Invariant metrics, measures and derivatives

j=1

(1 + d (αj, o))^1−pδαj is a Bp(Tⁿ)-Carleson measure,

is sufficient for ^∞ interpolation of the multiplier spaces MBp(Bn) on {z^j}^∞j=1 for all 1 < p < ∞, and necessary provided 1 < p < 2 + _n−1¹ . More precisely, for the sufficiency, we need (3) taken over all unitary rotations of the Bergman tree Tⁿ, since on average over the unitary group Uⁿ, tree distance is comparable to Bergman distance.

We are however able to show that (2) is sufficient for ^∞ interpolation of the multiplier spaces MBp(Bⁿ)for p > 2n, and that (2) is necessary for ^∞interpolation of the multiplier spaces MBp(Bn) for all 1 < p < ∞. Since a measure µ is a Bp(Tⁿ)-Carleson measure if and only if it satisfies the tree condition (1), we see that one obstacle to obtaining a characterization of ^∞ interpolation of the multiplier spaces MBp(Bn) in the exceptional range 2 + _n−1¹ , 2n is our failure to find a characterization of Carleson measures for Bp(Bⁿ) when p ≥ 2 + _n−1¹ . We consider mostly Besov spaces Bp(Bⁿ) on the unit ball, and for convenience in notation, we will suppress the dependence on the ball by writing simply Bp for Bp(Bⁿ).

2.1 Invariant metrics, measures and derivatives

We recall some basic definitions and properties from W. Rudin’s book [29], K.

Zhu’s book [38] and our paper [8]. For a ∈ Bⁿlet Pa denote orthogonal projection onto the one-dimensional complex subspace Ca generated by a, i.e.

Paz = z· a

|a|²a, (4)

and let Qa = I−P^adenote orthogonal projection onto the orthogonal complement of Ca. Define an involutive automorphism of the ball Bⁿ by ([29], page 25)

ϕ_a(z) =a− P^az− 1 − |a|²

1 2 Qaz

1− z · a , (5)

=

a− _|a|^z·a2a− 1 − |a|²

1

2 z− _|a|^z·a2a

1− z · a ,

for z ∈ Bⁿ. Then Aut (Bⁿ), the group of automorphisms of Bⁿ, consists of all maps U ϕ_a where U is a unitary transformation and a ∈ Bⁿ.We have ϕ_a(0) = a, ϕ_a(a) = 0 and ϕ_a◦ ϕa = I. We also have the following identities ([29], Theorem 2.2.2),

ϕ_a(0) =− 1 − |a|² Pa− 1 − |a|²

1

2 Qa, (6)

ϕ_a(a) =− 1 − |a|^{2 −1}Pa− 1 − |a|^{2 −}

1 2 Qa, 1− ϕa(w)· ϕa(z) =(1− a · a) (1 − w · z)

(1− w · a) (1 − a · z), 1− |ϕa(z)|²= 1− |a|² 1− |z|²

|1 − a · z|² , and ([29], Theorem 2.2.6)

Jϕ_a(z) =|det ϕa(z)|² = 1− |a|²

|1 − a · z|²

n+1

,

where Jϕ_a(z) denotes the real Jacobian of ϕ_a at z. For example, in dimension n = 1, ϕ_a(z) = _1−az^a−z, and we have

1− |ϕa(z)|²=|1 − az|²− |z − a|²

|1 − az|²

= 1− 2 Re (az) + |a|²|z|² − |z|²− 2 Re (az) + |a|²

|1 − az|²

= 1− |a|² 1− |z|²

|1 − a · z|² .

An invariant measure on Bⁿ is given by ([29], Theorem 2.2.6) dλn(z) = 1− |z|^{2 −n−1}dz.

The invariance of dλn follows from the above Jacobian formula and the last identity in (6).

An invariant metric on Bⁿ is the Bergman metric β (z, w) given by ([38], Proposition 1.21)

β (z, w) = 1

2log1 +|ϕz(w)|

1− |ϕz(w)|, z, w ∈ Bⁿ. (7) By invariance, the Bergman metric balls Bβ(a, r) of radius r at the point a ∈ Bⁿ satisfy

Bβ(a, r) = ϕ_a(Bβ(0, r)) ,

and if t > 0 is such that Bβ(0, r) = B (0, t) (note from (7) that Bergman metric balls centered at the origin are Euclidean balls), then the β-balls are the ellipsoids ([29], page 29)

Bβ(a, r) = z ∈ Bⁿ: |P^az− c^a|²

t²ρ²_a + |Q^az|²

t²ρ_a < 1 , where

ca = (1− t²) a

1− t²|a|², ρ_a = 1− |a|² 1− t²|a|².

We have the reproducing formula of Bergman ([29], Theorem 3.1.3), f (z) = n!

πⁿ _Bn

f (w)

(1− w · z)ⁿ⁺¹dw, f ∈ L¹(dλn)∩ H (Bⁿ) , (8) and the following variants ([29], Theorem 7.1.2)

f (z) = n!

πⁿ

n + s

n _Bn

1− |w|^{2 s}

(1− w · z)^s+n+1f (w) dw, Re s >−1, (9) valid for all f ∈ H (Bⁿ)for which the integrand is in L¹.

We now recall the invertible “radial” operators R^γ,t : H (Bⁿ)→ H (Bⁿ) given in [38] by

R^γ,tf (z) =

∞

k=0

Γ (n + 1 + γ) Γ (n + 1 + k + γ + t)

Γ (n + 1 + γ + t) Γ (n + 1 + k + γ)fk(z) ,

provided neither n + γ nor n + γ + t is a negative integer, and where f (z) =

∞

k=0fk(z) is the homogeneous expansion of f . Note that Γ (n + 1 + γ) Γ (n + 1 + k + γ + t)

Γ (n + 1 + γ + t) Γ (n + 1 + k + γ) ≈ (1 + k)^t.

If the inverse of R^γ,t is denoted Rγ,t, then Proposition 1.14 of [38] yields

R^γ,t 1

(1− w · z)^n+1+γ = 1

(1− w · z)^n+1+γ+t, (10) Rγ,t

1

(1− w · z)^n+1+γ+t = 1

(1− w · z)^n+1+γ,

for all w ∈ Bⁿ. Thus for any γ, R^γ,t is approximately differentiation of order t.

From Theorem 6.1 and Theorem 6.4 of [38] we have that the derivatives R^γ,mf (z) are “L^p norm equivalent” to ^m−1_k=0 ∇^kf (0) +∇^mf (z) for m large enough.

Proposition 1 (Theorem 6.1 and Theorem 6.4 of [38]) Suppose that 0 < p < ∞, n+γ is not a negative integer, and f ∈ H (Bⁿ). Then the following four conditions are equivalent:

1− |z|^{2 m}∇^mf (z)∈ L^p(dλn) for some m > n

p, m∈ N, 1− |z|^{2 m}∇^mf (z)∈ L^p(dλn) for all m > n

p, m∈ N, 1− |z|^{2 m}R^γ,mf (z)∈ L^p(dλn) for some m > n

p, m + n + γ /∈ −N, 1− |z|^{2 m}R^γ,mf (z)∈ L^p(dλn) for all m > n

p, m + n + γ /∈ −N.

Moreover, with σ (z) = 1 − |z|², we have for 1 < p < ∞,

C⁻¹ σ^m1R^γ,m1f _Lp(dλn) (11)

≤

m2−1

k=0

∇^kf (0) +

Bⁿ

1− |z|^{2 m2}∇^m2f (z) ^pdλn(z)

1 p

≤ C σ^m1R^γ,m1f _Lp(dλn)

for all m1, m2 > ⁿ_p, m1+ n + γ /∈ −N, m² ∈ N, and where the constant C depends only on m1, m2, n, γ and p.

Definition 2 We define the analytic Besov spaces Bp(Bⁿ) on the ball Bⁿ by taking γ = 0 and m = ⁿ_p + 1 and setting

Bp = Bp(Bⁿ) = f ∈ H (Bⁿ) : σ^mR^0,mf _Lp(dλn) <∞ . (12) We will indulge in the usual abuse of notation by using f _Bp(Bn) to denote any of the norms appearing in (11).

2.1.1 Duality and reproducing kernels

For α > −1, let ·, · α denote the inner product for the weighted Bergman space A²_α:

f, g _α =

Bn

f (z) g (z)dνα(z) , f, g∈ A²α,

where dνα(z) = 1− |z|^{2 α}dz. Recall that K_w^α(z) = K^α(z, w) = (1− w · z)^{−n−1−α} is the reproducing kernel for A²_α (Theorem 2.7 in [38]):

f (w) = f, K_{w α}^α =

Bn

f (z) K_w^α(z)dνα(z) , f ∈ A²α.

This formula continues to hold as well for f ∈ A^pα, 1 < p < ∞, since the polynomials are dense in A^p_α.

Corollary 6.5 of [38] states that R^γ,n+1+αp is a bounded invertible operator from Bp onto A^p_α, provided that neither n + γ nor n + γ +^n+1+α_p is a negative integer.

It turns out to be convenient to take γ = α −^n+1+α_p here (with this choice we can explicitly compute certain derivatives and Bp norms of our reproducing kernels - see (15) and (22) below), and thus we single out the special operators

R^αt = R^α−t,t.

Note that the operators R^αt and their inverses (R^αt)⁻¹=R_α−t,tare self-adjoint with respect to ·, · α since the monomials are orthogonal with respect to ·, · α (see (1.21) and (1.23) in [38]), and the operators act on the homogeneous expansion of f by multiplying the homogeneous coefficients of f by certain positive constants.

The next definition is motivated by the fact that R^αn+1+α

p

is a bounded invertible operator from Bp onto A^p_α, and that R^αn+1+α

p

is a bounded invertible operator from Bp onto A^p_α, provided that neither n + α, n + α − ^n+1+α_p nor n + α −^n+1+α_p is a negative integer. Note that this proviso holds in particular for α > −1.

Definition 3 For α > −1 and 1 < p < ∞, we define a pairing ·, · α,p for Bp

satisfies the following reproducing formula for Bp: f (w) = f, k_w^α,p _α,p= Thus we have the following theorem.

Theorem 4 Let 1 < p < ∞ and α > −1. Then the dual space of B^p can be identified with Bp under the pairing ·, · α,p, and the reproducing kernel k_w^α,p for this pairing is given by (13).

From (13) and (10) we have R^αn+1+α

p

k_w^α,p(z) = R^αn+1+α

p

−1K_w^α(z) (15)

= R_α−^n+1+α

p ,^n+1+α_p (1− w · z)^−(n+1+α)

= (1− w · z)^−n+1+αp .

Using this formula we will show in (22) below that the Bp norm of the reproducing kernel k_w^α,p is comparable to 1 + log ¹

1−|w|²

1 p .

We now state our analogue of Böe’s interpolation theorem in two separate statements.

Theorem 5 Let 1 < p < ∞, α > −1 and k^α,pw (z) be the reproducing kernel for Bp relative to the pairing ·, · α,p given in Theorem 4 above. Let {z^j}^∞j=1 be a sequence in the unit ball Bⁿ. Then the following conditions are equivalent.

1. {z^j}^∞j=1 interpolates Bp:

The map f → f (zj) kz^α,pj B_p

∞

j=1

takes Bp boundedly into and onto ^p. (16) 2. The following norm equivalence holds:

∞

j=1

aj

k^α,p_zj kz^α,pj B_p

B_p

≈

∞

j=1

|a^j|^p

1 p

. (17)

3. The following separation condition and Carleson embedding hold:

β (zi, 0)≤ Cβ (zⁱ, zj) , i = j and (18)

∞

j=1

k^α,p_zj ^−p

B_p δzj is a Bp-Carleson measure.

Theorem 6 Let 1 < p < ∞, α > −1 and k^α,pw (z) be the reproducing kernel for Bp relative to the pairing ·, · α,p given in Theorem 4 above. Let {z^j}^∞j=1 be a sequence in the unit ball Bⁿ. If p ∈ 1, 2 + _n−1¹ , then each of conditions (19) and (21) below is equivalent to the three conditions in Theorem 5. In general, for 1 < p < ∞, (21) implies (19) implies (20). For p > 2n (18) implies (19). If p∈ 1, 1 + _n−1¹ ∪ [2, ∞), we also have that (20) implies (18):

1. {z^j}^∞j=1 interpolates MBp:

The map f → {f (z^j)}^∞j=1 takes MBp boundedly into and onto ^∞. (19) 2. k_z^α,p_j ⁿ

j=1 is an unconditional basic sequence in Bp:

∞

Note in particular that for p ∈ 1, 2 +_n−1¹ ∪ (2n, ∞), multiplier interpolation (19) is characterized by the separation condition and Carleson embedding in (18).

The parameter α > −1 appearing in condition (18) is not essential, as evi-denced by the following calculation.

Lemma 7 For α > −1 and 1 < p < ∞, we have

by Theorem 1.12 of [38]. Note also that 1 + log ¹

1−|w|² ≈ d (α) if w ∈ K^α. Thus we can restate condition (18) in the equivalent form,

β (zi, 0)≤ Cβ (zⁱ, zj) and (23)

∞

j=1

1 + log 1 1− |z^j|²

1−p

δzj is a Bp-Carleson measure.

Remark 1 Our proofs show that the interpolations in (19) and (16) can be taken to be linear, i.e. there are bounded linear maps R : ^∞ → M^Bp and S : ^p → B^p that yield right inverses to the restriction maps in (19) and (16) respectively. In dimension n = 1 Böe has shown [12] the stronger result that there are functions fk ∈ M^Bp such that f_{k M}

Bp ≤ C, f^k(zj) = δ^j_k and _k|f^k(z)| ≤ C for all z ∈ D (compare Theorem 2.1 in chapter 7 of [22]). It seems likely that this extends to 1 < p < 2 + _n−1¹ for n > 1, but we will not pursue this here.

Remark 2 We do not know if (18) is sufficient for (21) when p ∈ 2 +_n−1¹ ,∞ . Note that (3) and (21) are equivalent.

Proof. Here we only prove the following “soft” implications: that (19) implies (20), that (16) and (17) are equivalent, that (16) implies (18), and that if p ∈

1, 1 +_n−1¹ ∪ [2, ∞), then (20) implies (18).

在文檔中 NCTS 2005 Summer School on Harmonic Analysis : Lecture Note(IV) (頁 24-31)