Fatou and Korányi-Vági type theorems on the minimal ball

Publ. Mat. 46 (2002), 49–75 FATOU AND KOR´ ANYI-V´ AGI TYPE THEOREMS ON THE MINIMAL BALL Nguyˆ en Viˆ et Anh Abstract In this paper we develop the Hp(p≥1) theory on the minimal ball. After identifying the admissible approach regions, we establish theorems of Fatou and Kor´anyi-V´agi type on this ball. 1. Introduction and statement of our main results It is well-known from the work of Stein [17] that holomorphic functions of Hpclass on a bounded domain in Cnwith C2-boundary converge almost everywhere to their boundary values, provided the limit is taken inside certain natural approach regions. Boundary behavior of Hpfunctions on smooth domains is rather well understood, see for example [6], [9], [10], [11], etc. In this paper we are interested in Fatou type theorems for Hp(p≥1) functions and Kor´anyi-V´agi type theorems on a non piecewise smooth domain: The minimal ball B∗. This is the convex circular domain defined for n≥2by B∗:= {z∈Cn:|z|2+|z•z|<1}, where z•w:= n j=1 zjwj. This is the unit ball with respect to the norm N∗(z):=|z|2+|z•z|,z∈Cn. The norm N:= N∗ √2was introduced by Hahn and Pflug [4], and was shown to be the smallest complex norm in Cnwith the following properties N(x)=|x|for x∈Rnand N(z)≤|z|for z∈Cn. Set V:= {z∈Cn\{0}:z•z=0}. The singular part of the boundary of B∗is obviously the set ∂B∗∩V. The regular part ∂B∗\Vconsists of all strictly pseudoconvex points. Moreover B∗is neither homogeneous nor Reinhardt (see [5], [13]). Function theory on the minimal ball was 2000 Mathematics Subject Classification. Primary 32A40, 32A35. Key words. Admissible approach regions, Fatou type theorem, Kor´anyi-V´agi type theorem, the minimal ball. 50 N. Viˆ et Anh studied by several authors (see [12], [14], [15], [8], [7], [18]). In his recent work [19], E. H. Youssfi developed a method for computing the Bergman and Szeg¨o kernel of a new class of pseudoconvex domains including the minimal ball. His paper is the main motivation for the present work. As in [8], [19], our method consists of two steps. At the first step we study the problem on an auxiliary complex manifold M. At the second step we transplant the results obtained on the complex manifold Mto B∗by means of a proper holomorphic mapping. This paper is organised as follows: In the first section we define the admissible approach regions for the minimal ball B∗and state our main results. In Section 2 we give some properties of the system of admissble approach regions. In Section 3 we discuss theorems of Fatou type and Kor´anyi-V´agi type on the complex manifold M. We next transplant these results to B∗in Section 4 in order to prove our main Theorems A, B and C below. We now identify the admissible approach regions. Definition 1.1. For α>1 and ζ∈∂B∗, we let the “admissible approach region” D∗ α(ζ) be the set of all z∈B∗such that min ∈{−1,1}1−z,ζ−(z•z)(ζ•ζ)<α 2(1 −N2 ∗(z)), where ., .denotes the standard Hermitian inner product. Definition 1.2. A function fdefined on B∗is said to have admissible limit λat a point ζ∈∂B∗if fconverges to λalong D∗ α(ζ) for every α>1. The question naturally arises how the system of admissible approach regions is different from the system of approach regions defined by E. Stein (see [17, p. 32]). Part (i) of Proposition 2.1 below asserts that on any compact set in the regular part of ∂B∗, these two systems are, in a sense, equivalent to each other. Furthermore, part (ii) of that proposition says that the admissible approach regions are tangential to the regular part ∂B∗\Vin the complex tangential directions. Note that the Kor´anyi approach regions for the Euclidean balls of Cnalso have this geometric property. Fatou and Kor´ anyi-V´ agi Type Theorems 51 In order to state our main results we need some notations. Let µbe a positive Borel measure on ∂B∗and p>0. Then the Hardy space Hp(B∗,µ) is defined by Hp(B∗,µ):=  f∈H(B∗),fp Hp(B∗,µ):= sup 0<r<1 ∂B∗ |fr|pdµ < ∞  , where frdenotes the dilated function defined for N∗(z)<1 rby fr(z):= f(rz). Let θbe the Lebesgue surface measure of ∂B∗\V. Throughout the paper, SB∗and PB∗denote the Szeg¨o and Poisson-Szeg¨o projection of B∗ (with respect to the measure θ) respectively. The letter Cwill denote a finite constant that is not necessarily the same at each occurence. Our first main result is the following Theorem A. If f∈Hp(B∗,|ζ•ζ|sdθ),p≥1and s∈R, (i) then fhas finite admissible limits f∗a.e. [θ]on ∂B∗and f∗∈ Lp(∂B∗,|ζ•ζ|sdθ); (ii) if moreover p>1and −2<s<2p−2, then f=SB∗[f∗]=PB∗[f∗]. Theorem A (or Fatou type theorem for the minimal ball) is only proved here for p≥1. It seems to be of interest to know whether part (i) of this theorem holds for all p>0. The Hardy spaces Hp(B∗,|ζ•ζ|p−2 2dθ) appear naturally in studying the Hptheory associated to the minimal ball (see the works [8], [7], [19]). Next, if u∈C(B∗) and α>1, the maximal function Mαu:∂B∗→ [0,∞] is defined by (Mαu)(ζ) := sup {|u(z)|:z∈D∗ α(ζ)}. We now state the second main result. Theorem B. If 1<p<∞and −2<s<∞, then for every f∈ Hp(B∗,|ζ•ζ|sdθ),  ∂B∗ |(Mαf)(ζ)|p|ζ•ζ|sdθ(ζ)≤C(s, α, p) ∂B∗ |f∗(ζ)|p|ζ•ζ|sdθ(ζ),(i) (ii) lim r→1−f∗−frLp(∂B∗,|ζ•ζ|sdθ)=0 and fHp(B∗,|ζ•ζ|sdθ)=f∗Lp(∂B∗,|ζ•ζ|sdθ). Theorem B should be compared with the analogous results (Theorems 5.6.5 and 5.6.6 of [16]) in the case of the Euclidean unit ball. Finally our third main result is the following 52 N. Viˆ et Anh Theorem C. Let 1<p<∞and α>1.Ifp 2−2<s<3p 2−2, then there exists C(s, α, p)<∞such that  ∂B∗ |MαSB∗[f](ζ)|p|ζ•ζ|sdθ(ζ)≤C(s, α, p) ∂B∗ |f(ζ)|p|ζ•ζ|sdθ(ζ) for all f∈Lp(∂B∗,|ζ•ζ|sdθ). We note that an analogue of Theorem C in the case of the Euclidean unit ball is the classical Kor´anyi-V´agi theorem (Theorem 6.3.1 of [16]). 2. Some properties of the system of admissible approach regions We have the following Proposition 2.1. (i) For α>1and ζ∈∂B∗\V, let Aα(ζ)be the classical approach region [17, p. 32] defined by Aα(ζ):=z∈B∗:|z−ζ,νζ|<αδ(z),|z−ζ|2<(α−1)δ(z).(2.1) Here νζdenotes the unit outward normal at ζand δ(z)is the distance from zto ∂B∗. (Notice that since B∗is convex, δ(z)is smaller than the distance from zto the tangent space at ζ.) Let Kbe a compact in ∂B∗\V. Then there exist β,γ >1such that Aα(ζ)⊂D∗ β(ζ)and D∗ α(ζ)⊂A γ(ζ),∀ζ∈K.(2.2) (ii) For α>1and ζ∈∂B∗\V, the admissible approach region D∗ α(ζ) is tangential to ∂B∗\Vin the direction of TC ζ(∂B∗\V). Proof: The proof of assertion (ii) is postponed until Section 4. Here we only prove assertion (i). A little calculation gives that νζ=1 √2ζ1+ζ1 ζ•ζ |ζ•ζ|,...,ζ n+ζn ζ•ζ |ζ•ζ|and ζ,νζ=1 √2.(2.3) For z∈B∗, let w∈∂B∗such that z−w=δ(z). A geometric argument shows that w∈∂B∗\Vand w−z=δ(z)νw. This, combined with (2.3), implies zk=wk−δ(z) √2wk+wk w•w |w•w|,∀1≤k≤n. Next, substituting the latter equation into the expression of N2 ∗(z) and using the equality N2 ∗(w) = 1, we obtain, after some simplifications, 1−N2 ∗(z) = min 2|w•w|,√2δ(z)(2 −√2δ(z)).(2.4) Fatou and Kor´ anyi-V´ agi Type Theorems 53 Now we prove the first assertion in (2.2). Consider ζ∈Kand z∈A α(ζ). By (2.1) and (2.3), we get 1− n  k=1 zkζk+ζk ζ•ζ |ζ•ζ|<α √2δ(z).(2.5) Thus for zclose enough to ζ, 1−z,ζ−(z•z)(ζ•ζ)<1− n  k=1 zkζk+ζk ζ•ζ |ζ•ζ| +(z•z)(ζ•ζ)− n  k=1 zkζk ζ•ζ |ζ•ζ|=I+II. From (2.5), we have I<α √2δ(z). We now estimate II: II =ζ•ζ(√z•z−ζ•ζ)−ζ•ζ· n  k=1 (zk−ζk)∂√z•z ∂zkz=ζ =O(|z−ζ|2)<√2M(α−1)δ(z), where the last inequality follows from (2.1) and the assumption that z∈A α(ζ), Mis a constant that depends only on K. Suppose zis close enough to the boundary of B∗and βsatisfies the following condition α+M(α−1) <β1−√2 2δ(z).(2.6) Then we obtain 1−z,ζ−(z•z)(ζ•ζ)<α √2δ(z)+√2M(α−1)δ(z) =√2δ(z)(α+M(α−1)) <β 2(1 −N2 ∗(z)), where the last inequality follows from (2.4) and (2.6). 54 N. Viˆ et Anh We have shown that if z∈A α(ζ) and zis close to ζ, then z∈D∗ β(ζ). This, combined with the identity  β>1 D∗ β(ζ)=B∗, proves that there exists βlarge enough which verifies Aα(ζ)⊂D∗ β(ζ), ∀ζ∈K. To prove the second assertion in (2.2) let ζ∈Kand z∈D∗ α(ζ), zis close to ζ. Then |z−ζ|2≤|z−ζ|2+|√z•z−ζ•ζ|2 =1+N2 ∗(z)−2Rez,ζ+(z•z)(ζ•ζ). Since z∈D∗ α(ζ), we get Re z,ζ+(z•z)(ζ•ζ)>1−α 2(1 − N2 ∗(z)). Thus |z−ζ|2≤1+N2 ∗(z)−21−α 2(1 −N2 ∗(z))=(α−1)(1 −N2 ∗(z)). On account of (2.4) and the last inequality, it follows that |z−ζ|2< 2√2(α−1)δ(z). Hence we can choose γsuch that γ−1>2√2(α−1). It now remains to show that |z−ζ,νζ|<γδ(z). The rest of our proof is similar to the previous proof of the first assertion in (2.2), this completes the proposition. Remark 2.2.Assertion (i) can not be sharpened. None of the two assertions in (2.2) holds if Kis replaced by the whole regular part of ∂B∗.In other words, the two systems of approach regions are not globally equivalent. This result can be shown by slightly modifying the arguments in the proof of assertion (i). 3. Analysis on the complex manifold M The complex manifold Mis defined by M=Mn:= z∈Cn+1 \{0}:z•z= 0 and |z|<1. The manifold ∂M:= z∈Cn+1 :z•z= 0 and |z|=1 is endowed with the unique probability O(n+1,R)-invariant measure σ. This measure is induced by Haar measure of O(n+1,R) (see [8]). Set M:= M∪∂M. From the work in [19], the Szeg¨o kernel of Mis given by SM(z,w)= 1+z,w (1 −z,w)n,for z∈Mand w∈∂M.(3.1) Fatou and Kor´ anyi-V´ agi Type Theorems 55 By exploiting this explicit formula, we shall establish the theorems of Fatou, Kor´anyi and Kor´anyi-V´agi type on Musing the standard techniques for the unit ball in Rudin’s book [16]. The work of Stein [17] would not give these results directly since an analogue of the potential theory on Euclidean spaces has not been available yet in the context of the manifold M. Definition 3.1. For a∈M,b∈M, d(a, b):=|1−a, b|1 2. For w∈∂M,δ>0, Q(w,δ):={η∈∂M:d(w, η)<δ}. It is clear that for every U∈O(n+1,R), d(Ua,Ub):=d(a, b),and U(Q(w, δ)) = Q(Uw,δ). Observe that ∂Mis a submanifold of the unit sphere of Cn+1. Then by Proposition 5.1.2 of [16], dis a metric on ∂M. Now we define a system of approach regions for M. Definition 3.2. For α>1 and w∈∂M, we let the approach region Dα(w) be the set of all z∈Msuch that |1−z,w|<α 2(1 −|z|2). The following proposition will be very useful. Proposition 3.3. There exist two constants 0<C 1,C 2<∞such that ∀a∈∂Mand 0<δ<√2:C1<σ(Q(a, δ)) δ2n<C 2. Proof: Since O(n+1,R) acts transitively on ∂Mand σ, d, Q(., δ) are O(n+1,R)-invariant, we may suppose without loss of generality that a:= 1 √2,i √2,0,...,0∈∂Mand δis sufficiently small. Consider the function H=(H1,...,H n): ∂M−→ S2n−1defined by H1(z):= z1 √2−iz2 √2 h(z),H 2(z):= z3 h(z),...,H n(z):=zn+1 h(z),(3.2) where S2n−1is the unit sphere of Cnand h(z):= z1 √2−iz2 √2 2 +|z3|2+···+|zn+1|2,z∈∂M. We now prove that His locally diffeomorphic near the point a. 56 N. Viˆ et Anh Write w=H(z). Since z∈∂M, we have 2H1(z)·z1 √2+iz2 √2 h(z)+H2 2(z)+···+H2 n(z)=z2 1+···+z2 n+1 h2(z)=0. Thus z1 √2+iz2 √2 h(z)=−w2 2−···−w2 n 2w1 . This, combined with equation (3.2), gives the following system                |z1|2+···+|zn+1|2=1 z1 h(z)=2w2 1−w2 2−···−w2 n 2√2w1 z2 h(z)=i(2w2 1+w2 2+···+w2 n) 2√2w1 z3 h(z)=w2,...,zn+1 h(z)=wn. It follows easily that the equation w=H(z) has a unique solution for every wnear (1,0,...,0). Therefore His locally diffeomorphic near a. Now, let Qδdenote the standard nonisotropic ball of radius δin S2n−1 centered at (1,0,...,0) (See [16, p. 65].) We shall prove the following fact H(Q(a, δ)) ⊂Q√3δand Qδ⊂H(Q(a, √3δ)),(3.3) provided δis sufficiently small. If z∈Q(a, δ),then by Definition 3.1 we get 1−z1 √2+iz2 √2<δ 2. This implies (3.4) |1−w1|≤1−z1 √2+iz2 √2 +1 h(z)−1· z1 √2−iz2 √2<δ 2+1 h(z)−1. On the other hand (3.5) 0 ≤1 h(z)−1= 1−h(z)2 h(z)(h(z)+1) ≤1−z1 √2+iz2 √21+z1 √2−iz2 √2 h(z)(h(z)+1) <2δ2, for z≈a, because of h(a)=1. By virtue of (3.4) and (3.5), we conclude that |1−w1|<3δ2, which proves the first assertion in (3.3). Fatou and Kor´ anyi-V´ agi Type Theorems 57 To prove the second assertion in (3.3) let w∈Qδ. Since His locally diffeomorphic near a, we can write w=H(z). Then |H1(z)|>1−δ2. By virtue of the definition of H1in (3.2) the last inequality implies that  z1 √2−iz2 √2 2 >(1 −δ2)2 1−(1 −δ2)2·|z3|2+···+|zn+1|2. Using this estimate, we obtain 1=|z1|2+|z2|2+···+|zn+1|2≥ z1 √2−iz2 √2 2 +|z3|2+···+|zn+1|2 >|z3|2+···+|zn+1|2 1−(1 −δ2)2. Thus |z3|2+···+|zn+1|2<1−(1 −δ2)2and |z1|2+|z2|2>(1 −δ2)2.(3.6) On the one hand, using the second estimate in (3.6) we get  z1 √2−iz2 √2 2 =|z1|2+|z2|2− z1 √2+iz2 √2 2 >(1 −δ2)2−|z1+iz2|2 2.(3.7) On the other hand, using the first estimate in (3.6) we have for z≈a, |z1+iz2|2 2=z2 1+z2 22 2|z1−iz2|2=z2 3+···+z2 n+12 2|z1−iz2|2 ≤|z3|2+···+|zn+1|22 2|z1−iz2|2≤1−(1 −δ2)22 2≤2δ4. Putting this estimate into (3.7), we obtain  z1 √2−iz2 √2 2 >(1 −δ2)2−2δ4>1−3δ2. This implies 0≤1 h(z)−1= 1−h(z)2 h(z)(h(z)+1) ≤ 1−z1 √2−iz2 √2 2 h(z)(h(z)+1) <2δ2, for z≈a. 64 N. Viˆ et Anh Lemma 4.1. For each p>−1, there exist two constants 0<C 4(p), C5(p)<∞such that C4< Q(a,δ)|wn+1|2pdσ(w) δ2n+2p<C 5,(4.5) where a=(a1,...,a n+1)∈∂Mwith an+1 =0and 0<δ<√2. Proof: Observe that ∂Mn−1={a∈∂Mn:an+1 =0}. Since SO(n, R) acts transitively on ∂Mn−1, we may suppose without loss of generality that a:= 1 √2,i √2,0,...,0∈∂Mas in the proof of Proposition 3.3. Using the local diffeomorphism Hconstructed in (3.2) together with its properties (3.3), inequality (4.5) is reduced to proving the following estimate C4< Qδ|ζn|2pdτ(ζ) δ2n+2p<C 5.(4.6) We now prove (4.6). Applying formula 1.4.4(1) in [16], we get  Qδ |ζn|2pdτ(ζ) =n−1 2 B2 (1 −|(λ1,λ 2)|2)n−3χ{|1−λ1|<δ2}|λ2|2pdν2(λ1,λ 2), where χis the characteristic function, B2is the unit ball of C2and ν2 is the Lebesgue measure on C2so normalized that ν2(B2)=1. Fubini’s theorem shows that the right side of the last equation equals C·C χ{|λ1|<1,|1−λ1|<δ2}dm(λ1) · |λ2|≤√1−|λ1|2 (1 −|λ1|2−|λ2|2)n−3|λ2|2pdm(λ2), where mis the ordinary Lebesgue measure of C. Fatou and Kor´ anyi-V´ agi Type Theorems 65 Using the beta function, since p>−1, the inner integral of the expression above is equal to C(1 −|λ1|2)n+p−2B(n−2,p+1). Hence we get  Qδ |ζn|2pdτ(ζ)=C· E(δ) (1 −|λ1|2)n+p−2dm(λ1),(4.7) where E(δ):={λ1∈C:|λ1|<1 and |1−λ1|<δ 2}. In view of (4.7), the arguments which have been used in Proposition 5.1.4 of [16] establish (4.6). Therefore, the proof of the lemma is complete. Proof of Theorem A: Take any k∈Nsuch that kp > 2s+ 2. Consider the function g∈H(M) defined by g(z):=zk n+1f(F(z)),∀z∈M.(4.8) By virtue of formula (4.2) we have, for 0 <r<1,  ∂M |g(rw)|pdσ(w)≤ ∂M |wn+1|2s+2|f(rF(w))|pdσ(w) =C3· ∂B∗ |f(rζ)|p|ζ•ζ|sdθ(ζ)<∞, since f∈Hp(B∗,|ζ•ζ|sdθ). The latter estimate gives that g∈Hp(M,σ). By part (ii) of Theorem 3.13 the boundary value g∗(w) exists almost everywhere with respect to the measure σon ∂M. This, combined with formula (4.2), identity (4.4) and equation (4.8) gives that fhas admissible limits almost everywhere with respect to the measure θon ∂B∗. Since rζ ∈D∗ α(ζ) for 0 ≤r<1 and α>2, the admissible convergence of fimplies that the dilated functions frconverge to f∗θ-almost everywhere on ∂B∗. Therefore it follows from Fatou’s lemma that f∗∈Lp(∂B∗,|ζ•ζ|sdθ). This completes the proof of part (i). 66 N. Viˆ et Anh Taking Theorem B for granted, we now prove part (ii). By part (ii) of Theorem B, we have lim r→1− ∂B∗ |f(rζ)−f∗(ζ)|p|ζ•ζ|sdθ(ζ)=0.(4.9) Using (4.9) and applying the H¨older’s inequality, we see that for each z∈B∗, there exists a constant C(z)<∞such that |SB∗[f∗](z)−f(z)|= lim r→1−|SB∗[f∗−fr](z)| ≤  ∂B∗ |SB∗(z,ζ)|p p−1|ζ•ζ|−s p−1dθ(ζ)  p−1 p ·lim r→1−  ∂B∗ |f(rζ)−f∗(ζ)|p|ζ•ζ|sdθ(ζ)  1 p ≤C(z)  ∂M |wn+1|2−2s p−1dσ(w)  p−1 p ·lim r→1−  ∂B∗ |f(rζ)−f∗(ζ)|p|ζ•ζ|sdθ(ζ)  1 p ≤C·C(z) lim r→1−  ∂B∗ |f(rζ)−f∗(ζ)|p|ζ•ζ|sdθ(ζ)  1 p =0, (4.10) where the second inequality follows from formula (4.2) and the third one comes from Lemma 4.1 and the hypothesis on s. This yields f=SB∗[f∗]. The identity f=PB∗[f∗] can be proved in the same way. In order to prove Theorems B and C we need some lemmas. Fatou and Kor´ anyi-V´ agi Type Theorems 67 We recall from the work of Calder´on [2] that the weight function µ(w), µ(w)≥0, is said to belong to the class Apof Muckenhoupt (1 <p<∞) if sup a∈∂M,0<δ<√2 M(µ, a, δ)<∞, where (4.11) M(µ, a, δ):=     1 σ(Q(a, δ))  Q(a,δ) µ(w)dσ(w)     ·     1 σ(Q(a, δ))  Q(a,δ) µ(w)−1 p−1dσ(w)     p−1 . Lemma 4.2. Consider the weight function µs(w):=|wn+1|2s,s∈R, in the space of homogeneous type (∂M,σ,d). Then µsbelongs to the class Apof Muckenhoupt if −1<s<p−1. Proof: Take two sequences {ak}⊂∂Mand {δk}⊂R+such that •sup a∈∂M,0<δ<√2 M(µs,a,δ) = lim k→∞M(µs,a k,δ k); •lim k→∞ak=a0, lim k→∞δk=δ0. There are three cases to consider. Case (1): δ0>0. Then sup a∈∂M,0<δ<√2 M(µs,a,δ)=M(µs,a 0,δ 0). Since s>−1 and −s p−1>−1, it follows from Lemma 4.1 that  Q(a0,δ0) µ(w)dσ(w)<∞and  Q(a0,δ0) µ(w)−1 p−1dσ(w)<∞. Hence M(µs,a 0,δ 0)<∞. Case (2): δ0= 0 and a0∈∂Mn−1. Then sup a∈∂M,0<δ<√2 M(µs,a,δ)≤lim sup δ→0 M(µs,a 0,δ). Applying Lemma 4.1 to the equation (4.11), we obtain lim sup δ→0 M(µs,a 0,δ)≤C5(s)$C5−s p−1%p−1 <∞. 68 N. Viˆ et Anh Case (3): δ0= 0 and a0∈ ∂Mn−1. It is easy to see that lim δ→0M(µs,a 0,δ)=|a0 n+1|2s|a0 n+1|−2s p−1p−1=1. In any case we always have sup a∈∂M,0<δ<√2 M(µs,a,δ)<∞, this establishes the lemma. If u∈C(M), the radial maximal function Mradu:∂M→[0,∞] is defined by (Mradu)(w) := sup 1 2<r<1|u(rw)|,∀w∈∂M.(4.12) Lemma 4.3. To every p>0and α>1corresponds a constant C(α, p)< ∞such that |(Mαu)(w)|p<C(α, p)·M(|Mradu|p)(w), for every u∈H(M)and w∈∂M. Proof: For z∈M, let Tzbe the complex tangent space to Mat zand let πzbe the orthogonal projection of Cn+1 onto Tz. Write z=rζ,ζ∈∂M. Pick the vectors ζ2,...,ζ n,ζ n+1 so that {ζ,ζ2,...,ζ n}is an orthonormal basis for Tzand {ζ,ζ2,...,ζ n,ζ n+1}is an orthonormal basis for Cn+1. For any δ>0, consider P(z,δ):=  w=rζ+λζ + n+1  j=2 λjζj:|λ|<δ, |λj|<δ1 2,j=2,...,n+1   , & P(z,δ):=  w=rζ+λζ + n  j=2 λjζj:|λ|<δ, |λj|<δ1 2,j=2,...,n   ⊂Tz. The polydiscs P(z,δ) were considered in the work of Ahern-Bruna [1, p. 132]. Since ∂Mis a subset of the unit sphere of Cn+1, it follows from Lemma 3.5 of [1] that for each α,β,1<α<β, there is an 0>0 such that if η,ζ ∈∂Mand z=rζ ∈Dα(η) then (4.13) P(z,0(1 −r2)) ∩M ⊂'z∈Dβ(η):1 2(1 −r2)<1−|z|2<2(1 −r2)(. Fatou and Kor´ anyi-V´ agi Type Theorems 69 Fix a point z0∈∂M. In a sufficiently small compact neighborhood U of z0in M, we can choose the vectors ζ2,...,ζ n,ζ n+1 so that they all depend smoothly on z∈U. By shrinking Uwe see that there exists an δ0>0 such that & P(z,δ)⊂πz(P(z,2δ)∩M),∀z∈U,0<δ<δ 0.(4.14) By Lemma 2.5 of [1], we have (4.15) |u(z)|p≤C δn+1  & P(z,δ) |(u◦(πz)−1)(ζ)|pdVz(ζ), for z∈Uand 0 <δ<δ 0. Here dVzis the Lebesgue measure on the hyperplane Tzand the constant Cis independant of z∈U. Choosing := min{0,δ 0}, it follows from (4.14) and (4.15) that |u(z)|p≤C (1 −r)n+1  P(z,(1−r2))∩M |u(w)|pdV (w),for z∈U.(4.16) Here dV is the surface measure on the complex manifold Mand the constant Cis independant of z∈U. Since the group O(n+1,R) acts transitively on ∂Mand dV is O(n+ 1,R)-invariant, we conclude that (4.16) also holds for all z∈Msufficiently close to ∂M. On the other hand this estimate is semi-trivial if z∈Msatisfies |z|<r 0, for some r0<1 fixed. Hence (4.16) is valid for all z∈M. On account of (4.13), (4.16) and by Proposition 3.3, the lemma is proved exactly as in Lemma 4.4 of [1]. We now come to the the proof of Theorem B. Proof of Theorem B: We begin the proof of part (i) by choosing m∈N such that mp 2−2<s<m 2+1 p−2. Then by Lemma 4.2, the weight function µ(w):=|wn+1|2s−mp+2 belongs to the class Ap. Fix some f∈Hp(B∗,|ζ•ζ|sdθ) and set g(z):=zm n+1f(F(z)),∀z∈M.(4.17) 70 N. Viˆ et Anh Then we obtain g∈Hp(M,|wn+1|2+2s−mp dσ) by virtue of formula (4.2). From the choice of m, it follows that there is an d∈Rsuch that 1 <d<p and (2 + 2s−mp)d<2(p−d). Therefore, by H¨older’s inequality, we obtain, for every 0 <r<1,  ∂M |g(rw)|ddσ(w)≤  ∂M |g(rw)|p|wn+1|2+2s−mp dσ(w)  d p ·  ∂M |wn+1|−(2+2s−mp)d p−ddσ(w)  p−d p ≤Cgd Hp(M,|wn+1|2+2s−mp dσ)<∞, (4.18) where the second estimate holds by using Lemma 4.1. Thus g∈Hd(M,σ). Applying part (i) of Theorem 3.14, we obtain g=PM[g∗].(4.19) Next, we first apply the Kor´anyi type inequality (Theorem 3.7), then Theorem 3 of Calder´on’s work [2] and formula (4.2), and obtain  ∂M |MαPM[g∗](w)|p|wn+1|2+2s−mp dσ(w) ≤C· ∂M |M(g∗)(w)|p|wn+1|2+2s−mp dσ(w) ≤C· ∂M |g∗(w)|p|wn+1|2+2s−mp dσ(w)<∞. (4.20) On the other hand, in view of (4.17) and (4.19), we see that for every w∈∂Mand α>2, |MαPM[g∗](w)|=|Mαg(w)|≥C|wn+1|msup 1 2<r<1|f(F(rw))|.(4.21) Fatou and Kor´ anyi-V´ agi Type Theorems 71 Combining the estimates (4.20), (4.21) and formulas (4.12) et (4.17), we obtain (4.22)  ∂M |Mrad(f◦F)(w)|p|wn+1|2s+2 dσ(w) ≤C· ∂M |(f◦F)∗(w)|p|wn+1|2s+2 dσ(w). Choose q∈Rsuch that q>max{1,s+2}.By Lemma 4.3, we get (Mα(f◦F)(w))p q≤CM |Mrad(f◦F)|p q(w),∀w∈∂M. Since by Lemma 4.2 the weight function µ(w):=|wn+1|2s+2 belongs to the class Aq, using the latter estimate and applying again Theorem 3 of [2], it follows that  ∂M (Mα(f◦F)(w))p|wn+1|2s+2 dσ(w) ≤C· ∂MM|Mrad(f◦F)|p q(w)q |wn+1|2s+2 dσ(w) ≤C· ∂M |Mrad(f◦F)(w)|p|wn+1|2s+2 dσ(w). (4.23) We deduce from estimates (4.22) and (4.23) that  ∂M |Mα(f◦F)(w)|p|wn+1|2s+2 dσ(w) ≤C· ∂M |(f◦F)∗(w)|p|wn+1|2s+2 dσ(w). Part (i) of the theorem now follows immediately from (4.2), (4.4) and the latter estimate. We now turn to prove part (ii). We have already observed in the proof of part (i) of Theorem A that for every 0 ≤r<1 and α>2, rζ ∈D∗ α(ζ) and the dilated functions frconverge to f∗θ-almost every where on ∂B∗ as r→1−. Hence in view of part (i) of Theorem B and applying the dominated convergence theorem, we see that lim r→1−f∗−frLp(∂B∗,|ζ•ζ|sdθ)=0.(4.24) 72 N. Viˆ et Anh To prove the second equality of part (ii), we remark that lim r→1−frp Lp(∂B∗,|ζ•ζ|sdθ) =Clim r→1− ∂M |(fr◦F)(w)|p|wn+1|2s+2 dσ(w) =Clim r→1− ∂M |wn+1|2s+2 dσ(w)1 2π π  −π |(fr◦F)(eiηw)|pdη. (4.25) Since the function |fr◦F|pis subharmonic, we deduce from (4.24) and (4.25) that f∗Lp(∂B∗,|ζ•ζ|sdθ)= lim r→1−frLp(∂B∗,|ζ•ζ|sdθ)=fHp(B∗,|ζ•ζ|sdθ). This completes the proof of part (ii). We now arrive at the proof of Theorem C. Proof of Theorem C: We remark that if −2<s<3p 2−2 and f∈ Lp(∂B∗,|ζ•ζ|sdθ), then by virtue of formula (4.2) and Lemma 4.1 we have that,  ∂B∗ |ζ•ζ|sdθ(ζ)= 1 C3 ∂M |wn+1|2s+2 dσ(w)<∞, and  ∂M |Tf|dσ ≤  ∂M |wn+1|p−(2s+2) p−1dσ(w)  p−1 p ·  ∂M |(f◦F)(w)|p|wn+1|2s+2 dσ(w)  1 p ≤CfLp(∂B∗,|ζ•ζ|sdθ), where the second estimate follows from applying H¨older’s inequality. Therefore, equation (4.3) holds for every f∈Lp(∂B∗,|ζ•ζ|sdθ) with −2<s<3p 2−2. Fatou and Kor´ anyi-V´ agi Type Theorems 73 For every 0 <r<1, consider the integral operator f−→ Sr[f] given by Sr[f](ζ):= ∂M SM(rζ, w)f(w)dσ(w),∀ζ∈∂M,∀f∈L1(∂M,σ). Observe that by virtue of estimate (3.12) and Theorem 3.15, the integral operator Sris a singular integral in the space of homogeneous type (∂M,d,σ). Consequently, we may apply the weighted theory of singular integral in [3]toSr. From the hypothesis on sand by Lemma 4.2, the weight function µ(w):=|wn+1|2s−p+2 belongs to the class Ap. Then it follows from [3] that (4.26) sup 0<r<1 ∂M |SM[Tf](rw)|p|wn+1|2s−p+2 dσ(w) ≤C ∂M |Tf(w)|p|wn+1|2s−p+2 dσ(w), for every f∈Lp(∂B∗,|ζ•ζ|sdθ). Using (4.1)–(4.3) and the remark made at the beginning of our proof, we obtain (4.27)  ∂M |SM[Tf](rw)|p|wn+1|2s−p+2 dσ(w) =Crp ∂B∗ |SB∗[f](rζ)|p|ζ•ζ|sdθ(ζ), and  ∂M |Tf(w)|p|wn+1|2s−p+2 dσ(w)=C ∂B∗ |f(ζ)|p|ζ•ζ|sdθ(ζ).(4.28) Combining (4.26)–(4.28), we get sup 1 2<r<1 ∂B∗ |SB∗[f](rζ)|p|ζ•ζ|sdθ(ζ)≤C ∂B∗ |f(ζ)|p|ζ•ζ|sdθ(ζ).(4.29) Consider the function g∈H(B∗) given by g:= SB∗[f]. By virtue of (4.29), gis in Hp(B∗,|ζ•ζ|sdθ). Therefore, Theorem B, applied to g, gives that  ∂B∗ |Mαg(ζ)|p|ζ•ζ|sdθ(ζ)≤Csup 1 2<r<1 ∂B∗ |g(rζ)|p|ζ•ζ|sdθ(ζ).(4.30)