Compact composition operators on Hardy-Orlicz and weighted Bergman-Orlicz spaces on the ball

0.16 MB·English

by  Stéphane Charpentier

COMPACT COMPOSITION OPERATORS ON HARDY-ORLICZ AND WEIGHTED
BERGMAN-ORLICZ SPACES ON THE BALL
STÉPHANECHARPENTIER
ABSTRACT. UsingrecentcharacterizationsofthecompactnessofcompositionoperatorsonHardy-
Orlicz andBergman-Orliczspacesonthe ball([2, 3]), we first showthata compositionoperator
¥
which is compact on every Hardy-Orlicz (or Bergman-Orlicz) space has to be compact on H .
Then, although it is well-known that a map whose range is contained in some nice Korányiap-
proachregioninducesacompactcompositionoperatoronHp(BN)oronAap(BN),weprovethat,
1 foreachKorányiregionG ,thereexistsamapf :BN G suchthat,Cf isnotcompactonHy (BN),
1 wheny growsfast. Finally,weextend(andsimplify→theproofof)aresultbyK.Zhuforclassical
0 weightedBergmanspaces,byshowingthat,underreasonableconditions,acompositionoperator
2 Cf iscompactontheweightedBergman-OrliczspaceAay (BN),ifandonlyif
n
a y 1 1/(1 f (z))N(a )
−
J lim −| | =0.
9 z 1 y (cid:16)1 1/(1 z)N(a ) (cid:17)
||→ − −| |
1
In particular,we deducethatthe compactn(cid:16)essofcomposit(cid:17)ionoperatorsonAya (BN)doesnotde-
] pendona anymorewhentheOrliczfunctiony growsfast.
A
F
.
h
at 1. INTRODUCTION
m
LetB = z=(z ,...z ) CN,(cid:229) N z 2 <1 denotetheopenunitballofCN. Givenaholo-
[ N 1 N ∈ i=1| i|
1 morphicmapnf :BN BN,thecompositionoperaotorCf ofsymbolf isdefinedbyCf (f)= f f ,
v for f holomorphicon→B . Compositionoperators havebeen extensivelystudiedoncommon◦Ba-
N
7 nachspacesofanalyticfunctions,inparticularontheHardyspacesHp(B )andontheBergman
7 N
6 spaces Ap(BN), 1 p<¥ . The continuity and compactness of these operators have been char-
≤
3 acterized in terms of Carleson measures ([5]). In dimension one, the boundedness ofCf for any
.
1 f : D D is a consequence of the Littlewood subordination principle ([16]). In CN, N > 1,
0 it is w→ell-known that there exists some map f : B B such that the associated composition
1 N N
1 operator is not bounded on Hp(B ). Whatever the d→imension, it appears that both boundedness
N
v: and compactness of Cf on Hp(BN) (resp. Ap(BN)) are independent of p. On the other hand,
Xi everycompositionoperatoris obviouslyboundedon H¥ and itis not difficultto check thatCf is
compactonH¥ ifand onlyif f <1. Thusthereisa“break”between H¥ andHp(B )(resp.
r ¥ N
a Ap(B )), forthecompactnesskinkdimensionone, andevenfortheboundedness,whenN >1.
N
These observations first motivated P. Lefèvre, D. Li, H. Queffélec and L. Rodríguez-Piazza
to study composition operators on Hardy-Orlicz spaces Hy (D) (resp. Bergman-Orlicz spaces
Ay (D)) of the disc ([7, 10, 8, 9]), and then the author of [2, 3] to look at these questions in
CN. These spaces both provide an intermediate scale of spaces between H¥ and Hp(B ) (resp.
N
Ap(B ))andgeneralizethelatter. Inparticular,in[10],theauthorswereinterestedinthequestion
N
of whether there are some Hardy-Orlicz spaces on which the compactness ofCf is equivalentto
¥
that on H . In fact, they answer this question in the negative, by proving ([10, Theorem 4.1])
that, for every Hardy-Orlicz space Hy (D), one can construct a surjectivemap f :D D which
→
2000MathematicsSubjectClassification. Primary:47B33-Secondary:32C22;46E15.
Keywordsandphrases. Carlesonmeasure-Compositionoperator-Bergman-Orliczspace-Hardy-Orliczspace
-angularderivative.
1
2 COMPACTCOMPOSITIONOPERATORSONHARDY-ORLICZANDBERGMAN-ORLICZSPACESONBN
induces a compact composition operatorCf on Hy (D). This result extends that obtained by B.
MacCluer and J. Shapiro for Hp(D) ([13, Example 3.12]). The same problem in the Bergman-
Orlicz case has not yet been completely solved. In several variables, the situation is much more
surprizing, as we show in [2, 3] that there exist some Hardy-Orlicz and Bergman-Orlicz spaces,
¥
“close”enoughtoH , onwhichevery compositionoperatorisbounded.
In this paper, we are mainly interested in the possibility to extend some known results about
compactnessofcompositionoperators on classicalHardy orBergman spaces, tothecorrespond-
ing Orlicz spaces. We think that this study may outlinesome interesting phenomena and precise
thelinkbetween thebehaviorofCf and that off .
Firstofall,wecomebacktothe“break”betweenH¥ andHp,1 p<¥ ,forthecompactness
of Cf . There is no difference between being compact for Cf on≤one Hp(BN) and on every
Hp(B ), while this property clearly depends on the Orlicz function y in Hy (B ). Therefore,
N N
we can wonder if the above question answered by [10] was the good one; indeed, the study of
Cf on Hardy-Orlicz spaces arises the following question: what can we say about a composition
operator which is compact on every Hardy-Orlicz space? It turns out that such an operator has
¥
to be compact on H , which seems to us to be a positive result, because it confirms that Hardy-
Orlicz spaces covers well the “gap” between every Hp and H¥ . This result also stands when we
replaceHardy-Orliczspaces by Bergman-Orliczspaces.
Moreover,inHardyorBergmanspaces,compactness(andboundedness)ofcompositionoper-
ators is handled in terms of geometric conditions, emphasizing the importance of the manner in
whichthesymbolf approachestheboundaryofB . Tobeprecise,ifwedenotebyG (z ,a) B ,
N N
forz S and a>1, theKorányiapproach region ⊂
N
∈
a
G (z ,a)= z B , 1 z,z < 1 z 2 ,
N
∈ | −h i| 2 −| |
n (cid:16) (cid:17)o
it is known ([12]) that if f takes the unit ball into a Korányi region G (z ,a) with a small enough
angular opening a, then Cf is compact on Hp(BN) and on Aap(BN). When N = 1, the Korányi
regions are just non-tangential approach regions. In this paper, we show that this result does not
holdanymoreforHardy-OrliczspacesonB ;forBergman-Orliczspaces,weobtainsucharesult
N
indimensiononeonly.
In [13], the authors related the compactness of the composition operator Cf on Hp(D) or
Aap(D) to the existence of angular derivative for f at the boundary. We say that the angular
derivativeoff existsat apointz T ifthereexistsw Tsuch that
∈ ∈
f (z) w
−
z z
−
has a finite limit as z tends non-tangentially to z through D. The Julia-Caratheodory Theorem
thenasserts thatthenon-existenceofan angularderivativeforf at somez Tisequivalentto
∈
1 z
(1.1) lim −| | =0.
z z 1 f (z)
→ −| |
ShapiroandTaylor[17]pointedoutthatifCf istobecompactonHp(D),thenf cannothavean
angularderivativeat evenasinglepointin T, whichmay bewritten:
1 z
(1.2) lim −| | =0.
z 11 f (z)
| |→ −| |
In[13],itisproventhat(1.1)isnotsufficienttothecompactnessofCf onHardyspacesoftheunit
disc in general, yet it is when f is univalent. However, this condition is necessary and sufficient
forCf to be compact on every weighted Bergman spaces of the disc. The last main goal of this
STÉPHANECHARPENTIER 3
paper is to extend some of these results to Hardy-Orlicz and Bergman-Orlicz spaces of the unit
ball.
In severalvariables, wecan also define theangularderivativeoff :B B at a pointin the
N N
unitsphereS andtheJulia-CaratheodoryTheoremalsoholdsinB ([15,T→heorem8.5.6]). Here,
N N
aswealreadysaid,thesituationiscomplicatedbythefactthatsomecompositionoperatorsarenot
bounded on Hardy or Bergman spaces, and the fact that even the boundedness ofCf on Aap (BN)
dependsona . In[18],K.ZhuprovesthatCf iscompactonAap(BN)ifandonlyifCondition(1.2)
is satisfied, wheneverCf is bounded on some Abp(BN), for some 1<b <a . This assumption
−
is somehowjustified by the aboveobservation and by [13, Section 6], in which theauthors show
that, for any a > 1 and any 0< p<¥ , there existsf :B B with no angular derivativeat
N N
any point of SN, s−uch thatCf is bounded on Aap(BN) but not→compact. There even exists such a
map f such thatCf is not bounded on Aap (BN). In the present paper, we generalize Zhu’s result
toweightedBergman-Orliczspacesontheball,byusingrecentcharacterizationsofboundedness
and compactnessofcompositionoperators on thesespaces ([2]). Weshow that,ifCf is bounded
y y
onsomeAb (BN), 1<b <a , then itiscompact onAa (BN)ifand onlyif
−
y 1 1/(1 f (z) )N(a )
−
−| |
(1.3) lim =0,
z 1 y (cid:16)1 1/(1 z )N(a ) (cid:17)
| |→ −
−| |
(cid:16) (cid:17)
where N(a )=N+a +1, under a mild and usual regularity condition on the Orlicz function y .
Our proof is quite simple, while that of K. Zhu uses a Schur test in H2(B ) and the fact that the
N
compactness of composition operators on Hp(B ) does not depend on p. Combining this result
N
y
with the automatic boundedness of every composition operator on Aa (BN) when y satisfies the
y
D 2-Condition, we get that the compactness on such Aa (BN) does not depend on a anymore. To
y
beprecise,Cf iscompact onAa (BN) ifandonlyif
y 1(1/(1 f (z) ))
−
lim −| | =0,
z 1 y −1(1/(1 z ))
| |→ −| |
whenevery satisfies theD 2-Condition.
We have to mention that Condition (1.3) is, in any case, necessary. Moreover, the authors of
[7] obtained such a result in dimension one, as announced in [11]. However, their proof uses the
characterizationofthecompactnessofcompositionoperatorsintermsoftheNevanlinnacounting
functionandis morecomplicated.
We organize our paper as follows: a first preliminary part is devoted to the definitions and the
statementsofthealreadyknownresultsweneed. Themainpartcontainsthethreemostimportant
resultsmentionnedabove.
Notation. Throughout this paper, we will denote by ds the normalized invariant measure on
N
a
the unit sphere SN = ¶ BN, and by dva = ca 1 z 2 dv, a > 1, the normalized weighted
−| | −
Lebesguemeasureon theball. (cid:16) (cid:17)
Given two points z,w CN, the euclidean inner product of z and w will be denoted by z,w ,
∈ h i
that is z,w = (cid:229) N z w; the notation will stand for the associated norm, as well as for the
h i i=1 i i |·|
modulusofacomplexnumber.
Ifa > 1 is areal number,wewilldenotebyN(a ) thequantityN+a +1.
−
4 COMPACTCOMPOSITIONOPERATORSONHARDY-ORLICZANDBERGMAN-ORLICZSPACESONBN
2. PRELIMINARIES
2.1. Hardy-Orlicz and Bergman-Orlicz spaces - Definitions. A strictly convex function y :
y (x)
R R is called an Orlicz function if y (0)=0, y is continuous at 0 and +¥ .
+ +
→ x −x−−+−→¥
If (W ,P) is a probability space, the Orlicz space Ly (W ) associated to the Orlicz func→tion y on
(W ,P)isthesetofall(equivalenceclassesof)measurablefunctions f onW suchthatthereexists
someC>0,suchthat W y |Cf| dPisfinite. Ly (W )isavectorspace,whichcanbenormedwith
theso-called LuxembuRrgno(cid:16)rm (cid:17)defined by
f
f =inf C>0, y | | dP 1 .
y
k k W C ≤
(cid:26) Z (cid:18) (cid:19) (cid:27)
Itis well-knownthat Ly (W ), isa Banach space(see [14]).
y
k·k
Taking W = S an(cid:16)d dP = ds , t(cid:17)he Hardy-Orlicz space Hy (B ) on B is the Banach space
N N N N
of analytic functions f : BN C such that f Hy := sup0<r<1 fr y < ¥ , where fr Ly (SN)
is defined by f (z)= f (rz). →Every functionkf kHy (B ) admitska rkadial boundary lim∈it f such
r N ∗
that f =sup f < ¥ ([3, Section∈1.3]). For simplicity, we will sometimes denote
∗ y 0<r<1 r y
by k y kthenormonHky (BkN),emphasizingthatHy (BN)can beseenas asubspaceofLy (SN).
k·k
WithW =BN anddP=dva ,a > 1,theweightedBergman-OrliczspaceAay (BN)isLy (BN)
− y ∩
H(BN), where H(BN) stands forthevectorspace ofanalyticfunctionson theunitball. Aa (BN)
isaBanach space.
From thedefinitions,it iseasy toverify thatthefollowinginclusionshold:
H¥ Hy (BN) Hp(BN) and H¥ Aya (BN) Aap(BN)
⊂ ⊂ ⊂ ⊂
foreveryOrliczfunctiony andany1 p<¥ . Moreover,ify (x)=xp,forsome1 p<¥ and
foreveryx 0, thenHy (BN)=Hp(B≤N) and Aya (BN)=Aap (BN). ≤
≥
2.2. Four classes of Orlicz functions. Let y be an Orlicz function. In order to distinguish the
Orlicz spaces and to get a significant scale of intermediate spaces between L¥ and Lp(W ), we
definefourclasses ofOrliczfunctions.
– Thetwofirstconditionsareregularityconditions: wesaythaty satisfiesthe(cid:209) -conditionif
0
itsatisfiesoneofthefollowingtwoequivalentconditions:
y (Bx) y (C By)
B
(i) For any B>1, there exists some constantC 1, such that for any
B ≥ y (x) ≤ y (y)
x ylargeenough;
≤ y (Bx)n y (C y)
n
(ii) Foranyn>0,thereexistsC >0suchthat foranyx ylargeenough.
n y (x)n ≤ y (By) ≤
Let us notice that (2) (1) is obvious, while an easy induction allows to prove (1) (2); the
⇒ ⇒
detailsareleft tothereader.
IftheconstantC canbechosenindependentlyofB,theny satisfiestheuniform(cid:209) -Condition.
B 0
– The(cid:209) -classconsistsofthoseOrliczfunctionsy suchthatthereexistsomeb >1andsome
2
x >0, suchthat y (b x) 2by (x), forx x .
0 0
≥ ≥
– The third one is a condition of moderate growth: y satisfies the D -Condition if there exist
2
x >0 and aconstantK >1, suchthat y (2x) Ky (x) forany x x .
0 0
≤ ≥
– Thefourthconditionisaconditionoffast growth: y satisfies theD 2-Conditionifitsatisfies
oneofthefollowingequivalentconditions:
(i) ThereexistC>0 andx >0, suchthaty (x)2 y (Cx) foreveryx x ;
0 0
≤ ≥
STÉPHANECHARPENTIER 5
(ii) Thereexistb>1,C>0 andx >0 suchthaty (x)b y (Cx), foreveryx x ;
0 0
≤ ≥
(iii) For every b > 1, there exist C > 0 and x > 0 such that y (x)b y (C x), for every
b 0,b b
≤
x x .
0,b
≥
Finally,wementionthat theseconditionsare notindependent(see[7, Proposition4.7]):
Proposition2.1. Let y beanOrliczfunction.
(1) Ify satisfiestheuniform(cid:209) -Condition,thenitsatisfiesthe(cid:209) -Condition;
0 2
(2) Ify satisfiestheD 2-Condition,then itsatisfiestheuniform(cid:209) -Condition.
0
For any 1 < p <¥ , every function x xp is an Orlicz function which satisfies the uniform
(cid:209) -Condition,(so(cid:209) and(cid:209) -conditions7−t→oo)and theD -Condition. Attheoppositeside, forany
0 2 0 2
a>0andb 1,x eaxb 1belongstotheD 2-Class(andthentotheuniform(cid:209) -Class),yetnot
0
≥ 7−→ −
totheD -one. Inaddition,theOrliczfunctionswhichcanbewrittenx exp a(ln(x+1))b 1
2
→ −
fora>0 and b 1, satisfythe(cid:209) and (cid:209) -Conditions,butdonot belongtot(cid:16)heD 2-Class. (cid:17)
2 0
≥
For a complete study of Orlicz spaces, we refer to [6] and [14]. We can also find precise and
usefulinformationin[7],suchasotherclassesofOrliczfunctionsandtheirlinkswitheachother.
2.3. Background results. All the results of the present paper are based on characterizations
of the boundedness and compactness of composition operators on Hardy-Orlicz and Bergman-
Orliczspaces([2,3]). AsIalreadysaid,thesecharacterizationsessentiallydependonthemanner
inwhichtheOrliczfunctiongrows.
The characterizations of the boundedness and compactness of Cf involve adapted Carleson
measures, and then geometric notions. For z S and 0 <h <1, let us denote by S(z ,h) and
N
∈
S (z ,h)thenon-isotropic“balls”, respectivelyin B and B , defined by
N N
S(z ,h)= z B , 1 z,z <h and S (z ,h)= z B , 1 z,z <h .
N N
{ ∈ | −h i| } ∈ | −h i|
We say that a finite positive Borel measure m on BN is a(cid:8)y -Carleson measure, y (cid:9)an Orlicz
function,if
1
m (S (z ,h))=O ,
h→0(cid:18)y (Ay −1(1/hN))(cid:19)
uniformly in z S and for some constant A > 0. m is a vanishing y -Carleson measure if the
N
∈
aboveconditionis satisfied forevery A>0 and withthebig-Oh conditionreplaced by alittle-oh
condition.
A finitepositiveBorel measure m on B isa(y ,a )-Bergman-Carleson measureif
N
1
m (S(z ,h))=O ,
h→0 y Ay −1 1/hN(a ) !
uniformly in z SN and for some constant A >(cid:0)0. m is(cid:0)a vanish(cid:1)in(cid:1)g (y ,a )-Bergman-Carleson
∈
measureiftheaboveconditionissatisfiedforeveryA>0andwiththebig-Ohconditionreplaced
byalittle-ohcondition.
When y satisfies the D -Condition, a (vanishing) y -Carleson measure (resp. (vanishing)
2
(y ,a )-Bergman-Carlesonmeasure)isa(vanishing)Carlesonmeasure(resp. (vanishing)Bergman-
Carleson measure)(see[2, 3,Sections 3]).
For f :BN BN, we denote by m f the pull-back measure of s N by the boundary limit f ∗ of
→
f ,and by m f ,a thatofdva byf . To beprecise, foranyE BN (resp. E BN),
⊂ ⊂
m f (E)=s N (f ∗)−1(E) and m f ,a (E)=va f −1(E) .
(cid:16) (cid:17)
(cid:0) (cid:1)
6 COMPACTCOMPOSITIONOPERATORSONHARDY-ORLICZANDBERGMAN-ORLICZSPACESONBN
2.3.1. Resultsfor Hardy-Orliczspaces(see[3, Section 3]). Themaintheoremis thefollowing:
Theorem 2.2. Lety beanOrliczfunctionwhichsatisfiesthe(cid:209) -Conditionandletf :B B
2 N N
→
beholomorphic.
(1) If y satisfies the uniform (cid:209) 0-Condition, then Cf is bounded from Hy (BN) into itself if
andonlyif m f isa y -Carleson measure.
(2) Ify satisfiesthe(cid:209) 0-Condition,thenCf iscompactfromHy (BN)intoitselfifandonlyif
m f isa vanishingy -Carleson measure.
(3) If y satisfies the D 2-Condition, then Cf is bounded (resp. compact) from Hy (BN) into
itselfif andonlyif m f isa Carlesonmeasure(resp. a vanishingCarlesonmeasure).
(4) If y satisfiestheD 2-Condition,thenCf isboundedon Hy (BN).
The first two points are contained in [3, Theorem 3.2]; according to [5, Theorem 3.35], the
third point means that, if y satisfies the D 2-Condition, then Cf is bounded (resp. compact) on
Hy (B ) if and only if it is on Hp(B ) (see [3, Corollary 3.4]). The last point is [3, Theorem
N N
3.7].
Due to the non-separability of small Hardy-Orlicz spaces, [3, Theorem 3.2] is not a direct
consequence of Carleson-type embedding theorems obtained in [3, Section 2]; however, if we
follow the proofs of these embedding theorems directly for composition operators, by working
on spheres of radius 0<r <1, then we get the followingcharacterizations of both boundedness
andcompactnessofcompositionoperators:
Theorem 2.3. Let y be an Orlicz function satisfying the (cid:209) -Condition and let f :B B be
2 N N
→
holomorphic.
(1) If y satisfies the uniform (cid:209) 0-Condition, then Cf is bounded on Hy (BN) if and only if
thereexistssomeA>0 suchthat
1
(2.1) 0<sur<p1m f r(S (z ,h))=Oh→0(cid:18)y (Ay −1(1/hN))(cid:19)
uniformlyin z S .
N
(2) If y satisfies th∈e (cid:209) 0-Condition, thenCf is compact on Hy (BN) if and only if, for every
A>0,
1
(2.2) 0<sur<p1m f r(S (z ,h))=Oh→0(cid:18)y (Ay −1(1/hN))(cid:19)
uniformlyin z S .
N
∈
In the further, we will see how these two characterizations are useful depending on the situa-
tions.
2.3.2. Results for Bergman-Orlicz spaces (see [2, Section 3]). The main result is similar to that
statedinthepreviousparagraph:
Theorem 2.4. Let y beanOrliczfunction,leta > 1 and letf :B B beholomorphic.
N N
− → y
(1) If y satisfies the uniform (cid:209) 0-Condition, then Cf is bounded from Aa (BN) into itself if
andonlyif m f isa (y ,a )-Bergman-Carlesonmeasure.
y
(2) If y satisfiesthe (cid:209) 0-Condition, thenCf is compact from Aa (BN) into itself if and only if
m f isa vanishing(y ,a )-Bergman-Carlesonmeasure.
y
(3) If y satisfies the D 2-Condition, then Cf is bounded (resp. compact) from Aa (BN) into
itself if and only if m f is a Bergman-Carleson measure (resp. a vanishing Bergman-
Carlesonmeasure).
y
(4) If y satisfiestheD 2-Condition,thenCf isboundedon Aa (BN).
STÉPHANECHARPENTIER 7
According to [5, Theorem 3.37], the third point means that, if y satisfies the D -Condition,
2
y
thenCf is bounded (resp. compact) on Aa (BN), ifand only ifCf is bounded (resp. compact)on
Aap(BN).
3. MAIN RESULTS
3.1. Compactness of Cf on every Hardy-Orlicz or Bergman-Orlicz spaces. The following
theoremcompletesbothTheorem2.2 and Theorem2.4.
Theorem 3.1. Letf :B B beaholomorphicmap. Thefollowingassertionsareequivalent:
N N
→
(1) Cf iscompact onHy (BN), foreveryOrliczfunctiony ;
y
(2) Forsomea > 1,Cf is compacton Aa (BN), forevery Orliczfunctiony ;
(3) Cf iscompact−onH¥ (BN);
(4) f <1.
¥
k k
Proof. Itiswell-knownthatCf iscompactonH¥ (BN)ifandonlyif f ¥ <1. Usingthefactthat
a compositionoperator is compact on Hy (BN) (resp. Aya (BN)) if akndkonly if for every bounded
sequence (fn)n ⊂ Hy (BN), kfnky ≤ 1, (resp. (fn)n ⊂ Aya (BN), kfnkAay ≤ 1) which tends to 0
uniformlyoneverycompactsubsetofBN,thenkfn◦f ky −n−−→¥ 0(resp. kfn◦f kAya −n−−→¥ 0),itis
notdifficulttoshowthatif f ¥ <1,thenCf is compacton→Hy (BN)(resp. Aya (BN)→.)
k k
Itremainstoprove(1) (4)and(2) (4). Wefirstdealwiththeproofof(1) (4). Wewilluse
thenecessarypartofThe⇒orem2.3. Let⇒usassumethatf inducesacompactcom⇒positionoperator
oneveryHardy-Orliczspace. According to(2.2), thismeansthat
1
0<sur<p1zsuSpN m f r(S (z ,h)) =oh→0(cid:18)y (Ay −1(1/hN))(cid:19),
∈ (cid:0) (cid:1)
foreveryA>0 and everyOrliczfunctiony , which inturn implies
1
(3.1) sup sup m f (S (z ,h)) ,
0<r<1z SN r ≤ y (Ay −1(1/hN))
∈ (cid:0) (cid:1)
foreveryA>0, foreveryOrliczfunctiony andforh sufficientlysmall. Weintendtoshowthat
sup sup m f (S (z ,h)) =0,
r
0<r<1z S
N
∈ (cid:0) (cid:1)
forall0<h≤h0,h0 ∈(0,1). Bycontradiction,wesupposethat sup sup m f r(S (z ,h)) 6=0
0<r<1z S
N
foreveryh>0, since ∈ (cid:0) (cid:1)
h sup sup m f (S (z ,h))
7−→0<r<1z S r
N
∈ (cid:0) (cid:1)
is an increasing function on (0,1). A straightforward computation shows that inequality (3.1) is
satisfied for every A >0, for every Orlicz function y and for h small enough, if and only if we
have,by puttingx=1/h,
y 1 xN 1
−
(3.2) ,
≤ A
y 1 1/ sup sup (cid:0)m f (cid:1)(S (z ,1/x))
− r
0<r<1z S !
N
∈ (cid:0) (cid:1)
foreveryA>0,foreveryOrliczfunctiony andforxlargeenough. Thefollowinglemmaensures
thatthiscannotoccur:
8 COMPACTCOMPOSITIONOPERATORSONHARDY-ORLICZANDBERGMAN-ORLICZSPACESONBN
Lemma 3.2. Let f,g:[0,+¥ [ [0,+¥ [ be two increasing functions which tend to +¥ at +¥ .
There exist d > 0 and a conti→nuous increasing concave function n : [0,+¥ [ [0,+¥ [, with
n (f (x)) →
lim n (x)=+¥ ,suchthat d >0,for everyxlargeenough.
x +¥ n (g(x)) ≥
→
We assume for a while that this lemma has been proven, and we finish the proof of Theorem
3.1. Withthenotationsofthelemma,weput
f (x)=xN and
g(x)=1/ sup sup m f (S (z ,1/x)) .
r
( 0<r<1z S
N
∈ (cid:0) (cid:1)
It is clear that limx +¥ f (x)=+¥ ; sinceCf is supposed to be compact on every Hy (BN), it is
in particular bound→ed on Hp(B ) ([3, Corollary 3.5]), then we have g(x) +¥ (Theorem
N
−x−−+−→¥
2.2, 3). Now, the above lemma provides a constant d > 0 and a continuous→increasing concave
functionn , tendingtoinfinityat infinity,suchthat
1
n xN /n d >0,
 sup m f Sf (x ,1/x) ≥
(cid:0) (cid:1) x SN 
 ∈ (cid:0) (cid:0) (cid:1)(cid:1)
for every x large enough. It is not difficult to check that n can be constructed such that y =n 1
−
x
isanOrliczfunction,i.e. suchthat +¥ (thatiswhatwearedoingintheproofofthe
n (x) −x−−+−→¥
→
lemmabelow). Therefore, weget acontradictionwithCondition(3.2), sowe musthave
sup sup m f r Sf (x ,h) =0,
0<r<1x S
N
∈ (cid:0) (cid:0) (cid:1)(cid:1)
foreveryh>0 smallenough. It followsthatthere existssome0<r <1such that
0
(3.3) sup m f r(C (r0,1))=0,
0<r<1
where C (r ,1) = z B ,r < z <1 . We intend to show that f 1(C (r ,1)) = 0/, which
0 N 0 − 0
{ ∈ | | }
shouldgivetheresult. Let0<r<1and let uslookat theset
f 1(C (r ,1)) S = z S ,f (z ) C (r ,1) .
r− 0 N N r 0
∩ { ∈ ∈ }
Condition(3.3)implies
s f 1(C (r ,1)) S =0.
N r− 0 N
∩
Since f r is continuous on BN, f r−1((cid:0)C (r0,1)) SN must b(cid:1)e an open subset of SN and then must
∩
beempty. So wehaveproventhat, foranyr (0,1),
∈
z S ,f (z ) C (r ,1) =f 1(C (r ,1)) rS =0/,
N r 0 − 0 N
{ ∈ ∈ } ∩
whererS = z B , z =r , hence
N N
∈ | |
(cid:8) f 1(C(cid:9)(r ,1))= f 1(C (r ,1)) rS =0/.
− 0 − 0 N
∩
0<r<1
[
(cid:0) (cid:1)
The proof in the Bergman-Orlicz context is much easier. Proceeding as above and using the
necessarypartofthesecondpointofTheorem2.4, wegetthatCondition m f (C (r0,1))=0must
hold, for some 0 < r < 1. By continuity of the map f on B , f 1(C (r ,1)) cannot be but
0 N − 0
empty.
To finishtheproof, wehaveto dothatofLemma3.2:
STÉPHANECHARPENTIER 9
ProofofLemma 3.2. The proof will be constructive. Let f and g be given as in the statement of
the lemma. We are going to build by induction a sequence (a ) which will be of interest in the
n n
construction of the desired function n . We put a =0, a =1, and we deduce a from a and
0 1 n+2 n
a in thefollowingway: wedefine
n+1
b =sup g(x), f (x) a
n+2 n+1
{ ≤ }
and
a =max b ,a +(a a ) .
n+2 n+2 n+1 n+1 n
{ − }
Weobservethat:
(1) If f (x) a , then g(x) a ;
n+1 n+2
≤ ≤
(2) a a a a 1.
n+2 n+1 n+1 n
− ≥ − ≥
Wenowconstructtheconcavefunctionn asacontinuousaffineone,whosederivativeisequalto
1
e = ontheinterval(a ,a ),andwithn (0)=0. Ofcoursen isincreasingand
n n n+1
√n(a a )
n+1 n
then maps [0,+−¥ [ into itself. Since e is decreasing, because of 2) above, n is concave. In order
n
tocheck that n tendstoinfinityat infinity,wecomputen (a ):
n
1
n (a )=n (a )+e (a a )=n (a )+ .
n+1 n n n+1 n n
− √n
Thereforen (an+1)=(cid:229) nk=+11 √1k whichshowsthatlimx→+¥ n (x)=+¥ , sincean →+¥ .
n f (x)
Wenowcheck that ◦ isboundedbelowby someconstantd >0, when xisbig enough.
n g(x)
Let x [0,+¥ [, and let n◦ be an integer such that a f (x) a ; we have n (f (x)) n (a ).
n n+1 n
Using∈thefirstpropertyofthesequence(a ) above,≤weget n≤(g(x)) n (a ). Thisy≥ields,for
n n n+2
≤
n 1,
≥
n (f (x)) n (an) = (cid:229) nk=1 √1k d >0,
n (g(x)) ≥ n (a ) (cid:229) n+2 1 ≥
n+2
k=1 √k
hencetheresult. (cid:3)
(cid:3)
3.2. Korányi regions and compactness ofCf on Hardy-Orlicz and Bergman-Orlicz spaces.
Forz S anda>1,werecallthattheKorányiapproachregionG (z ,a)ofangularopeningais
N
∈
defined by
a
G (z ,a)= z B , 1 z,z < 1 z 2 .
N
∈ | −h i| 2 −| |
n (cid:16) (cid:17)o
[5, Theorem 6.4]and thethirdpartofTheorem 2.2yieldsthefollowingresult:
Theorem3.3. Lety beanOrliczfunctionsatisfyingtheD (cid:209) -Condition. Letalsof :B B
2 2 N N
∩ →
beholomorphic. WeassumethatN >1and wefixb =(cos(p /(2N))) 1.
N −
(1) Iff (BN) G (z ,bN),thenCf is boundedonHy (BN);
(2) If f (BN)⊂ G (z ,b), for some z SN and for some 1 < b <bN, then Cf is compact on
Hy (B ).⊂ ∈
N
(3) Both of the above results are sharp in the following sense: given c > b , there exists
N
f with f (BN) G (z ,c), for some z SN, andCf not bounded on Hy (BN); there also
existssomef w⊂ithf (BN) G (z ,bN),∈forsomez SN,andCf notcompactonHy (BN).
⊂ ∈
10 COMPACTCOMPOSITIONOPERATORSONHARDY-ORLICZANDBERGMAN-ORLICZSPACESONBN
Remark 3.4. (1) If N =1, the two first point of the previous theorem are true if we put b =+¥
1
and G (z ,+¥ ) = D. Indeed, the first point is nothing but the continuity of every composition
operator on the disc, and the second one is contained in [5, Proposition 3.25] which says that,
whenever f (D) is contained in some nontangential approach region in D, then Cf is Hilbert-
SchmidtinH2(D), hencecompacton everyHp(D), 1 p<¥ .
≤
(2) Followingthe proof of[4, Theorem 3.3], it is not difficult to showthat theboundedness or
compactness ofCf on Hy (BN) implies that on Aya (BN), for any a > 1, as soon as the Orlicz
function y satisfies the D -Condition. Thus, the first two points of th−e previous theorem also
2
holdsfor Bergman-Orliczspaces.
ThefollowingresultshowsthatTheorem3.3doesnotholdassoonastheOrliczfunctiongrows
fast.
Theorem 3.5. Let y be an Orlicz function satisfying the D 2-Condition. Then, for every z S
N
and every b > 1, there exists a holomorphic self-map f taking BN into G (z ,b), such thatC∈f is
notcompacton Hy (B ).
N
Remark 3.6. Observethat thereisno assumptiononN.
ProofofTheorem 3.5. Theproof will usethenecessary part ofthe second point ofTheorem 2.2.
First ofall, werecall thatD 2-Conditionimplies(cid:209) -Condition(see Paragraph 2.2). We denoteby
2
e thevector(1,0...0)inCN. Itisclearlyenoughtoprovethetheoremforz =e . Foranyb>1,
1 1
weset
2cos 1(1/b)
(3.4) b = −
p
in(0,1). We need alemmawhoseproofisincludedin thatof[5, Theorem6.4]:
Lemma 3.7. Let b>1 and let b be defined by (3.4). There exists a holomorphicmap f :B
N
B ,with f (B ) G (e ,b),suchthat →
N N 1
⊂
(3.5) s f 1(S (e ,h)) Ch1/b ,
N − 1
≥
forsomeconstantC>0 dependingonlyonf andb.
Proof. Withoutgoingintodetails,webrieflygivetheideasoftheproof. Itusesthedeep Alexan-
drov’sresultwhichgivestheexistenceofnon-constantinnerfunctionsinB ([1]). Therefore,we
N
considerafunctionf which can bewritten
f = k j ,0 ,
′
◦
where 0 is the (n 1)-tuple (0,...,0), j is (cid:0)an inner f(cid:1)unction with j (0) =0, and where k is a
′
biholomorphic ma−p from D onto the non-tangential approach region G (1,b) in the disc, defined
by
b
G (1,b)= z D, 1 z < 1 z 2 .
∈ | − | 2 −| |
(cid:26) (cid:27)
(cid:16) (cid:17)
Onecanshowthatthelower-estimate(3.7)holdsforthismapf ,usingthefactthatinnerfunctions
j aremeasurepreserving mapsofS intoT(see[15, p. 405])inthefollowingsense:
N
s N (j ∗)−1(E) =s 1(E),
(cid:16) (cid:17)
foranyBorel set E inT. (cid:3)