) +d(z, w).ρ(τ) +d(z, w).

On the other hand,

ρ(τ) +d(z, w).ρ(τ) + (d(z, π(τ)) +d(π(τ), w)).(1 +η)ρ(τ) +d(π(τ), w) .ρ(τ) +d(π(τ), w).d(τ, w).

4 The method of pseudoanalytical continuation 4.1 Definition of pseudoanalytical continuation

The main tool of this paper is the method of continuation of function f ∈ H(Ω) outside the domain Ω. Let f ∈ H¹(Ω) and let the boundary values of f almost everywhere coincide with the boundary values of some function f ∈ C_loc¹ (Cⁿ\Ω) such that

∂f¯

∈ L¹(Cⁿ \Ω). Then by Stokes formula for z∈Ωwe have

f(z) = lim

r→0+

1 (2πi)ⁿ

Z

∂Ωr

f(ξ)∂ρ(ξ)∧( ¯∂∂ρ(ξ))ⁿ⁻¹ h∂ρ(ξ), ξ−ziⁿ =

r→0+lim 1 (2πi)ⁿ

Z

Cⁿ\Ωr

∂f(ξ)¯ ∧∂ρ(ξ)∧( ¯∂∂ρ(ξ))ⁿ⁻¹ h∂ρ(ξ), ξ−ziⁿ

= 1

(2πi)ⁿ Z

Cⁿ\Ω

∂f¯ (ξ)∧∂ρ(ξ)∧( ¯∂∂ρ(ξ))ⁿ⁻¹ h∂ρ(ξ), ξ−ziⁿ , (12) since (for details see [12])

dξ

∂ρ(ξ)∧( ¯∂∂ρ(ξ))ⁿ⁻¹ h∂ρ(ξ), ξ−ziⁿ

= 0, z∈Ω, ξ ∈Cⁿ\Ω.

This formula allows us to study properties of function f ∈H(Ω) relying on estimates of its continuation. Note that it is not necessary for the function f to be a continuation in terms of coincidence of boundary values. We call the function f ∈ C_loc¹ (Cⁿ\Ω) satisfying (12) the pseudoanalytic continuation of the functionf.

4.2 Continuation by symmetry

Theorem 4.1 Let f ∈H_p¹(Ω)and1< p <∞, m∈N.There exist a pseudo- analytical continuation f ∈C_loc¹ (Cⁿ\Ω)of functionf such thatsuppf ⊂Ωε,

∂f¯ (z)

∈L^p(Ωε\Ω)and

∂f¯ (z)

. max

|α|=m|∂^αf(z^∗)|ρ(z)^m−1, z∈Ωε\Ω, (13) wherez^∗ is a symmetric to z with respect to∂Ω.

Proof Define

f0(z) = X

|α|≤m−1

∂^αf(z^∗)(z−z^∗)^α

α! , z∈Ωε\Ω. (14) Let α±ek = (α1, . . . , αk±1, αn) and define (z−z^∗)^α−ek = 0 if αk = 0.In these notations we have

∂¯jf0=

∞

X

k=1

X

|α|≤m−1

∂^α+ekf(z^∗)(z−z^∗)^α

α! −∂^αf(z^∗)(z−z^∗)^α−ek (α−ek)!

∂¯jz^∗_k

=

∞

X

k=1

X

|α|=m−1

∂^α+ekf(z^∗)(z−z^∗)^α

α! ∂¯jz_k^∗, (15) hence,

∂f¯0(z)

. max

|α|=m|∂^αf(z^∗)|ρ(z)^m−1, z∈Cⁿ\Ω.

Consider functionχ∈C^∞(0,∞) such thatχ(t) = 1 fort≤ε/2 andχ(t) = 0 for t ≥ ε. The function f(z) = f0(z)χ(ρ(z)) satisfies the condition (14) and suppf ⊂ Ωε.

Letd= dist(z^∗, ∂Ω)/10,then for every mutiindexαsuch that|α|=mby Cauchy maximal inequality we have

|∂^αf(z^∗)|.d^−m+1 sup

|τ−z^∗|<d|∂f(τ)|.ρ(z)^−m+1 sup

τ∈Dⁱ(π(z),c0d)|∂f(τ)|, for somec0>0.Finally, by theorem 2.1 from [7] we get

Z

Ωε\Ω

∂f¯ (z)

pdµ(z). Z

Ω\Ω−ε

dµ(z) sup

τ∈Dⁱ(π(z),c0d)|∂f(τ)|

!p

.k∂fk^pH^p(Ω).

4.3 Continuation by global approximations.

Letf ∈ H¹(Ω) and consider a polynomial sequenceP1, P2, . . . converging to f inL¹(∂Ω).Define

λ(z) =ρ(z)⁻¹|P₂^k+1(z)−P₂^k(z)|, 2^−k< ρ(z)<2^−k+1.

Theorem 4.2 Assume thatλ∈L^p(Cⁿ\Ω)for somep≥1.Then there exist a pseudoanalytical continuationf of function f such that

∂f(z)¯

.λ(z), z∈Cⁿ\Ω. (16)

Proof Consider function χ ∈ C^∞(0,∞) such that χ(t) = 1 for t ≤ 1 and χ(t) = 0 fort≥2.We define continuation of functionf by formula

f(z) =P2^k(z) +χ(2^kρ(z))(P2^k+1(z)−P2^k(z)), 2^−k< ρ(z)<2^−k+1. Nowf isC¹-function onCⁿ\Ω¯ and

∂f¯ (z)

.λ(z). We define a function Fk(z) asFk(z) =f(z) forρ(z)>2^−k and asFk(z) =P2^k+1(z) forρ(z)<2^−k. The the functionFk is smooth and holomorphic inΩ2^−k,and

∂F¯ k(z) .λ(z) forz∈Cⁿ\Ω2^−k.Thus similarly to12we get

P₂^k+1(z) =Fk(z) = 1 (2πi)ⁿ

Z

Cⁿ\Ω

∂F¯ k(ξ)∧∂ρ(ξ)∧( ¯∂∂ρ(ξ))ⁿ⁻¹

h∂ρ(ξ), ξ−ziⁿ , z∈Ω,

We can pass to the limit in this formula by the dominated convergence the- orem; hence, function f satisfies the formula (12) and is a pseudoanalytical continuation of functionf.

4.4 Pseudoanalytical continuation of Hardy-Sobolev spaces

Theorem 4.3 Let Ω be a strongly convex domain, 1 < p < ∞, l ∈ N and f ∈H^p(Ω). Then f ∈H_p^l(Ω) if and only if there exists such pseudoanalytical continuation f that

Z

∂Ω

dσ(z)





 Z

D^e(z,α)

∂f¯ (τ)ρ(τ)^−l

2dν(τ)







p/2

<∞, (17)

wheredν(τ) = _ρ(τ)^dµ(τ)n−1.

Proof Let f ∈ H_p^l(Ω). By theorem 4.3 we could construct pseudoanalytical continuationf such that

∂f(z)¯

. max

|α|=m+1|∂^αf(z^∗)|ρ(z)^m, z∈Cⁿ\Ω.

Note that the symmetry with respect to ∂Ω maps the external sector D^e(z, η) into some internal Kor´anyi sector. Indeed, for every α > 0 there existsη1=η1(η)>0 such that

{τ^∗:τ∈D^e(z, η)} ⊆Dⁱ(z, η1).

Applying area-integral inequality (10) we obtain

Z

∂Ω

dσ(z)





 Z

D^e(z,η)

∂f¯ (τ)ρ(τ)^−l

2dν(τ)







p/2

. max

|α|=m

Z

∂Ω

dσ(z)





 Z

D^e(z,η)

|∂^αf(τ^∗)|²dν(τ)







p/2

. max

|α|=m

Z

∂Ω

dσ(z)





 Z

Dⁱ(z,η1)

|∂^αf(τ)|² dµ(τ) (−ρ(τ))ⁿ⁻¹







p/2

<∞

To prove the sufficiency, assume that functionf ∈H¹(Ω) admits the pseu- doanalytical continuationf with estimate (17.) We will prove that for every function g∈L^p′(∂Ω), ¹_p+_p¹′ = 1,and every multiindex α, |α| ≤l,

Z

∂Ω

g(z)∂^αf(z)dS(z)

≤c(f)kgkL^p′(∂Ω).

Assume, without loss of generality, that α = (l,0, . . . ,0). By representa- tion (12) we have

f(z) = Z

Cn\Ω

∂f¯ (ξ)∧ω(ξ) h∂ρ(ξ), ξ−ziⁿ

and withCnl=^(n+l−1)!_(n−1)!

Z

∂Ω

g(z)∂^αf(z)dS(z) =Cnl

Z

∂Ω

g(z)





 Z

Cⁿ\Ω

∂ρ(ξ)

∂ξ1

l ∂f(ξ)¯ ∧ω(ξ) h∂ρ(ξ), ξ−zi^n+l





dS(z)

=Cnl

Z

Cⁿ\Ω

∂ρ(ξ)

∂ξ1

l

∂f¯ (ξ)∧ω(ξ) Z

∂Ω

g(z)dS(z) h∂ρ(ξ), ξ−zi^n+l.

DefineΦl(ξ) = R

∂Ω

g(z)dS(z)

h∂ρ(ξ), ξ−zi^n+l.Applying H¨older inequality twice we have

Z

∂Ω

g(z)∂^αf(z)dS(z)

. Z

Cⁿ\Ω

∂f¯ (ξ)

|Φl(ξ)|dµ(ξ) .

Z

∂Ω

dS(ξ) Z

D^e(ξ,α)

∂F¯ (τ)

|Φl(τ)|dµ(τ) ρ(τ)ⁿ

. Z

∂Ω

dσ(ξ)





 Z

D^e(ξ,α)

∂F¯ (τ)

2ρ(τ)^−2l dµ(τ) ρ(τ)ⁿ⁻¹







1/2

×

×





 Z

D^e(ξ,α)

|Φl(τ)|²ρ(τ)^2l−2 dµ(τ) ρ(τ)ⁿ⁻¹







1/2

.







 Z

∂Ω

dS(ξ)





 Z

D^e(ξ,α)

∂F¯ (τ)

2ρ(τ)^−2ldν(τ)







p/2







1/p

×

×







 Z

∂Ω

dS(ξ)





 Z

D^e(ξ,α)

|Φl(τ)|²ρ(τ)^2l−2dν(τ)







p^′/2







1/p^′

.

The first product term is bounded by (17), and the second one by the area- integral inequality (30), that we will prove in the appendixAin theoremA.1.

5 Constructive description of Hardy-Sobolev spaces

Theorem 5.1 Let f ∈H¹(Ω) and1 < p < ∞, l ∈N. Then f ∈H_p^l(Ω)iff there exists a polynomial sequenceP₂^k such that

Z

∂Ω

dσ(z)

∞

X

k=1

|f(z)−P2^k(z)|²2^2lk

!p/2

<∞. (18)

Proof Assume that condition (18) holds, then polynomials P2^k converge to function f inL^p(∂Ω) and by the theorem 4.2we could construct pseudoana- lytical continuationf such that

∂f¯ (z)

.|P2^k+1(z)−P2^k(z)|ρ(z)⁻¹, z∈Cⁿ\Ω, 2^−k ≤ρ(z)<2^−k+1.

Consider the decomposition of region D^e(z, η) to sets Dk(z) = {τ ∈ D^e(z, η) : 2^−k ≤ρ(τ)<2^−k+1},and define functions

ak(z) =|P2^k+1(z)−P2^k(z)|2^−kl,

bk(z) =





 Z

Dk(z)

∂f¯ (τ)ρ(τ)^−l

2dν(τ)







1/2

.

Lemma 5.1 bk(z).M ak(z)

Assume, that this lemma holds, then by Fefferman-Stein maximal theorem we have

Z

∂Ω

∞

X

k=1

bk(z)²

!p/2

dσ(z). Z

∂Ω

∞

X

k=1

ak(z)²

!p/2

dσ(z).

The right-hand side of this inequality is finite by the condition 18, also we have

∞

X

k=1

bk(z)²≍ Z

S(z)

∂f(ξ)ρ(ξ)¯ ^−l

2dν(ξ),

which completes the proof of the sufficiency in the theorem..

Prove the necessity. Now f ∈H_p^l(Ω) with 1< p <∞andl∈N.By theo- rem4.3we could construct continuationfof functionfwith estimate (17). Ap- plying the approximation of Cauchy-Leray-Fantappi`e kernel from lemma 3.3 to functionf we define polynomials

P2^k(z) = Z

Cn\Ω

∂f¯ (ξ)∧ω(ξ)K₂^globk (ξ, z).

We will prove that these polynomials satisfy the condition (18). From lemma3.3 we obtain

|f(z)−P₂^k(z)|. Z

Cⁿ\Ω

∂f(ξ)¯

1

h∂ρ(ξ), ξ−zi^d −K₂^globk (ξ, z)

dµ(ξ)

=U(z) +V(z) +W1(z) +W2(z),

where

U(z) = Z

d(τ,z)<2^−k

∂f¯ (τ)

|h∂ρ(τ), τ−zi|ⁿdµ(τ), V(z) = 2^kn

Z

d(τ,z)<2^−k

∂f¯ (τ)

dµ(τ),

W1(z) = 2^−kr Z

d(τ,z)>2^−k ρ(τ)<2^−k

∂f¯ (τ)

|h∂ρ(τ), τ−zi|^n+rdµ(τ),

W2(z) = 2^−kr Z

ρ(τ)>2^−k

∂f¯ (τ)

|h∂ρ(τ), τ−zi|^n+rdµ(τ).

The parameterr >0 will be chosen later.

Note thatV(z).cU(z) and estimate the contribution ofU(z) to the sum.

For somec1, c2>0 we have

U(z)≤ Z

d(w,z)<c12^−k w∈∂Ω

dσ(w) X

j>c2k

Z

Dj(w)

∂f¯ (τ)

|h∂ρ(τ), τ−zi|ⁿ dν(τ)

ρ(τ)

≤ Z

d(w,z)<c12^−k w∈∂Ω

dσ(w) X

j>c2k





 Z

Dj(w)

∂f¯ (τ)ρ(τ)^−l

2dν(τ)







1/2

×

×





 Z

Dj(w)

ρ(τ)^2(l−1)dν(τ)

|h∂ρ(τ), τ−zi|ⁿ







1/2

= X

j>c2k

Z

d(w,z)<c12^−j

bj(w)mj(w)dσ(w)

Consider the integralmj(w).Sinceτ ∈Dj(w) then by estimates from lemma3.4 d(τ, z)≍ρ(τ) +d(w, z)>2^−j and

mj(w) =





 Z

Dj(w)

ρ(τ)^2(l−1)dν(τ)

|h∂ρ(τ), τ−zi|ⁿ







1/2

.2^−j(l−1)

2^−jn 2^−j= 2^jn−jl. (19) Thus

2^klU(z). X

j>c1k

2^−(j−k)l2^jn Z

d(w,z)<c22^j

bj(w)dσ(w). X

j>c1k

2^−(j−k)lM bj(z) (20)

Now estimate the valueW1(z).Similarly to the previous we have W1(z)≤2^−krX

j>k

Z

d(w,z)≥c12^−k

bj(w)m^r_j(w)dσ(w),

where

m^r_j(w) =





 Z

Dj(w)

ρ(τ)^2(l−1)dν(τ)

|h∂ρ(τ), τ−zi|^2(n+r)







1/2

. (21)

Applying the estimated(τ, z)≍ρ(τ) +d(w, z)&2^−j,we obtain

m^r_j(w).2^j(n+r−l). (22)

Finally

2^klW1(z).2^kl−krX

j>k

2^j(r−l)+jn Z

d(w,z)≤2^−j

bj(w)dσ(w)

.X

j>k

2−(j−k)(r−l)M bj(z). (23) Similarly, estimating the contribution of W2(z),we obtain

2^klW2(z).2^−k(r−l)

k

X

j=0

Z

∂Ω

bj(w)m^r_j(w)dσ(w). (24)

Sinced(τ, z)&2^−j+d(w, z) forτ∈∂Ω, τ ∈Dj(z) then m^r_j(w). 2^−jl

(2^−j+d(w, z))^n+r ≤min

2^j(n+r−l),2^−jld(w, z)^−n−r . Thus

Z

∂Ω

bj(w)m^r_j(w)dσ(w). Z

d(w,z)≤2^−j

2^−jl

2^−j(n+r)bj(w)dσ(w)

+

j−1

X

t=1

Z

2^−t−1≤d(w,z)≤2^−t

2^−jl

2^−t(n+r)bj(w)dσ(w)

.

j

X

t=1

2^−jl2^trM bj(z).2^−jl2^jrM bj(z).

Choosingr=l+ 1,we have W2(z)2^kl.

k

X

j=1

2−(k−j)(r−l)M bj(z)≤

k

X

j=1

2^−(k−j)M bj(z). (25)

Combining the estimates (20,23,25) we finally obtain

|f(z)−P2^k(z)|.

k

X

j=1

2^−(k−j)M bj(z) +X

j>k

2^−(j−k)lM bj(z),

which similarly to [5] implies

∞

X

k=1

|f(z)−P2^k(z)|².

∞

X

k=1

(M bk(z))²= Z

D^e(z,η)

∂f¯ (τ)

2ρ(τ)^−2ldµ(τ)<∞.

This completes the proof of the theorem and it remains to prove lemma5.1.

Proof (of the lemma 5.1). Define gk(z) := 2^−kl(P2^k+1(z)−P2^k(z)).

Letz∈∂Ω andτ∈Sk(z).Consider complex normal vectorn(z) =_|∂ρ(z)|^∂ρ(z) at z, complex tangent hyperplaneTz ={w ∈Cⁿ :h∂ρ(z), w−zi= 0} and complex planeT_z,τ^⊥ , orthogonal toTz and containing the pointτ

T_z,τ^⊥ :={τ+sn(z) :s∈C}. Projection of vectorτ∈Cto∂ΩT

T_z,τ^⊥ we will denote asπz(τ).

Define Ωz,τ =ΩTT_z,τ^⊥ andγz,τ =∂Ωz,τ.There exist a conformal map ϕz,τ :T_z,τ^⊥ \Ωz,τ →C\{w∈C:|w|= 1}such that ϕz,τ(∞) =∞, ϕ^′_z,τ(∞)>

0,and we could consider analytical inT_z,τ^⊥ \Ωz,τ function Gk(s) := ^gk(s)

ϕ^2k+1_z,τ (s). Applying to function Gk Dyn’kin maximal estimate from [4] for domain T_z,τ^⊥ \ Ω(z, τ) we obtain the estimate

|Gk(τ)|. 1 ρ(τ)

Z

s∈Iz,τ

|Gk(s)| |ds|+ Z

∂Ωz,τ\Iz,τ

|Gk(s)| ρ(τ)^m

|s−πz(τ)|^m+1|ds|, whereIz,τ ={s∈γz,τ :|s−πz(τ)|<dist(τ, ∂Ωz,τ)/2},andm >0 could be chosen arbitrary large.

Note that |ϕz,τ(s)| −1≍dist(s, ∂Ωz,τ)≍2^−k,thus|gk(s)| ≍ |Gk(s)|for s∈Dk(z)T

T_z,τ^⊥. Hence,

|gk(τ)|.

∞

X

j=1

2^−jm 1 2^jρ(τ)

Z

s∈∂Ωz,τ

|s−πz(τ)|<2^jρ(τ)

|gk(s)| |ds|. (26)

Since the boundary of the domainΩisC³-smooth, we can assume that the constant in this inequality (26) does not depend onz∈∂Ω andτ∈Ωε\Ω.

Note that functiongk(τ+z−w) is holomorphic inw∈Tz,then estimating the mean we obtain

|gk(τ)| ≤ 1 ρ(τ)ⁿ⁻¹

Z

|w−z|<√

ρ(τ)

|gk(τ+z−w)|dµ2n−2(w)

.

∞

X

j=1

2^−jm 1 ρ(τ)ⁿ⁻¹

Z

|w−z|<√

ρ(τ)

dµ2n−2(w) 2^jρ(τ)

Z

s∈∂Ωz,τ

|s−πz(τ+z−w)|<2^jρ(τ)

|gk(s)| |ds|

.

∞

X

j=1

2^{−j(m−n+1)} Z

B(z,2^jρ(τ))

|gk(w)|dσ(w), (27)

wheredµ2n−2is Lebesgue measure in Tz

Assume that m > n−1, then |gk(τ)| . M gk(z), z ∈ ∂Ω, τ ∈ Dk(z).

Finally,

bk(z) =





 Z

Dk(z)

∂f¯ (τ)ρ(τ)^−l

2dν(τ)







1/2

.





 Z

Dk(z)

gk(τ)ρ(τ)^−l−1

2dν(τ)







1/2

.M ak(z)





 Z

Dk(z)

dν(τ) ρ(τ)²







1/2

.M ak(z) (28)

and the lemma is proved.

A Area-integral inequality for external Kor´anyi region

LetΩ⊂Cⁿbe a strongly convex domain and η >0. For functiong∈L¹(∂Ω) andl∈N we define a function

I_l(g, z) =





 Z

D^e(z,η)

Z

∂Ω

g(w)dS(w) h∂ρ(τ), τ−wi^n+l

2

dν_l(τ)







1/2

, (29)

wheredνl(τ) =_ρ(τ)^dµn−2l−1^2n(τ) .

Theorem A.1 LetΩbe strongly convex domain andg∈L^p(∂Ω), 1< p <∞,Then Z

∂Ω

I_l(g, z)^pdσ(z). Z

∂Ω

|g(z)|^pdσ(z). (30)

Note that in the one-variable case the integral29gives the holomorphic function and the result of the theorem follows from [5].

Definition 1 Assume, that defining functionρfor strongly convex domainΩhas the fol- lowing form nearz0∈∂Ω

ρ(z) = Im(zn) +

n

X

j,k=1

A_jkzjz¯_k+O(|z|³). (31)

Define a set

D0(η, ε) ={τ∈Cⁿ\Ω:|τ1|²+. . .+|τn−1|²< ηIm(τn),

|Re(τn)|< ηIm(τn), |Im(τn)|< ε1}. (32) Theorem A.2 There exists such covering of the setΩε\Ω−ε by open setsΓj such that for everyξ∈Γj we can find a holomorphic change of coordinatesϕj(ξ,·) :Cⁿ→Cⁿ such that

1. The mappingϕj(ξ,·)transforms functionρto the type (31) and could be expressed as follows

ϕj(ξ, z) =Φj(ξ)(z−ξ) +i(z−ξ)^⊥Bj(ξ)(z−ξ)en, (33) where matricesΦj(ξ), Bj(ξ)depend smooth onξ∈Γj,anden= (0, . . . ,0,1).

2. Letψj(ξ,·)be an inverse map of ϕj(ξ,·),and letJj(ξ,·)be a complex Jacobian ofψ.

Note that real Jacobian is then equal to|Jj(ξ,·)|²=Jj(ξ,·)Jj(ξ,·), Then sup

τ∈Ωε\Ωε

Jj(ξ,·)−Jj(ξ^′,·) .

ξ−ξ^′

, (34) sup

τ∈Ωε\Ωε

ψj(ξ,·)−ψj(ξ^′,·) .

ξ−ξ^′

. (35) 3. There exist constants c >0, 0< η0, ε1<1such that

ϕj(ξ, D^e(ξ, η))⊆D0(cη, cε), ψj(ξ, D⁰_η)⊆D^e(ξ, cη), for0< η < η0. (36) Proof Let ξ∈ ∂Ω,by linear change of coordinates z^′ = (z−ξ)Φ(ξ) we could obtain the following form for functionρ

ρ(z) =ρ(ξ+Φ⁻¹(ξ)z^′) = Im(z_n^′) +

n

X

j,k=1

A¹_jk(ξ)z^′_j¯z^′_k+ Re

n

X

j,k=1

A²_jk(ξ)z_j^′z_k^′ +O(

z^{′ 3}).

Settingz^′′_n=z_n^′ +iA²_jkz_j^′z_k^′ andz^′′_j =z_j^′, 1≤j≤n−1,we have

ρ(z^′′) = Im(z^′_n) +

n

X

j,k=1

A¹_jk(ξ)z_j^′′z¯_k^′′+O(

z^′′

³).

DenoteB(ξ) =Φ(ξ)^⊥A¹(ξ)Φ(ξ),then

ϕ(ξ, z) =Φ(ξ)(z−ξ) +i(z−ξ)^⊥B(ξ)(z−ξ)en.

We choose Γj such that the matrixΦ(ξ) could be defined onΓj smoothly, this choice we denote asΦj,and the change corresponding to this matrix asϕj

ϕj(ξ, z) =Φj(ξ)(z−ξ) +i(z−ξ)^⊥Bj(ξ)(z−ξ)en.

Thus mappingsϕjsatisfy the first condition. Easily, the second condition also holds.

To prove the last condition (36) it is sufficient to show, that if function ρ has the form (31), then for somec, η0>0 we have

D^e(0, η)⊆D0(cη), D0(η)⊆D^e(0, cη),for0< η < η0.

Indeed, for the functionρof the form (31) the Kor´anyi sector could be expressed as follows D^e(0, η) ={τ∈Cⁿ\Ω:|τ1|²+. . .+|τn−1|²≤(ηρ(τ))², |Re(τn)| ≤ηρ(τ), ρ(τ)< ε}

Assumeτ∈D^e(0, η),then

ρ(τ) = Im(τn) +

n

X

j,k=1

Ajkτjτ¯k≤Im(τn) +c0|Im(τn)|²

+c0(|τ1|²+. . .+|τn−1|+|Re(τn)|²)≤Im(τn) +c0|Im(τn)|²+c0ηρ(τ) +c0(ηρ(τ))². (37) By choosingη0<1/(4c0) we haveρ(τ)≤cIm(τn) forτ∈D^e(0, η) and 0< η < η0.Hence, D^e(0, η)⊆D0(cη) for 0< η < η0 andε1=cε.Note, that

ρ(τ) = Im(τn) +

n

X

j,k=1

A_jkτj¯τ_k>Im(τn) forτ∈D0(η),

since matrix (Ajk) is strictly positive definite. Hence,D0(η)⊆D^e(0, η).This ends the proof of the theorem.

Further we will assume, that the coveringΩε\Ω−ε⊂ S^N

j=1

Γjand mapsϕj, ψjare chosen by the theoremA.2. For covering{Γj}we consider a smooth decomposition of identity on

∂Ω:

χj∈C^∞(Γj), 0≤χj≤1, suppχj⊂Γj,

N

X

j=1

χj(z) = 1, z∈∂Ω.

Fix the parameterη < η0and denoteD0=D0(η).Then by (36) Il(g, z)²

=

N

X

j=1

χj(z) Z

ϕj(z,D^e(z,η/c))

Z

∂Ω

g(w)Jj(z, τ)dS(w) h∂ρ(ψj(z, τ)), ψj(z, τ)−wi^n+l

2

dµ(τ) Im(τn)^n−2l+1

.

N

X

j=1

Z

D0

Z

∂Ω

g(w)χ^1/2_j (z)Jj(z, τ)dS(w) h∂ρ(ψj(z, τ)), ψj(z, τ)−wi^n+l

2

dµ(τ)

Im(τn)^n−2l+1. (38) We will consider the function

Kj(z, w)(τ) = χ^1/2_j (z)Jj(z, τ)

h∂ρ(ψj(z, τ)), ψj(z, τ)−wiⁿ⁺¹ (39) as a map∂Ω×∂Ω →L(C, L²(D0, dνl)),such that its values are operator of multiplica- tion fromCto L²(D0, dνl),where dνl(τ) = _Im(τ^dµ(τ)

n)^n−2l+1 is a measure on the regionD0. Throughout the proof of the theoremA.1j, lwill be fixed integers and the norm of function F in the spaceL²(D0, dνl) will be denoted askFk.

We will show that integral operator defined by kernel Kj is bounded onL^p.To prove this we applyT1-theorem for transformations with operator-valued kernels formulated by Hyt¨onen and Weis in [8], taking in account that in our case concerned spaces are Hilbert.

Some details of the proof are similar to the proof of the boundedness of operator Cauchy- Leray-Fantappi`eKΩfor lineally convex domains introduced in [13]. Below we formulate the T1-theorem, adapted to our context.