Multiplier space sufficiency

j=1

bjk_z^α,p_j

B_p

≤ C

∞

j=1

ajk_z^α,p_j

B_p

, whenever |b^j| ≤ |a^j| . (5)

3. {z^j}^∞j=1 = cαj

∞

j=1 where {α^j}^∞j=1 is a sequence in a Bergman tree Tⁿ sat-isfying

β (zi, 0)≤ Cβ (zⁱ, zj) , i = j and (6)

∞

j=1

(1 + d (αj, o))^1−pδαj satisfies the tree condition (??).

1.1 Multiplier space sufficiency

Here we prove that (6) implies (4) for 1 < p < ∞, and also that (3) implies (4) for p > 2n, beginning with the proof that the multiplier interpolation property (4) follows from (6). We generalize the main ideas in Böe’s one-dimensional proof to the unit ball Bⁿ. First we give the following characterization of multipliers in terms of Carleson measures.

Theorem 3 Let ϕ ∈ H^∞(Bⁿ)∩ Bp^σ(Bⁿ) and m + σ > ⁿ_p. Then ϕ is a pointwise multiplier on B_p^σ(Bⁿ), i.e. ϕf _Bσ

p ≤ C f B^σ_p for all f ∈ Bp^σ(Bⁿ), if and only if 1− |z|^{2 m+σ}ϕ^(m)(z)

p

dλn(z) is a B_p^σ(Bⁿ)-Carleson measure on Bⁿ.

Proof. Using

ϕf ^p_Bσ p =

Bn

1− |z|^{2 m+σ}(ϕf )^(m)(z) ^pdλn(z) , together with

(ϕf )^(m)(z) = ϕ^(m)(z) f (z) + ... + ϕ (z) f^(m)(z) ,

we see that the last term ϕ (z) f^(m)(z)can be handled using the boundedness of ϕ, while the first term is handled by the Carleson embedding. The intermediate terms are handled by an interpolation argument - see [8].

For z ∈ Bⁿ and β < 1, define the region V_z^β by

V_z^β = w∈ Bⁿ:|1 − w · P z| ≤ (1 − |z|)^β ,

where P z denotes the radial projection of z onto the sphere ∂Bⁿ. The intersection of V_z^β with the complex line Cz through z and the origin is

w∈ Bⁿ∩ Cz : |w − P z| ≤ (1 − |z|)^β ,

and the intersection of V_z^β with the sphere ∂Bⁿ is an “ellipse” with radius (1− |z|)^β in the radial tangential direction, and radius (1 − |z|)^β2 in the com-plex tangential directions. Using arguments in Marshall and Sundberg [24], the separation condition in (3) implies the following geometric separation conditions.

Lemma 4 Suppose the separation condition in (3) holds. Then there are con-stants 0 < β < 1 < βη < η such that if V_z^β_i ∩ Vz^βj = φ and |z^j| ≥ |zⁱ|, then zⁱ ∈ V/ z^βj

and

(1− |z^j|) ≤ (1 − |zⁱ|)^η. (7) We now fix constants β and η as in Lemma 4, and write Vz = V_z^β.

Lemma 7 below is the key construction in the sufficiency proof and is moti-vated by the formula ((1.35) in [39])

R^s−n,n 1

(1− w · z)^1+s = 1

(1− w · z)^n+1+s, valid for s not a negative integer. The point is that if we define

Γsg (z)≡

Bn

g (w) 1− |w|^{2 s}

(1− w · z)^1+s dw (8)

for a given (not necessarily holomorphic) function g, then with ϕ (z) = Γsg (z), so that ϕ is essentially an n-fold antiderivative of g, we have

R^s−n,nϕ (z) = R^s−n,nΓsg (z) =

Bn

g (w) 1− |w|^{2 s}

(1− w · z)^n+1+sdw, (9) and by the reproducing formula valid for Re s > −1, we also have that

R^s−n,nϕ (z) = cn,s Bⁿ

R^s−n,nϕ (w) 1− |w|^{2 s} (1− w · z)^n+1+s dw.

Thus R^s−n,nϕ (w)behaves morally like g (w), and this provides flexibility in choos-ing g so that ϕ has desirable algebraic multiplier properties on the one hand, while

controlling the multiplier norm of ϕ on the other hand. Indeed, by Theorem 3, the multiplier norm is equivalent to the Carleson norm of 1− |z|^{2 n}∇ⁿϕ (z)

p

dλn(z), which can in turn be dominated by the “tree condition” norm of 1− |z|^{2 n}g (z)

p

dλn(z) by the lemma in the next subsubsection.

Finally, we note that the construction of an approximate zero-one Dirichlet multiplier in Lemma 7 below (which results in a holomorphic function that is close to one on a Carleson tent and close to zero away from it) is a substitute for Jones’ clever construction of an exact zero-one Hardy multiplier in the disk using Blaschke products. In the analogous result on the Bergman tree, the Dirichlet multiplier construction is given simply by defining a sequence h^j ={h^j(α)}_α∈Tn

to satisfy h^j(αj) = 1and to decrease linearly to 0 on a sufficiently small stretch of the geodesic preceding α, and then “linearly” to 0 off the geodesic as well. See [7] for such constructions.

Terminology We say that a measure µ on Bⁿ satisfies the tree condition if its discretization µ does.

It is here that we first use the tree condition in a significant way.

1.1.1 Transformation of Carleson measures

Lemma 5 (analogue of Lemma 2.4 in [12]) Suppose that g satisfies the following reverse Hölder condition on Bergman kubes,

Kα

|g (z)|^pdλn(z)

1 p

≤ C⁰

Kα

|g (z)| dλⁿ(z) , α ∈ Tⁿ, (10) and that the measure

dµ (z) = 1− |z|^{2 n}g (z)

p

dλn(z)

satisfies the tree condition with norm C1. Then for s sufficiently large, both 1− |z|^{2 n}R^s−n,nΓsg (z)

p

dλn(z) and

1− |z|^{2 n}∇ⁿΓsg (z)

p

dλn(z) satisfy the tree condition with norms at most C (C0+ C1).

Note that it then follows that both measures in the conclusion of the lemma are Bp-Carleson measures. The proof of the lemma uses only Schur’s test and lengthy calculations involving discretization of functions and measures, and uni-tary rotations of the Bergman tree. We emphasize again that this is where the

tree condition is used in a significant way. The multiplier norm of ϕ (z) = Γsg (z) is the same as the Carleson measure norm of

1− |z|^{2 m+σ}∇^mΓsg (z) ^pdλn(z) ,

and the tree condition norm of this latter expression is estimated by the tree condition norm of

1− |z|^{2 n}g (z)

p

dλn(z) .

Following Böe [12] one can prove the following alternate version of Lemma 5, where the tree condition is replaced by the Bp-Carleson measure condition, for the range p > 2n with s > n − p¹. This version will be instrumental in proving the implication (3) implies (4) for p > 2n below.

Lemma 6 (another analogue of Lemma 2.4 in [B]) Suppose that sup

ζ∈Bⁿ

1− |ζ|^{2 n}g (ζ) ≤ C⁰, (11) and that the measure

dµ (z) = 1− |z|^{2 n}g (z)

p

dλn(z)

is a Carleson measure for Bp with norm C1. Then for p > 2n and s > n − p¹, both

1− |z|^{2 n}R^s−n,nΓsg (z)

p

dλn(z) and

1− |z|^{2 n}∇ⁿΓsg (z)

p

dλn(z)

are also Carleson measures for Bp, and with norms at most C (C0+ C1).

The simpler Lemma 6 uses the characterization of Bp given by Theorem 6.28 of [39], which states that

f ^p_Bp ≈

Bn Bn

|f (z) − f (w)|^p

|1 − w · z|^2(n+1+t)dνt(z) dνt(w) , (12) provided t > −1 and p > 1, n = 1

2n n > 1. In order to obtain the full range 1 < p < ∞ when n > 1, we instead need to use the the tree condition to obtain Lemma 5.

1.1.2 Multiplier approximations

The next lemma constructs a holomorphic function that is close to 1 on the Carleson region associated to a point w ∈ Bⁿ, and decays appropriately away from the Carleson region. We follow Böe’s proof in [12], which adapts a real-variable argument of Marshall and Sundberg in [24] to produce a holomorphic multiplier approximation. Given β < ρ < α < 1, we will use the cutoff function cρ,α defined by

cρ,α(γ) =





0 for γ < ρ

γ−ρ

α−ρfor ρ ≤ γ ≤ α 1 for α < γ

. (13)

Lemma 7 (analogue of Lemma 4.1 in [12]) Suppose s > −1. There are ρ and α satisfying β < ρ < α < 1 such that for every w ∈ Bⁿ, we can find a function gw

so that

ϕ_w(z) = Γsgw(z) =

Bn

gw(ζ) 1− |ζ|^{2 s} 1− ζ · z ^1+s dζ satisfies











ϕ_w(w) = 1

ϕ_w(z) = cρ,α(γ_w(z)) + O log ¹

1−|w|²

−1 , z ∈ V^w

|ϕw(z)| ≤ C log_1−|w|^{1 2 1−p}, z /∈ V^w

, (14)

where γ_w(z) is defined by

|1 − z · P w| = 1 − |w|^{2 γw(z)} and cρ,α is as in (13). Furthermore we have the estimate

Bⁿ

1− |ζ|^{2 n}gw(ζ) ^pdλn(ζ) dζ ≤ C log 1 1− |w|²

1−p

. (15)

The final estimate (15) will lead to the Carleson measure estimate for 1− |z|^{2 n}g (z)

p

dλn(z) .

Remark 1 The proof of Lemma 7 shows that the third estimate in (14) can be vastly improved, and also holds for a larger range of z; namely there is β < β₁ < ρ such that

|ϕw(z)| ≤ C log 1 1− |w|²

−1

1− |w|^{2 (ρ−β1)(1+s)}, z /∈ Vw^β1. This fact will be used in the proof of Lemma 8 below.

Proof. Define gw(ζ)by gw(ζ) 1− |ζ|^{2 s}

1− ζ · w ^1+s = K log 1 1− |w|²

−1

1− ζ · P w ⁻ⁿ⁻¹, (16) when ζ lives in the annular sector S centred at P w given as the intersection of the annulus

A = A^w = ζ ∈ Bⁿ: 1− |w|^{2 α}≤ 1 − ζ · P w ≤ 1 − |w|^{2 ρ} (17) and the cone

C = C^w = ζ ∈ Bⁿ: Im ζ· P w + ζ − ζ · P w P w ² ≤ c 1 − ζ · P w , where c is a suitably small constant. Define gw(ζ) = 0 otherwise. The following observation will be used repeatedly.

Remark 2 The cone C^w corresponds to the geodesic in the Bergman tree Tⁿ joining the root to the “boundary point” P w. To see this, consider the case w = (t, 0, ..., 0) and ζ = re^iθ, ζ with re^iθ = x + iy, so that Im ζ · P w = y, ζ− ζ · P w P w = (0, ζ ) and 1 − ζ · P w = 1 − r.

Now choose K so that ϕ_w(w) = 1, i.e.

K = log 1

1− |w|² S

1− ζ · P w ⁻ⁿ⁻¹dζ

−1

, which satisfies

K ≈ K^α,ρ,n = cn

α− ρ (18)

since the annular sector

E^a = ζ ∈ Bⁿ: a≤ 1 − ζ · P w ≤ 2a ∩ C

is comparable to a Bergman ball of radius one, 1 − ζ · P w ⁻ⁿ⁻¹dζ is comparable to invariant measure dλn(ζ)on E^a, and S ≈ ∪^Jj=0E₂^j(1−|w|²)^α where

J = log 1− |w|^{2 ρ}

1− |w|^{2 α} = (ρ− α) log 1 − |w|² . Note also that

1− ζ · P w ≈ 1 − ζ · w ≈ 1 − |ζ|², ζ ∈ S, (19)

and so gw satisfies the estimate

|g^w(ζ)| ≤ C log 1 1− |w|²

−1

1− ζ · P w ⁻ⁿ, ζ ∈ Bⁿ. (20) Now fix z ∈ V^w and set

E1= ζ ∈ Bⁿ: 1− ζ · P w ≤ 1 − |w|^{2 γw(z)} ,

E2= Bⁿ\ E¹ = ζ ∈ Bⁿ: 1− ζ · P w > 1 − |w|^{2 γw(z)} .

Thus the common boundary of E1 and E2 passes through z. The main contribu-tion to ϕ_w(z) will come from integration over E2. Thus we write

ϕ_w(z) =

E1

gw(ζ) 1− |ζ|^{2 s} 1− ζ · z ^1+s dζ +

E2

gw(ζ) 1− |ζ|^{2 s}

1− ζ · z ^1+s dζ = I + II.

By (19), (20) and the definition of γ_w(z), term I is dominated by C log ¹

1−|w|²

−1

times

{(^1−|w|2)^α≤|^{1−ζ·P w}|^{≤|1−z·P w|}}^∩C

1− |ζ|² 1− ζ · z

1+s

dλn(ζ) , which is at most a constant C since

1− ζ · z ≈ |1 − z · P w| , ζ ∈ C ∩ E¹. Thus we have

|I| ≤ C log 1 1− |w|²

−1

. We now write

II =

E2∩S

gw(ζ) 1− |ζ|^{2 s} 1− ζ · z ^1+s

dζ

=

E2∩S

gw(ζ) 1− |ζ|^{2 s}

1− ζ · z ^1+s −gw(ζ) 1− |ζ|^{2 s} 1− ζ · w ^1+s dζ +

E2∩S

gw(ζ) 1− |ζ|^{2 s} 1− ζ · w ^1+s

dζ

= III + IV.

The point of isolating term IV is that the variable z occurs there only in the exponent γ_w(z), and this leads to the following exact calculation. Using (18),

and that gw is supported in S, we calculate that term IV is log_1−|w|^{1 2 −1} times K

(1−|w|²)^γw(z)≤|^{1−ζ·P w}|^≤(1−|w|²)^ρ ∩C

1− ζ · P w ⁻ⁿ⁻¹dζ = γ_w(z)− ρ

α− ρ log 1 1− |w|² in the case ρ < γ_w(z) < α. We also have IV = 0 in the case γ_w(z) < ρ, and IV = log ¹

1−|w|² in the case α < γ_w(z). This gives the estimate IV = cρ,α(γ_w(z)) + O log 1

1− |w|²

−1

, z ∈ V^w. Using

1

1− ζ · z ^1+s − 1

1− ζ · w ^1+s ≤ C |z − w|

1− |ζ|^{2 2+s} together with (19) and (20), we obtain that

|III| ≤ C

E2∩S|g^w(ζ)| 1 − |ζ|^{2 −2}|z − w| dζ

≤ C |z − w| log 1 1− |w|²

−1

E2∩S

1− |ζ|^{2 −1}dλn(ζ)

≤ C log 1 1− |w|²

−1

,

as required. This completes the proof of the second estimate in (14).

We now turn to the third estimate in (14). For z /∈ V^w, we have 1 − ζ · z ≥ c 1− |w|^{2 β} for ζ ∈ S, and thus

|ϕw(z)| ≤ log 1 1− |w|²

−1

S

1− |ζ|² 1− ζ · z

1+s

dλn(ζ)

≤ C log 1 1− |w|²

−1

1− |w|2 (ρ−β)(1+s)

≤ C^p log 1 1− |w|²

1−p

.

Finally, the estimate (15) is a calculation using (19), (20) the definition of the support of gw. Indeed, the left side of (15) is at most

C

S

log 1 1− |w|²

−p

dλn(ζ)≤ C log 1 1− |w|²

1−p

.

The next lemma uses Lemma 5 to construct inductively a holomorphic func-tion whose restricfunc-tion to the sequence {z^j}^∞j=1 approximates an arbitrarily pre-scribed bounded sequence ξ_j ^∞_j=1.

Lemma 8 (analogue of Lemma 4.2 in [12]) Suppose s > −1, that ξj

∞

j=1 ∈ ^∞ and let 0 < δ < 1. Let ϕ_j, gj and γ_j correspond to zj as in Lemma 7 and with the same s. Then there is {aⁱ}^∞i=1 ∈ ^∞ such that ϕ = ^∞_i=1aiϕ_i satisfies

ξ_j− ϕ (z^j) ^∞_j=1

∞ < δ ξ_j ^∞_j=1

∞ (21)

and

{aⁱ}^∞i=1 ^∞, ϕ _H∞(Bn) ≤ C ξ_j ^∞_j=1

∞. (22)

Remark 3 The series ^∞_i=1aiϕ_i in Lemma 8 converges absolutely for each z ∈ Bⁿ. In fact, the proof below will show that (using #G ≤ Cβ (0, z ))

∞

i=1

|ϕi(z)| ≤ C 1 + log 1

1− |z|² , z ∈ Bⁿ.

Remark 4 The construction in the proof below shows that both the sequence {aⁱ}^∞i=1 and the function ϕ depend linearly on the data ξ_j ^∞_j=1.

Proof. We follow the proof of Lemma 4.2 in [12]. Let ξ_j ^∞_j=1

∞ = 1. We first choose J so large that

sup

j≥J

log 1 1− |z^j|

−1

+

∞

j=J

log 1 1− |z^j|

1−p

< ε, (23) where ε > 0 will be determined later. Note that the series above converges by the Carleson embedding. By standard arguments in [24], we may discard the finitely many points {z^j}^J−1j=1. Thus we may assume that J = 1 in (23).

Order the points {z^j}^∞j=1 so that 1 − |z^j+1| ≤ 1 − |z^j| for j ≥ 1. We now define a “forest structure” on the index set N by declaring that j is a child of i (or that i is a parent of j) provided that

i < j, (24)

Vzj⊂ V^zi,

Vzj# V^zk for i < k < j.

Note that a child j chooses the “nearest” parent i if we have competing indices i and i with Vzj ⊂ V^zi∩ V^zi . We define a partial order associated with this parent-child relationship by declaring that j is a successor of i (or that i is a predecessor

of j) if there is a “chain” of indices {i = k¹, k2, ..., km = j} ⊂ N such that k⁺¹ is a child of k for 1 ≤ < m. Under this partial ordering, N decomposes into a disjoint union of trees. Thus associated to each index ∈ N, there is a unique tree containing and, unless is the root of the tree, a unique parent P ( ) of in that tree. Denote by G the unique geodesic joining the root of the tree to . We will now define the coefficients {aⁱ}^∞i=1 of ϕ = ^∞_i=1aiϕ_i, where ϕ_i is the function ϕ_zi in Lemma 7 with w there replaced by zi, by considering separately the indices in each tree of the forest N.

Let Y be a tree in the forest N with root k⁰. For each k ∈ Y \ {k⁰}, define β_k ∈ [0, 1] by

β_k= c γ_{P (k)}(zk) ,

where the functions c = cρ,α and γ_j = γ_zj are defined as in the statement of Lemma 7 with w there replaced by zj. Note that by Lemma 7 with w = zP (k), we have the estimate

ϕ_{P (k)}(zk) = c γ_{P (k)}(zk) + O



 log 1

1− z^{P (k) 2}

−1



= β_k+ O



 log 1

1− z^{P (k) 2}

−1

 ,

which can serve as motivation for the definition of the coefficients given below in (26). Indeed, with gross oversimplification, what we want is

ξ_k= ϕ (zk)≈ a^kϕ_k(zk) + aP (k)ϕ_{P (k)}(zk) + ...

≈ a^k+ aP (k)β_k+ ..., which leads to (26).

We will now define numbers {a^k}_k∈Y by induction on the linear ordering in Y induced from the natural ordering of N, so that

|a^k| ≤ 2

i∈Gk\{k0}β_iaP (i) ≤ 1 (25) holds for all k ∈ Y. First define a^k0 = ξ_k0. Now fix ∈ Y \ {k⁰} and assume that ak has been defined for all k ∈ Y for which k < so that (25) holds for all k ∈ Y for which k < . We now define a by

a = ξ −

i∈G \{k0}

β_iaP (i). (26)

Of crucial importance is the observation that the geodesics G and G^{P ( )}are related by

G^{P ( )} =G \ { } ,

i.e. if G = [k⁰, k1, ..., k_m−1, km] with km = , then k_m−1 = P ( ) and G^{P ( )} = [k0, k1, ..., k_m−1]. At this point the reader should draw a picture. The region Vz_kj = V_z^β

kj is essentially a Carleson tent with vertex not at zkj, but rather at the point z_k^β_j lying on the ray through zkj and having distance 1 − z^kj

β to the boundary, much larger than the distance 1 − z^kj from zkj to the boundary. Note that zkj can have infinitely many children in the tree Y, one of which is the point zkj+1 having Vz_kj+1 ⊂ V^z_kj with 1 − z^kj+1 1− z^kj . Finally, note that β_j+1 equals 1 if zkj+1 lies within the smaller Carleson tent V_z^α

kj, β_j+1 equals 0 if zkj+1

lies outside the Carleson tent V_z^ρ

kj, and β_j+1 is defined linearly in (0, 1) for zkj+1

within the annulus of Carleson tents V_z^ρ

kj \ Vz^α_kj. By the induction assumption and the fact that P ( ) ∈ Y and P ( ) < , we have

i∈GP ( )\{k⁰}

β_iaP (i) ≤ 1.

We have from (26) and the above that

i∈G \{k⁰}

β_iaP (i) =





i∈GP ( )\{k⁰}

β_iaP (i)



 + β



ξ_{P ( )}−

i∈GP ( )\{k⁰}

β_iaP (i)





= β ξ_{P ( )}+ (1− β )

i∈GP ( )\{k0}

β_iaP (i)

≤ β ξP ( ) + (1− β )

i∈GP ( )\{k0}

β_iaP (i) ≤ 1.

From this and (26) once more it immediately follows that |a | ≤ 2, which shows that (25) holds for k = as well. This completes the inductive definition of the sequence {a^k}_k∈Y satisfying (25) on the tree Y, and hence defines the entire sequence {aⁱ}^∞i=1.

We omit the proof that both (21) and (22) hold for the function ϕ = ^∞_i=1aiϕ_i. 1.1.3 The proof of multiplier interpolation

Using Lemma 8, we first complete the proof that (6) implies (4) for 1 < p < ∞.

Fix s > −1, 0 < δ < 1 and ξj

∞

j=1 with ξ_j ^∞_j=1

∞ = 1. Then by Lemma 8 there is f1 = ^∞_i=1a¹_iϕ_i ∈ H^∞(Bⁿ) such that ξ_j − f¹(zj) ^∞_j=1

∞ < δ and {a¹i}^∞i=1 ∞, f_{1 H}∞(Bⁿ) ≤ C where C is as in (22). Now apply Lemma 8 to the sequence ξ_j − f¹(zj) ^∞_j=1 to obtain the existence of f2 = ^∞_i=1a²_iϕ_i ∈ H^∞(Bⁿ)

such that ξ_j − f¹(zj)− f²(zj) ^∞_j=1

∞ < δ²and {a²i}^∞i=1 ∞, f_{2 H}∞(Bn) ≤ Cδ where C is as in (22). Continuing inductively, we obtain fm = ^∞_i=1a^m_i ϕ_i ∈ H^∞(Bⁿ) such that

ξ_j −

m

i=1

fi(zj)

∞

j=1 _∞

< δ^m, {a^mi }^∞i=1 ∞, f_{m H}∞(Bn)≤ Cδ^m−1. If we now take ϕ = ^∞_m=1fm, we have

ξ_j= ϕ (zj) , 1≤ j < ∞, (27) ϕ _H∞(Bn)≤ C^δ,

as well as ϕ = ^∞_i=1aiϕ_i with {aⁱ}^∞i=1 ^∞ ≤ C^δ. Recall that the series ϕ =

∞

i=1aiϕ_i converges absolutely by Remark 3, and depends linearly on the data ξ_j ^∞_j=1 by Remark 4 and the linear construction in this paragraph. Thus ϕ ∈ H^∞(Bⁿ) linearly interpolates the values ξ_j ^∞_j=1 on the sequence {z^j}^∞_j=1, and it remains to prove that ϕ ∈ M^Bp. Recall that our function ϕ depends on our choice of s > −1.

By Theorem 3, ϕ ∈ M^Bp will follow if we show that 1− |z|^{2 n}∇ⁿϕ (z)

p

dλn(z)

Carleson ≤ C. (28)

Since

ϕ =

∞

i=1

aiϕ_i =

∞

i=1

aiΓsgi = Γsg

where g = ^∞_i=1aigi with sup_i≥1|aⁱ| ≤ C^δ, (28) will follow from Theorem ?? and Lemma 5 for s sufficiently large provided we show that (10) holds and that

1− |z|^{2 n}g (z)

p

dλn(z)

satisfies the tree condition. From the definition of gi in (16), and the fact that the supports of the gi are pairwise disjoint by the separation condition, we may assume that the reverse Hölder condition on Bergman balls in (10) holds. The tree condition estimate will follow from the next lemma.

Lemma 9 With s > n − p¹ and g = ^∞_i=1aigi as above, we have 1− |z|^{2 n}g (z)

p

dλn(z)

tree condition≤ C. (29)

Proof. Inequality (29) follows from the estimate (15) as follows. If we discretize (15), we obtain with w = zi and S (α) ≈ V^zi,

β∈Tⁿ:β≥α

1− |c^β|^{2 np}gzi(β)^p ≤ C log 1 1− |zⁱ|²

1−p

. (30)

Denote the Carleson tent at cβ by S (β) = Tβ =∪γ≥βKγ. We are assuming that the tree condition holds for the measure ν = ^∞_j=1 log ¹

1−|zj|²

1−pδzj, i.e.

β∈Tn:β≥α





zj∈Tβ

log 1 1− |z^j|²

1−p



p

(31)

=

β∈Tn:β≥α

I^∗ν (β)^p

≤ C^pI^∗ν (α) = C^p

zj∈T^α

log 1 1− |z^j|²

1−p

<∞, α ∈ Tⁿ.

If we now also discretize (29), we see that we must prove

β∈Tⁿ:β≥α

I^∗µ (β)^p ≤ C^pI^∗µ (α) <∞, α ∈ Tⁿ,

where

µ (β) = 1− |c^β|^{2 n}g (β)

p

, g (β) =

Kα

|g| dλⁿ, I^∗µ (α) =

β∈Tⁿ:β≥α

µ (β) ,

for all g = ^∞_i=1aigi as above. Since ν = ^∞_j=1 log ¹

1−|zj|²

1−pδzj satisfies the tree condition, it suffices to prove

β∈Tn:β≥α

I^∗µ (β)^p ≤ C^pI^∗ν (α) , α∈ Tⁿ. (32)

Indeed, (32) shows that µ+ν satisfies the tree condition , and now the equivalence with the Carleson embedding on the tree shows that µ satisfies the tree condition as well (since the Carleson embedding is preserved for smaller measures).

Now g = ^∞_i=1aigi where the supports of the gi = gzi are pairwise disjoint by

Now we use (30) to dominate the last sum above by

C

where the final inequality follows from (31). This establishes (32), and completes the proof of Lemma 9.

With this done, we have completed the proof that (6) implies (4) for 1 < p <

∞.

We now prove that (3) implies (4) for p > 2n. For this we will argue as above but with Lemma 5 replaced by Lemma 6, and with Lemma 9 replaced by the following analogue.

Proof. Inequality (33) follows from estimate (15),

Bn

as follows. Fix an index i. From Remark 2 we see that the support of gzi is essentially the union of a geodesic segment of Bergman kubes K₁ⁱ, K₂ⁱ, ..., K_Mⁱ _i where

Mi ≈ (α − ρ) log 1 1− |zⁱ|².

Indeed, recall that the support gzi is contained in the intersection of the cone C^zi

and the annulus A^zi. Now for ζ in the cone C^zi, we have 1 − ζ · P zⁱ ≈ 1 − |ζ|², and thus for ζ in the annulus A^zi as well, we have approximately

log 1

1− |ζ|² ∈ ρ log 1

1− |zⁱ|², α log 1 1− |zⁱ|² .

Thus ζ ∈ supp g^zi lies in the union of those kubes in Tⁿalong the geodesic joining the root to the “boundary point” P zi, and having tree distance from the root lying roughly between ρβ (0, zi) and αβ (0, zi). Moreover, this segment can be contin-ued to a longer sequence of adjacent Bergman kubes K₁ⁱ, K₂ⁱ, ..., K_Mⁱ _i, ...K_Jⁱ_i = Kzi

connecting the support of gzi to the kube Kzi containing zi, and where Ji ≈ log 1

1− |zⁱ|². (35)

Choose wj ∈ Kjⁱ for 1 ≤ j < Jⁱ. Then we have for z ∈ Kmⁱ , 1 ≤ m ≤ Mⁱ, and f ∈ B^p(Bⁿ),

|f (z)|^p= [f (z)− f (w^m)] +

Ji−1

j=m

[f (wj)− f (w^j+1)] + [f (wJi)− f (zⁱ)] + f (zi)

p

 |f (z) − f (w^m)|^p+ (Ji)^p−1

Ji−1

j=1

|f (w^j)− f (w^j+1)|^p +|f (w^Ji)− f (zⁱ)|^p+|f (zⁱ)|^p

≤ C (Jⁱ)^p−1

Ji

j=1

max

z1,z2∈(^Kjⁱ)^∗|f (z¹)− f (z²)|

p

+ C|f (zⁱ)|^p.

Using this together with (34), (35) and the fact that the supports of the gzi are pairwise disjoint, we obtain

Bn

|f (z)|^p 1− |z|^{2 n}g (z) ^pdλn(z)

≤ C

i Ji

j=1

max

z1,z2∈(^Kjⁱ)^∗|f (z¹)− f (z²)|

p

+C

i

|f (zⁱ)|^p log 1 1− |zⁱ|²

1−p

.

Since the kubes K_jⁱ are pairwise disjoint by Lemma 4, the first term on the right is dominated by

α∈Tⁿ

z1max,z2∈K^α|f (z¹)− f (z²)|

p

≤ C f ^pBp(Bn)

by Theorem 6.30 of [Zhu]. The second term is dominated by C f ^p_B

p(Bn) since we are assuming in (3) that _i log ¹

1−|zi|²

1−pδzi is a Bp(Bⁿ)-Carleson measure.

This completes the proof of Lemma 10.

Arguing as above, the proof that (3) implies (4) will follow from the char-acterization of Carleson measures on trees, together with Lemma 6 provided we show that (11) holds and that

1− |z|^{2 n}g (z)

p

dλn(z)

is a Bp(Bⁿ)-Carleson measure. From the definition of gi in (16), and the fact that the supports of the gi are pairwise disjoint by the separation condition, we have that (11) holds. Lemma 10 above shows that 1− |z|^{2 n}g (z) ^pdλn(z) is a Bp(Bⁿ)-Carleson measure for p > 2n, and this completes the proof that (3) implies (4) for p > 2n.

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在文檔中 NCTS 2005 Summer School on Harmonic Analysis : Lecture Note(IV) (頁 67-85)