An integral representation for Besov and Lipschitz spaces

0.14 MB·English

by  Kehe Zhu

AN INTEGRAL REPRESENTATION FOR
BESOV AND LIPSCHITZ SPACES
1
1 KEHEZHU
0
2
ABSTRACT. ItiswellknownthatfunctionsintheanalyticBesovspace
n B1ontheunitdiskDadmitsanintegralrepresentation
a
J z−w
f(z)= dµ(w),
5 ZD 1−zw
1 whereµisacomplexBorelmeasurewith|µ|(D) < ∞. Wegeneralize

this result to all Besov spaces B with 0 < p ≤ 1 and all Lipschitz
] p
V spacesΛ with t > 1. We also obtainaversionforBergmanandFock
t
C spaces.
.
h
t
a
m
[ 1. INTRODUCTION

1 LetDdenotetheunitdiskinthecomplexplaneC,H(D)denotethespace
v of all analytic functions in D, and dA denote the normalized area measure
5
9 on D. For 0 < p < ∞ we consider the analytic Besov space B consisting
p
9 offunctionsf ∈ H(D)with theproperty that(1−|z|2)kf(k)(z) belongsto
2
. Lp(D,dλ),where
1
0 dA(z)
dλ(z) =
1
(1−|z|2)2
1
: istheMo¨biusinvariantareameasureonDandk isanypositiveintegersuch
v
i thatpk > 1. ThespaceB is independentoftheintegerk used.
X p
Itiswellknownthatananalyticfunctionf inDbelongstoB ifandonly
r 1
a if there exists a complex Borel measure µ on the D such that |µ|(D) < ∞
and
z −w
f(z) = dµ(w), z ∈ D. (1)
Z 1−zw
D
See [1, 2, 10]. The purpose of this paper is to generalize the above result
toseveralotherspaces,includingBesovspaces,Lipschitzspaces,Bergman
spaces, and Fock spaces. We state our main results as Theorems A and B
below.
2000MathematicsSubjectClassification. 30H20,30H25,32A36.
Key words and phrases. Bergmanspaces, Besov spaces, Fock spaces, Carleson mea-
sures,atomicdecomposition,Berezintransform.
1
2 KEHEZHU
Theorem A. Suppose0 < p ≤ 1, 0 < r < 1, and f is analyticin D. Then
f ∈ B ifandonlyifitadmitsa representation
p
z −w
f(z) = dµ(w), z ∈ D,
ZD 1−zw
where µ is a complex Borel measure on D such that the localized function
z 7→ |µ|(D(z,r))belongstoLp(D,dλ),where
z −w
D(z,r) = w ∈ D : < r
(cid:26) (cid:12)1−zw(cid:12) (cid:27)
(cid:12) (cid:12)
(cid:12) (cid:12)
isthepseudo-hyperbolicdiskat z withra(cid:12)diusr. (cid:12)
Recall that for any real number t, the analytic Lipschitz space Λ on the
t
unit disk consists of functions f ∈ H(D) such that (1 − |z|2)k−tf(k)(z) is
bounded,wherek isanynonnegativeintegergreater thant.
Theorem B. Suppose t > 1, 0 < r < 1, and f is analytic in D. Then
f ∈ Λ if andonlyif
t
z −w
f(z) = dµ(w), z ∈ D,
ZD 1−zw
forsomecomplexBorel measureµwith thepropertythat
|µ|(D(z,r))
sup < ∞.
z∈D (1−|z|2)t
In addition to Besov and Lipschitz spaces in dimension 1, where the in-
tegralrepresentationlooksparticularlynice,wewillalsoconsiderBergman
andFock spaces inhigherdimensions.
2. PRELIMINARIES ON MEASURES
Suppose µ is a complex Borel measure on D and r ∈ (0,1). We can
definetwofunctionson Das follows.
µ(D(z,r))
µ (z) = µ(D(z,r)), µ (z) = .
r r
(1−|z|2)2
Itis wellknownthatthearea ofthepseubdo-hyperbolicdiskD(z,r) is
1−|z|2 2
πr2 ,
(cid:18)1−r2|z|2(cid:19)
which is comparable to (1 − |z|2)2 whenever r is fixed. That is why we
think of µ as an averaging function for the measure µ. We will call µ a
r r
localizedfunction forµ. Thebehaviorofµ and µ isoften independentof
r r
theparticbularradiusr beingused.
b
INTEGRALREPRESENTATION 3
Another averaging function for µ is the so-called Berezin transform of
µ. We need the assumption |µ|(D) < ∞ in order to define the Berezin
transform:
(1−|z|2)2
µ(z) = dµ(w), z ∈ D.
Z |1−zw|4
D
See[10]forbasiecinformationabout theseaveragingoperations.
Wewillneedtodecomposetheunitdiskintoroughlyequal-sizedpartsin
thepseudo-hyperbolicmetric. Morespecifically,wewillneedthefollowing
result.
Lemma 1. For any 0 < r < 1 there exists a sequence {z } in D and a
n
sequenceofBorel subsets{D } ofD with thefollowingproperties:
n
(a) D = D ∪D ∪···∪D ∪···.
1 2 n
(b) ThesetsD aremutuallydisjoint.
n
(c) D(z ,r/4) ⊂ D ⊂ D(z ,r)forevery n.
n n n
Proof. Thisiswell known. See [10]forexample. (cid:3)
Any sequence {z } satisfying the three conditions above will be called
n
an r-latticeinthepseudo-hyperbolicmetric.
Lemma 2. Suppose µ is a positive Borel measure on D and 0 < p ≤ ∞.
If r and s are two radii in (0,1) and {z } is an r-lattice in the pseudo-
n
hyperbolicmetric. Then thefollowingconditionsareequivalent.
(a) Thefunctionµ (z) belongsto Lp(D,dλ).
s
(b) Thesequenceµ (z ) belongstolp.
r n
b
If1/2 < p ≤ ∞,then theaboveconditionsarealsoequivalentto
b
(c) Thefunctionµ(z) belongstoLp(D,dλ).
Proof. Thisisalso weellknown. See[10]forexample. (cid:3)
As a consequence of the above lemma on the averaging function µ we
r
obtainseveralequivalentconditionsforthelocalized functionµ .
r
b
Corollary 3. Suppose 0 < p ≤ ∞ and µ is a positive Borel measure on
D. If r and s are two radii in (0,1) and {z } is an r-lattice in the pseudo-
n
hyperbolicmetric. Then thefollowingconditionsareequivalent.
(a) Thesequence{µ (z )}belongsto lp.
r n
(b) Thefunctionµ (z) belongsto Lp(D,dλ).
s
Ifp 6= ∞,theaboveconditionsarealsoequivalentto
(c) Thefunctionµ (z) belongsto Lp(D,dA ), where
s 2(p−1)
dA (z) = (1−|z|2)2(p−1)dA(z).
b2(p−1)
4 KEHEZHU
Proof. ConsiderthepositiveBorel measure
dν(z) = (1−|z|2)2dµ(z).
Itiswellknownthat(1−|z|2)2iscomparableto(1−|z |2)2forzinD(z ,t),
n n
where t is any fixed radius in (0,1). See [10] for example. Thus ν (z) is
t
comparableto µ (z), and ν (z) is also comparableto (1−|z|2)2µ (z). The
t t t
desiredresult thenfollowsfrom Lemma2. b (cid:3)
b b
In view of the equivalence of conditions (a) and (c) in Lemma 2, it is
temptingto conjecture that condition (c) in Corollary 3 aboveis equivalent
to µ ∈ Lp(D,dA ) whenever p > 1/4. It turns out that this is not
2(p−1)
true. This already fails at p = 1. In fact, if p = 1, the condition µ ∈
Lp(eD,dA ) means
2(p−1)
e
+∞ > µ(z)dA(z)
Z
D
e dµ(w)
= (1−|z|2)2dA(z)
ZD ZD |1−zw|4
(1−|z|2)2dA(z)
= dµ(w) .
ZD ZD |1−zw|4
ThistogetherwithLemma3.10in[10]showsthat,forp = 1, thecondition
µ ∈ Lp(D,dA )is thesameas
2(p−1)
e 1
log dµ(w) < ∞.
ZD 1−|w|2
On the other hand, it is easy to see that condition (a) in Corollary 3, for
p = 1, simply means µ(D) < ∞, which is obviously different from the
integralconditionabove.
TheBerezin transformwillnotreallybeusedintherestofthepaper,but
itisalways interestingand insightfultocomparethebehaviorofµ and µ.
r
Another notion critical to the integral representation of Lipschitz spaces
isthatofCarleson measures. b e
Let t > 0. We say that a positive Borel measure µ on the unit disk D is
t-Carleson if
µ(D(z,r))
sup < ∞
z∈D (1−|z|2)t
for some r ∈ (0,1). It is well known that if the above condition holds for
somer ∈ (0,1),thenitholdsforeveryr ∈ (0,1). Thusbeingt-Carleson is
independentoftheradiusr used inthedefinition.
INTEGRALREPRESENTATION 5
If t > 1, then every t-Carleson measure is finite. In fact, in this case, we
useLemma1to get
∞ ∞
µ(D) = µ(D ) ≤ µ(D(z ,r))
n n
Xn=1 Xn=1
∞ ∞
≤ C (1−|z |2)t ≤ C′ (1−|z|2)t−2dA(z)
n
Z
Xn=1 Xn=1 Dn
= C′ (1−|z|2)t−2dA(z) < ∞.
ZD
It is clear from the above argument that not every t-Carleson measure is
finitewhen t ≤ 1.
Ift > 1,thenµist-Carelsonifandonlyifforsome(orevery)0 < p < ∞
thereexistsaconstantC = C > 0 suchthat
p
|f(z)|pdµ(z) ≤ C |f(z)|p(1−|z|2)t−2dA(z)
Z Z
D D
for all f ∈ H(D). Because of this, such measures are also called Carleson
measures for Bergman spaces. When t > 1, it is also known that µ is t-
CarlesonifandonlyifthereisaconstantC > 0suchthatµ(S ) ≤ Cht for
h
all“Carleson squares”S ofsidelengthh. See [11, 8].
h
We warn the reader that there is a fine distinction between 1-Carleson
measures defined above and the classical Carleson measures (for Hardy
spaces). This can be seen by considering an arbitrary Bloch function f
intheunitdisk. In fact, forsuchafunctionifwedefinethemeasureµ by
dµ(z) = (1−|z|2)|f′(z)|2dA(z),
thenµ is1-Carleson, because
µ(D(z,r)) = (1−|w|2)|f′(w)|2dA(w)
Z
D(z,r)
1
∼ (1−|w|2)2|f′(w)|2dA(w)
1−|z|2 Z
D(z,r)
≤ C(1−|z|2).
But µ is not a classical Carleson measure, because being so would mean
thatf isinBMOA.See[5]. ItiscertainlywellknownthatBMOAisstrictly
contained in the Bloch space. A classical Carleson measure is 1-Carleson,
butnottheotherway around.
Similarly, for 0 < t < 1, there is a subtle difference between measures
satisfying the condition µ(S ) ≤ Cht and those satisfying the condition
h
µ(D(z,r)) ≤ C(1−|z|2)t.
6 KEHEZHU
3. BESOV SPACES IN THE UNIT DISK
We prove Theorem A in this section. For this purpose we need to make
useofweightedBergman spaces. Thus foranyα > −1 welet
dA (z) = (α+1)(1−|z|2)αdA(z)
α
denotetheweightedarea measureonD. Thespaces
Ap = H(D)∩Lp(D,dA ), 0 < p < ∞,
α α
arecalled weightedBergman spaces withstandard weights.
We will see that Theorem A can be thought of as an extension of the
following atomic decomposition for weighted Bergman spaces, which can
befoundin[10]forexample.
Theorem 4. Suppose0 < p < ∞, α > −1, and
b > max(1,1/p)+(α+1)/p. (2)
There exists some positive number δ such that for any r-lattice {z } with
n
r < δ the weighted Bergman space Ap consists exactly of functions of the
α
form
∞ (1−|z |2)(pb−2−α)/p
f(z) = c n , (3)
n
(1−zz )b
Xn=1 n
where{c } ∈ lp.
n
Atomicdecompositionfor Bergman spaces was first obtained in [4]. We
willfollowtheproofoftheabovetheorem as found in[10]. We beginwith
an explictconstructionforameasurein(1)when f is apolynomial.
Lemma 5. If f is a polynomial and 0 < p ≤ ∞, there exists a complex
Borel measure µ such that the localized function |µ| (z) is in Lp(D,dλ)
r
and
z −w
f(z) = dµ(w)
ZD 1−zw
forallz ∈ D.
Proof. Iff isanonzero constantfunction,weusethemeasure
|w|
dµ(w) = c (1−|w|2)N dA(w),
w
wherecisanappropriateconstantandN issufficientlylarge(dependingon
p). Iff(w) = wn withn ≥ 1, weusethemeasure
dµ(w) = cwn−1(1−|w|2)N dA(w),
wherecisanappropriateconstantandN issufficientlylarge(dependingon
p). Thisfollowsfrom theTaylorexpansionofthefunction1/(1−zw) and
polarcoordinates. (cid:3)
INTEGRALREPRESENTATION 7
We now proceed to the proof of Theorem A. First assume that f ∈ B
p
for some 0 < p ≤ 1. Let k be any positive integer such that pk > 1. Let
b = k + 1 and α = pk − 2. Then b satisfies the condition in (2). In fact,
since0 < p ≤ 1, wehave
1 α+1 1 pk −1
max 1, + = + = k < b.
(cid:18) p(cid:19) p p p
Also, f ∈ B if and only if its k-th order derivative f(k) is in Ap and the
p α
exponentin thenumeratorof(3)is
pb−2−α
= 1.
p
Itfollowsfrom Theorem4that wecan find an r-lattice{z }in thepseudo-
n
hyperbolicmetricandasequence{c } ∈ lp suchthat
n
∞ 1−|z |2
f(k)(z) = c n , z ∈ D.
n
(1−zz )k+1
Xn=1 n
There is considerable freedom in the choice of the r-lattice in Theorem 4.
So we may assume that z 6= 0 for each n and |z | → 1 as n → ∞. We
n n
thenconsiderthefunction
∞
z −z
g(z) = c′ n , z ∈ D,
n1−zz
Xn=1 n
where
c
c′ = n , n ≥ 1.
n k!zk−1
n
Clearly, the sequence {c′ } is still in lp ⊂ l1. This is where we use the
n
assumption 0 < p ≤ 1 in a critical way to ensure that the infinite series
defining g actually converges. Differentiating term by term shows that the
functiong satisfiesg(k) = f(k). Thusthere isapolynomialP(z) suchthat
∞
z −z
f(z) = P(z)+ c′ n .
n1−zz
Xn=1 n
By Lemma 5, there is a measure ν such that the localized function |ν| (z)
r
isinLp(D,dλ)and
z −w
P(z) = dν(w), z ∈ D.
ZD 1−zw
Ifwedefine
∞
µ = ν + c′ δ ,
n zn
Xn=1
8 KEHEZHU
where δ denotes the unit point mass at z , we obtain the desired repre-
zn n
sentation for f with the localized function |µ| (z) belonging to Lp(D,dλ).
r
ThisprovesonedirectionofTheoremA.
ToprovetheotherdirectionofTheoremA,letusassumethatµisacom-
plex Borel measure such that the function |µ| (z) is in Lp(D,dλ). Let r be
r
asufficientlysmallradiusand{z }beanr-latticeinthepseudo-hyperbolic
n
metric. By Corollary 3 thesequence|µ| (z ) is inlp. Nowif
r n
z −w
f(z) = dµ(w), z ∈ D,
Z 1−zw
D
then
(1−|w|2)wk−1
f(k)(z) = k! dµ(w), z ∈ D.
Z (1−zw)k+1
D
We use Lemma 1 to decompose D into the disjoint union of {D } and
n
rewrite
∞ (1−|w|2)wk−1
f(k)(z) = k! dµ(w),
Z (1−zw )k+1
Xn=1 Dn n
sothat
∞ 1−|w|2
|f(k)(z)| ≤ k! d|µ|(w).
Z |1−zw|k+1
Xn=1 Dn
Foreach n ≥ 1 andz ∈ D thereissomepointw (z) ∈ D such that
n n
1−|w (z)|2 1−|w|2
n
= sup .
|1−zw (z)|k+1 |1−zw|k+1
n w∈Dn
Thus,
∞ 1−|w (z)|2
|f(k)(z)| ≤ c n
n|1−zw (z)|k+1
Xn=1 n
for all z ∈ D, where c = k!|µ|(D ) is a sequence in lp as |µ|(D ) ≤
n n n
|µ|(D(z ,r)). ByLemma4.30of[10],thereexistsaconstantC > 0(inde-
n
pendentofnand z) suchthat
1−|w (z)|2 1−|z |2
n n
≤ C
|1−zw (z)|k+1 |1−zz |k+1
n n
foralln ≥ 1 and allz ∈ D. Therefore,
∞ 1−|z |2
|f(k)(z)| ≤ C c n
n|1−zz |k+1
Xn=1 n
forallz ∈ D. Since 0 < p ≤ 1,weapplyHo¨lder’sinequalitytoget
(1−|z |2)p
|f(k)(z)|p ≤ Cp |c |p n .
n |1−zz |p(k+1)
Xn=1 n
INTEGRALREPRESENTATION 9
Integrate term by term and apply Proposition 1.4.10 of [7]. We obtain an-
otherconstantC > 0 (independentoff)suchthat
∞
|f(k)(z)|p(1−|z|2)pk−2dA(z) ≤ C |c |p < ∞,
n
ZD Xn=1
whichshowsthat f ∈ B and completestheproofofTheoremA.
p
It is clear that Theorem A cannot possibly be true when p > 1. This is
because any function f represented by the integral in Theorem A must be
bounded,whilethereareunboundedfunctionsinB when p > 1.
p
4. BERGMAN TYPE SPACES ON THE UNIT BALL
In this section we show how to extend Theorem A to Bergman type
spaces on the unit ball B in Cn. The main reference for this section is
n
[8]. Whenα > −1, all backgroundinformationcan alsobefoundin [11].
Foranyreal parameterα weconsidertheweighted volumemeasure
dv (z) = (1−|z|2)αdv(z),
α
wheredv is theLebesguevolumemeasureonB .
n
Forrealαand0 < p < ∞weuseAp todenotethespaceofholomorphic
α
functionsf inB suchthat(1−|z|2)kRkf(z)isinLp(B ,dv ),wherek is
n n α
a nonnegativeintegersatisyingpk +α > −1 and Rf is thestandard radial
derivativedefined by
∂f ∂f
Rf(z) = z +···+z .
1 n
∂z ∂z
1 n
It is well known that the space Ap is independent of the integer k used in
α
thedefinition.
Various names exist in theliteraturefor thespaces Ap: Bergman spaces,
α
Besov spaces, and Sobolev spaces, among others. We follow [8] and call
them Bergman spaces here. When α > −1, Ap are indeed the weighted
α
Bergman spaceswithstandardweights. Forα = −(n+1),Ap becomethe
α
so-calleddiagonalBesov spaces.
If p is fixed, all the spaces Ap are isomorphic as Banach spaces for 1 ≤
α
p < ∞ and as complete metric spaces for 0 < p < 1. The isometry can be
realized by certain fractional radial differential operators. Because of this,
itisoftenenoughforusjusttoconsiderthecaseα = 0andobtaintheother
cases byfractional differentiationorfractional integration.
Ontheunitballthereexistsauniquefamilyofinvolutiveautomorphisms
ϕ (z) thatare highdimensionalanalogsoftheMo¨biusmaps
a
a−z
ϕ (z) =
a
1−az
10 KEHEZHU
on the unit disk. See [7] and [11]. The pseudo-hyperbolic metric on B
n
is still the metric defined by d(z,w) = |ϕ (w)|. For any complex Borel
z
measure µ on B the localized function µ and the averaging function µ
n r r
aredefined in exactlythesameway as before.
Wecan nowextendTheoremA toall thespaces Ap as follows. b
α
Theorem 6. Supposeα isreal, 0 < p < ∞,0 < r < 1, and
1 α+1
b > max 1, + .
(cid:18) p(cid:19) p
Then afunctionf ∈ H(B ) belongstoAp ifand onlyif
n α
(1−|w|2)(pb−n−1−α)/p
f(z) = dµ(w) (4)
Z (1−hz,wi)b
Bn
forsomecomplexBorelmeasureµonB withthelocalizedfunction|µ| (z)
n r
belongingtoLp(D,dλ),where
dv(z)
dλ(z) =
(1−|z|2)n+1
istheMo¨biusinvariantvolumemeasureon B .
n
Proof. That every function f ∈ Ap has the desired integral representation
α
in (4), with µ being atomic, follows from Theorem 32 in [8], which is the
atomicdecompositionforthesespaces.
On the other hand, if f is a function represented by (4), we follow the
secondhalfoftheproofofTheorem A toobtainthefollowingestimate,
∞ (1−|z |2)(pb−n−1−α)/p
|RNf(z)| ≤ C c k , z ∈ B ,
k |1−hz,z i|b+N n
X k
k=1
where C is some positiveconstant, {c } ∈ lp, and N is a sufficiently large
k
positive integer. Using the arguments on pages 92-93 of [10] (the proof
for atomic decompositionof Bergman spaces) we can then show that RNf
belongsto theBergman spaceAp , whichmeans thatf ∈ Ap. (cid:3)
Np+α α
One particular case is worth mentioning. If p = 1 and µ is a pos-
itive Borel measure on B , then we use Fubini’s theorem, the fact that
n