THE DUREN-CARLESON THEOREM IN TUBE DOMAINS OVER SYMMETRIC CONES 6 1 0 DAVID BE´KOLLE´, BENOIT F. SEHBA, AND EDGAR L. TCHOUNDJA 2 n Abstract. In the setting of tube domains over symmetric cones, we study the a characterizationofthe positiveBorelmeasuresµ forwhichthe HardyspaceHp is J continuously embedded into the Lebesgue space Lq(TΩ,dµ), 0 0, y > 0 ; 0 n { 1 ··· n ∈ 1 −···− n 1 } 6 its determinant function is given by the Lorentz form 1 : ∆(y) = y2 y2. v 1 −···− n i rX For 0 < q < ∞ and ν ∈ R, let Lqν(TΩ) = Lq(TΩ,∆ν−nr(y)dxdy) denote the space a of measurable functions f satisfying the condition 1/q kfkq,ν = ||f||Lqν(TΩ) := |f(x+iy)|q∆ν−nr(y)dxdy < ∞. (cid:18)ZTΩ (cid:19) ItsclosedsubspaceconsistingofholomorphicfunctionsinT istheweightedBergman Ω space Aq(T ). This space is not trivial i.e Aq(T ) = 0 only for ν > n 1 (see ν Ω ν Ω 6 { } r − [10], cf. also [1]). The Bergman projector P is the orthogonal projector from ν the Hilbert-Lebesgue space L2(T ) to its closed subspace A2(T ). The usual (un- ν Ω ν Ω weighted) Bergman space Aq(T ) corresponds to the case ν = n. Ω r As usual, we write in the sequel V = Rn. By Hp(T ), 0 < p < , we denote Ω ∞ the holomorphic Hardy space on the tube domain that is the space of holomorphic Key words and phrases. Symmetric cones, Hardy spaces, Bergman spaces. 1 2 D. BE´KOLLE´, B. F. SEHBA, ANDE. L. TCHOUNDJA functions f such that 1/p f = sup f(x+it) pdx < . Hp k k | | ∞ (cid:18)t∈Ω ZRn (cid:19) Let 0 < p < q < . Our purpose is to characterize those positive Borel measures µ ∞ onT forwhich theHardyspace Hp(T )iscontinuously imbedded intotheLebesgue Ω Ω space Lq(T ,dµ).We recall that given two Banach spaces of functions X and Y with Ω respective norms and , it is said that X embeds continuously into Y X Y k · k k · k (X ֒ Y), if there exists a constant C > 0 such that for any f X, → ∈ f C f . Y X k k ≤ k k When we test on the functions z w¯ G(z) = Gw(z) := [∆−ν−nr( − )]1q, 2i with w = u+iv T , a necessary condition is the existence of a positive constant Ω ∈ C such that p,q,µ z w¯ (1) ∆−ν−nr( − ) dµ(z) Cp,q,µ∆−(ν+nr)+nrpq(v) | 2i | ≤ ZTΩ whenever n p 2n (ν + ) > 1. r q r − Ourfirstresultreducesourproblemtothestandardmeasuresdµ(x+iy) = ∆nr(pq−2)(y)dxdy when q > 2 r. We generalize to tube domains over symmetric cones a result due p − n to O. Blasco [6] (cf. also [7]) for the unit disc. Theorem 1.1. Let 0 2 r. Let ν R be such that ∞ p − n ∈ (ν + n)p > 2n 1. The following two assertions are equivalent. r q r − (i) Hp(T ) ֒ Lq(T ,dµ) if (and only if) there exists a positive constant C Ω Ω p,q,µ → such that estimate (1) holds for every w = u+iv T . Ω (ii) Hp(T ) is continuously embedded into Aq (T ∈). Ω n(q−1) Ω r p Remark 1.2. For n = r = 1 (the case of the upper half-plane, Ω = (0, )), ∞ assertion (i) of the theorem was proved by P. Duren [12] (cf. also [11]), using a modification of the argument given by L. Carleson [9] in the case p = q = 2; assertion (ii) was proved earlier by Hardy and Littlewood [14]. In section 3, we shall prove Theorem 1.1 in a more general form where ν is a vector of Rr. Our next result is the following Hardy-Littlewood Theorem. Theorem 1.3. Let 4 p < . Then H2(T ) ֒ Ap (T ). ≤ ∞ Ω → nr(p2−1) Ω In the case where r = 2, it is possible to go below the power p = 4. We have exactly the following. DUREN-CARLESON THEOREM 3 Theorem 1.4. Let r = 2 and n = 3,4,5,6. Then (1) H2(T ) ֒ Ap (T ) for all 8 < p < 4. Λ3 → 3p−3 Λ3 3 4 2 (2) H2(T ) ֒ Ap (T ) for all 3 < p < 4. Λ4 → p−2 Λ4 (3) H2(T ) ֒ Ap (T ) for all 16 < p < 4. Λ5 → 5p−5 Λ5 5 4 2 (4) H2(T ) ֒ Ap (T ) for all 10 < p < 4. Λ6 → 3p−3 Λ6 3 2 Remark 1.5. For every positive integer m 2, it is easy to see that the continuous embedding H2(T ) ֒ Ap (T ) implies≥the continuous embedding H2m(T ) ֒ Amp (T ). TΩ → (p2−1)nr Ω TΩ → (p−1)n Ω 2 r Recallthatgiventwo BanachspacesofanalyticfunctionsX andY withrespective norms and , we say an analytic function G is a multiplier from X to Y, X Y k·k k·k if there exists a constant C > 0 such that for any F X, ∈ FG C F . Y X k k ≤ k k We denote by (X,Y) the set of multipliers from X to Y. Let α R. WMe denote by H∞(T ), the Banach space of analytic functions F on ∈ α Ω T such that Ω F := sup ∆( z)α F(z) < . α,∞ k k ℑ | | ∞ z∈TΩ In particular, for α = 0, the space H∞(T ) is the space H∞ of bounded holomorphic 0 Ω functions on T . The above results allow us to obtain the following characterization Ω of pointwise multipliers from H2(T ) to Ap(T ). Ω ν Ω Theorem 1.6. Let 4 p < , ν > n 1. Define γ = 1(ν+ n) n. Then for any ≤ ∞ r − p r − 2r integer m 1, the following assertions hold. ≥ (a) If γ > 0, then (H2m(T ),Apm(T )) = H∞(T ). M Ω ν Ω mγ Ω (b) If γ = 0, then (H2m(T ),Amp(T )) = H∞(T ). M Ω ν Ω Ω (c) If γ < 0, then (H2(T ),Ap(T )) = 0 M Ω ν Ω { } For p < 4, we have under further restrictions the following. Theorem 1.7. Let 2(2−nr) nr −1. Assume that P(p2−1)nr is bounded on Lp (T ). Define γ = 1(ν + n) n. Then for any integer m 1, the following (p2−1)nr Ω p r − 2r ≥ assertions hold. (a) If γ > 0, then (H2m(T ),Amp(T )) = H∞(T ). M Ω ν Ω mγ Ω (b) If γ = 0, then (H2m(T ),Amp(T )) = H∞(T ). M Ω ν Ω Ω (c) If γ < 0, then (H2m(T ),Amp(T )) = 0 M Ω ν Ω { } Finally we particularize the previous problems to the tube domain over the light cone Λ . We take advantage of the geometry of this cone to prove the following n restricted Hardy-Littlewood Theorem. The point here is that the exponent p is no more restricted to the set of even positive integers and the exponents p and q are 4 D. BE´KOLLE´, B. F. SEHBA, ANDE. L. TCHOUNDJA just related by the inequality 1 p < q < . We say that a subset B of the Lorentz ≤ ∞ cone Λ is a restricted region with vertex at the origin O if the Euclidean distance n of any point of B from O is less that a multiple of the Euclidean distance of that point from the boundary of Λ . We denote T the tube domain over B. n B Theorem 1.8. Let 1 p < q < . Then given each restricted region B of the ≤ ∞ Lorentz cone Λ with vertex O, there exists a positive constant C such that n p,q |F(z)|q∆n2(pq−2)(y)dxdy ≤ Cp,q,γ||F||qHp ZTB for all F Hp(T ). ∈ Λn The plan of this paper is as follows. In section 2, we present some preliminary results. The Blasco Theorem 1.1 is proved in section 3. The Hardy-Littlewood Theorems 1.3 and 1.4 are established in section 4. Section 5 is devoted to the proof of Theorem 1.6. The proof of the restricted Hardy-Littlewood Theorem 1.8 is given in section 6 while in section 7, we pose some opem questions related this work. 2. Preliminaries and useful results Materials of this section are essentially from [13]. We give some definitions and useful results. In this section, Ω is an irreducible open cone of rank r in Rn. We recall that Ω induces a structure of Euclidean Jordan algebra in V Rn, in which the closure of ≡ Ω is exactly the set x2 : x V . Let us denote by e the identity element in V and { ∈ } by G(Ω) the group of transformations of Ω. The canonical inner product in V is given by (x/y) = tr(xy). We denote by G the identity component in G(Ω). Since Ω is homogeneous, G acts transitively on Ω and there is a subgroup H of G which acts simply transitively on Ω. That is for every y Ω, there exists a unique h H such that y = he. ∈ ∈ Fix a Jordan frame c , ,c in V; that is a systems of primitive idempotents 1 r { ··· } such that c + +c = e and c c = 0, i = j. 1 r j j ··· 6 Then the induced Pierce decomposition of V is V = V . 1≤i≤j≤r i,j ⊕ For x V, we denote by ∆ (x) the determinant of the projection P x of x, in k k ∈ the Jordan subalgebra V(k) = V . Note that ∆ (x), ,∆ (x) are the 1≤i≤j≤k i,j 1 r ⊕ ··· principal minors of x V with respect to the above Jordan frame. It is known that ∈ for x Ω, ∆ (x) > 0, k = 1, ,r. We observe that ∆ = ∆ . The generalized k r ∈ ··· power function on Ω is defined as ∆s(x) = ∆s11−s2(x)∆s22−s3(x)···∆srr(x), s(s1,s2,··· ,sr) ∈ Cr, x ∈ Ω. DUREN-CARLESON THEOREM 5 We now recall the definition of the generalized gamma function on Ω: ΓΩ(s) = e−(e|ξ)∆s(ξ)∆−n/r(ξ)dξ (s = (s1, ,sr) Cr). ··· ∈ ZΩ We set d := nr−1. The above integral converges if and only if s > (j 1)d, 2 r−1 ℜ j − 2 for all j = 1, ,r. Being in this case it is equal to: ··· r n−r d ΓΩ(s) = (2π) 2 Γ(sj (j 1) ) − − 2 j=1 Y (see Chapter VII of [13]). In the same way we denote by ∆⋆(x), ,∆⋆(x) the principal minors of x V with respect to the fixed Jordan fram1e c ,·c·· , r ,c and for s = (s ,...,s )∈ Cr, we r r−1 r 1 r { ··· } ∈ let ∆⋆(x) = ∆⋆(x)s1−s2∆⋆(x)s2−s3, ,∆⋆(x)sr. s 1 2 ··· r We refer to [13, Proposition VII.1.2 and Proposition VII.1.6] for the following result on the Laplace transform of the generalized power function. Lemma 2.1. Let s = (s1,...,sr) ∈ Cr with ℜsj > (j−1)d2 for all j = 1,...,r. Then, for all y Ω we have ∈ e−(y|ξ)∆s(ξ)∆−nr(ξ)dξ = ΓΩ(s)∆⋆−s⋆(y) ZΩ with s⋆ = (s ,...,s ). r 1 We extend the definition of the generalized power function to T as follows. Ω Definition 2.2. Fors = (s ,...,s ) Cr such that s > (r j)d. for all j = 1,...,r. 1 r ∈ ℜ j − 2 We define a holomorphic extension to TΩ of the function ∆s(y), y Ω, by ∈ z ∆−s( ) := e−(zi|ξ)∆⋆s⋆(ξ)∆−nr(ξ)dξ. i ZΩ We will be using the following definition of the β-function of the symmetric cone Ω. BΩ(p,q) = ∆p−nr(x)∆q−nr(e x)dx, − ZΩ∩(e−Ω) where p and q are in Cr. The above integral converges absolutely if p > n 1 and ℜ r − q > n 1, and in this case, ℜ r − Γ (p)Γ (q) Ω Ω B (p,q) = Ω Γ (p+q) Ω (see [13, Theorem VII.1.7]). We refer also to [13, Theorem VII.1.7] for the following result. 6 D. BE´KOLLE´, B. F. SEHBA, ANDE. L. TCHOUNDJA Lemma 2.3. Let p,q C with p > n 1 and q > n 1. Then, for all y Ω ∈ ℜ r − ℜ r − ∈ we have ∆p−n(x)∆q−n(u x)dx = BΩ(p,q)∆p+q−nr(u). r r − ZΩ∩(u−Ω) The following is [1, Proposition 3.5]. Lemma 2.4. Let 1 p < and ν > n 1. Then there is a constant C > 0 such ≤ ∞ r − that for any f Ap(T ) the following pointwise estimate holds: ∈ ν Ω (2) f(z) C∆−p1(ν+nr)( z) f p,ν, for all z TΩ. | | ≤ ℑ k k ∈ We refer to [10] for the following, whose proof relies on the previous lemma. Lemma 2.5. Let 1 p,q < , α,β > n 1. Then Ap(T ) ֒ Aq(T ) if and only ≤ ∞ r − α Ω → β Ω if 1(α+ n) = 1(β + n). p r q r From the above lemma, we deduce that to prove Theorem 1.3, it is enough to do this for p = 4. We will make use of Paley-Wiener theory in the next section to prove Theorem 1.3 and Theorem 1.4. The following can be found in [13]. Theorem 2.6. For every F H2(T ) there exists f L2(Ω) such that Ω ∈ ∈ 1 F(z) = ei(z|ξ)f(ξ)dξ, z T . (2π)n ∈ Ω 2 ZΩ Conversely, if f L2(Ω) then the integral above converges absolutely to a function ∈ F H2(T ). In this case, F = f . Ω H2 L2(Ω) ∈ || || || || In the sequel, we write V = Rn. For the proofs of the following two lemmas, cf. e.g. [15]. Lemma 2.7. Let s = (s ,...,s ) Rr and define 1 r ∈ x+iy Is(y) := ∆−s( ) dx for y R. | i | ∈ Rn Z Then Is(y) is finite if and only if ℜsj > (r − j)d2 + nr. In this case, Is(y) = C(s)(∆−s∆nr)(y). Furthermore, the function F(z) = Fw(z) = ∆−s(z−2iw¯) (w = u+iv fixedin TΩ) is in Hp(TΩ) wheneverℜsj > (r−j)pd2+nr .In thiscase, wehave||F||Hp = C(s,p)(∆−s∆rnp)(v). Lemma 2.8. Let v T and s = (s ,...,s ),t = (t ,...,t ) Cr. The integral Ω 1 r 1 r ∈ ∈ ∆−s(y +v)∆t(y)dy ZΩ converges if t > (j 1)d n et (s t ) > n +(r j)d. In this case this integral ℜ j − 2− r ℜ j− j r − 2 n is equal to Cs,t(∆−s+t∆r)(v). DUREN-CARLESON THEOREM 7 We denote as in [1] L2 (Ω) = L2(Ω;∆−ν(2ξ)dξ). −ν The following Paley-Wiener characterization of the space A2(T ) can be found in ν Ω [13]. Theorem 2.9. For every F A2(T ) there exists f L2 (Ω) such that ∈ ν Ω ∈ −ν 1 F(z) = ei(z|ξ)f(ξ)dξ, z T . n Ω (2π) ∈ 2 ZΩ Conversely, if f L2 (Ω) then the integral above converges absolutely to a function ∈ −ν F A2(T ). In this case, F = f . ∈ ν Ω || ||p,ν || ||L2−ν The (weighted) Bergman projection P is given by ν P f(z) = K (z,w)f(w)dV (w), ν ν ν ZTΩ where K (z,w) = c ∆−(ν+n)(z−w) is the Bergman kernel, i.e the reproducing kernel ν ν r 2i of A2(T ) (see [13]). Here, we use the notation dV (w) := ∆ν−n(v)dudv, where ν Ω ν r w = u + iv is an element of T . For ν = n, we simply write dV(w) instead Ω r of dVn(w).The positive Bergman operator P+ is defined by replacing the kernel ν r function by its modulus in the definition of P . ν In the particular case of the tube domain over the Lorentz cone Λ on Rn, the n following theorem is a consequence of results of [2] and the recent l2-decoupling theorem of [8]. Theorem 2.10. Let ν > n 1. Then the Bergman projector P of T admits a 2 − ν Λn bounded extension on Lp(T ) if and only if ν Λn ν +n 1 (1 ν) p′ 2. Let ν > n 1. r − Then the Bergman projector P of T admits a bounded extension on Lp(Ω) if ν Ω ν ν q′ < p < q := 2+ . ν ν n 1 r − We will sometimes face situations where the weight of the projection differs from the weight associated to the space. We then need the following result (see [16]). Proposition 2.12. Let 1 p < , ν R, and µ > n 1. Then P+ is bounded on ≤ ∞ ∈ r − µ Lp(T ) if and only if 1 n 1 max 1,p 1 . ν Ω ν − r − { − } (cid:0) (cid:1) 8 D. BE´KOLLE´, B. F. SEHBA, ANDE. L. TCHOUNDJA Definition 2.13. The generalized wave operator ✷ on the cone Ω is the differential operator of degree r defined by the equality ✷ [ei(x|ξ)] = ∆(ξ)ei(x|ξ) where ξ Rn. x ∈ When applied to a holomorphic function on T , we have ✷ = ✷ = ✷ where Ω z x z = x+iy. We observe with [3, 4, 1] the following. Theorem 2.14. Let 1 n 1. ∞ r − (1) There exists a positive constant C such that for every F Ap, ∈ ν ✷F C F . p,ν+p p,ν k k ≤ k k (2) If moreover p 2, the following two assertions are equivalent. ≥ (i) P is bounded on Lp(T );. ν ν Ω (ii) For some positive integer m, the differential operator ✷(m) := ✷ ... ✷ (m times) : Ap Ap is a bounded isomorphism. ◦ ◦ ν → ν+mp Let us finish this section by the following result on complex interpolation of Bergman spaces of this setting. Proposition 2.15. Let 1 p n 1. Assume that for some ≤ 0 1 ∞ 0 1 r − µ > n 1, the projection P is bounded on both Lp0(T ) and Lp1(T ). Then for r − µ ν0 Ω ν1 Ω any θ (0,1), the complex interpolation space [Ap0,Ap1] coincides with Ap with ∈ ν0 ν1 θ ν equivalent norms, where 1 = 1−θ + θ and ν = 1−θν + θ ν . p p0 p1 p p0 0 p1 1 (cid:3) Proof. Consult e.g. [5]. 3. Proof of the Blasco Theorem. 3.1. Proof of Theorem 1.1. By Lemma 2.7, the function −ν1−ν2 −νr−1−νr −νr+nr z w¯ G(z) = G (z) := (∆ q ...∆ q ∆ q )( − ) w 1 r−1 r 2i with w = u+iv T , belongs to Hp(T ) if and only if (ν +n)p > (r j)d+n (j = ∈ Ω Ω j r q − 2 r 1,...,r). Moreover −ν1−ν2 −νr−1−νr −νr+nr +n G = C (∆ q ...∆ q ∆ q rp)(v). || ||Hp(TΩ) p,q 1 r−1 r So for these ν, a necessary condition for the continuous embedding p(T ) ֒ Ω H → Lq(T ,dµ) is the existence of a positive constant C such that Ω p,q,µ (3) (∆−(ν1−ν2)...∆−(νr−1−νr)∆−νr−nr)(z −w¯) dµ(z) C (∆−(ν1−ν2)...∆−(νr−1−νr)∆−νr−nr+nrpq)(v) | 1 r−1 r 2i | ≤ p,q,µ 1 r−1 r ZTΩ for every w = u+iv T . Ω ∈ We state Theorem 1.1 in the following more general form. DUREN-CARLESON THEOREM 9 Theorem 3.1. Let 0 2 r. Let ν = (ν ,...,ν ) Rr ∞ p − n 1 r ∈ be such that (ν + n)p > (r j)d + n (j = 1,...,r). The following two assertions are j r q − 2 r equivalent. (i) Hp(T ) ֒ Lq(T ,dµ) if ( and only if) there exists a positive constant C Ω Ω p,q,µ → such that the estimate (3) holds for every w = u+iv T . Ω (ii) Hp(T ) is continuously embedded into Aq (T ). ∈ Ω n(q−1) Ω r p Proof. We first show the implication (i) (ii). We must prove that the measure dµ(x+iy) = ∆nr(qp−2)(y)dxdy satisfies the⇒estimate (3), i.e. L := (∆−(ν1−ν2)...∆−(νr−1−νr)∆r−νr−nr))(z −w¯) ∆nr(pq−2)dxdy | 1 r−1 2i | ZTΩ C (∆−(ν1−ν2)...∆−(νr−1−νr)∆−νr−nr+nrpq)(v). ≤ p,q,µ 1 r−1 r By Lemma 2.7, we have (∆−(ν1−ν2)...∆−(νr−1−νr)∆−νr−nr)(x+iy −w¯) dx | 1 r−1 r 2i | Rn Z = C ∆−(ν1−ν2)...∆−(νr−1−νr)∆−νr(y +v) ν 1 r−1 r whenever ν > (r j)d (j = 1,...,r). Moreover by Lemma 2.8, we have j − 2 (∆−(ν1−ν2)...∆−(νr−1−νr)∆−νr(y +v))∆nr(pq−2)dy 1 r−1 r ZΩ = (∆−(ν1−ν2)...∆−(νr−1−νr)∆−νr+nr(pq−1))(v) 1 r−1 r whenever q > 2 r and ν > n(q 1)+(r j)d (j = 1,...,r). p − n j r p − − 2 We next show the implication (ii) (i). We shall use the following lemma. ⇒ Lemma 3.2. Let q > 0 and let ν = (ν ,...,ν ) Rr. There exists a positive constant 1 r ∈ C such that for every F ol(T ) we have q,ν Ω ∈ H F(u+iv) q(∆ν1−ν2...∆νr−1−νr∆νr−nr(v) F(z) q C | | 1 r−1 r dudv. | | ≤ q,νZTΩ |(∆1ν1−ν2...∆rν−r−11−νr∆rνr+nr)(x+i2yi−w¯)| Proof of the lemma. We denote B(ζ,ρ) the Bergman ball with centre ζ and radius ρ. Since F q is plurisubharmonic, we have | | dudv F(ie) q C F(u+iv) q . | | ≤ ZB(ie,1)| | ∆2rn(v) 10 D. BE´KOLLE´, B. F. SEHBA, ANDE. L. TCHOUNDJA Recall that dudv is the invariant measure on T . Let z T and let g be an affine ∆2rn(v) Ω ∈ Ω automorphism of T such that g(ie) = z. We have Ω F(z) q = (F g)(ie) q | | | ◦ | dudv C (F g)(u+iv) q ≤ | ◦ | ∆2n(v) ZB(ie,1) r dudv = C F(u+iv) q . | | ∆2n(v) ZB(z,1) r We recall that ∆ (z−w¯) ∆ (v) for all w = u+iv B(z,1). This implies that | j 2i | ≃ j ∈ F(z) q C |F(u+iv)|q(∆ν11−ν2...∆νrr−−11−νr∆rνr−nr)(v)dudv | | ≤ Cq,νRB(z|,F1)(u+|(i∆v)ν1|1q−(∆ν2ν1.1..−∆ννr2r−.−.1.∆1−νrν−rr−1∆1−rνrν+r∆nrrν)r(−x+nri2)yi(−vw)¯d)|udv. ≤ q,ν TΩ |(∆1ν1−ν2...∆νrr−−11−νr∆νrr+nr)(x+i2yi−w¯)| R (cid:3) Let us set dµ(z) I(w) := ZTΩ |(∆1ν1−ν2...∆rν−r−11−νr∆rνr+nr)(x+i2yi−w¯)| and recall that for ν = (ν , ,ν ) Rr, 1 r ··· ∈ ∆ (v) = (∆ν1−ν2...∆νr−1−νr∆νr−nr)(v). ν 1 r−1 r Using the Fubini-Tonelli Theorem, it follows from the previous lemma that F(z) qdµ(z) C I(u+iv) F u+iv) q∆ (v)dudv q,ν ( ν | | ≤ | | ZTΩ ZTΩ Cp,q,µ ∆nr(pq−2)(v) F(u+iv) qdudv. ≤ | | ZTΩ An application of the assertion (ii) of the theorem implies that F(z) qdµ(z) C F q . | | ≤ p,q,ν|| ||Hp ZTΩ This finishes the proof of the implication (ii) (i). ⇒ (cid:3) 4. Proofs of the Hardy-Littlewood Theorems. 4.1. Proof of Theorem 1.4. We now give a proof of the following result which is sufficient in proving Theorem 1.3 as remarked in section 2. Theorem 4.1. We have that 2(T ) ֒ A4(T ). Ω Ω H →

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