THESIS FOR THE DEGREE OF LICENTIATE OF PHILOSOPHY
Carleson measures for Hardy Sobolev spaces and Generalized Bergman spaces
EDGAR TCHOUNDJA
Department of Mathematical Sciences Chalmers University of Technology and G¨oteborg University SE - 412 96 G¨oteborg, Sweden
G¨oteborg, March 2007 Carleson measures for Hardy Sobolev spaces and Generalized Bergman spaces Edgar Tchoundja ISSN:1652-9715 /NO 2007:7
c Edgar Tchoundja, 2007
Department of Mathematical Sciences Division of Mathematics Chalmers University of Technology and G¨oteborg University SE-412 96 G¨oteborg Sweden Telephone +46 (0)31 772 10 00
Printed in G¨oteborg, Sweden 2007 Abstract
Carleson measures for various spaces of holomorphic functions in the unit ball have been studied extensively since Carleson’s original result for the disk. In the first paper of this thesis, we give, by means of Green’s formula, an alternative proof of the characterization of Carleson measures for some Hardy Sobolev spaces (including Hardy space) in the unit ball. In the second paper of this thesis, we give a new characterization of Carleson measures for the Generalized Bergman spaces on the unit ball using singular integral techniques.
Keywords: Carleson measures, Hardy Sobolev spaces, Generalized Bergman spaces, Besov Sobolev spaces, Arveson’s Hardy space, T(1)-Theorem.
AMS 2000 Subject Classification: 26B20, 32A35, 32A37, 32A55, 46E35.
i Acknowledgments
First of all I would like to thank my advisor Bo Berndtsson, at Chalmers, for introducing me to the topic of this thesis. I have since then benefited from his valuable explanations and comments. He has taught me the mathe- matical spirit. He has greatly contributed to making my stay here enjoyable, with his friendly attitude. I would like to thank my co-advisor Genkai Zhang, at Chalmers, for stimulating discussions on this topic, as well as many others topics. Thanks for always taking time, even when busy, to answer my questions. I would like to thank my advisor David B´ekoll´e, at Yaound´e I, for being the one who introduced me to the complex analysis, and for his constant solicitude. I have benefited from his numerous comments on the singular integral techniques. I deeply thank the International Science Program (ISP), Uppsala, Swe- den for the financial support through a fellowship which gave me the oppor- tunity to carry out part of my posgraduate studies here at Chalmers. In addition, I thank the Chalmers University of Technology for providing me with an excellent academic environment and good logistics. In particu- lar, I thank the complex analysis group at the department of Mathematical sciences. I express my gratitude to everyone at the Mathematical depart- ment of the Faculty of Sciences at the University of Yaound´e I. Especially to my fellows at the classical analysis group. At last but not least, I would like to thank my family for their love, and numerous friends I have met for their kind support.
ii This thesis consists of an introduction and the following two papers.
Paper I. E. Tchoundja: Carleson measures for Hardy Sobolev spaces.
Paper II. E. Tchoundja: Carleson measures for the Generalized Bergman spaces via a T(1)- Theorem type.
iii CARLESON MEASURES FOR HARDY SOBOLEV SPACES AND GENERALIZED BERGMAN SPACES
EDGAR TCHOUNDJA
INTRODUCTION Let Ω be a region in Cn and X a Banach space of continuous functions in Ω. We raise the following problem: Characterize positive measures µ in Ω such that there exists a constant C = C(µ) with the property that
p p |f(z)| ≤ CkfkX ZΩ for all f ∈ X. Such problems are now known in the literature as Carleson mea- sures problem for X. The purpose of this work is to study Carleson measures for Hardy Sobolev spaces, see Paper I, and also Carleson measures for the Generalized Bergman spaces, see Paper II. In this introduction, we briefly give a literature overview and motivation of this question. We outline our work in the last two sections . 0.1. Overview and motivation. Carleson [CA58] was the first to study this question in the case of the unit disk U of C and for the Hardy space Hp(U) which is the space of holomorphic functions f in U with the property that: π sup |f(reiθ)|pdθ < ∞. r<1 Z−π Carleson measures arise in many questions involving analysis in functions spaces. We could mention here the problem of multipliers. That is, given a space, X, of functions on a set Ω, we want to describe those functions f for which the map of multiplication by f is a continuous map of X to itself. We mention also the interpolating sequences problem. That is, to characterize those sequences {yi} in Ω with the property that, as f ranges over X, the set of sequences of values, {f(yi)} ranges over the space of all sequences which satisfy a natural growth condition. Carleson measures have played an important role in the solution of some famous problems such as the Corona problem and the duality theory for H1. See Fefferman and Stein [FS72], Carleson [CA62]. This explains why Carleson measures have been studied extensively since the characterization obtained by Carleson [CA62] in 1962. 1 2 EDGAR TCHOUNDJA
In 1967, H¨ormander [H67] extended Carleson’s result to the unit ball of Cn. Since then, for various spaces, this problem as well as its applications have attracted many authors. Among them, we will mention Stegenga [ST80] who characterized Carleson measures for the space, Dα, of analytic functions f in the unit disk for which ∞ 2 α 2 (1 + n ) |an| < ∞ Xn=0 n where f(z)= n≥0 anz is the Taylor expansion of f. Cima and WogenP [CW82] gave the characterization of Carleson measures p Cn for the weighted Bergman spaces, Aα (α > −1), in the unit ball, Dn, of . This is the space of holomorphic functions f in Dn such that
|f(z)|p(1 − |z|2)αdλ(z) < ∞. ZDn p The Hardy Sobolev space Hβ(Dn) is the space of holomorphic functions f in Dn such that
p sup (I + R)βf(rξ) dσ(ξ) < ∞, ZS 0≤r<1 n β β where (I + R) f(z)= (1+ k) fk(z) if f = fk is the homogeneous expan- sion of f. P.Ahern [A88]P characterized CarlesonP measures for this space when 0
1, Cascante and Ortega [CO95] studied the same question and gave some necessary and some sufficient conditions. They also studied Carleson measures for the holomorphic Besov Sobolev space. The Carleson measures for the analytic Besov spaces in the unit ball have been character- ized by Arcozzi, Rochberg and Sawyer [ARS02, ARS06]. We mention finally p R the generalized Bergman space Aα (α ∈ ), introduced in [ZZ05], that is the space of holomorphic functions f in the unit ball such that for some integer m with mp + α> −1
|(I + R)mf(z)|p (1 − |z|2)pm+αdλ(z) < ∞. ZDn Zhu and Zhao [ZZ05] characterized Carleson measures for this space when 0 < p ≤ 1 and Arcozzi, Rochberg and Sawyer [ARS062] recently obtained results for the case p = 2 and α ∈ (−n − 1, −n]. 0.2. Carleson measures on Hardy Sobolev spaces. We say that a positive p p Borel measure µ on Dn is a Carleson measure for Hβ = Hβ(Dn) if there exists a constant C = C(µ) such that: CARLESON MEASURES ON SOME HOLOMORPHIC SPACES 3
(1) kfk p ≤ CkfkHp L (µ) β p for all f ∈ Hβ. Observe that when β = 0 we have the usual Hardy space. For 0 < p ≤ n 1 and 0 <β < p , Ahern[A88] described Carleson measures using atomic decomposition for functions in Hp (see [A88, Lemma 1.1]). For β = 0 and p > 0, H¨ormander [H67] solved the problem using a covering lemma and boundedness of a maximal function. In paper I, inspired by results in [An97, B87], we give a new proof of these results. We use Green’s formula with respect to an appropriate positive closed (1.1) form to reduce the problem to pointwise estimates for functions in these spaces. Moreover we show, by giving a counterexample, that the natural growth condition is no longer sufficient for p> 1 and β > 0. 0.3. Carleson measures on the Generalized Bergman space. We re- 2 strict ourselves to p = 2. The Generalized Bergman spaces Aα [ZZ05] consist of all holomorphic functions f in the unit ball Dn with the property that:
2 m 2 2 2m+α kfkα = |(I + R) f(z)| (1 − |z| ) dλ(z) ZDn where 2m + α> −1. Note that this definition is independent of m. 2 In this case a positive Borel measure µ on Dn is a Carleson measure for Aα if there exists a constant C = C(µ) such that:
2 2 2 |f| dµ ≤ Ckfkα, f ∈ Aα. ZD One can easily observe that when α > −1 we have the usual weighted Bergman spaces (take m = 0), and when α = −1, we have the usual Hardy space. Thus Carleson type embedding problems have been settled in these cases by Carleson, H¨ormander, Stegenga, Cima and Wogen. The range α ∈ (−n − 1, −1) was unsolved until 2006 when Arcozzi, Rochberg and Sawyer [ARS062] obtained results for the range α ∈ (−n − 1, −n]. Their characteriza- tion used certain tree condition. They have transformed the question, equiva- 2 lently to a question of boundedness of an operator Tα in L (µ) where
Tαf(z)= f(w)Kα(z,w)dµ(w), z ∈ D, ZD 1 with Kα defined by Kα(z,w)= ℜ n+1+α . n (1−zw) o 4 EDGAR TCHOUNDJA
In paper II, we use singular integral techniques to characterize measures µ for which this operator is bounded. To be precise, recall that for a topological space X with a pseudo distance d, a kernel K(x, y) is called an n Calder´on- Zygmund kernel with respect to the pseudo distance d if C1 a) |K(x, y)| ≤ d(x,y)n and b) There exists 0 < δ ≤ 1 such that d(x, x′)δ |K(x, y) − K(x′, y)| + |K(y, x) − K(y, x′)| ≤ C 2 d(x, y)n+δ ′ ′ if d(x, x ) ≤ C3d(x, y), x, x , y ∈ X. Given a Calder´on-Zygmund kernel K, we can define (at least formally) a Calder´on-Zygmund operator (CZO) associated with this kernel by
T f(x)= K(x, y)f(y)dµ(y). ZX One important question in the Calder´on-Zygmund theory is to find a crite- rion for boundedness of the CZO in L2(µ). (In this context, such problem is called a T(1)-Theorem problem) Many authors studied this problem. In fact, when X = Rm, µ = dx (the usual Lebesgue measure) and d is the Euclidean distance, a famous criterion called ”T(1)-Theorem” was obtained by Journ´eand G. David [DJ84]. This criterion states that a CZO is bounded in L2(dµ) if and only if it is weakly bounded (in some sense), and the operator and its adjoint send the function 1 in BMO. This result was extended to space of homogeneous type in an un- published work by R. Coifman. It was then an interesting question to extend this T(1)-Theorem in the case where the space is not of homogeneous type (This essentially means that the measure µ does not satisfy the doubling con- dition). Several authors such as Tolsa, Nazarov, Treil, Volberg and Verdera [T99, NTV97, V00] treated this situation in the setting of Rm with the Eu- clidean distance. One good example of such an operator was the Cauchy integral operator. We say that the Cauchy integral operator is bounded in L2(dµ) whenever for some positive constant C, one has for every ǫ> 0
|C (fµ)|2dµ ≤ C |f|2dµ, f ∈ L2(dµ), Z ǫ Z where f(ζ) Cǫ(fµ)(z)= dµ(ζ), z ∈ C. Z|ζ−z|>ǫ ζ − z CARLESON MEASURES ON SOME HOLOMORPHIC SPACES 5
Their result is that the Cauchy integral operator is bounded in L2(dµ) if and only if i) µ(D) ≤ Cr(D), for each disc D with radius r(D). 2 ii) D |Cǫ(χDµ)| dµ ≤ Cµ(D), for each disc D, ǫ> 0. R Similarly, when we consider the kernel Kα associated with the operator Tα, we first show that this kernel is an n +1+ α Calder´on Zygmund kernel in the unit ball associated with a pseudo distance d. To characterize the boundedness of this operator, we adapt the idea used by Verdera [V00] to give an alternative proof of the T (1)− Theorem for the Cauchy integral operator. For this, we introduce a sort of Menger curvature [M95] in the unit ball and establish a good λ inequality.
References
[A88] P.Ahern: Exceptional sets for holomorphic sobolev functions. Michigan Math.J. 35 (1988), pp 29-41. [An97] M. Andersson: Topic in complex analysis, Springer-Verlag New York, (1997). [ARS02] N. Arcozzi, R. Rochberg & E. Sawyer: Carleson measures for analytic Besov spaces, Rev. Mat. Iberoamericana 18 (2002), pp 443 -510. [ARS06] N. Arcozzi, R. Rochberg & E. Sawyer: Carleson measures and interpolating se- quences for Besov spaces on complex balls. Mem. Amer. Math. Soc.182 (2006), no. 859, vi+163. [ARS062] N. Arcozzi, R. Rochberg & E. Sawyer: Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls. Preprint. (2006) sub- mitted. [B87] B. Berndtsson: ∂b and Carleson type inequalities. Complex analysis, II (College Park, Md 1985-86), 42-54, Lecture Notes in Math, 1276, Springer, Berlin, 1987. [CA58] L. Carleson: An interpolation problem for bounded analytic functions, Amer. J. Math., 80 (1958), pp 921-930. [CA62] L. Carleson: Interpolation by bounded analytic functions and the corona prob- lem, Ann. of Math., 76 (1962), pp 547-559. [CO95] C. Cascante & J. M. Ortega : Carleson measures on spaces of Hardy-sobolev type, Can. J. Math., Vol 47 (6) 1995, pp 1177 - 1200. [CW82] J. A. Cima & W. Wogen: A Carleson measure theorem for the Bergman space on the Ball, J. Operator theory, 7 (1982) pp 157-165. [DJ84] G. David & J.L. Journ´e:A boundedness criterion for generalized Calder´on-Zygmund operators, Annals of Math., 120 (1984), pp 371-397. [FS72] C. Fefferman & E. M. Stein: Hp spaces of several variables, Acta Math. 129 (1972), pp 137-193. [H67] L. H¨ormander: Lp-estimates for (pluri)subharmonic functions, Math. Scand. 20 (1967,) p 65 - 78. 6 EDGAR TCHOUNDJA
[M95] M. S. Melnikov: Analytic capacity: a discrete approach and the curvature of a mea- sure, (Russian), Mat. Sb. 186(6) (1995), 57-76; translation in Sb. Math., 186: 6 (1995), pp 827-846. [NTV97] F. Nazarov, S. Treil & A. Volberg: Cauchy Integral and Calder´on-Zygmund opera- tors on non-homogeneous spaces, Internt. Math. Res. Notices (1997), n◦15, pp 703-726. [ST80] D. A. Stegenga: Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), pp 113-139. [T99] X. Tolsa: L2-boundedness of the Cauchy integral operator for continuous mea- sures, Duke Math. J., 98 (1999) pp 269-304. [V00] J. Verdera: On the T(1)-Theorem for the Cauchy integral. Ark. Mat., 38 (2000) pp 183-199. [ZZ05] R. Zhao & K. Zhu: Theory of Bergman spaces in the unit ball of Cn, Univ. Albany. Preprint. (2005). CARLESON MEASURES FOR HARDY SOBOLEV SPACES
EDGAR TCHOUNDJA
Abstract. The main goal of this paper is to present an alternative proof of Carleson measures for Hardy Sobolev spaces in the unit ball (Hardy spaces Hp included) β. We simply use Green’s formula and pointwise estimates for functions in these spaces. An example shows that for p > 1, the natural necessary condition is no longer sufficient.
1. Introduction In this note, we give a new proof of the characterization of Carleson measures for some Hardy Sobolev spaces in the unit ball. Our method avoids some tech- nicalities used by previous methods such as maximal functions, covering lemma and atomic decomposition; see [7, 1]. It seems therefore easy and can be used for other purposes. We just use Green’s formula associated to an appropriate positive closed (1.1) form. This method is inspired by results in [4, 2]. Let D = z Cn : z −< 1 be the unit ball of Cn; and S = ∂D its boundary. We will denote{ ∈ by dλ |the| normalized} Lebesgue measure on D, and by dσ the normal- ized Lebesgue measure on S. For β R and 0 (1) f Lp(µ) C f Hp k k ≤ k k β p for all f Hβ. We give a new proof of the following theorem. Theorem∈ 1.1. Let µ be a positive Borel measure on D. Assume that 0 0 and β = 0. Then the following are equivalent: 2000 Mathematics Subject Classification. 26B20, 32A35. Key words and phrases. Carleson measures, Hardy Sobolev spaces, Arveson’s Hardy space, Green’s formula. 1 2 EDGAR TCHOUNDJA p i) µ is a Carleson measure for Hβ. ii) There exists a constant C such that m µ (Qδ(ξ)) Cδ ≤ for any ξ S and δ > 0, ∈ where m = n βp and − Qδ(ξ)= z D : 1 zξ < δ . ∈ | − | The sets Qδ are the high dimensional analogues of Carleson boxes in the unit disk. They are also called non isotropic balls. See [8] for more informations. 2. Preliminary results Let Ω be a smooth pseudo-convex domain in Cn and ρ a function such that Ω= z Cn : ρ(z) < 0 { ∈ } with dρ > 0 in ∂Ω= ρ(z) = 0 , the boundary of Ω. If ω| is| a positive closed{ (1, 1) }form in Ω, the form ω defined a K¨ahler metric − ωn in Ω and the volume form with respect to this metric is given by dV = n! . Definition 2.1. Suppose that ω is a positive closed (1, 1) form in Ω. The Lapla- cian with respect to the metric induced by ω is defined by:− n n 1 ω ω − ∆ωu = i∂∂u , n! ∧ (n 1)! − for any function u of class 2 in Ω. C An application of Stoke’s formula gives this Green’s formula. Theorem 2.2. Suppose that ω is a positive closed (1, 1) form in Ω. Let u,v − ∈ ∞ Ω , and u,v real. Then C n 1 ω − (v∆ωu u∆ωv) dV = i v∂u u∂v . ℜ − ℜ − ∧ Γ(n) ZΩ Z∂Ω Recall that the Euclidean volume form is determined by the closed (1, 1) form βn − β = i∂∂ z 2; precisely, dλ = . | | n! In the sequel, in view of our objective, we suppose that Ω is contained in D. CARLESON MEASURES FOR HARDY SOBOLEV SPACES 3 1 We consider now the closed (1, 1) form on D given by ω = i∂∂ log 1 z 2 . − −| | Observe that: β γ ω = + 1 z 2 (1 z 2)2 − | | − | | where β = i∂∂ z 2 is the usual Euclidean closed (1, 1) form and | | − 2 2 γ = i zjzkdzj dzk = i∂ z ∂ z . ∧ | | ∧ | | j,k X We can see that γ2 = 0 and β γ = γ β. Therefore for any integer p with 1 p p 1 p β pβ − γ (2) ω = p + ∧ . (1 z 2) (1 z 2)p+1 − | | − | | In particular, βn (3) ωn = (1 z 2)n+1 − | | and n 1 n 2 n 1 β − (n 1)β − γ ω − . (4) = n 1 + − 2 n∧ (1 z 2) − (1 z ) − | | − | | Using the definition of the Laplacian together with (3) and (4), we obtain by straightforward computation, the next lemma which gives us a formula to compute the Laplacian with respect to the above closed form ω. 1 2 Lemma 2.3. Suppose that ω = i∂∂ log 1 z 2 . Then for a function u on D, −| | C the Laplacian with respect to ω is given by : 2 (5) ∆ωu(z)= 1 z Lu(z) − | | ∂2u ∂2u where Lu(z)= zjzk. k ∂zk∂zk − j,k ∂zj ∂zk From this weP see that theP Laplacian with respect to this form is the same as the Bergman Laplacian or the so called invariant Laplacian. So ∆ω (u ϕa) = ◦ (∆ωu) ϕa, where ϕa is the involutive automorphism of D that interchanges the ◦ points 0 and a D. This invariance also follows from the fact that ϕa∗(ω) = ω; just use the identity∈ (9) and the definition of the Laplacian to see this. The 4 EDGAR TCHOUNDJA Green’s formula in this case is given by the following result (see [10, p28]and the references therein) . Proposition 2.4. Suppose 0 2 s 1 2 2 ∆ωφ(z) = (1 z ) − sn(1 z )+ s(s 1) z − | | − − | | − | | 2 s 1 2 2 4 +(1 z ) − s(1 z ) z s(s 1) z − | | − | | | | − − | | 2 2 2 2 2 s 1 = ( s(n s)(1 z ) s (1 z ) )(1 z ) − . − − − | | − − | | − | | 2 From Proposition 2.4 we obtain the following result. Theorem 2.6. There exists a positive constant C such that for all u ∞ D , u real and positive, we have ∈ C (7) Ludλ C u dσ. D ≤ S Z Z Proof: Let u ∞ D , fix δ ]0, 1[ small and apply Proposition 2.4 with ∈ C ∈ r2 = 1 δ. Let v(z)=(1 z 2)n, by (6), we have ∆ v(z)= n2 (1 z 2)n+1. ω Proposition− 2.4 gives − | | − − | | n ∂u Ludλ + n2 udλ = δ + 2n(1 δ)nu dσ. z 2<1 δ z 2 2 From this, we obtain an important estimate in the following theorem. Theorem 2.7. There exists a positive constant C such that, if g,ψ ∞ D ,ψ is real, and g is analytic in D, then for 0 (8) Lψ(z) g peψdλ C g peψdσ. D | | ≤ S | | Z Z Proof: Let g,ψ be as in the theorem. Fix a positive ǫ small. A straightforward computation shows that L (log g 2 + ǫ) 0. Since for any smooth function h, we | | ≥ p have L(eh) ehLh, if we apply (7) to u =( g 2 + ǫ) 2 eψ, we obtain ≥ | | p p Lψ g 2 + ǫ 2 eψdλ C g 2 + ǫ 2 eψdσ D | | ≤ S | | Z Z for any ǫ. Letting ǫ tend to 0, this gives (8) and this ends the proof of the theorem. 2 We will also need the following lemma. s 1 z 2 ( −| | ) Lemma 2.8. Let s> 0 and a D. Consider the function ξ(z)= 1 za 2s ; then ∈ | − | s s 1 s2 (1 z 2) (1 a 2) s (n s) (1 z 2) − Lξ(z)= − | | − | | − − | | − 1 za 2(s+1) − 1 za 2s | − | | − | Proof: Let ξ be as in the lemma. Recall that 2 2 2 (1 z ) (1 a ) (9) 1 ϕa(z) = − | | − | | − | | 1 za 2 | − | 2 s 2 s and observe then that ξ(z) (1 a ) = (1 ϕa(z) ) . By the invariance of the Laplacian, we have − | | − | | 2 s 2 s 1 a ∆w (ξ(z)) = ∆w ξ(z) 1 a − | | − | | 2 = ∆w 1 ϕa(z) − | | 2 s = ∆w 1 z ϕa(z). − | | ◦ We use now (6) and (9) to obtain the result. 2 6 EDGAR TCHOUNDJA 3. Carleson measures for Hardy sobolev In this section, we will give a proof of Theorem 1.1. For β R and 0 p p β f Hp (D) = sup (I + ) f(rξ) dσ(ξ) < . k k β r<1 S R ∞ Z By the definition, we can see that the operator (I + )β is an invertible opera- β R tor with inverse (I + )− . For β > 0, by simple change of variables, we see that 1 R β 1 β 1 1 − k (1+ k)− = Γ(β) 0 log t t dt, where k is an integer. Thus for a holomorphic function f and β > 0 we have R 1 β 1 1 1 − (10) f(z)= log (I + )β f(tz)dt. Γ(β) t R Z0 On the other hand, using integration by parts, we observe that 1 β 1 1 β 1 − 1+ k 1 log tkdt = log tkdt. t β t Z0 Z0 So we can extend the formula (10) for β > 1 and β = 0. We obtain − 6 1 1 1 β f(z)= log (I + )β+1 f(tz)dt. Γ(β + 1) t R Z0 We iterate the procedure n times and obtain that for a holomorphic function f and β > n, β = 0, 1, 2, ... − 6 − − 1 β+n 1 1 1 − (11) f(z)= log (I + )β+n f(tz)dt. Γ(β + n) t R Z0 From this discussion we will obtain the following estimates. Lemma 3.1. Let m,s be two positive real numbers such that m s> 0. Consider 1 − the function f(z)= (1 zw)m . Then the following holds: − i) There exists a constant C such that 1 (I + )s f(z) C , | R | ≤ 1 zw m+s | − | for all w D. ∈ CARLESON MEASURES FOR HARDY SOBOLEV SPACES 7 ii) There exists a constant C such that s 1 − (I + ) f(z) C m s , R ≤ 1 zw − | − | for all w D. ∈ Proof: Suppose that s is an integer, then a straightforward computation shows that P (zw) (I + )s f(z)= s , R (1 zw)m+s − where Ps is a polynomial of degree s. We then obtain i) in this case. Suppose now that s is not an integer, write s = n + γ where n is an integer and γ a real number such that γ ( 1, 0). If we take β = s, we observe that β ( n; n + 1). Using (11) we have∈ − − ∈ − − 1 γ 1 1 1 − − (I + )s f(z)= log (I + )+n f(tz)dt. R Γ( γ) t R − Z0 Hence 1 γ 1 1 − − (I + )s f(z) C log (I + )+n f(tz) dt | R | ≤ t | R | Z0 1 γ 1 (1 t)− − C − dt, ≤ 1 tzw m+n Z0 | − | by the first case of i). We then obtain part i) of the lemma because 1 r (1 t) k+r+1 − dt 1 z − , 1 tz k ≃ | − | Z0 | − | for k r 1 > 0 and 1 s Qs(zw) (I + )− f(z)= m s , R (1 zw) − − where Qs is a polynomial of degree s. We then obtain ii) in this case. Suppose now that s is not an integer and write s = n + γ, where n is an integer and γ a real number such that γ ( 1, 0). Substituting− − β = n γ in (11), we have ∈ − − − 8 EDGAR TCHOUNDJA 1 γ 1 s 1 1 − − n (I + )− f(z)= log (I + )− f(tz)dt. R Γ( γ) t R − Z0 The proof of ii) follows as in i). 2 Definition 3.2. We say that a positive Borel measure µ on D is a Carleson p measure for Hβ if there exists a constant C = C(µ) such that: (12) f Lp(µ) C f Hp k k ≤ k k β for all f Hp . ∈ β Recall also that, for any ξ S and δ > 0, the Carleson box is the set ∈ Qδ(ξ)= z D : 1 zξ < δ . ∈ | − | The following theorem, which is proved in [10], gives a characterization of mea- sures which satisfy certain conditions on the Carleson boxes. Theorem 3.3. Let t be a strictly positive real and µ be a positive Borel measure on D. Then the following conditions are equivalent: a) There exists a positive constant C such that t µ (Qδ(ξ)) Cδ ≤ for all ξ S and all δ > 0. b) For each∈s> 0 there exists a positive constant C such that (1 z 2)s dµ w C < . (13) sup − | | t+s ( ) z D D 1 zw ≤ ∞ ∈ Z | − | c) For some s> 0 there exists a positive constant C such that the inequality in (13) holds. From this we obtain the following necessary condition. Lemma 3.4. Suppose m = n βp > 0. Let µ be a positive Borel measure on D. − p If µ is a Carleson measure for Hβ, then there exists a positive constant C such that CARLESON MEASURES FOR HARDY SOBOLEV SPACES 9 m (14) µ (Qδ(ξ)) Cδ ≤ for all ξ S and all δ > 0. ∈ Proof: p Suppose that µ is a Carleson measure for Hβ. If for sufficiently large k we take f(z)= 1 , then by Lemma 3.1 part i), there is an absolute constant C such (1 zw)k p− C that f Hp (1 z 2)kp+βp−n . k k β ≤ −| | This fact together with the Carleson condition gives kp+βp n (1 z 2) − sup − | | kp dµ(w) C, z D D 1 zw ≤ ∈ Z | − | which by Theorem 3.3 is equivalent to (14). 2 We are ready to give the Proof of Theorem 1.1. It is enough to proof the Carleson inequality for f holomorphic in D, and smooth enough up to the boundary. For such f, we consider the functions (1 z 2)m g z I β f z ψ z dµ w . ( )=( + ) ( ) and ( )= − | | 2m ( ) R − D 1 zw Z | − | Since µ satisfies (14), by Theorem 3.3, we get that eψ 1. By Theorem 2.7 and Lemma 2.8 we have ≃ m m 1 m2 (1 z 2) (1 w 2) m (n m) (1 z 2) − dµ w g z pdλ z − | | 2m+2− | | + − −2m | | ( ) ( ) ( ) D 1 zw 1 zw | | ZZ | − | | − | ! (15) C g(ξ) pdσ(ξ). ≤ S | | Z Therefore in view of Fubini’s theorem, (12) follows if we can prove that m2 (1 w 2)m (1 z 2) f z p C g w pdλ w (16) ( ) − | | 2m+2− | | ( ) ( ) | | ≤ D 1 zw | | Z | − | m 1 m (n m) (1 w 2) − C g w pdλ w + − −2 |m | ( ) ( ) D 1 zw | | Z | − | ! for all z D. We proceed to prove (16). Observe that for n m > 0, (16) is equivalent∈ to − 10 EDGAR TCHOUNDJA m 1 (1 w 2) − f z p C g w pdλ w . (17) ( ) − | | 2m ( ) ( ) | | ≤ D 1 zw | | Z | − | For 0 2 s β g(w) (1 w ) (I + ) f(z)= g(z) = Cs.m −n |+1+| s dλ(w). R D (1 zw) Z − Hence by Lemma 3.1 part ii), we have g(w) (1 w 2)s f z C dλ w . (18) ( ) | | n−+1+ | s| β ( ) | | ≤ D 1 zw Z | − | − Assume s is sufficiently large so that we can set 2m s = + β (n + 1). p − Thus (18) becomes g(w) 2 s f(z) C 2m 1 w dλ(w). | | ≤ D (1 zw) p − | | Z − n+1+m 1 We observe that s = − (n + 1); therefore we apply Lemma 2.15 in [10] p − to obtain m 1 (1 w 2) − f z p C g w pdλ w , ( ) − | | 2m ( ) ( ) | | ≤ D 1 zw | | Z | − | which is (17). f(z) For p > 0 and β = 0, by the same reproducing formula applied to (1 za)α for some a D and α> 0 to be chosen, we have − ∈ f(z) (1 w 2)s f(w) α = Cs,m − | n|+1+s α dλ(w). (1 za) D (1 zw) (1 wa) − Z − − Hence we take a = z and obtain f(z) (1 w 2)s C f w dλ w . | |2 α − |n+1+| s+α ( ) ( ) (1 z ) ≤ D 1 zw | | − | | Z | − | CARLESON MEASURES FOR HARDY SOBOLEV SPACES 11 If 0 f(z) p (1 w 2)n C f w pdλ w , | | 2 − | |2n+2 ( ) ( ) (1 z ) ≤ D 1 zw | | − | | Z | − | which is (16). 1 Finally if p > 1 and α > p , we simply apply H¨older estimates and Theorem 1.12 in [10] to obtain (16). This ends the proof. 2 4. Comments and further results p As we can see in our approach, the space Hβ(D) is involved just when we want to obtain the key estimate (16). Therefore to obtain some result for p > 1 and β > 0, the difficulty is to prove (16). It turns out that for this case, the estimate n 1 (16) is in general not true. We can see this by taking p = 2, β = −2 (n> 2) and α ! zα f(z,w)= g(z)= | | , |α| s α! 2 α Nn−2 (n 2) (1 + α ) ln (1 + α ) ∈X − | | | | Cn 2 C2 1 where (z,w) − and 2 2 2 (1 w ) n−1 g z 2 C I 2 g w 2dλ w , ( ) − | | 2 ( + ) ( ) ( ) | | ≤ D 1 zw | R | Z | − | n 2 where D is now a unit ball of C − . This also implies that n−1 g(z) 2 C (I + ) 2 g(w) 2dλ(w). | | ≤ D | R | Z A straightforward computation using Taylor expansion finally shows that the function g must satisfy 1 g(z) 2 C . | | ≤ 2s m (1 + m) ln (1 + m) X 12 EDGAR TCHOUNDJA 1 This means that g must be bounded since 2 βp p p (19) sup (I + ) f (rξ) dσ(ξ) C f Hp r<1 S R ≤ k k β Z holds, for all f Hp . In those cases as before, we apply the same reproducing ∈ β formula to (I + )βp f p(z) to obtain R βp p 2 m 1 βp p (I + ) f (w) (1 w ) − (I + ) f (z) = Cn,m R n+−m | | dλ(w). R D (1 zw) Z − Hence by Lemma (3.1) part ii), we will have βp m 1 (I + ) f p(w) (1 w 2) − f z p C dλ w , ( ) R ||n+m −βp | | ( ) | | ≤ D 1 zw Z | − | − which is (17). Thus (14) will still characterize Carleson measures on such cases. However, we don’t know the cases for which (19) is true apart from the trivial case p = 1. Therefore, if we introduce the space p (D) as the space of all holomorphic Hb,β functions f in D such that (when terms involved make sense) p βp p f p = sup (I + ) f (rξ) dσ(ξ), k kHb,β r<1 S R Z Hp D then for the analogue question of Carleson measures for b,β( ), with our approach, we obtain the following theorem. Theorem 4.1. Let µ be a positive Borel measure on D. Suppose that m = n βp > 0. Then the following are equivalent: − p i) µ is a Carleson measure for Hb,β(D) m ii) There exists a constant C such that µ (Qδ(ξ)) Cδ for any ξ S and δ > 0. ≤ ∈ We note that our approach easily gives the following closed, computable and sufficient condition. CARLESON MEASURES FOR HARDY SOBOLEV SPACES 13 Proposition 4.2. Let µ be a positive Borel measure on D and assume that m = n βp > 0. If for some ǫ> 0, the measure µ satisfies − m+ǫ µ (Qδ(ξ)) Cδ ≤ for any ξ S and δ > 0, then µ is a Carleson measure for Hp . ∈ β Proof: In this case, we simply need to prove that m 1+ǫ (1 w 2) − f z p C I β f w pdλ w . ( ) − | | 2m+2ǫ ( + ) ( ) ( ) | | ≤ D 1 zw | R | Z | − | It is enough to we just use the same reproducing formula once more and H¨older’s inequality. 2 Finally, the following result shows that when p > 1, the necessary condition (14) is no longer sufficient. This means that we can not expect to remove ǫ in the n 1 previous proposition. This result is proved here for p = 2 and β = −2 . However, it can be extended to other cases in the same manner. n 1 Theorem 4.3. Suppose p = 2 and β = −2 (n> 1). There exists a finite positive measure µ in D such that µ satisfies the growth condition (14), but µ is not a 2 Carleson measure for H n−1 . 2 2 Before giving the proof of this theorem, let us note that the space H n−1 is 2 special. It is now known in the literature as the Arveson Hardy space. Arveson [3] has studied extensively this space in connection with applications in multivariable operator theory. We add a change on notations here to emphasize the dimension of the space we are dealing with. So in this proof, we will denote the unit ball of Cn by n Dn = z C : z < 1 , { ∈ | | } and the Carleson boxes in Dn by n Q (ξ)= z Dn : 1 z.ξ < δ . δ ∈ | − | Proof of Theorem 4.3: We will adapt the idea used in [1, p 34] to prove an analogue problem for exceptional sets. First assume n = 2 and consider functions of the form f(z,w)= g(2zw), where g is holomorphic in the unit disk D1. By means of Taylor expansion and Stirling’s 14 EDGAR TCHOUNDJA 2 2 formula, one can show that f H 1 (D2) if and only if g H 1 (D1). Carleson ∈ 2 ∈ 4 2 measures for H 1 (D1) have been characterized by Stegenga using a capacity condi- 4 tion; see [9, Theorem 2.3]. The proof of this uses the fact that Carleson measures 2 for H 1 (D1) coincide with Carleson measures for the space of Poisson transforms 4 of Bessel potentials of L2 functions. If we now use Proposition 2.1 in [5] by ap- plying it to the function ϕ(x)= √x, we conclude that there exists a finite Borel 2 measure ν in D1∗ (= D1 0 ) and a function g H 1 (D1) such that \{ } ∈ 4 1 ν Q1(ξ) Cδ 2 , δ ≤ and g(z) 2dν = . D | | ∞ Z 1 Let T denote the unit circle (T = ∂D1) with its arclength measure dθ. Define Ψ : D∗ T D by 1 × → 2 2 is iθ √2 iθ √2 i(s θ) Ψ(r e ,e )= r e , r e − . 2 2 ! √2 iθ √2 i(s θ) If we set D = r e , r e − D : 0 f(z,w) 2dµ = f(z,w) 2dµ D | | D | | Z 2 Z = e g(r2eis 2dν dθ D∗ T | | × Z 1× = 2π g(z) 2dν = . D | | ∞ Z 1 The theorem will be proved in this case if we show that µ satisfies (14). We proceed to prove this. It is enough to prove (14) for ξ ∂D. Moreover, by √2 √∈2 invariance under rotation, we may assume that ξ = 2 , 2 . It is then enough e CARLESON MEASURES FOR HARDY SOBOLEV SPACES 15 to show that iθ is e e iθ (20) 1 r r e− < δ − 2 − 2 is iθ 1 implies that 1 r2e < cδ and 1 e < Cδ 2 . First observe that (20) implies that 1 r <| δ.− On the| other hand,| − (20)| is the same as − s r s s i i(θ 2 ) i( 2 θ) e− 2 e − + e − < δ, − 2 that is s 2 s s 2 sin + cos r cos θ < δ2. 2 2 − 2 − Thus s s s sin < δ and cos r cos θ < δ. 2 2 − 2 − Since we may also assume δ small enough, the first of these inequalities implies that s< √2δ; it also implies that 1 r2eis = 1 r2 + r2 1 eis | − | − − is < 2δ + 1 e | − s | = 2δ + 2sin < 4δ. 2 The second inequality implies that s s r 1 cos θ < δ + r cos − 2 − − 2 s < δ + δ + 1 cos < 3δ. − 2 Consequently, since δ small implies r large, we get s s θ 1 cos θ = 2 sin2 < 6δ. − 2 − 4 − 2 s 1 1 So θ < Cδ 2 and then θ iθ θ 1 1 e = 2sin < Cδ 2 . | − | 2 This finishes the case n = 2. The case n > 2 follows in the following way. We identify the following subset D2 = z Dn : zj = 0, j 3 of the unit ball of n 2 { ∈ ≥ } C with the unit ball D2 of C . We then consider the measure µ in Dn obtained by transporting the measure fµ in D2 we have just constructed. The extension of this measure to Dn satisfies the growth condition (14). Fore the functions 16 EDGAR TCHOUNDJA 2 2 f(z) = F (z1, z2) we have that, f H n−1 (Dn) if and only if F H 1 (D2). We ∈ 2 ∈ 2 2 then obtain from the previous construction a function f H n−1 (Dn) such that ∈ 2 2 D f dµ = . This finishes the proof of the theorem. n | | ∞ R References e [1] P.Ahern: Exceptional sets for holomorphic sobolev functions. Michigan Math.J. 35 (1988), pp 29-41. [2] M. Andersson: Topic in complex analysis, Springer-Verlag New York, (1997). [3] W. Arveson: Subalgebras of ∗ algebras III; multivariable operator theory. Acta Math. 181 (1998), p 159-228 C − [4] B. Berndtsson: ∂b and Carleson type inequalities. Complex analysis, II (College Park, Md 1985-86), 42-54, Lecture Notes in Math, 1276, Springer, Berlin, 1987. [5] C. Cascante, J. M. Ortega : Carleson measures on spaces of Hardy-sobolev type, Can. J. Math., Vol 47 (6) 1995, pp 1177 - 1200. [6] W.S. Cohn, I.E. Verbitsky: On the trace inequalities for Hardy- Sobolev functions in the Unit Ball of n. Indiana Univ. Math. J. Vol 43. n◦4 (1994), pp 1079-1097. [7] L. H¨ormander:C Lp-estimates for (pluri)subharmonic functions, Math. Scand. 20 (1967,) p 65 - 78. [8] W.Rudin: Function theory on the Unit Ball of n, Springer-verlag, New-York, 1980. [9] D. A. Stegenga: Multipliers of the Dirichlet space,C Illinois J. Math. 24 (1980), pp 113-139. [10] K.Zhu: Spaces of holomorphic functions in the Unit Ball, Springer-Verlag, new-York, 2004. Department of Mathematical Sciences, Division of Mathematics, Chalmers University of Technology and Goteborg¨ University, SE-412 96 Goteborg,¨ Swe- den E-mail address: [email protected] CARLESON MEASURES FOR THE GENERALIZED BERGMAN SPACES VIA A T(1)-THEOREM TYPE EDGAR TCHOUNDJA Abstract. In this paper, we give a new characterization of Carleson mea- sures on the Generalized Bergman spaces. We show first that this problem is equivalent to a T(1)-Theorem problem type. Using Verdera idea (see [V]), we introduce a sort of curvature in the unit ball adapted to our kernel and we establish a good λ inequality which then yields to the solution of this T(1)- Theorem problem. Contents 1. Introduction 2 2. Preliminary results 5 2.1. Definition and Properties of a space of homogeneous type 5 2.2. Properties of the pseudo distance d 9 2 3. Bergman-Sobolev type space Aα. 15 4. Proof of the equivalence (b) (c) of Theorem 1.1 20 4.1. Proof of 1) 3) ⇔ 21 4.2. Related maximal⇒ functions 21 4.3. Curvature in the unit ball 27 4.4. A good λ inequality 30 4.5. Proof of the implication 3) 2) of Theorem 4.1 33 4.6. Proof of the implication 2) ⇒ 1) in Theorem 4.1 34 5. Comments ⇒ 34 References 34 2000 Mathematics Subject Classification. 32A37, 32A55, 46E35. Key words and phrases. Carleson measures, Generalized Bergman spaces, Arveson’s Hardy space, T(1)-Theorem. 1 2 EDGAR TCHOUNDJA 1. Introduction Let n be a positive integer and let Cn = C C ×···× denote the n dimensional complex Euclidean space. n For z =(z , , zn) and w =(w , ,wn) in C , we write 1 ··· 1 ··· zw = z,w = z w + + znwn h i 1 1 ··· and 2 2 z = z1 + + zn . n | | | | ··· | | The open unit ball in C is thep set D = z Cn : z < 1 . { ∈ | | } We use H (D) to denote the space of all holomorphic functions in D. Let S = ∂D be the boundary of D. For α R,α> n 1, we define the Generalized 2 ∈ − − Bergman spaces Aα [ZZ] to consist of all holomorphic functions f in the unit ball D with the property that Γ(n +1+ α)m! f 2 = c(m) 2 < , k kα | | Γ(n +1+ m + α) ∞ m Nn X∈ | | where f(z)= c(m)zm is the Taylor expansion of f. m∈Nn For β R, weP define the fractional radial derivative of order β by ∈ (I + )β f(z) := (1 + m )β c(m)zm. R m | | X One then easily observes, by means of Taylor expansion and Stirling’s formula, that 2 m 2 2 2m+α (1) f α = (I + ) f(z) (1 z ) dλ(z) k k ∼ D | R | − | | Z where 2m+α> 1. One also observes that the right hand side of (1) is indepen- − 2 σ dent of the choice of m. If we let 2σ = α+n+1 then we see by (1) that Aα = B2 , σ where B2 is the analytic Besov-Sobolev spaces defined in [ARS]. Thus this scale 2 of spaces includes the Drury-Arveson Hardy space A−n, the usual Hardy space 2 2 H (D)= A−1 and the weighted Bergman spaces when α> 1. An interesting question about these spaces is to find their− Carleson measures, that is CARLESONMEASURESANDT(1)-THEOREM 3 characterize positive measures µ on D such that 2 2 2 (2) f dµ C(µ) f α, f Aα. D | | ≤ k k ∈ Z 2 (A measure µ which satisfies (2) is called a Carleson measure for Aα or simply 2 an Aα Carleson measure.) 2 Viewing the space Aα as defined by the relation (1), we see that the literature is now rich with solutions of this question for various values of α. Indeed the first case of interest was the case α = 1 (the usual Hardy space). In [CA], Carleson gave the result when n = 1 and− later in 1967, H¨ormander [H] gave a solution for n > 1. Stegenga [ST] (when n = 1), Cima and Wogen [CW] (when n > 1) characterized Carleson measures for α> 1. The range α ( n 1, 1) was unsolved− until 2006, when Arcozzi, Rochberg and Sawyer [ARS],∈ − obtained− − results for the range α ( n 1, n]. Their results use certain tree conditions and seem difficult to handle.∈ − − − The purpose of this note is to present an alternative characterization of Car- leson measures in the same range α ( n 1, n] as in [ARS]. We obtain a result which seems simple in the sense∈ of− applications.− − To obtain our characterization, we show first that this problem is equivalent to a kind of T(1)-Theorem problem associated with a Calder´on-Zygmund type kernel and then we solve the T(1)-Theorem problem type which occurs. To be precise, recall that for a topological space X with a pseudo distance d, a kernel K(x, y) is called an n Calder´on-Zygmund kernel (or simply a Calder´on-Zygmund kernel) with respect to the pseudo distance d if a) K(x, y) C1 , and | | ≤ d(x,y)n b) There exists 0 < δ 1 such that ≤ d(x, x′)δ K(x, y) K(x′, y) + K(y, x) K(y, x′) C | − | | − | ≤ 2 d(x, y)n+δ if d(x, x′) C d(x, y), x, x′, y X. ≤ 3 ∈ Given a Calder´on-Zygmund kernel K, we can define (at least formally) a Calder´on-Zygmund operator (CZO) associated with this kernel by T f(x)= K(x, y)f(y)dµ(y). ZX One important question in the Calder´on-Zygmund theory is to find a criterion for boundedness of a CZO in L2(µ). (We will call such a problem a T(1)-Theorem problem) 4 EDGAR TCHOUNDJA Many authors studied this problem. When X = Rm, µ = dx (the usual Lebesgue measure) and d is the Euclidean distance, a famous criterion called ”T(1)-Theorem” was obtained by Journ´eand G. David [DJ]. This criterion states that a CZO is bounded in L2(dµ) if and only if it is weakly bounded (in some sense), and the operator and its adjoint send the function 1 in BMO. This result was extended to space of homogeneous type in an unpublished work by R. Coif- man. Later it was an interesting question to extend this T(1)-Theorem in the case where the space is not of homogeneous type (This essentially means that the measure µ does not satisfy the doubling condition). Several authors such as Tolsa, Nazarov, Treil, Volberg and Verdera [T1, NTV, V] treated this situation in the setting of Rm with the Euclidean distance. One good example of such an operator is the Cauchy integral operator. We say that the Cauchy integral operator is bounded in L2(dµ) whenever for some positive constant C, one has for every ǫ> 0, 2 2 2 ǫ(fµ) dµ C f dµ, f L (dµ), |C | ≤ | | ∈ Z Z where f(ζ) ǫ(fµ)(z)= dµ(ζ), z C. C ζ z ∈ Z|ζ−z|>ǫ − Their result is that the Cauchy integral operator is bounded in L2(dµ) if and only if i) µ(D) Cr(D), for each disc D with radius r(D); ≤ 2 ii) ǫ(χDµ) dµ Cµ(D), for each disc D, ǫ> 0. D |C | ≤ We now return back to the problem (2). We consider the kernel K defined by R α 1 Kα(z,w)= . ℜ (1 zw)n+1+α − For a positive Borel measure µ in D, we consider the operator Tα associated with this kernel defined by Tαf(z)= f(w)Kα(z,w)dµ(w), z D. D ∈ Z We will prove that if on D we consider as in [B] the pseudo distance d defined by z w d(z,w)= z w + 1 , || | − | || − z w | | | | K n α D the kernel α is an ( +1+ ) Calder´on-Zygmund kernel in the unit ball with respect to the pseudo distance d. Let B = B (z, r) = w D; d(z,w) < r be { ∈ } CARLESONMEASURESANDT(1)-THEOREM 5 a ”pseudo ball” or simply a ball of center z and radius r. We are now ready to state our result. Theorem 1.1 (Main theorem). Suppose α ] n 1, n] and µ be a positive Borel measure in D. Then the following conditions∈ − are− equivalent.− 2 a) µ is a Carleson measure for Aα ; 2 b) Tα is bounded in L (µ); c) There exists a constant C such that i) µ(B(z, r)) Crn+1+α, ≤2 ii) B Tα(χB) dµ Cµ(B), for| each ball| B ≤= B(z, r) which touches the boundary of D. R This theorem is a T(1)-Theorem type result with respect to the CZO Tα (b c) and it shows the equivalency of this T(1)-Theorem problem with Carleson⇔ 2 measures for Aα (a b). Observe that the equivalency a b is proved in [ARS, Lemma 24, p 42] in⇔ a more general situation. Nevertheless⇔ we have included the proof here for the sake of completeness. Thus to prove Theorem 1.1 , we will essentially prove the hard part b c. To prove the hard part, we will adapt to the unit ball the idea used by J.⇔ Verdera [V] to give an alternative proof of the T(1)-Theorem for the Cauchy integral operator. The paper is organized as follows. In section 2 we gather some preliminaries including a key covering lemma, terminology and background. Section 3 is de- 2 voted to the study of the generalized Bergman spaces Aα and the proof of a b. Section 4 contains the proof of the hard part of the main theorem. Section⇔ 5 deals with some extensions, comments and opens questions. 2. Preliminary results We collect in this section few results which will be useful to our purpose. These concern results on general homogeneous spaces and results for the special case of the unit ball D. 2.1. Definition and Properties of a space of homogeneous type. Definition 2.1. A pseudo distance on a set X is a map ρ from X X to R+ such that × 1) ρ(x, y) = 0 x = y 2) ρ(x, y)= ρ(⇔y, x) 3) there exists a positive constant K (K 1) such that, for all x, y, z X ≥ ∈ ρ(x, y) K(ρ(x, z)+ ρ(z, y)). (Quasi triangular inequality) ≤ 6 EDGAR TCHOUNDJA For x X and r > 0, the set B(z, r) = y X : ρ(x, y) < r is called a pseudo ball∈ or simply a ball of center x and radius{ ∈ r. } Definition 2.2. A space of homogeneous type is a topological space X with a pseudo distance ρ and a positive Borel measure µ on X such that: 1) The balls B(x, r) form a basis of open neighborhoods of x. 2) (Doubling property) There exists a constant A > 0 such that, for all x X and r> 0, we have ∈ 0 <µ(B(x, 2r)) Aµ(B(x, r)) < . ≤ ∞ The triplet (X,ρ,µ) is called a space of homogeneous type or simply a homo- geneous space. We will often abuse by calling X a homogeneous space instead of (X,ρ,µ). Homogeneous spaces have been treated by several authors such as Coifman and Weiss [CWe], Stein [S]. We refer to them for further details. We will use the following lemma to prove a key type of covering lemma, Lemma 2.4 below. It will be crucial in our argument later. Lemma 2.3. There exists a constant C1 such that if B(x1, r1) and B(x2, r2) are two non disjoint balls and if r r then 1 ≤ 2 B(x , r ) B(x ,C r ). 1 1 ⊂ 2 1 2 Proof: Let y B(x , r ) B(x , r ). We have for x B(x , r ) ∈ 1 1 ∩ 2 2 ∈ 1 1 ρ(x, x ) K(ρ(x, y)+ ρ(y, x )) 2 ≤ 2 K(K(ρ(x, x )+ ρ(x , y)) + ρ(y, x )) ≤ 1 1 2 < K(2Kr1 + r2) < K(2K + 1)r2. We obtain the desired result if we set C1 = K(2K + 1). 2 Lemma 2.4. Let (X,d,µ) be an homogeneous space. There exists positive con- stants K1,K2,K3 with K3 >K2 >K(C1 + 1)K1 such that: for an open set O of X (O ( X), there exists a collection of balls Bk := B(xk,ρk) ∗ ∗∗ ∗∗∗ so that, if Bk = B(xk,K1ρk), Bk = B(xk,K2ρk) and Bk = B(xk,K3ρk), a) the balls Bk are pairwise disjoint ∗ b) O = Bk ∪k ∗∗ c) O = Bk ∪k d) for each k, B∗∗∗ Oc = k ∩ 6 ∅ CARLESONMEASURESANDT(1)-THEOREM 7 e) a point x O belongs to at most M balls B∗∗ (bounded overlap property). ∈ k Moreover, the constant M depends only on constants K1,K2,A and K. 1 Proof: Let O be an open set of X (O ( X). Let ǫ = 2 2 where C1 16K C1 (1+C1) is the constant defined in Lemma 2.3. Consider the covering of O by the balls B(x, ǫδ(x)) where δ(x)= d(x, Oc), x O. We have d(x, Oc) > 0 since Oc is a∈ closed set. d(x, Oc) > 0 is finite. We now select a maximal disjoint subcollection of B(x, ǫδ(x)) x O; for this subcollection . { } ∈ B1, B2,...,Bk,... with Bk = B(xk, ǫδ(xk)) = B(xk,ρk), we shall prove assertions a), b), c), d), and e) above. We set 1 1 2 K , K and K . 1 = 2 2 = 3 = 4K (C1 + 1)ǫ 2ǫK ǫ Observe that our choice makes these constants satisfy our hypothesis. Observe that a) and d) hold automatically by our choice of Bk. It is also clear that ∗ δ(xk) δ(xk) ∗∗ B = B x , 2 B x , = B O; k k 4K (C1+1) k 2K k ⊂ ∗ ⊂ what remains to be shown is that O Bk (in this case b) and c) will be valid) ⊂∪k and that e) is true. ∗ Let us prove that O Bk. ⊂∪k Let x O; by the maximality of the collection Bk, ∈ B(xk, ǫδ(xk)) B(x, ǫδ(x)) = for some k. ∩ 6 ∅ δ(x) 1 We claim that δ(xk) . If not, since ǫ< < 1, we have ≥ 4C1 2C1 δ(x) B(xk, 2δ(xk)) B x, = . ∩ 2C 6 ∅ 1 δ(x) δ(x) Since 2δ(xk) < , by the Lemma 2.3, B(xk, 2δ(xk)) B x, , which gives 2C1 ⊂ 2 c δ(x) a contradiction since B(xk, 2δ(xk)) meets O , while B x, O. Using the 2 ⊂ fact that 4C1ǫδ(xk) ǫδ(x), Lemma 2.3 gives ≥ 2 ∗ x B(x, ǫδ(x)) B(xk, 4ǫC δ(xk)) = B . ∈ ⊂ 1 k This proves b) and c). We proceed to prove e). M M ∗∗ Assume that x Bk = B(xk,K2ρk). We have ∈ k∩=1 k∩=1 c c c d(xk,O ) K(d(x, O )+ d(x, xk)) K(d(x, O )+ K ρk); ≤ ≤ 2 8 EDGAR TCHOUNDJA c 1 c this implies that d(x, O ) (d(xk,O ) KK ρk). But ≥ K − 2 d(x ,Oc) KK ρ = KK ǫd(x ,Oc)= k 2 k 2 k 2 c c d(xk,O ) ρk c and thus d(x, O ) 2K = 2Kǫ . Hence ρk 2Kǫd(x, O ). On the other hand,≥ ≤ c c d(x, O ) K(d(xk,O )+ d(x, xk)) ≤ ρk −1 K K ρk + = K(K + ǫ )ρk. ≤ 2 ǫ 2 ∗∗ So if x B , the radius ρk satisfies ∈ k d(x, Oc) ρ Kǫd x, Oc . −1 k 2 ( ) K(K2 + ǫ ) ≤ ≤ From this, we have c B(xk,ρk) B(x, C d(x, O )) ⊂ 2 2 where C2 = 2K (K2 + 1)ǫ. We also have, for each k c B(x, C d(x, O )) B(xk,C ρk) 2 ⊂ 3 −1 where C3 = K(C2K(K2 + ǫ )+ K2). Thus c B(xk,ρk) B(x, C2d(x, O )) ∪k ⊂ and c B(x, C d(x, O )) B(xk,C ρk) for each k. 2 ⊂ 3 Therefore, by the doubling property and the disjointness of Bk, we have M c µ(B(xk,ρk)) µ(B(x, C d(x, O ))) ≤ 2 k X=1 µ(B(xk,C ρk)) ≤ 3 Cµ(B(xk,ρk)). ≤ Thus M C and we are done. ≤ 2 One key result in the real variable theory is that, by means of Besicovitch cov- ering Lemma, the usual central Hardy-Littlewood maximal function is bounded in Lp(Rm,dµ) (1 [KR], we will obtain the same Lp estimates for a certain ”contractive” central Hardy-Littlewood maximal function to be defined later, via the following lemma Lemma 2.5 (ǫ-Besicovitch). Let (X,d,µ) be an homogeneous space. Let E be a bounded set, fix a positive number M and denote by the family of balls B(a, r) with center a E and radius r M. Then there existsF a countable subfamily ∞ ∈ ≤ B(ak, rk) k=1 of with the following properties. { } ∞ F i) E B(ak, rk) ⊂ k∪=1 rk ∞ 7K ii) the family B(ak, α ) k=1 is disjoint, where α = 3 , and K is the constant in the quasi{ triangle} inequality for d; ∞ iii) for all 0 <ǫ< 1, the family ǫ = B(ak, (1 ǫ)rk) k=1 has bounded overlaps, namely F { − } ∞ 1 χ (x) C log , B(ak,(1−ǫ)rk) ≤ ǫ k X=1 where C depends only on constants of X and χA denotes the characteristic function of the set A. Proof: See [FGL, Lemma 3.1]. 2 We now turn our attention to the special domain of interest, the unit ball D = z Cn : z < 1 . { ∈ | | } In [B] it is defined a map d on D D by × z w + 1 z w if z,w D∗, d(z,w)= || | − | || − |z| |w| ∈ ( z + w otherwise , | | | | where D∗ = D 0 . \{ } 2.2. Properties of the pseudo distance d. Lemma 2.6. The following assertions hold: i) d is a pseudo distance on D ii) d is invariant under rotation. Proof: 1 Assertion i) will follow essentially from the fact that the map (ξ, ζ) 1 ξζ 2 is a distance in ∂D (see [R, Proposition 5.1.2] ). 7−→ | − | 10 EDGAR TCHOUNDJA Indeed, the first property of a pseudo distance follows easily from this assump- tion; the second property is obvious. For the third property, let z,w and ζ D. We have ∈ w z d(z,w) = z w + 1 || | − | || − w z | | | | w ζ ζ z z ζ + ζ w + 2 1 + 1 ≤ || | − | || || | − | || − w ζ − ζ z | | | | | | | | (by the triangular inequality for the distance in the boundar y). So d(z,w) 2(d(z, ζ)+ d(ζ,w)). ≤ Assertion ii) follows from the fact that the inner product zw is invariant under rotation. 2 The pseudo balls associated with this pseudo distance satisfy this important observation. Lemma 2.7. The pseudo ball B(z, r) = w D : d(z,w) < r touches the boundary of D if and only if r> 1 z . { ∈ } − | | Proof: Fix a pseudo ball B(z, r). Let ǫ = r (1 z ). Since we are interested in − −| | points which touch the boundary, we have to find conditions on points w B(z, r) ∈ such that w > z . For such w, we have d(z,w)= w z + 1 z w . So | | | | | | − | | − |z| |w| z w (3) d(z,w) < r 1 < ǫ + 1 z w + z = ǫ + 1 w . ⇔ − z w − | | − | | | | − | | | | | | From this we have our result. In fact, (3) shows that B(z, r) touches the boundary of D if and only if ǫ> 0. 2 These pseudo balls have close relations with the so called Kor´anyi balls. Pre- cisely, for ξ ∂D = S and δ > 0, if we set ∈ Qδ(ξ) := Q(ξ, δ)= z D : 1 zξ < δ , { ∈ | − | } then we have the following proposition. CARLESONMEASURESANDT(1)-THEOREM 11 Proposition 2.8. There exists positive constants a1 and a2 such that, for every pseudo ball B(z, r) which touches the boundary of D, z Q , r B(z,a r), z ⊂ 1 | | and z B(z, r) Q ,a r . ⊂ z 2 | | Proof: Let B(z, r) be a pseudo ball which touches the boundary of D. Let w B(z, r). Since z w < r we have for w z , ∈ || | − | || | | ≤ | | z w < r 1 w < 2r | | − | | ⇒ − | | and for w > z | | | | 1 w < 1 z < r. − | | − | | So if w B(z, r), 1 w < 2r. Therefore for w B(z, r), ∈ − | | ∈ z z 1 w = 1 w + w w − z − | | | | − z | | | | w z 1 w + w 1 ≤ − | | | | − w z | | | | w z < 2r + 1 < 3r. − w z | | | | So B(z, r) Q z , 3r . Take a = 3. ⊂ |z| 2 For the first inclusion, we can suppose that 0 If z w , then w z < 1 z < r. So Q z , r B(z, 5r). Take a = 6. | | ≤ | | | | − | | − | | |z| ⊂ 1 2 2 α For α > 1, let dλα(z) = (1 z ) dλ(z) where dλ(z) is the usual Lebesgue measure of −Cn R2n. We then− have | | the following result. ∼ Lemma 2.9. For each fixed α > 1, the triplet (D,d,dλα) is an homogeneous space. − Proof: Since d is already a pseudo distance on D, we need only to prove that dλα is a doubling measure. One can prove that for 0 Remark 2.10. This lemma shows that we can apply Lemmas 2.4 and 2.5 in the unit ball D. We will make use of these others properties of d. Lemma 2.11. For every z D and r0, 0 < r0 < 1, if we denote by z =(r , 0, , 0) we have ∈ 0 0 ··· 1) 1 z r 1 d(z, z ) | − 1 0| ≥ 3 0 2) z1 r0 d(z, z0) | n − | ≤ 2 3) zk 2d(z, z0) k=2| | ≤ 2 4) P1 zz0 1 r + d(z, z0). | − | ≤ − 0 Proof: 1) We have z z 1 1 1 z r + z r 1 − z ≤ | − 1 0| 1 0 − z | | | | 1 z r + 1 r z ≤ | − 1 0| − 0| | 2 1 z r . ≤ | − 1 0| If z r then | | ≤ 0 z r = r z 1 z 1 z r . || | − 0| 0 − | | ≤ − | 1| ≤ | − 1 0| If z > r0 then | | z r = z r 1 r 1 z r . || | − 0| | | − 0 ≤ − 0 ≤ | − 1 0| CARLESONMEASURESANDT(1)-THEOREM 13 Hence d(z, z )= 1 z1 + z r 3 1 z r . 0 − |z| || | − 0| ≤ | − 1 0| 2) We have z r z z + z r | 1 − 0| ≤ | 1 − | || || | − 0| z z 1 1 + z r d(z, z ). ≤ | | − z || | − 0| ≤ 0 | | 3) We have n 2 2 2 zk = z z 2 z z | | | | − | 1| ≤ || | − | 1|| k X=2 z 2 z z 2 1 1 2d(z, z ). ≤ | 1 − | || ≤ − z ≤ 0 | | 4) We have 2 2 1 zz0 = 1 z1r0 1 r0 + r0 z1r0 | − | | − | ≤ − 2 | − | 2 1 r0 + r0 z1 1 r0 + d(z, z0). ≤ − | − | ≤2) − 2 For α> n 1 fixed, set k = n +1+ α. We consider the following kernel − − 1 Kα(z,w)= ℜ (1 zw)k − and obtain the following this important result about this kernel. Proposition 2.12. 1) There exists a constant C such that for all z,w D, 3 ∈ C3 Kα(z,w) . | | ≤ d(z,w)k 2) There exists two constants C ,C such that for all z,w,ζ D satisfying 1 2 ∈ d(z, ζ) >C1d(w, ζ), we have 1 d(w, ζ) 2 Kα(z,w) Kα(z, ζ) C 1 . 2 k | − | ≤ d(z, ζ) + 2 14 EDGAR TCHOUNDJA Proof: Assertion 1) follows from Lemma 2.11 and the invariance under rotation. In- deed for z,w D, if ∆ is the rotation such that ∆(w)=( w , 0, , 0) then we have ∈ | | ··· (5) d(z,w)= d(∆(z), ∆(w)) 3 1 ∆(z)∆(w) 3 1 zw . ≤ | − | ≤ | − | k k k k So d(z,w) 3 1 zw . Take C3 = 3 . Let us prove≤ | 2).− By| the invariance under rotation, we can suppose ζ = (r , 0, , 0). We use the identity 0 ··· 1 kz(w ζ) Kα(z,w) Kα(z, ζ)= − dt k+1 − 0 ℜ (1 zw tz(ζ w)) Z − − − to obtain 1 k z(w ζ) (6) Kα(z,w) Kα(z, ζ) | − | dt. k+1 | − | ≤ 0 1 zw tz(ζ w) Z | − − − | We have 1 1 n 2 n 2 2 2 z(w ζ) z (w r ) + zk wk | − | ≤ | 1 1 − 0 | | | | | k=2 ! k=2 ! X 1 X 1 n 2 n 2 2 2 w r + zk wk . ≤ | 1 − 0| | | | | k ! k ! X=2 X=2 So by Lemma 2.11 and (5), we have 1 1 1 z(w ζ) 2d(w, ζ) 2 d(w, ζ) 2 + d(z, ζ) 2 | − | ≤ 4 1 1 d(w, ζ) 2 d(z, ζ) 2 ≤ √C1 C 1 1 (7) d(w, ζ) 2 1 zζ 2 . ≤ √C1 | − | 1 This shows that for C1 large enough, we have z(w ζ) 2 1 zζ . On the other hand, observe that | − | ≤ | − | 1 zw tz(ζ w) = 1 zη | − − − | | − | where η = (1 t)w + tζ. Since − (1 zζ) (1 zη) = z(η ζ) | − − − | | − | CARLESONMEASURESANDT(1)-THEOREM 15 and (1 zζ) (1 zη) = (1 t) z(w ζ) z(w ζ) , | − − − | − 1 | − | ≤ | − | we conclude that for large C1, 1 zη > 2 1 zζ . Therefore, from (6), (7) and| (5),− we| have| − | 1 1 2 2 k+1 C 1 zζ d(w, ζ) Kα(z,w) Kα(z, ζ) 2 k | − | k+1 | − | ≤ √C1 1 zζ 1 | − | d(w, ζ) 2 C . 2 k 1 ≤ d(z, ζ) + 2 2 Remarks 1. This Proposition shows that the kernel Kα is a k Calder´on-Zygmund kernel with respect to the pseudo distance d. We can observe from the proof of Proposition 2.12 that the assertions are still 1 true if we replace the kernel Kα by the kernel (1−zw)k . 2 3. Bergman-Sobolev type space Aα. 2 In this section, we define the space Aα. We give some properties of this space. Finally we show that the Carleson measures problem for these space is equivalent to the T(1)-Theorem problem associated with the Calder´on-Zygmund kernel Kα. 2 Definition 3.1. Let α R,α > n 1. We denote by Aα the space of all holomorphic functions f ∈in the unit− ball−D with the property that Γ(n +1+ α)m! f 2 = c(m) 2 < , k kα | | Γ(n +1+ m + α) ∞ m∈N n X | | where f(z)= c(m)zm is the Taylor expansion of f. m∈Nn P 2 Theorem 3.2. The space Aα is equipped with an inner product such that the associated reproducing kernel is given by 1 B (z,w)= . α (1 zw)n+1+α − Proof: For f(z)= c(m)zm and g(z)= d(m)zm, define the product by m m P P Γ(n +1+ α)m! f,g = c(m)d(m) . h iα Γ(n +1+ m + α) m X | | 16 EDGAR TCHOUNDJA 2 2 m This clearly defines an inner product in Aα. Let f Aα with f(z)= c(m)z . ∈ m Since P Γ(n +1+ m + α) B (z,w)= | | zmwm, α Γ(n +1+ α)m! m X we have for w D, ∈ Γ(n +1+ m + α) m Γ(n +1+ α)m! f, Bα(., w) = c(m) | | w h iα Γ(n +1+ α)m! Γ(n +1+ m + α) m X | | = c(m)wm = f(w). m X We are done. 2 2 Remark 3.3. The space A is a Hilbert space with the Hilbert norm . α. α k k 1 Proposition 3.4. Let w Bn and set f(z) = s . If 2s > n +1+ α then ∈ (1−zw) f A2 . Moreover, ∈ α 1 f 2 = . k kα ∼ (1 w 2)2s−n−1−α − | | Proof: It is enough to verify the last assertion of the proposition. Let 1 Γ( m + s) f(z)= = | | zmwm. (1 zw)s Γ(s)m! m∈N n − X CARLESONMEASURESANDT(1)-THEOREM 17 we have by Stirling’s formula, Γ2( m + s) Γ(n +1+ α)m! f 2 = | | wm 2 k kα ∼ Γ2(s)(m!)2 | | Γ(n +1+ m + α) m X | | Γ2(k + s)Γ(n +1+ α) Γ(k + 1) = wm 2 Γ2(s)Γ(n +1+ k + α)Γ(k + 1) m! | | k X |mX|=k Γ2(k + s)Γ(n +1+ α) = w 2k Γ2(s)Γ(n +1+ k + α)Γ(k + 1)| | k X = (1 + k)2s−2−α−n w 2k ∼ | | k X Γ(2s n 1 α + k) = − − − w 2k ∼ Γ(2s n α 1)Γ(k + 1)| | k X − − − 1 = . (1 w 2)2s−n−1−α − | | 2 For β R, we define the fractional radial derivative of order β by ∈ (I + )β f(z) := (1 + m )β c(m)zm, R | | m X where f(z)= c(m)zm is the Taylor expansion of f. m∈Nn The followingP lemma follows by the use of Taylor’s expansion and Stirling’s formula. Lemma 3.5. 2 m 2 2 2m+α (8) f α = (I + ) f(z) (1 z ) dλ(z) k k ∼ D | R | − | | Z where 2m + α> 1. − We also observe that the right hand side of (8) is independent of the choice of m. Remarks 2. p The space Aα are introduced in [ZZ] for general values of p, we refer there • for further details about this space. 18 EDGAR TCHOUNDJA For α = 1, the space A2 is the usual Hardy space • − α 2 2 2 H (D)= f H(D) : f H2(D) = sup f(rξ) dσ(ξ) < . ∈ k k r< S | | ∞ 1 Z 2 For α > 1, the space Aα is the usual weighted Bergman space. When • −2 α = n, Aα is the so called Arveson’s Hardy space; for other values of α, we obtain− analytic Besov-Sobolev spaces. We recall that we want to characterize positive Borel measures µ on D such that 2 2 2 (9) f dµ C(µ) f α, f Aα. D | | ≤ k k ∈ Z 2 (A measure µ which satisfies (9) is called a Carleson measure for Aα or simply 2 an Aα Carleson measure.) As we have mentioned in the introduction the solution of this question is well known for α 1. The result from these cases is the following theorem. ≥ − Theorem 3.6 (Carleson, H¨ormander, Stegenga, Cima and Wogen). Let α 1 and µ be a positive Borel measure on D. The following conditions are equivalent.≥ − a) There exists a positive constant C such that n+1+α (10) µ(Qδ(ξ)) Cδ ≤ for all ξ S and all δ > 0. ∈ 2 b) The measure µ is an Aα Carleson measure. The range α ( n 1, 1) is difficult. In this note our approach yields a new characterization∈ for− Carleson− − measures for this space in the range α ( n 1, n]. A characterization of Carleson measures in this range has been previously∈ − − obtained− by Arcozzi, Rochberg and Sawyer [ARS]. It seems likely that our characterization could be extended to the remaining range α ( n, 1). However, we have not yet succeeded to do this. ∈ − − Nevertheless, observe that for α> n 1, condition (10) remains a necessary −2 − condition for Carleson measures for Aα. This can be seen by using Proposition 3.4, (9) and the following result [ZZ, Theorem 45]. Theorem 3.7. Let α be a real such that n +1+ α and µ be a positive Borel measure on D . Then the following conditions are equivalent. a) There exists a positive constant C such that n+1+α µ(Qδ(ξ)) Cδ ≤ CARLESONMEASURESANDT(1)-THEOREM 19 for all ξ S and all δ > 0. b) For each∈s> 0 there exists a positive constant C such that (1 z 2)s C < . (11) sup − |n+1+| α+s z D D 1 zw ≤ ∞ ∈ Z | − | c) For some s> 0 there exists a positive constant C such that the inequality in (11) holds. We end this section by proving, in a general setting, the equivalence a b of ⇔ Theorem 1.1. Recall that the CZO Tα is defined by Tαf(z)= f(w)Kα(z,w)dµ(w), z D, D ∈ Z where the kernel Kα is defined by 1 Kα(z,w)= . ℜ (1 zw)n+1+α − Proposition 3.8 (cf [ARS]). Suppose n +1+ α> 0 and let µ be a positive Borel measure on D. Then the following conditions are equivalent. 2 a) The measure µ is an Aα Carleson measure. 2 b) The operator Tα is bounded in L (µ). Proof: Let be the linear map defined by: I : A2 L2(µ) α . I f → f 7→ The adjoint of is given by I f(w)dµ(w) ∗f z . ( )= n+1+α I D (1 zw) Z − Indeed for f L2(µ) with ∗f A2 we have, by the reproducing property, ∈ I ∈ α ∗ ∗ f(z) = f, Bα(., z) I hI iα = f, (Bα(., z)) h I iµ f(w)dµ(w) f, B ., z . = ( α( )) µ = n+1+α h i D (1 zw) Z − On the other hand, observe that is bounded if and only if the measure µ is 2 I an Aα Carleson measure. It is well known that is bounded ∗ is bounded. I ⇔ I 20 EDGAR TCHOUNDJA Note that ∗ is bounded means that I (12) ∗f 2 = ∗f, ∗f C f 2 . kI kα hI I iα ≤ k kµ Then ∗f 2 = ∗f, ∗f kI kα hI I iα = Bα(., w)f(w)dµ(w), Bα(., z)f(z)dµ(z) Z Z α = Bα(., w), Bα(., z) f(w)dµ(w)f(z)dµ(z) h iα ZZ = Bα(z,w)f(w)dµ(w)f(z)dµ(z). ZZ Having (12) for general f is equivalent to having it for real f. We now suppose f is real. In this case we continue with ∗ 2 f = Bα(z,w) f(w)dµ(w)f(z)dµ(z) kI kα ℜ{ } ZZ = Tαf, f 2 . h iL (µ) The last quantity satisfies the required estimates exactly if Tα is bounded. The proof is complete. 2 4. Proof of the equivalence (b) (c) of Theorem 1.1 ⇔ This section is devoted to the proof of the T(1)-Theorem. That is the charac- terization of positive Borel measures µ on D such that the operator Tα is bounded in L2(µ). To get the equivalence (b) (c) in Theorem 1.1, it suffices to prove the following theorem. ⇔ Theorem 4.1. Let k = n +1+ α and µ be a positive Borel measure on D. Then the following conditions are equivalent. 2 1) The operator Tα is bounded in L (µ). p 2) The operator Tα is bounded in L (µ) for some p> 2. 3) i) There exists a constant C > 0 such that (13) µ(B(z, r)) Crk ≤ for all pseudo balls B(z, r) which touch the boundary, and CARLESONMEASURESANDT(1)-THEOREM 21 ii) There exists a constant C > 0 such that 1 2 dµ(w) dµ(z) Cµ(B) ℜ (1 zw)k ≤ ZB ZB − for all pseudo balls B which touch the boundary. 4.1. Proof of 1) 3). Assertion i) follows from the discussion after Theorem ⇒ z 3.6 and the fact that the sets B(z, r) and Qr |z| are comparable when B(z, r) touches the boundary (in the sense of Proposition 2.8), and also by Proposition 3.8. Assertion ii) is easily obtained by testing the boundedness on the character- istic function f = χB. 2 4.2. Related maximal functions. Definition 4.2. We say that a measure µ satisfies the growth condition when µ satisfies inequality (13). We proceed now to prove that i) and ii) are sufficient for the boundedness of Tα for some p > 2 that is the proof of the implication 3) 2). We focus our ⇒ attention first to the special case α = n. We then set T = T−n and K = K−n. As we have mentioned in the introduction,− we follow the same idea as in [V]. Indeed, we first introduce a sort of curvature which plays the role of the Menger curvature. This curvature is adapted to our domain and has a close relation with our operator. Next, we proceed to construct for every ball which touches the boundary, a ”big piece” associated with this ball. This is the first crucial step of our proof. Finally, the next crucial step is to prove an appropriate good λ inequality without resorting to a doubling property on µ. Lemma 2.4 is used in those steps. We suppose that a measure µ satisfies the growth condition. In our estimates we use two variants of the central Hardy-Littlewood maximal operator acting on a complex Radon measure ν, namely, ν (B(z, r)) Mν(z) = sup | | , r>1−|z| r and for a positive constant ρ 1 ≥ ρ ν (B(z, r)) Mµν(z) = sup | | z supp µ , r>1−|z| µ(B(z,ρr)) ∈ 22 EDGAR TCHOUNDJA where B(z, r) is the pseudo ball centered at z of radius r which touches the boundary and suppµ is the closed support of µ. Proposition 4.3. Let µ be a positive Borel measure which satisfies the growth condition. For every ρ > 1 , there exists a positive constant C(ρ) such that for any f Lp(µ), ∈ ρ p p (14) Mµ(fµ) dµ C f dµ, 1 f(w)dµ(w) Rβ CM fµ z 0 1+β ( )( ) 0 d(z ,w) ≤ Zd(z ,w)>R for all z B(z0,R). In particular, we have ∈ dµ(w) µβ B z0,R C. ( ( )) 0 1+β 0 1 z w ≤ Zd(z ,w)>R | − | Proof: Fix β > 0, z0, R> 1 z0 and a positive function f. Let z B(z0,R). We have − | | ∈ f(w)dµ(w) f(w)dµ(w) 0 1+β 0 1+β 0 d(z ,w) ≤ k 0 k+1 d(z ,w) d(z ,w)>R k 2 R T ∗(fν)2dµ C f 2dν, f L2(ν). ≤ | | ∈ ZΩ Z 24 EDGAR TCHOUNDJA Proof: It is enough to prove that for some η > 0, there exists ρ> 1, γ > 0 and C > 0 such that (17) µ( z Ω : T ∗(fν)(z) > (1 + η)t ) µ( z Ω : M ρ(fν)(z) > γt ). { ∈ } ≤ { ∈ µ } Indeed if (17) is true, then +∞ T ∗(fν)2dµ = µ( z Ω : T ∗(fν)(z) >t )2tdt Ω 0 { ∈ } Z Z +∞ = (1+ η)2 µ( z Ω : T ∗(fν)(z) > (1 + η)t )2tdt 0 { ∈ } +∞ Z C µ( z Ω : M ρ(fν)(z) > γt )2tdt ≤ { ∈ µ } Z0 C M ρ(fν)2(z)dµ(z) ≤ µ ZΩ C f 2dν, by (14). ≤ | | Z We prove (17) using Lemma 2.4 applied to the open set ∗ Et = z Ω : T (fν)(z) >t . { ∈ } We obtain (17) once we prove that for each j (18) µ( z B∗ : T ∗(fν)(z) > (1 + η)t, M ρ(fν)(z) γt ) = 0, { ∈ j µ ≤ } ∗ where Bj is a term of the first decomposition of the open set Et with respect to Lemma 2.4. In fact we will have CARLESONMEASURESANDT(1)-THEOREM 25 µ( z Ω : T ∗(fν)(z) > (1 + η)t ) { ∈ } µ( z B∗ : T ∗(fν)(z) > (1 + η)t ) ≤ { ∈ j } j X µ( z B∗ : T ∗(fν)(z) > (1 + η)t, M ρ(fν)(z) γt ) ≤ { ∈ j µ ≤ } j X + µ( z B∗ : M ρ(fν)(z) > γt ) { ∈ j µ } j X µ( z B∗ : M ρ(fν)(z) > γt ) ≤ { ∈ j µ } j X Cµ( z Ω : M ρ(fν)(z) > γt ). ≤ { ∈ µ } by the bounded overlap property. So it remains to prove (18). B ∗ ′ B ∗∗ Set B = B z ,K1̺ = Bj and B = B z ,K2̺ = Bj . Suppose without loss of generality that there exists ξ0 B such that ∈ M ρ(fν)(ξ0) γt. µ ≤ Let z0 be a point in Ec B zB,K ̺ . Set B to be a ball centered at z0 whose t ∩ 3 radius is equal to max (2(1 z0 ),C̺), where C is a constant greater than or − | | equal to K3 to be precised later. Then B touches the boundary of D. Let f = fχ and f = f f = fχ c . There exists a constant A such that 1 B 2 − 1 B 1 (19) T ∗(fν)(z) T ∗ (fνχ )(z)+(1+ A γ)t, z B. ≤ B 1 ∈ To prove (19), let z B. Then ∈ f(w) dν(w) ∗ c T (fνχB )(z) = | | c 1 zw ZB | − | f(w) dν(w) 1 1 | | + f(w) dν(w) ≤ c 1 z0w c | | 1 zw − 1 z0w ZB | − | ZB − − 0 1 d(z, z ) 2 t + f(w) dν(w) 3 c ≤ B | | d w, z0 2 Z ( ) provided that C is chosen large enough so that we can use Proposition 2.12. Hence by Lemma 4.4 we have ∗ T (fνχ c )(z) (1 + A γ)t. B ≤ 1 26 EDGAR TCHOUNDJA This proves (19). 0 ′ Set B = B(ξ ,C1K1̺) and observe that B B B Et. Now, if C̺ 2(1 z0 ), there exists a constant⊂ ⊂A >⊂0 such that for z B ≥ − | | 2 ∈ e e (20) T ∗ (fνχ )(z) T ∗ fνχ (z)+ A γt. B ≤ B 2 To prove (20), we have e f(w) dν(w) T ∗ (fνχ )(z) T ∗ fνχ (z)+ | | B ≤ B 1 zw ZB\B | − ∗ e = T fνχB (z)+ I, e |f(w)|dν(w) where I = B\B |1−zw . e By (5), R 1 e C , and on the other hand, for w B B we have |1−zw| ≤ d(z,w) ∈ \ C K ̺ d(ξ0,w) K(d(zB, ξ0)+ d(zB,w)) T ∗(fν)(z) (1 + Aγ)t, z B. ≤ ∈ From this we have µ( z B : T ∗(fν)(z) > (1 + η)t, M ρ(fν)(z) γt ) { ∈ µ ≤ } µ( z B : (1+ Aγ)t> (1 + η)t, M ρ(fν)(z) γt ); ≤ { ∈ µ ≤ } η so if we choose 0 <γ 2A , we obtain (18). This ends the proof of the Lemma. ≤ 2 CARLESONMEASURESANDT(1)-THEOREM 27 4.3. Curvature in the unit ball. Definition 4.6. Given three points z , z , z D, we define their curvature 1 2 3 ∈ c(z1, z2, z3) by 2 c (z1, z2, z3)= K zσ(2), zσ(1) K zσ(3), zσ(1) σ X where the sum is taken over the six permutations of 1, 2, 3. For a positive Borel measure ν the quantity 2 2 c (ν)= c (z1, z2, z3)dν(z1)dν(z2)dν(z3) ZZZ is called the Total curvature of ν. One important fact about this curvature is 2 that c (z1, z2, z3) > 0. Indeed for z and w in D (1 zw) 1 (zw) K(z,w)= ℜ − = − ℜ > 0 1 zw 2 1 zw 2 | − | | − | since (zw) < 1. Theℜ next Lemma gives a relation between this curvature and our operator T . Lemma 4.7. Let νj, j = 1, 2, 3 be three Borel measures. Then 2 T νσ(1) T νσ(2) dνσ(3) = c (z1, z2, z3)dν1(z1)dν2(z2)dν3(z3). σ Z ZZZ X Proof: We have T νσ(1) T νσ(2) dνσ(3) Z = K zσ(3), zσ(1) K zσ(3), zσ(2) dνσ(1) zσ(1) dνσ(2) zσ(2) dνσ(3) zσ(3) . ZZZ Since for each σ dνσ(1) zσ(1) dνσ(2) zσ(2) dνσ(3) zσ(3) = dν1(z1)dν2(z2)dν3(z3), summing over the six permutations we obtain 2 T νσ(1) T νσ(2) dνσ(3) = c (z1, z2, z3)dν1(z1)dν2(z2)dν3(z3). σ Z ZZZ X 2 28 EDGAR TCHOUNDJA 2 We apply Lemma 4.7 to ν1 = ν2 = fµ with f (a real function) in L (µ) and ν3 = χBµ with B a fixed pseudo ball which touches the boundary. We then have 2 (21) 2 T (fµ) dµ + 4 T (fµ)T (χBµ) fdµ | | ZB Z 2 = c (z,w,ζ)f(z)f(w)χB(ζ)dµ(z)dµ(w)dµ(ζ). ZZZ In particular taking f = χB, one gets 6 T (fµ) 2dµ = c2(z,w,ζ)dµ(z)dµ(w)dµ(ζ), | | 3 ZB ZZZB and thus (22) c2(z,w,ζ)dµ(z)dµ(w)dµ(ζ) Cµ(B), 3 ≤ ZZZB provided µ satisfies condition ii) of 3) in Theorem 4.1 (case k = 1). We are now ready to produce a ”big piece” inside a given pseudo ball B which touches the boundary. As in [V], set 2 2 cB(z)= c (z,w,ζ)dµ(w)dµ(ζ), z B. 2 ∈ ZZB By Chebyschev’s inequality, condition ii) of 3) in Theorem 4.1 and (22) µ( z B : cB(z) > t or T (χBµ)(z) >t ) { ∈ | | } 1 2 2 c (z)dµ(z)+ T (χBµ)(z) dµ(z) ≤ t2 B | | ZB ZB µ(B) (23) C . ≤ t2 From this we have the following lemma. Lemma 4.8. Given 0 <θ< 1, there exists a set E B such that ⊂ 2 C 2 C c (z) and T (χBµ)(z) , z E B ≤ θ | | ≤ θ ∈ and µ(B E) θ(µ(B)). \ ≤ CARLESONMEASURESANDT(1)-THEOREM 29 Proof: Fix 0 <θ< 1. If µ(B) = 0, there is nothing to do. If µ(B) = 0, set 6 2 C 2 C E = z B : c (z) and T (χBµ)(z) . ∈ B ≤ θ | | ≤ θ It is then easy to verify that this set satisfies our requirements. 2 2 We set k(z,w)= B c (z,w,ζ)dµ(ζ) so that C (24) Rk(z,w)dµ(w)= c2 (z) (z E). B ≤ θ ∈ ZE Since k(z,w)= k(w, z) we obtain the following lemma. Lemma 4.9. There exists a constant C = C(θ) which does not depend on B such that 2 2 c (z,w,ζ)f(z)f(w)χB(ζ)dµ(z)dµ(w)dµ(ζ) C f dµ, ≤ ZZZ Z where f L2(E)= L2(E,µ), with f real. ∈ Proof: The result follows from Schur’s test since C k(z,w)dµ(w) (z E). E ≤ θ ∈ Z 2 Therefore from (21), Lemma 4.8 and Lemma 4.9, for any f L2(E), we get 1 1 ∈ 2 2 T (fµ) 2dµ C T (fµ) 2dµ f 2dµ + C f 2dµ | | ≤ | | ZB ZB Z Z and consequently T (fµ) 2dµ C f 2dµ, f L2(E). | | ≤ ∈ ZB ZE By duality this implies T (gµ) 2dµ C g2dµ, g L2(B). | | ≤ ∈ ZE ZB So by Chebyshev’s inequality C (25) µ( z E : T (gµ)(z) >t ) g2dµ, g L2(B). { ∈ | | } ≤ t2 ∈ ZB 30 EDGAR TCHOUNDJA Now, for every h L2(D,µ), Lemma 4.5 and (25) give ∈ µ( z E : T (hµ)(z) >t ) { ∈ | | } t µ z E : T (hχBµ)(z) > ≤ ∈ | | 2 t +µ z E : T (hχBc µ)(z) > ∈ | | 2 C 2 C 2 h dµ + T (hχBc µ)(z) dµ ≤ t2 t2 | | ZB ZB C C h2dµ h2dµ 2 + 2 ≤ t t c ZB ZB C h2dµ. (26) 2 ≤ t D Z 4.4. A good λ inequality. We will establish in this section the next crucial argument in the proof of the implication 3) 2) of Theorem 4.1. The result is the following theorem. ⇒ Theorem 4.10. Let µ be a positive Borel measure on D with satisfies i) and ii). Then for each η > 0, there exists γ = γ(η) > 0 small enough so that ρ 1 µ z D : T (fµ)(z) > (1 + η)t and M (f 2µ) 2 (z) γt ∈ | | µ ≤ n 1 o µ ( z D : T (fµ)(z) >t ) . ≤ 2 { ∈ | | } Proof: Let Ω = z D : T (fµ)(z) >t . The set Ω is open. By Lemma 2.4 applied to this set,{ the∈ theorem| will follow| } if we can prove the following lemma. Lemma 4.11. Let η > 0 and 0 <α< 1. There exists γ = γ(η, α) > 0 such that ρ 1 (27) µ z B∗ : T (fµ)(z) > (1 + η)t and M (f 2µ) 2 (z) γt ∈ j | | µ ≤ n αµ oB∗∗ , ≤ j ∗ ∗∗ where Bj and Bj are respectively the first and the second decompositions of the open set Ω with respect to Lemma 2.4. CARLESONMEASURESANDT(1)-THEOREM 31 Indeed, if the lemma is true, then ρ 1 µ z D : T (fµ)(z) > (1 + η)t and M (f 2µ) 2 (z) γt ∈ | | µ ≤ n ρ 1 o = µ z Ω : T (fµ)(z) > (1 + η)t and M (f 2µ) 2 (z) γt ∈ | | µ ≤ n ρ 1 o µ z B∗ : T (fµ)(z) > (1 + η)t and M (f 2µ) 2 (z) γt ≤ ∈ j | | µ ≤ j X n o α µ B∗∗ ≤ j j X αMµ(Ω) (by the bounded overlap property). ≤ 1 We then have to choose α so that αM = 2 . We obtain the result. Let us turn our attention to the proof of the Lemma. B ∗ ′ B ∗∗ Set B = B z ,K1̺ = Bj and B = B z ,K2̺ = Bj . We follow with a little change, the proof of the Lemma 4.5. Suppose without loss of generality that there exists ξ0 B such that ∈ ρ 1 M (f 2µ) 2 (ξ0) γt. µ ≤ Let z0 be a point in Ωc B zB,K ̺ . Let B be a ball centered at z0 whose ∩ 3 radius is equal to max (2(1 z0 ),C̺), where C is a constant greater than or − | | equal to K3 to be precised later. Then B touches the boundary of D. Let f = fχ and f = f f = fχ c . As in the proof of (19) there exists a 1 B 2 − 1 B constant A1 such that (28) T (fµ)(z) T (fµχ )(z) +(1+ A γ)t, z B. | | ≤ | B | 1 ∈ 0 On the other hand, if we set B = B(ξ ,C1K1̺), we observe that B ρB B′ Ω e⊂ ⊂ ⊂ for some ρ> 1. e Now, if C̺ 2(1 z0 ), there exists a constant A > 0 such that for z B, ≥ − | | 2 ∈ (29) T (fµχ )(z) T fµχ (z) + A γt. | B | ≤ | B | 2 We obtain (29) as in the proof of (20). e 32 EDGAR TCHOUNDJA From (28), we have ρ 1 µ z B : T (fµ)(z) > (1 + η)t and M (f 2µ) 2 (z) γt ∈ | | µ ≤ n ρ o1 µ z B : T (fµχ )(z) > (η A γ)t and M (f 2µ) 2 (z) γt . ≤ ∈ | B | − 1 µ ≤ n o If C̺ < 2(1 z0 ), then as in the proof of the Lemma 4.5 we have − | | T (fµχ )(z) C”γt | B | ≤ so that for γ small enough ρ 1 z B : T (fµχ )(z) > (η A γ)t and M (f 2µ) 2 (z) γt = . ∈ | B | − 1 µ ≤ ∅ n o Thus (27) is satisfied in this case. If C̺ 2(1 z0 ), then from (29), we have ≥ − | | ρ 1 µ z B : T (fµ)(z) > (1 + η)t and M (f 2µ) 2 (z) γt ∈ | | µ ≤ n oρ 1 µ z B : T (fµχ )(z) > (η (A + A )γ)t and M (f 2µ) 2 (z) γt . ≤ ∈ | B | − 1 2 µ ≤ n o If we choose γ small enough (0 <γ η will do), we finally have ≤ 2(A1+A2) ρ 1 (30) µ z B : T (fµ)(z) > (1 + η)t and M (f 2µ) 2 (z) γt ∈ | | µ ≤ n η ρ 1 o µ z B : T (fµχ )(z) > t and M (f 2µ) 2 (z) γt . ≤ ∈ | B | 2 µ ≤ n e o We distinguish two cases. If B does not touch the boundary then we easily obtain, T (fµχ )(z) C∗Mf(ξ0) Cγt, | B | ≤ ≤ e η such that for γ small enough (0 <γ 4C ), (27) is satisfied. Finally, suppose that B touches the≤ boundary. Let E be a ”big piece” associ- ated with the ball B and the number θ. From (30) and (26) we have CARLESONMEASURESANDT(1)-THEOREM 33 ρ 1 µ z B : T (fµ)(z) > (1 + η)t and M (f 2µ) 2 (z) γt ∈ | | µ ≤ n η ρ 1 o µ z B : T (fµχ )(z) > t and M (f 2µ) 2 (z) γt ≤ ∈ | B | 2 µ ≤ n η o µ(B E)+ µ z Ee : T (fµχ )(z) > t ≤ \ ∈ | B | 2 C n o θµ(B)+ f 2dµ e ≤ η2t2 | | ZB C ρ θµ(B)+ µ e ρB M (f 2µ)(ξ0) ≤ η2t2 µ C θµ(B)+ µ ρBe γ2t2 ≤ η2t2 θ + Cη−2γ2 µ ρeB ≤ αµ(B′), ≤ e provided θ and γ are chosen small enough so that (θ + Cη−2γ2) α. This completes the proof of the lemma and consequently the proof of the≤ theorem. 2 4.5. Proof of the implication 3) 2) of Theorem 4.1. Let f Lp(µ), p> 2. We have ⇒ ∈ T (fµ) pdµ | | Z +∞ = µ ( z D : T (fµ)(z) >t ) dtp 0 { ∈ | | } Z +∞ = (1+ η)p µ ( z D : T (fµ)(z) > (1 + η)t ) dtp 0 { ∈ | | } Z +∞ 1 p ρ 2 2 p (1 + η) µ z D : T (fµ)(z) > (1 + η)t; Mµ(f µ) (z) γt dt ≤ 0 ∈ | | ≤ Z n +∞ o 1 p ρ 2 2 p +(1 + η) µ z D : Mµ(f µ) (z) γt dt 0 ∈ ≥ +∞ Z n o (1 + η)p µ ( z D : T (fµ)(z) >t ) dtp ≤ 2 { ∈ | | } Z0 p p (1 + η) ρ 1 + M (f 2µ) 2 (z) dµ γp µ Z 34 EDGAR TCHOUNDJA by Theorem 4.10. We then choose η small and use Proposition 4.3 to obtain (31) T (fµ) pdµ C f pdµ. | | ≤ | | Z Z 2 4.6. Proof of the implication 2) 1) in Theorem 4.1. Since Tα is self adjoint, then by duality, for 1 2 c (z1, z2, z3)= Kα zσ(2), zσ(1) Kα zσ(3), zσ(1) . σ X 5. Comments One reason why we could not carry out our argument in the remaining • range n<α< 1 is that the curvature we have defined is no longer strictly− positive in− this range. Nevertheless, we conjecture that conditions 3) in Theorem 4.1 are sufficient for boundedness in the remaining range. 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Department of Mathematical Sciences, Division of Mathematics, Chalmers University of Technology and Goteborg¨ University, SE-412 96 Goteborg,¨ Swe- den E-mail address: [email protected]