The Duren-Carleson theorem in tube domains over symmetric cones

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by  David Békollé

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<p<q < . Ex-
0 ∞
tendingaresultduetoBlascofortheunitdisc,wereducetheproblemtostandard
2
measures. We obtain that a Hardy space Hp, 1 p < , embeds continuously
≤ ∞
] in weighted Bergman spaces with larger exponents. Finally we use this result to
A characterize multipliers from H2m to Bergman spaces for every positive integer
C m.
.
h
t
a
m
1. Introduction and statements of the results
[

2 Our settings are tube domains TΩ = V + iΩ, where V is an Euclidean space of
v dimension n, and Ω is an irreducible symmetric cone in the complexification VC of
9
V. As in [13], r is the rank of the cone Ω while ∆ is the determinant function of V.
9
8 In the case V = Rn, a typical example of an irreducible symmetric cone of rank 2 is
4
the forward light cone Λ defined for n 3 by
0 n ≥
1. Λ = (y , ,y ) Rn : y2 y2 > 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 < p < q < be such that q > 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) < p < 4, ν > 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′ < p < p := − − +.
ν ν n 1 − n 1
2 − 2 −
For the other cases we recall the following partial result.
Theorem 2.11. [2], [3]. Let Ω be a symmetric cone of rank > 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 < p < q and µp ν > 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 < p < and ν > 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 < p < , ν ,ν > 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 < p < q < be such that q > 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 →