Constructive description of Hardy-Sobolev spaces in $\mathbb{C}^n$

0.26 MB·

by  Alexander Rotkevich

Constructive description of Hardy-Sobolev spaces in Cn
Alexander Rotkevicha
6 aDepartment of Mathematical analysis, Mathematics andMechanics Faculty, St. Petersburg
1 State University,198504, Universitetskyprospekt, 28, Peterhof, St. Petersburg, Russia
0
2

v
o
N Abstract

9
Inthis paperwe studythe polynomialapproximationsinHardy-Sobolevspaces

] onfor convexdomains. We use the method ofpseudoanalyticalcontinuationto
V
C obtain the characterization of these spaces in terms of polynomial approxima-
.
h tions.
t
a Keywords: Hardy-Sobolev Spaces, polynomial approximations,
m
pseudoanalytical continuation, Cauchy-Leray-Fantappi`eintegral
[

2010 MSC: 32E30,41A10
2
v
8
1 1. Introduction
2
3
0 ThepurposeofthispaperistogiveanalternativecharacterizationsofHardy-
.
1 Sobolev (see. [1]) spaces
0
6
1 Hl(Ω)= f H(Ω): f + ∂αf < (1)
p { ∈ k kHp(Ω) k kHp(Ω) ∞}
:
v |Xα|≤l
i
X on strongly convex domain Ω Cn.
⊂
r
a We continue the research started in [15] and devoted to description of ba-
sic spaces of holomorphic functions of several variables in terms of polynomial
approximationsandpseudoanalyticalcontinuation. In particular,we show that
for1<p< andl 1aholomorphiconastronglyconvexdomainΩfunction
∞ ≥
f is in the Hardy-Sobolev space Hl(Ω) if and only if there exist a sequence of
p
Email address: [email protected] (Alexander Rotkevich)
Preprint submitted toJournal of Approximation Theory November 10, 2016
2k degree polynomials P such that
− 2k
∞ p/2
dσ(z) f(z) P (z)222lk < . (2)
| − 2k | ! ∞
Z k=1
∂Ω X
Intheonevariablecasethisconditionfollowsfromthecharacterizationobtained
by E.M. Dynkin [5] for Radon domains.
The paper is divided into five sections with one appendix. In section 2 we
givemaindefinitions andpreliminariesofthis work. Section3 isdevotedto 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 [8] for external Kor´anyi regions established
in appendix A. Finally, section 5 contains the proof of characteristics (2).
2. Main notations and definitions
Let Cn be the space of n complex variables, n 2, z = (z ,...,z ), z =
1 n j
≥
x +iy ;
j j
∂f 1 ∂f ∂f ∂f 1 ∂f ∂f
∂ f = = i , ∂¯ f = = +i ,
j j
∂z 2 ∂x − ∂y ∂z¯ 2 ∂x ∂y
j (cid:18) j j(cid:19) j (cid:18) j j(cid:19)
n n
∂f ∂f
∂f = dz , ∂¯f = dz¯ , df =∂f +∂¯f.
k k
∂z ∂z¯
k k
k=1 k=1
X X
The notation
n
∂f(z)
∂f(z), w = w .
k
h i ∂z
k
k=1
X
is used to indicate the action of ∂f on the vector w Cn, and
∈
∂f ∂f
∂¯f = +...+ .
| | ∂z ∂z
(cid:12) 1(cid:12) (cid:12) n(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
2
The euclidean distance form the point z Cn to the set D Cn we denote
∈ ⊂
asdist(z, D)=inf z w :w D .LebesguemeasureinCn wedenoteasdµ.
{| − | ∈ }
For a multiindex α = (α ,...,α ) Nn we set α = α + ... + α
1 n 1 n
∈ | |
and α! =α1!...α2!, also zα =z1α1...znαn and ∂αf = ∂z¯1α∂1|.α..|∂fz¯nαn.
LetΩ= z Cn :ρ(z)<0 beastronglyconvexdomainwithaC3-smooth
{ ∈ }
defining function. We need to consider a family of domains
Ω = z Cn :ρ(z)<t
t
{ ∈ }
that arealso stronglyconvexfor each t <ε, where ε>0 is smallenough, that
| |
is d2ρ(z) is positive definite when ρ(z) ε. For z Ω Ω we denote the
ε −ε
| | ≤ ∈ \
nearest point on ∂Ω as pr (z). Then the mapping
∂Ω
pr :Ω Ω ∂Ω
∂Ω ε\ −ε →
is well defined, C2 smooth on Ω Ω and z pr (z) =dist(z, ∂Ω).
− ε\ | − ∂Ω |
For ξ ∂Ω we define the complex tangent space
t
∈
T = z Cn : ∂ρ(ξ), ξ z =0 .
ξ
{ ∈ h − i }
The space of holomorphic functions we denote as H(Ω) and consider the
Hardy space (see [18], [6])
Hp(Ω):= f H(Ω): f p = sup f(z)pdσ (z)< ,
 ∈ k kHp(Ω) | | t ∞
−ε<t<0
 ∂ZΩt 
where dσt is induced Lebesgue measure on the boundary of Ωt. We alsodenote
dσ =dσ . Hardy-Sobolev spaces Hl(Ω) are defined by (1).
0 p
Throughout this paper we use notations ., . We let f . g if f cg for
≍ ≤
some constant c > 0, that doesn’t depend on main arguments of functions f
and g and usually depend only on dimension n and domain Ω. Also f g if
≍
c−1g f cg for some c>1.
≤ ≤
3. Cauchy-Leray-Fantappi`e formula
In the context of theory of several complex variables there is no unique
reproducing formula formula, however we could use the Leray theorem, that
3
allows us to construct holomorphic reproducing kernels ([2], [12], [13]). For
convex domain Ω = z Cn :ρ(z)<0 this theorem brings us Cauchy-Leray-
{ ∈ }
Fantappi`e formula, and for f H1(Ω) and z Ω we have
∈ ∈
1 f(ξ)∂ρ(ξ) (∂¯∂ρ(ξ))n−1
f(z)=K f(z)= ∧ = f(ξ)K(ξ,z)ω(ξ),
Ω (2πi)n ∂ρ(ξ), ξ z n
Z h − i Z
∂Ω ∂Ω
(3)
where ω(ξ)= 1 ∂ρ(ξ) (∂¯∂ρ(ξ))n−1, and K(ξ,z)= ∂ρ(ξ), ξ z −n.
(2πi)n ∧ h − i
The(2n 1)-formωdefineson∂Ω Leray-LevymeasuredS,thatisequivalent
t
−
to Lebesgue surface measure dσ (for details see [2], [10], [11]). This allows us
t
to identify Lebesgue, Hardy and Hardy-Sobolev spaces defined with respect to
measures dσ and dS. Also note, that measure dV defined by the 2n-form
t
dω =(∂∂¯ρ)n is equivalent to Lebesgue measure dµ in Cn.
By [14] the integral operator K defines a bounded mapping on Lp(∂Ω) to
Ω
Hp(Ω) for 1<p< .
∞
The function d(w,z) = ∂ρ(w), w z defines on ∂Ω quasimetric, and
|h − i|
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 [14]. Therefore ∂Ω,d,σ is a space of homo-
≍ { }
geneous type.
Note also the crucial role in the forthcoming considerations of the following
estimate that is proved in [15].
Lemma 3.1. Let Ω be strongly convex, then
d(w,z) ρ(w)+d(pr (w),z), w Cn Ω, z ∂Ω.
≍ ∂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´ekernel based on theorem by V.K. Dzyadyk about estimates of
Cauchy kernel on domains on complex plane (theorem 1 in part 1 of section 7
in[3]). Theapproximationischoosedsimilarlyto[16]. Thisconstructionallows
us in theorem 5.1 to get polynomials that approximate holomorphic function
with desired speed.
4
Lemma 3.2. Let Ω be a strongly convex domain with 0 Ω, then for every
∈
ξ Ω Ω the value of λ = h∂ρ(ξ), zi for z Ω lies in domain L(t), bounded
∈ ε \ h∂ρ(ξ), ξi ∈
by the bigger arc of the circle λ = R = R(Ω) and the chord λ C : λ =
| | { ∈
1+eits, s R, λ R , where t= π arg( ∂ρ(ξ), ξ ).
∈ | |≤ } 2 − h i
Proof. For ξ ∂Ω define
∈
∂ρ(ξ), z
Λ(ξ)= λ C:λ= h i, z Ω .
∈ ∂ρ(ξ), ξ ∈
(cid:26) h i (cid:27)
The convexity of Ω with 0 Ω implies that
∈
∂ρ(ξ), ξ & ∂ρ(ξ) ξ &1, (4)
|h i| | || |
Re ∂ρ(ξ), z ξ 0, z Ω¯, ξ Ω Ω. (5)
ε
h − i≤ ∈ ∈ \
The domain Λ(ξ) C is also convex and contains 0, thus the equality
⊂
∂ρ(ξ), z ∂ρ(ξ), z ξ
h i =1+ h − i
∂ρ(ξ), ξ ∂ρ(ξ), ξ
h i h i
with estimates (4), (5) completes the proof of the lemma. (cid:3)
Lemma 3.3. Let Ω be a strongly convex domain and r > 0. Then for every
k N there exist function Kglob(ξ,z) defined for ξ Ω Ω and polynomial in
∈ k ∈ ε\
z Ω with degK (ξ, ) k and following properties:
k
∈ · ≤
1 1 1
K(ξ,z) Kglob(ξ,z) . , d(ξ,z) ; (6)
− k kr d(ξ,z)n+r ≥ k
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
1
Kglob(ξ,z) .kn, d(ξ,z) . (7)
k ≤ k
(cid:12) (cid:12)
Proof. Due to [3] an(cid:12)(cid:12)d [17] for a(cid:12)(cid:12)ny j N there exists a function Tj(t,λ)
∈
polynomial in λ with degT (t, ) j such that
j
· ≤
1 1 1
T (t,λ) . (8)
1 λ − j jr 1 λ1+r
(cid:12)(cid:12) − (cid:12)(cid:12) | − |
(cid:12) (cid:12)
for λ L(t) λ: 1 (cid:12)λ < 1 and coe(cid:12)fficients of polynomials T (t,λ) contin-
∈ \ | − | j j
uously dependnon t. Note alsoothat by maximum principle
1
T (t,λ).j, λ L(t) λ: 1 λ < . (9)
j
∈ | − | j
(cid:26) (cid:27)
\
5
Let t(ξ)= π arg( ∂ρ(ξ), ξ and for j N and (j 1)<k jn define
2 − h i ∈ − ≤
1 ∂ρ(ξ), z
Kglob(ξ,z)=Kglob(ξ,z)= Tn t(ξ),h i .
k jn ∂ρ(ξ), ξ n j ∂ρ(ξ), ξ
h i (cid:18) h i(cid:19)
Due to definition of T polynomials Kglob(ξ, ) satisfy relations (6), (7). (cid:3)
j k ·
3.2. Kora´nyi regions
For ξ ∂Ω and ε>0 we define the inner Kora´nyi region as
∈
Di(ξ,η,ε)= τ Ω:pr (τ) B(ξ, ηρ(τ)), ρ(τ)> ε .
{ ∈ ∂Ω ∈ − − }
The strongconvexityofΩ implies thatarea-integralinequality byS. Krantz
and S.Y. Li [8] for f Hp(Ω), 0<p< , could be expressed as
∈ ∞
p/2
dµ(τ)
dσ(z) ∂f(τ)2 c(Ω,p) f pdσ. (10)
 | | ( ρ(τ))n−1 ≤ | |
Z Z − Z
∂Ω Di(z,η,ε) ∂Ω
 
 
Consider the decomposition of vector τ Cn as τ = w + tn(ξ), where
∈
w T , t C, and n(ξ) = ∂¯ρ(ξ) is a complex normal vector at ξ. We define
∈ ξ ∈ ∂¯ρ(ξ)
| |
the external Kora´nyi region as
De(ξ,η,ε)= τ Cn Ω:τ =w+tn(ξ),
{ ∈ \
w T , t C, w < ηρ(τ), Im(t) <ηρ(τ), ρ(τ)<ε . (11)
ξ
∈ ∈ | | | | }
p
In appendix A we will proof the area-integral inequality similar to (10) for
external regions De(ξ,η,ε).
We point out two rules for integration over regions De(ξ,η,ε). First, for
every function F we have
dµ(τ)
F(z) dµ(z) dσ(ξ) F(τ) .
| | ≍ | | ρ(τ)n
Z Z Z
Ωε\Ω ∂Ω De(ξ,η,ε)
Second, if F(w)=F˜(ρ(w)) then
ε
F(τ) dµ(τ) F˜(t) tndt.
| | ≍
De(ξZ,η,ε) Z0 (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
6
Similar rules are valid for regions Di(ξ,η,ε).
We could clarify the estimate of d(τ,w) in lemma 3.1 for τ De(z,η,ε).
∈
Lemma 3.4. Let Ω be a strongly convex domain and ε,η >0, then
d(τ,w) ρ(τ)+d(z,w), z,w ∂Ω, τ De(z,η,ε). (12)
≍ ∈ ∈
Proof. For τ De(z,η,ε) we denote τˆ = pr (τ), then d(τˆ,z) . ηρ(τ) and
∈ ∂Ω
by lemma 3.1
d(τ,w).ρ(τ)+d(τˆ.ρ(τ)+d(τˆ,z)+d(z,w).ρ(τ)+d(z,w).
On the other hand,
ρ(τ)+d(z,w).ρ(τ)+(d(z,τˆ)+d(τˆ,w)).(1+η)ρ(τ)+d(τˆ,w)
.ρ(τ)+d(τˆ,w).d(τ,w).
(cid:3)
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
∈
C1 (Cn Ω) such that ∂¯f L1(Cn Ω). Then by Stokes formula for z Ω we
loc \ ∈ \ ∈
have (cid:12) (cid:12)
(cid:12) (cid:12)
1 f(ξ)∂ρ(ξ) (∂¯∂ρ(ξ))n−1
f(z)= lim ∧ =
r→0+(2πi)n ∂ρ(ξ), ξ z n
Z h − i
∂Ωr
1 ∂¯f(ξ) ∂ρ(ξ) (∂¯∂ρ(ξ))n−1
lim ∧ ∧
r→0+(2πi)n ∂ρ(ξ), ξ z n
CnZ\Ωr h − i
1 ∂¯f(ξ) ∂ρ(ξ) (∂¯∂ρ(ξ))n−1
= ∧ ∧ ,
(2πi)n ∂ρ(ξ), ξ z n
CnZ\Ω h − i
7
since (for details see [13])
∂ρ(ξ) (∂¯∂ρ(ξ))n−1
d ∧ =0, z Ω, ξ Cn Ω.
ξ ∂ρ(ξ), ξ z n ∈ ∈ \
(cid:18) h − i (cid:19)
This formula allows us to study properties of function f H(Ω) relying on
∈
estimates of its continuation.
Definition 4.1. We call the function f C1 (Cn Ω) the pseudoanalytic con-
∈ loc \
tinuation of the function f H(Ω) if
∈
1 ∂¯f(ξ) ∂ρ(ξ) (∂¯∂ρ(ξ))n−1
f(z)= ∧ ∧ , z Ω. (13)
(2πi)n ∂ρ(ξ), ξ z n ∈
CnZ\Ω h − i
Note that it is not necessary for the function f to be a continuationin terms of
coincidence of boundary values.
4.2. Continuation by symmetry
For z Ω Ω we define the symmetric along ∂Ω point z∗ Ω by
ε
∈ \ ∈
z∗ z =2(pr (z) z).
− ∂Ω −
Theorem 4.1. Let f H1(Ω) and 1<p< , m N. There exist a pseudo-
∈ p ∞ ∈
analytical continuation f C1 (Cn Ω) of function f such that supp f Ω ,
∈ loc \ ⊂ ε
∂¯f(z) Lp(Ω Ω) and
ε
∈ \
(cid:12) (cid:12)
(cid:12) (cid:12) ∂¯f(z) . max ∂αf(z∗) ρ(z)m−1, z Ω Ω. (14)
ε
|α|=m| | ∈ \
(cid:12) (cid:12)
Proof. Define(cid:12) (cid:12)
(z z∗)α
f (z)= ∂αf(z∗) − , z Ω Ω. (15)
0 ε
α! ∈ \
|α|X≤m−1
Letα ek =(α1,...,αk 1,αn)anddefine (z z∗)α−ek =0ifαk =0.Inthese
± ± −
notations we have
∞ (z z∗)α (z z∗)α−ek
∂¯ f = ∂α+ekf(z∗) − ∂αf(z∗) − ∂¯ z∗
j 0 α! − (α e )! j k
Xk=1|α|X≤m−1(cid:18) − k (cid:19)
∞ (z z∗)α
= ∂α+ekf(z∗) − ∂¯ z∗, (16)
α! j k
Xk=1|α|X=m−1
8
hence,
∂¯f (z) . max ∂αf(z∗) ρ(z)m−1, z Cn Ω.
0
|α|=m| | ∈ \
(cid:12) (cid:12)
Consider functi(cid:12)on χ (cid:12)C∞(0, ) such that χ(t) = 1 for t ε/2 and χ(t) = 0
∈ ∞ ≤
for t ε. The function f(z) = f (z)χ(ρ(z)) satisfies the condition (15) and
0
≥
supp f Ω .
ε
⊂
Let d = dist(z∗, ∂Ω)/10, then for every mutiindex α such that α = m by
| |
Cauchy maximal inequality we have
∂αf(z∗) .d−m+1 sup ∂f(τ) .ρ(z)−m+1 sup ∂f(τ) ,
| | | | | |
|τ−z∗|<d τ∈Di(pr∂Ω(z),c0d,ε)
for some c >0. Finally, by theorem 2.1 from [8] we get
0
p
∂¯f(z) pdµ(z). dµ(z) sup ∂f(τ)
ZΩε\Ω ZΩ\Ω−ε τ∈Di(pr∂Ω(z),c0d,ε)| |!
(cid:12) (cid:12)
(cid:12) (cid:12) . ∂f p < .
k kHp(Ω) ∞
(cid:3)
4.3. Continuation by global approximations.
Let f H1(Ω) andconsider a polynomialsequence P ,P ,... convergingto
1 2
∈
f in L1(∂Ω). Define
λ(z)=ρ(z)−1 P (z) P (z) , 2−k <ρ(z) 2−k+1.
| 2k+1 − 2k | ≤
Theorem 4.2. Assume that λ Lp(Cn Ω) for some p 1. Then there exist
∈ \ ≥
a pseudoanalytical continuation f of function f such that
∂¯f(z) .λ(z), z Cn Ω. (17)
∈ \
(cid:12) (cid:12)
Proof. Consider functio(cid:12)n χ (cid:12) C∞(0, ) such that χ(t) = 1 for t 5 and
∈ ∞ ≤ 4
χ(t)=0 for t 7. We let
≥ 4
f (z)=P (z)+χ(2kρ(z))(P (z) P (z)), 2−k <ρ(z)<2−k+1, k N,
0 2k 2k+1 − 2k ∈
and define the continuation of a function f by formula f =χ(2ρ(z))f (z).
0
9
Now f is C1-function on Cn Ω and ∂¯f(z) . λ(z). We define a function
\
Fk(z) as Fk(z) = f(z) for ρ(z) > 2−k and(cid:12)(cid:12)as Fk(cid:12)(cid:12)(z) = P2k+1(z) for ρ(z) < 2−k.
The the function F is smooth and holomorphic in Ω , and ∂¯F (z) . λ(z)
k 2−k k
for z Cn Ω . Thus similarly to 13 we get (cid:12) (cid:12)
∈ \ 2−k (cid:12) (cid:12)
1 ∂¯F (ξ) ∂ρ(ξ) (∂¯∂ρ(ξ))n−1
k
P2k+1(z)=Fk(z)= (2πi)n ∧∂ρ(ξ), ξ∧ z n , z ∈Ω,
CnZ\Ω h − i
Wecanpasstothelimitinthisformulabythedominatedconvergencetheorem;
hence, function f satisfies the formula (13) and is a pseudoanalytical continua-
tion of function f. (cid:3)
4.4. Pseudoanalytical continuation of Hardy-Sobolev spaces
Theorem 4.3. Let Ω be a strongly convex domain, 1 < p < , l N and
∞ ∈
f Hp(Ω). Then f Hl(Ω) if and only if there exists such pseudoanalytical
∈ ∈ p
continuation f that for some η >0
p/2
dσ(z) ∂¯f(τ)ρ(τ)−l 2dν(τ) < , (18)
 
∞
Z Z
∂Ω De(z,η,ε) (cid:12) (cid:12)
 (cid:12) (cid:12) 
 
where dν(τ)= dµ(τ) .
ρ(τ)n−1
Proof. Let f Hl(Ω). By theorem 4.1 we could construct pseudoanalytical
∈ p
continuation f such that
∂¯f(z) . max ∂αf(z∗) ρ(z)l, z Cn Ω.
|α|=l+1| | ∈ \
(cid:12) (cid:12)
Note that th(cid:12)e sym(cid:12)metry (z z∗) with respect to ∂Ω maps the external
7→
sector De(z,η,ε) into some internal Kor´anyi sector. Indeed, for every η > 0
there exists η ,ε >0 such that
1 1
τ∗ :τ De(z,η,ε) Di(z,η ,ε ).
1 1
{ ∈ }⊆
10