Taiwan lecture 1
Friday July 1 2005
1
Introduction
In these talks we consider mainly interpolation sequences for the analytic Besov-Sobolev spaces Bσ
p (Bn)on the unit ball Bnin Cn, consisting of those holomorphic
functions f on the ball such that
f Bσ p(Bn)= #] Bn 1− |z|2m+σf(m)(z) pdλn(z) + m−1[ k=0 f(k)(0)p $1 p <∞, where m + σ > np, dλn(z) =
1− |z|2−n−1dz is invariant measure on the ball with dz Lebesgue measure on Cn, and f(m) is the mth order complex derivative
of f . Thus Bpσ(Bn) consists of those holomorphic functions on the ball having
σ “invariant” derivatives in Lp with respect to invariant measure. This scale
of spaces includes the Hardy space on the disc H2
(D) = B 1 2 2 (D) with σ = 1 2,
the Dirichlet space B20(D) with σ = 0, and the various weighted Bergman and
Dirichlet-type spaces. In fact, for f (z) = S∞n=0anzn, z ∈ D, the orthogonality
relations 1 2π ] 2π 0 ei(n−m)θdθ = 1 if n = m 0 if n = m yield ∞ [ n=0 |an| 2 = sup 0<r<1 1 2π ] 2π 0 ∞ [ n=0 an reiθn 2 dθ = sup 0<r<1 1 2π ] 2π 0 freiθ2dθ ≡ f 2H2(D),
while the calculation ] 1 0 1− r2r2(n−1)dr = 1 2n− 1− 1 2n + 1 = 2 4n2− 1 yields f 2 B 1 2 2(D) = ] D 1− |z|21+ 1 2 f (z) 2 dz 1− |z|22 +|f (0)| 2 = 1 2π ] 2π 0 ] 1 0 ∞ [ n=1 nan reiθn−1 2 1− r2dr +|a0| 2 = ∞ [ n=1 |nan|2 ] 1 0 1− r2r2(n−1)dr +|a0|2 =|a0|2+ ∞ [ n=1 |an|2 2n2 4n2− 1 ≈ ∞ [ n=0 |an|2.
Finally, the Dirichlet norm squared of f satisfies f 2B0 2(D) = ] D |f (z)|2dz +|f (0)|2, where ] D |f (z)|2dz = ] D det uxuy vx vy dxdy = ] D Jfdxdy = ] f (D) dudv
is the area of the image f (D) of the disc under f by the Cauchy-Riemann equa-tions ux = vy, uy =−vx if f = u + iv.
The case σ = 12 and p = 2 is the Drury-Arveson Hardy space H2 n = B
1 2
2 (Bn)
that can be identified with the symmetric Fock space over Cn (see [10] and [19]),
and enjoys many universal operator-theoretic properties. An excellent survey of Hilbert space developments in this area up to now is the beautiful recent monograph of K. Seip [32].
1.1
Origins of interpolation in the Corona problem
The theory of Carleson measures and interpolating sequences has its roots in Lennart Carleson’s 1958 paper [15], the first of his papers motivated by the corona problem for the Banach algebra H∞(D) of bounded holomorphic functions in the
unit disk D: if {fj}Jj=1 is a finite set of functions in H∞(D) satisfying J
[
j=1
are there are functions {gj} J j=1 in H∞(D) with J [ j=1 fj(z) gj(z) = 1, z ∈ D,
i.e., is every multiplicative linear functional on H∞(D) in the closure of the point
evaluations, so that there is no “corona”? In [15], Carleson observed the following connection between the corona problem and interpolating sequences. A Blaschke product B0 has the “baby corona” property,
For all f1 ∈ H∞(D) satisfying inf
z∈D{|B0(z)| + |f1(z)|} > 0, (1)
there are g0, g1 ∈ H∞(D) with B0g0+ f1g1 ≡ 1,
if the zero set
Z0 ={z ∈ D : B0(z) = 0} = {zj}∞j=1
of B0 is an interpolating sequence for H∞(D):
The map f → {f (zj)}∞j=1 takes H∞(D) boundedly into and onto ∞(Z0),
(2) (if g1 ∈ H∞(D) satisfies f1(zj) g1(zj) = 1 for all j, then we can choose g0 =
1−f1g1
B0 ). Carleson solved this latter problem completely by showing that a
se-quence Z = {zj}∞j=1 is an interpolating sequence for H∞(D) if and only if
\ j:j=k 1zj− z− zkzkj ≥ c > 0, k = 1, 2, 3, ... (3)
The necessity of (3) is easy. The open mapping theorem shows that given ξ =
ξj ∞j=1 ∈ ∞, there is an interpolating f ∈ H∞(D) such that f
H∞(D) ≤
C ξ ∞. Let B (z) =T∞k=1 |zk|
zk
zk−z
1−zkz be the Blaschke product with zeroes {zk}
∞ k=1 and Bj(z) = T k=j |zk| zk zk−z
1−zkz. If fj is such that fj(zk) = δjk and fj H∞(D) ≤ C,
then B1 j(zj) = fj Bj(zj) = fj Bj
H∞(D) ≤ C, which is (3). The rest of Carleson’s proof
made crucial use not only of Blaschke products, but also of duality. In the same paper he showed implicitly that the characterizing condition (3) can be rephrased in modern language as 1zj− z− zkzkj ≥ c > 0 for j = k, and µ = ∞ [ j=1 1− |zj|2
δzj is a Carleson measure for H
p
where a positive Borel measure µ on the disk D is now said to be a Carleson measure for Hp(D) if the embedding Hp(D) ⊂ Lp(dµ) holds. Carleson later showed that µ is a Carleson measure if and only if
µ (S (I))≤ C |I| , for all arcs I ⊂ T, where S (I) = reiθ : θ
∈ I and 0 < 1 − r < |I| , and solved the corona problem affirmatively in [16] by demonstrating the absence of a corona in the maximal ideal space of H∞(D).
1.2
Peter Jones’ proof in the upper half plane
We follow the excellent exposition in Seip [32]. First we show that the analogue of (3) in the upper half plane,
\ j:j=k zzkk− z− zjj ≥ c > 0, k = 1, 2, 3, ... (4)
implies the following separation condition on Z, and the following Carleson con-dition on the associated measure µ = µZ =
S∞ j=1yjδzj where zj = xj+ iyj: zzkk− z− zjj ≥ c > 0 for j = k, and (5) µ (T (zk)) = [ zj∈T (zk) yj ≤ Cyk= C × height of T (zk) ,
where the tent T (zk)is the equilateral triangle with vertex zk and opposite side
on the x-axis. Clearly the separation condition in (5) is implied by (4). Fix k. If Bk(z) =
T
j:j=k z−zj
z−zj denotes the Blaschke product with zeroes Z \ {zk}, then
from − ln t ≥ 1 − t for t > 0 we have that 2 ln1 c ≥ − ln |Bk(zk)| 2 =−[ j=k ln zzkk− z− zjj 2 ≥[ j=k # 1− zzkk− z− zjj 2$ =[ j=k 4ykyj |zk− zj|2 .
Now if zj ∈ T (zk), then |zk− zj|2 ≤ 4y2k and so,
[ zj∈T (zk) yj ≤ yk [ zj∈T (zk) 4ykyj |zk− zj|2 ≤ 2 ln1 c yk.
Finally, we remark that a covering lemma shows that we have the extended inequality, µ (T (z)) = [ zj∈T (z) yj ≤ 4 ln 1 c
To see this, fix z, let Izj be the base of the tent T (zj), and let {Izk}k∈E be
a subcollection of the intervals Izj : zj ∈ T (z)
having union ∪zj∈T (z)Izj, and
finite overlap 2 (we may assume the sequence finite for this). Then [ zj∈T (z) yj≤ [ k∈E [ zj∈T (zk) yj ≤ [ k∈E 2 ln1 c yk = 2 ln1 c [ k∈E √ 3 2 |Izk| ≤ 4 ln 1 c √ 3 2 |Iz| = 4 ln1 c y. 1.2.1 A linear interpolation operator
We wish to construct bounded analytic functions fj in the upper half plane such
that fk(zj) = δj,k (7) ∞ [ k=1 |fk(z)| ≤ C < ∞, for all z.
With this done, the function f = S∞k=1akfk(z) takes the value ak at zk and
has f H∞ ≤ C {aj}∞j=1
∞, showing that Z is an interpolating sequence for the
upper half plane. A natural choice to try first is fk(z) =
Bk(z)
Bk(zk)
,
since fk ∈ H∞ satisfies the first condition in (7), fk(zj) = δj,k. However, the
Blaschke products have modulus one on the boundary R (again it suffices to consider only finite sequences Z), and so typically S∞k=1|fk(z)| → ∞ as z →
R. To remedy the problem at infinity, we introduce the bounded holomorphic function g (z) = (z+i)−42 that vanishes at ∞ (note g (i) = 1) and try the modification
fk(z) = Bk(z) Bk(zk) gk(z) , where gk(z) = g z−xk yk = −4y2k
(z−zk)2 is g rescaled and translated to take the value 1
at zk. Of course the modified fk’s still fail to satisfy the second condition in (7).
This could be fixed by further modifying the fkby multiplying by an exponential
factor of the form e−α
S yj ≤yk Bkgk(z)(zk)
holomorphic. Instead, we use a positive harmonic majorant u for |g| and the corresponding exponential factor
e
−αSyj ≤ykuj (z)+ivj (z)
|Bj(zj)| ,
where v is the harmonic conjugate of u in the upper half plane. Here are the details.
Let u (x, y) = x24(1+y)+(1+y)2 =
4(1+y)
|z+i|2 be 4 times the Poisson kernel for the half space
R×(−1, ∞) so that u is positive harmonic and coincides with |g (x, y)| = x2+(1+y)4 2
on the boundary y = 0. Obviously, u ≥ |g| (or more generally by the maximum principle). Define uk(z) = u z−xk yk = 4yk(yk+y) |z−zk|2 and Un(z) = [ yj≤yn uj(z) |Bj(zj)| .
Note that Un(z) is finite, and so a positive harmonic function, by the extended
Carleson inequality (6) with z ( ) = x + i2 y: Un(z)≤ c−1 [ zj∈T (z(2)) yj≤yn 4yj(yj+ y) |z − zj|2 + c−1 ∞ [ =2 [ zj∈T (z( ))\T (z( −1)) yj≤yn 4yj(yj+ y) |z − zj|2 ≤ Cc−1yn+ y y2 [ zj∈T (z(2)) yj≤yn yj+ Cc−1 ∞ [ =2 yn+ y (2 y)2 [ zj∈T (z( )) yj ≤ Cc−1 ∞ [ =1 yn+ y (2 y)2 4 ln1 c 2 y ≤Cc ln1 c yn+ y y <∞,
but that this bound is not uniform in z and n. With Vn the harmonic conjugate
of Un in the upper half plane and Gn= Un+ iVn, we set
fn(z) =
Bn(z)
Bn(zn)
gn(z) e−α{Gn(z)−Gn(zn)},
for a positive constant α to be chosen momentarily. Clearly, the fn are
holomor-phic in the upper half plane and satisfy the first condition in (7). We estimate the second condition in terms of the quantity
} (Z) = sup n≥1 Un(zn)≤ C c ln 1 c
to obtain (!!) ∞ [ n=1 |fn(z)| ≤ ∞ [ n=1 Bgnn(z(z)n) e−α{Un(z)−}(Z)} ≤ e}(Z) α [∞ n=1 αun(z) |Bn(zn)| e− S yj ≤yn αuj (z) |Bj(zj)| ≤ eα}(Z) α ] ∞ 0 e−tdt = e α}(Z) α ≤ e } (Z) ,
with α = }(Z)1 , since if the zjare ordered with decreasing yj and tn=
S
yj≤yn
αuj(z)
|Bj(zj)|
for any fixed z, then
] ∞ 0 e−tdt≥ ∞ [ n=1 ] tn tn+1 e−tdt≥ ∞ [ n=1 (tn− tn+1) e−tn.
Thus we have obtained (7) with constant C = e } (Z) (see [32] for a discussion of sharpness).
Remark 1 If B is the Blaschke product with zeroes Z and we write the above linear operator of interpolation as
f (z) = ∞ [ j=1 ajfj(z) = B (z) ∞ [ j=1 aj hj(z) B (zj) (z− zj) ,
then a computation reveals that the function u = Bf solves the equation ∂u ∂z ≡ 1 2 ∂ ∂x − i ∂ ∂x u = 2πi ∞ [ j=1 ajyj Bj(zj) δzj.
By approximation, the equation
∂u ∂z = µ
can then be solved with u ∈ L∞ for any complex measure µ such that |µ| is a
Carleson measure: |µ| (T (z)) ≤ Cy for all z in the upper half plane. See Peter Jones [23] on this topic.
1.3
Origins of interpolation in control theory
Here we follow the excellent expository lecture by John McCarthy [25]. Let P
denote an engineering plant that accepts an input u = {un}∞n=0 and produces an
1. Causality: un= 0 for n ≤ N implies yn= 0 for n ≤ N,
2. Time invariance: input {0, u1, u2, ...} has output {0, y1, y2, ...},
3. Stability: energy S∞n=0|yn|2 of output y is at most a constant times that
of the input u,
4. Linearity: input u1+ λu2 has output y1+ λy2.
The setup can be transferred to the unit disk D by viewing u as the sequence of coefficients in a power series about the origin: hu (z) =S∞n=0unzn. The energy
of u is then given by ∞ [ n=0 |un|2 = sup 0<r<1 ∞ [ n=0 |un|2r2n = sup 0<r<1 1 2π ] 2π 0 hureiθ2dθ ≡ hu 2H2(D).
Properties Stability and Linearity show that the plant P is a continuous (bounded) linear operator on H2
(D). Properties Causality, Time invariance and Linearity show that P is the operator Mϕ of multiplication by ϕ = P 1: if f =
SN n=0anzn is a polynomial, then P f (z) = N [ n=0 anP (zn1) = N [ n=0 anznP 1 (z) = f (z) ϕ (z) = Mϕf (z) ,
which by the density of polynomials in H2
(D), extends to all f ∈ H2
(D). Now the operator norm squared of Mϕ satisfies
Mϕ 2op = sup f 2 H2(D)≤1 sup 0<r<1 1 2π ] 2π 0 ϕreiθ2freiθ2dθ ≤ ϕ 2H∞(D),
and with f = 1 and Mϕ iterated n times, we have the reverse inequality
sup 0<r<1 1 2π ] 2π 0 ϕ reiθ2ndθ 1 2n =qMϕn1 2 H2(D) r1 2n ≤q Mϕ 2n op r1 2n = Mϕ op, which yields ϕ H∞(D) ≤ Mϕ op upon letting n → ∞.
Thus the plant P is identified with an element ϕ of H∞(D), and
viewed as a multiplier operator Mϕ on the Hilbert space H2(D).
Now we suppose there is another plant W that modulates a noise input e that is added to the output x of P to get y. In order to minimize the effects of the noise, the engineers create a feedback loop by subtracting the output y from the input u to get v and then modifying v by another plant C called a compensator,
to get w which is then fed back into the plant P to get x. See the diagram. Thus we have
y = P C (y− u) + W e, or solving for y, y = (I + P C)−1P Cu + (I + P C)−1W e.
The requirements that the internal signals x, v and w be stable (the plant C need not be assumed stable) leads, after a small calculation, to the requirement that F = C (I + P C)−1 satisfy the Stability property. i.e.
C (I + P C)−1 ∈ H∞(D) .
The compensator C that minimizes the effects of noise is that which minimizes the operator norm
(I + P C)−1W
op= (I− P F ) W op,
since a short computation reveals (I + P C) (I − P F ) = I. However, factoring P and W into their inner and outer factors PiPo and WiWo, we have (the following
can be made rigorous by assuming P and W are rational) inf F ∈H∞(D) (I− P F ) W op=F ∈Hinf∞(D) (Wo− PiPoF Wo) Wi op = inf F ∈H∞(D) Wo− PiPoF Wo op = inf G∈H∞(D) Wo− PiG op = inf H∈H∞(D) q H op: H (zj) = Wo(zj) , 1≤ j < ∞ r , where Z = {zj}∞j=1 is the zero set of P = ϕ.
Thus the noise reduction problem is equivalent to the interpolation problem of finding the bounded analytic function of least norm that takes the values of Wo on the sequence Z of zeroes of P . For a given
plant P , solving this problem for all stable plants W can be achieved by solving the interpolation problem (2) of Carleson.
1.4
Hilbert space methods
In 1961, H. Shapiro and A. Shields [35] demonstrated that the interpolation property (2) is equivalent to weighted interpolation for Hardy spaces Hp(D), The map f →q1− |zj|2 1 pf (z j) r∞ j=1 takes H p
The factor 1− |zj|2
1
p forces the map to be into ∞(Z
f), since if zj = rjeiθj, then |f (zj)| = 2π1 ] 2π 0 Prj(θj − t) f ∗eitdt ≤Prj Lp(T) f∗ Lp(T) ≤ C (1 − rj)− 1 p f Hp(D);
the Carleson measure condition ensures that it is into p(Z f).
We now recast the case p = 2 of this result in a way that will emphasize the analogy with what comes later. The Hardy space H2
(D) is a Hilbert space with reproducing kernel, i.e. the point evaluations f → f (z) are continuous linear functionals. This means that for each z ∈ D, there is kz ∈ H2(D), the reproducing
kernel for z, which is characterized by the fact that for any f ∈ H2
(D) we have f (z) = f, kz . A sequence Z = {zj}∞j=1 is an interpolating sequence for H2(D)
if one can freely assign the values of an H2
(D) function on Z, subject only to the natural size restriction. More precisely, Hilbert space basics ensure that if f ∈ H2
(D), then the function zi → f (zi) kzi =1− |zi| 212 f (zi)
is a bounded function on Z. The sequence Z is called an interpolating sequence for H2
(D) if all the functions on Z which are obtained in this way are in 2(Z),
and if furthermore, every function in 2(Z) can be obtained in this way. For any z ∈ D, set hkz = kkzz , and note that by the Cauchy-Schwarz inequality,
Gkhz, ikwH ≤ 1, z, w ∈ D.
If Z is an interpolating sequence, then it must be possible, given any i and j, to find f ∈ H2(D) so that 1− |zi| 212 f (zi) = 0, 1− |zj| 212 f (zj) = 1;
and to do this with control on the size of f : say f ≤ C. This implies a weak separation condition on the points of Z which is necessary for Z to be an interpolating sequence: there is ε > 0 so that for all i = j,
Gkizi, ikzj
H ≤ 1 − ε. Indeed, simply minimize over λ ∈ C the right side of
1 = f (zj) 2 kzj 2 = f, kzj − λkzi 2 kzj 2 ≤ C 2 kzj − λkzi 2 kzj 2 .
This states that there is a uniform lower bound on the angle between reproducing kernels associated to the points in Z. An equivalent geometric statement is that there is a uniform lower bound on the hyperbolic distances β (zi, zj), where the
hyperbolic distance is obtained by transporting the Euclidean Riemannian metric at the origin to points of the disk via the automorphism group z → eiθ w1−wz−z. The result of Shapiro and Shields can now be restated as saying that interpolating sequences for H2
(D) are characterized by the following two conditions: There is ε > 0 so that Gkizi, ikzj H ≤ 1 − ε for all i = j, (8) and ∞ [ j=1 kzj −2
δzj is a Carleson measure for H
2
(D) . (9)
Interpolation problems, multiplier questions and Carleson measure character-izations have been studied by various authors in other classical function spaces on the disk, including certain of the spaces Bα
p (D) normed by m−1[ k=0 f(k)(0) + ] D 1− |z|2m+αf(m)(z) p dλ (z) 1 p ,
where dλ (z) = 1− |z|2−2dz is invariant measure on the disk, and for fixed α and p, the norms are equivalent for (m + α) p > 1. This scale of spaces includes the Hardy space H2
(D) = B
1 2
2 (D) with α = 1
2, the weighted Bergman spaces with
α > 1p, and the weighted Dirichlet-type spaces with 0 < α < 1p. See for example the recent book by K. Seip [32], which contains an in depth discussion of the history of interpolating sequences for Hilbert spaces of functions of a single vari-able. Interpolation proved more difficult for the family of analytic Besov spaces Bp(D) = Bp0(D) on the disk, the prototypical Möbius invariant spaces, which do
not admit any infinite Blaschke products - the Dirichlet norm f B2(D) measures the square root of the area of the range of f counting multiplicities, and so is infinite for every infinite Blaschke product f . These Besov spaces are also distin-guished by being the limit of those spaces Bpα(D) with α < 0 that are too smooth
(they admit continuous extensions to the closed disk D) to contain any infinite interpolating sequences. Indeed, if ζ ∈ T is an accumulation point of an interpo-lating sequence Z, say zjk → ζ as k → ∞, then the subsequence {f (zjk)}
∞ k=1 has
limit f (ζ), and hence we cannot interpolate any bounded sequence ξj
∞ j=1 for
which the subsequence ξjk
∞
k=1 has no limit.
1.4.1 The Dirichlet space
In a revolutionary paper in 1994, D. Marshall and C. Sundberg [24] used Hilbert space methods (and independently C. Bishop [11] used different techniques) to
characterize interpolating sequences for the Dirichlet space B2(D) and its
multi-plier space MB2(D) (note the connection H∞(D) = MH2(D)) by the condition
β (zi, 0)≤ Cβ (zi, zj) for i = j and ∞ [ j=1 # 1 + log 1 1− |zj| 2 $−1 δzj is a B2(D) -Carleson measure,
where β is the Bergman metric, and a positive Borel measure µ is a B2
(D)-Carleson measure if the embedding B2(D) ⊂ L2(dµ) holds:
]
|f (z)|2dµ (z)≤ C f 2B2(D).
A crucial part of their argument used the Nevanlinna-Pick property of B2(D):
an important consequence of this property for any space X of analytic functions on the disk, is that X then has the same interpolating sequences as its multiplier algebra MX - see e.g. [32]. The above two conditions can be rewritten in exactly
the same form as (8) and (9) with only the natural changes; the hk smust now be normalized reproducing kernels for the Dirichlet space and the measure must be a Dirichlet space Carleson measure.
1.5
Interpolation in Besov spaces
More recently, in 2002 in [12], B. Böe has extended the above theorem to all 1 < p <∞ by a long and clever construction involving Carleson measures, that was in turn based on an earlier construction in [24] (see also the analogous construction on trees in Section 6 of [7]), together with, in Böe’s words, a “curious lemma” on unconditional basic sequences {fj}∞j=1 of positive functions in a Lebesgue space
Lq(dµ): ∞ [ j=1 |ajfj| Lq(dµ) ≈ Cq supj≥1|ajfj| Lq(dµ) .
The sequence {fj}∞j=1is an unconditional basic sequence if
S∞j=1bjfj ≤ CS∞j=1ajfj whenever |bj| ≤ |aj|. 1.5.1 Higher dimensions
In Arcozzi, Rochberg and Sawyer [8], Böe’s results were extended to the ana-lytic Besov spaces Bp(Bn) on the unit ball Bn in Cn for n > 1. We note that
the corresponding questions for the Hardy spaces on the ball remain open in higher dimensions, due in part to the lack of Blaschke products, but also since the relevant separation condition fails to be sparse enough to accommodate the “hands-on” type of construction used by Böe. The Nevanlinna-Pick property fails as well.
At least two difficulties arise immediately in higher dimensions. Böe makes use of Stegenga’s 1980 characterization [37] of B2(D)-Carleson measures by a
capacity condition, as well as later extensions to p > 1: µ (T (E))≤ C capp(E) ,
for all compact subsets E (or equivalently finite unions of arcs) of the circle T, and where T (E) denotes the Carleson tent associated to E, and
capp(E) = inf
] π −π feiθpdθ : f ≥ 0 and ] π −π fei(φ−θ)|θ|−12 dθ≥ χ E(φ) . This characterization is not yet available in higher dimensions, and as indicated in [12], seems difficult to check even in certain one-dimensional situations. Instead, the characterization in [7] involving the discrete Bergman tree condition,
[ β∈T :β≥α # [ γ∈T :γ≥β µ (γ) $p ≤ Cp [ β∈T :β≥α µ (α) <∞, α∈ T , (10)
is extended to higher dimensions where it plays a crucial role both as a substi-tute for a capacity condition, and in generalizing the clever Carleson measure construction of Böe in [12].
The second difficulty runs deeper. It is connected to the fact that the re-producing kernel kw(z) = log1−wz1 for Bp(D) has derivative w1−wz1 where 1−wz1
has positive real part, and that this positivity played a crucial role in part of Böe’s argument when p < 2. In particular his “curious lemma”, which deals with positive functions, is applied to those real parts. This property persists in dimension n only for 1 < p < 1 + n−11 , where the analogous derivative Rαn+1+α
p
of the reproducing kernel kα,p
w (z)is Rαn+1+α p kwα,p(z) = (1− w · z)− n+1+α p , α >−1,
which has positive real part only when n+1+α
p ≤ 1 for some α > −1, i.e. p <
1 + n−11 .
As a consequence, the aforementioned “curious lemma” of Böe only generalizes to prove the necessity of the discrete tree condition for MBp(Bn) interpolation in
the thin range 1 < p < 1 + n−11 (where reproducing kernels for Bp(Bn) have the
requisite positivity property). To combat the failure of this positivity property for larger p, we introduce “holomorphic” Besov spaces HBp(Tn) on Bergman trees
Tn whose reproducing kernels do enjoy a suitable positivity property, and such
that the restriction map from Bp(Bn) to HBp(BnTn), as well as the restriction
effort and is accomplished in the latter half of the paper [8]. Another consequence is that our one-dimensional proof of the characterization of Carleson measures by the discrete tree condition extends to dimension n only in the thin range of p given by 1 < p < 1 + n−11 . A T T∗ argument lifts the proof to the larger range
1 < p < 2 + n−11 , beyond which we are unable to proceed at this time.
1.5.2 The Drury-Arveson space
In [9], Carleson measures are characterized in particular for the Besov-Sobolev spaces Bσ
2 (Bn), 0 ≤ σ < 12, by the tree condition
[ β∈Tn:β≥α 22σd(β) # [ γ∈Tn:γ≥β µ (γ) $2 ≤ C [ β∈Tn:β≥α µ (α) <∞, α∈ Tn.
Combined with recent work of Böe [12] and Agler and McCarthy [1], the above
Carleson measure characterization yields that a sequence Z = {zj}∞j=1 in the ball
Bn is an interpolating sequence for B2σ(Bn) if and only if it is an interpolating
sequence for the multiplier algebra MBσ
2(Bn) if and only if Z is separated in the
sense that infi=jβ (zi, zj) > 0and the measure µ =S∞j=1
1− |zj|2
2σ
δzj satisfies
the above tree condition. This characterization of Carleson measures fails for the Drury-Arveson Hardy space B
1 2
2 (Bn) (the “endpoint” case), and is instead
replaced by the simple condition 2d(α)I∗µ (α) ≤ C, α ∈ Tn, together with the
“split” tree condition [ k≥0 [ γ≥α 2d(γ)−2k [ (δ,δ )∈G(k)(γ) I∗µ (δ) I∗µ (δ )≤ CI∗µ (α) , α∈ Tn.
The restriction (δ, δ ) ∈ G(k)(γ) in the sum above means that we sum over all pairs (δ, δ ) of grandk
-children of γ that have γ as their minimum in Tn, and do not
lie in a common ring in the quotient tree Rn, but whose immediate predecessors
do.
von Neumann’s inequality We can now give a sharp estimate for the gen-eralization of von Neumann’s celebrated inequality [26] to the complex ball by Drury [21]. Let A = (A1, ..., An)be an n-contraction on a complex Hilbert space
H, i.e. an n-tuple of linear operators on H satisfying AjAk = AkAj for all 1 ≤ j, k ≤ n, and
n [ j=1 Ajh 2 ≤ h 2 for all h ∈ H. Equivalently, the Aj commute and the row operator A = (A1, ..., An) is bounded
with norm one from Onj=1H to H: Snj=1Ajhj 2 ≤Snj=1 hj 2 . Drury showed in [21] that if f is a complex polynomial on Cn, then
sup
Aan n-contraction
f (A) = f M
where f (A) is the operator norm of f (A) on H, and f MK(Bn) denotes the
multiplier norm of the polynomial f on Drury’s Hardy space of holomorphic functions K (Bn) = + [ k akzk, z ∈ Bn : [ k |ak|2 k! |k|! <∞ , , denoted by H2
n in Arveson [10] (who also proves (11) in Theorem 8.1). The
original inequality of von Neumann in dimension n = 1 is sup
Aa contraction
f (A) = f M
H2(D) = f H∞(D).
Chen [19] has identified the Drury-Arveson Hardy space K (Bn) = Hn2 as the
Besov-Sobolev space B
1 2
2 (Bn) consisting of those holomorphic functions Skakzk
in the ball with coefficients ak satisfying
[ k |ak|2 |k| n−1(n − 1)!k! (n− 1 + |k|)! <∞.
Indeed, the coefficient multipliers in the definitions of K (Bn) and B
1 2
2 (Bn) are
easily seen to be comparable. It now follows that the multiplier norms are equiv-alent: f M K(Bn) ≈ f M B 1 2 2(Bn) .
We note in passing that a number of important operator-theoretic properties of the Hilbert space H2
n are developed by Arveson in [10] that establish its central
position in multivariable operator theory.
Ortega and Fabrega [28] have shown that f is a pointwise multiplier on B
1 2
2 (Bn)if and only if f is a bounded holomorphic function and the measure
dµf(z) =R(n+12 )f (z)
21− |z|2dz is a Carleson measure for the Drury-Arveson Hardy space B
1 2
2 (Bn). In fact, we
can replace dµf by any of the measures dµmf (z) =
f(m)
(z)21− |z|22m−ndz, m > n− 1
2 .
Using this we obtain the following estimate. Theorem 1 For any m > n−12 ,
sup Aan n-contraction f (A) ≈ f ∞+ sup α∈Tn t 2d(α)I∗µm f (α) (12) + sup α∈Tn y x x wI∗µm1 f (α) [ k≥0 [ γ≥α 2d(γ)−k [ δ,δ ∈G(k)(γ) I∗µm f (δ) I∗µmf (δ ),
for all polynomials f on Cn.
The right side of (12) can of course be transported onto the ball using that ∪β≥αKβ is an appropriate nonisotropic tent in Bn, and that 2−d(α) ≈
1− |z|2 for z ∈ Kα.
Complete Nevanlinna-Pick kernels The universal complete Nevanlinna-Pick property of the Drury-Arveson space H2
n = B
1 2
2 (Bn)provides another
appli-cation of Carleson measures for H2
n. We recall the theory of Hilbert spaces with
a complete Nevanlinna-Pick kernel k (x, y) in Agler and McCarthy [1], keeping in mind the classical model of the Szego kernel k (x, y) = 1−xy1 on the unit disk D. Let X be an infinite set and k (x, y) be a positive definite kernel function on X, i.e. for all finite subsets {xi}mi=1of X,
m
[
i,j=1
aiajk (xi, xj)≥ 0 with equality ⇔ all ai = 0.
Denote by Hk the Hilbert space obtained by completing the space of finite linear
combinations of kxi’s, where kx(y) = k (x, y), with respect to the inner product
- m [ i=1 aikxi, m [ j=1 bjkyj . = m [ i,j=1 aibjk (xi, yj) .
The kernel k is called a complete Nevanlinna-Pick kernel if the solvability of the matrix-valued Nevanlinna-Pick problem is characterized by the contractivity of a certain family of adjoint operators Rx,Λ (we refer to [1] for an explanation of
this generalization of the classical Pick condition).
Let an(x, y) = 1− y,x1 for x, y ∈ Bn, the unit ball in n-dimensional Hilbert
space 2
n of cardinality n, and denote the Hilbert space Han by H
2 n (so that H2 n= B 1 2
2 (Bn)when n is finite). Theorem 4.2 of [1] shows that if k is an irreducible
kernel on X, and if for any fixed point x0 ∈ X, the Hermitian form
F (x, y) = 1− k (x, x0) k (x0, y) k (x, y) k (x0, y0)
has rank n, then k is a complete Nevanlinna-Pick kernel if and only if there is an injective function f : X → Bn and a nowhere vanishing function δ on X such
that
k (x, y) = δ (x)δ (y) an(f (x) , f (y)) =
δ (x)δ (y) 1− f (x) , f (y) .
Moreover, if this happens, then the map kx → δ (x) (an)f (x)extends to an
that k is continuous on X × X, then the map f will be a continuous embedding of X into Bn.
As a result, the Carleson embedding norm µ Carleson of H2
n⊂ L2(µ) can be
used to give a necessary and sufficient condition for Carleson measures on any Hilbert space Hk with a complete continuous irreducible Nevanlinna-Pick kernel
k. To see this, consider first the case where the Hermitian form F above has finite rank (F is positive semi-definite if k is a complete Nevanlinna-Pick kernel by Theorem 2.1 in [1]). Denote by f∗ν the pushforward of a Borel measure ν on
X under the continuous map f . If µ is a positive Borel measure on X, then µ is Hk-Carleson, i.e.
]
X
|h (x)|2dµ (x)≤ C h 2H
k, h∈ Hk, (13)
if and only if the measure µ = f∗|δ|2µ is B
1 2 2 (Bn)-Carleson, i.e. ] Bn |G|2dµ ≤ C G 2 B 1 2 2(Bn) , G∈ B 1 2 2 (Bn) . (14)
Indeed, the functions h =Smi=1cikxi are dense in Hk and have norm squared
- m [ i=1 cikxi, m [ i=1 cikxi . = m [ i,j=1 cicjk (xi, xj) = m [ i,j=1 cicjδ (xi)an(f (xi) , f (xj)) δ (xj) ,
which coincides with the norm squared in H2
nof H = T h = Sm i=1ciδ (xi) (an)f (xi): - m [ i=1 ciδ (xi) (an)f (xi), m [ i=1 ciδ (xi) (an)f (xi) . = m [ i,j=1 ciδ (xi)cjδ (xj) an(f (xi) , f (xj)) .
The change of variable f yields ] X |h (y)|2dµ (y) = ] X m [ i=1 cik (xi, y) 2 dµ (y) = ] X m [ i=1 ciδ (xi)an(f (xi) , f (y)) 2 |δ (y)|2dµ (y) = ] f (X)|H| 2 dµ = ] Bn |H|2dµ , and it follows immediately that (14) implies (13).
For the converse, we observe that if G ∈ H2 n = B
1 2
2 (Bn), then we can write
Now since J is orthogonal to all functions δ (x) (an)f (x) with x ∈ X, and since δ
is nonvanishing on X, we obtain that J vanishes on the subset f (X) of the ball Bn. Since µ is carried by f (X) and orthogonal projections have norm 1, we then
have with H = T h, ] Bn |G|2dµ = ] Bn |H|2dµ = ] X |h|2dµ, and h H k= H Hn2 ≤ G H2n.
It follows immediately that (13) implies (14).
We can extend the above characterization to the case of infinite rank n, by characterizing Carleson measures on H2
∞ (where ∞ denotes any infinite cardinal)
as follows. Given a finite dimensional subspace L of C∞, let P
Ldenote orthogonal
projection onto L and set BL = B∞∩ L, which we identify with the complex ball
Bn, n = dim L. We say that a positive measure ν on BL is Hn2(BL)-Carleson if,
when viewed as a measure on Bn, n = dim L, it is Hn2(Bn)-Carleson.
Lemma 2 A positive Borel measure ν on B∞ is H2
∞-Carleson if and only if
(PL)∗ν is uniformly Hn2(BL)-Carleson, n = dim L, for all finite-dimensional
subspaces L of C∞.
Proof. Suppose that (PL)∗νis uniformly Hn2(BL)-Carleson for all finite-dimensional
subspaces L of C∞, n = dim L. Let
f (z) = m [ i=1 cia∞(wi, z) = m [ i=1 ci 1 1− z, wi (15) for a finite sequence {wi}mi=1⊂ B∞ (such functions are dense in H∞2 ). If we let L
be the linear span of {wi}mi=1 in C∞, then since f (PLz) = f (z), we can view f
as a function on both B∞ and BL, and from our hypothesis we have
] B∞ |f|2dν = ] BL |f|2d (PL)∗ν ≤ C f 2 H2 n(BL) = C f 2 H2 ∞, (16)
with a constant C independent of f . Since such functions f are dense in H∞2 , we conclude that ν is H2
∞-Carleson. Conversely, given a subspace L and a measure
ν that is H2
∞-Carleson, functions of the form (15) with {wi}mi=1 ⊂ BL are dense
in Hn2(BL) and so (16) shows that (PL)∗ν is a Hn2(BL)-Carleson measure on BL
with constant C independent of L, n = dim L.
The above lemma now yields the following characterization of Carleson mea-sures on any Hilbert space Hkwith a complete continuous irreducible
Nevanlinna-Pick kernel k. Note that the irreducibility assumption on k can be removed using Lemma 1.1 of [1].
Theorem 3 Let k be a complete continuous irreducible Nevanlinna-Pick kernel on a set X. With notation as above, and rank (F ) = m, a positive measure µ on X is Hk-Carleson if and only if there is a positive constant C such that
(PL)∗f∗
|δ|2µCarleson ≤ C, for all finite-dimensional subspaces L of Cm, and where ν
Carleson denotes the
norm of the embedding H2
n(BL) ⊂ L2(µ) with n = dim L. Note that (PL)∗f∗ =
(PL◦ f)∗.
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Taiwan lecture 2
Monday July 4 2005
1
Plan of the attack on interpolation in Besov spaces in
higher dimensions
We will introduce a tree structure for the unit ball Bn by choosing a set Tn
of points in the ball at roughly a fixed distance apart in the Bergman metric, and declaring a point β ∈ Tn to be a child of another point α ∈ Tn if the
Bergman ball around β lies just “beyond” the Bergman ball around α. This simple construction suffices for dealing with Carleson measures and sufficient conditions for interpolation. The construction must be significantly refined in order to deal with the holomorphic Besov spaces on trees. The refinement allows us to develop an effective discrete version of passing from spaces defined by a single derivative to spaces of functions defined using higher derivatives. We postpone the rigorous construction of Tn for now, since we only need it for the sake of
completeness in stating our interpolation theorem below.
We use this tree structure Tn to characterize Carleson embeddings for Besov
spaces Bp(Bn) on the ball,
Bn
|f (z)|pdµ (z)≤ C f (z) pB
p(Bn),
in terms of a discrete condition on the Bergman tree,
β∈Tn:β≥α
I∗µ (β)p ≤ CpI∗µ (α) <∞, α ∈ Tn, (1)
where I∗µ (β) =
γ∈Tn:γ≥βµ (γ) - we are however unable to obtain the necessity
of the tree condition when 2+n−11 ≤ p < ∞. It turns out that the one-dimensional methods in [7], using a positivity property of the reproducing kernels, generalize to obtain the characterization in the thin range 1 < p < 1 +n−11 . A standard T T∗
turns out to have appropriate “derivative” log 1−w·z1 , whose real part is positive. We combine the two techniques to obtain the larger range 1 < p < 2 + n−11 .
Interpolating sequences for both Besov spaces Bp(Bn) and their multiplier
spaces MBp(Bn) are the treated by following, for the most part, the development
in Böe [12]. In Theorem 5, weighted Bp(Bn) interpolation is characterized by
separation and Carleson embedding conditions for all 1 < p < ∞: β (zi, 0)≤ Cβ (zi, zj) , i = j and ∞ j=1 kzα,pj −p Bp δzj is a Bp-Carleson measure, where kα,p
w (z) is a reproducing kernel for Bp. In Theorem 6, the separation and
tree conditions, d (αi, o)≤ Cd (αi, αj) , i = j and ∞ j=1 1 + log 1 1− cαj 2 1−p
δcαj satisfies the tree condition (1),
are proved sufficient for MBp(Bn) interpolation, and the separation and Carleson
embedding conditions above are proved necessary, for all 1 < p < ∞. As well, in the range p > 2n, we prove that the separation and Carleson embedding conditions are sufficient. The necessity of the Carleson embedding condition is proved first for p in the two ranges 1, 1 + 1
n−1 and [2, ∞). The first range
exploits the positivity of a reproducing kernel on the ball, and the second range exploits the embedding of q spaces in connection with Khinchine’s inequality.
Necessity of the Carleson embedding condition for MBp(Bn) interpolation in
the remaining range 1 + n−11 ≤ p < 2 is much more difficult, and is the subject of later talks.
2
Interpolating sequences
Let {zj}∞j=1 be a sequence of points in the unit ball Bn, and 1 < p < ∞. Here we
will prove that weighted p interpolation for Besov spaces B
p(Bn) holds on the
sequence {zj}∞j=1 if and only if the following separation condition and Carleson
embedding hold; β (zi, 0)≤ Cβ (zi, zj) and (2) ∞ j=1 1 + log 1 1− |zj| 2 1−p δzj is a Bp(Bn)-Carleson measure.
We may assume without loss of generality that the points zj occur as the centers
(this requires only a much weaker notion of separation, β (zi, zj)≥ c > 0). Note that d (αi, o)≈ β (cαi, 0)≈ log 1 1− |cαi| 2,
where d denotes distance in the Bergman tree Tn. Furthermore, the separation
condition β (zi, 0)≤ Cβ (zi, zj) on the ball implies the tree separation condition
d (αi, o) ≤ Cd (αi, αj), but not conversely. We then show that the analogue of
condition (2) on the Bergman tree Tn,
β (zi, 0)≤ Cβ (zi, zj) and (3)
∞
j=1
(1 + d (αj, o))1−pδαj is a Bp(Tn)-Carleson measure,
is sufficient for ∞ interpolation of the multiplier spaces M
Bp(Bn) on {zj}
∞ j=1 for
all 1 < p < ∞, and necessary provided 1 < p < 2 + 1
n−1. 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 +
1
n−1, 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−Padenote orthogonal projection onto the orthogonal complement
of Ca. Define an involutive automorphism of the ball Bn by ([29], page 25)
ϕa(z) =a− Paz− 1 − |a| 2 12 Q az 1− z · a , (5) = a− z·a |a|2a− 1 − |a| 2 12 z− z·a |a|2a 1− z · a ,
for z ∈ Bn. Then Aut (Bn), the group of automorphisms of Bn, consists of all
maps U ϕa where U is a unitary transformation and a ∈ Bn.We have ϕa(0) = a,
ϕa(a) = 0 and ϕa◦ ϕa = I. We also have the following identities ([29], Theorem 2.2.2), ϕa(0) =− 1 − |a|2 Pa− 1 − |a|2 1 2 Qa, (6) ϕa(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) Jϕa(z) =|det ϕa(z)| 2 = 1− |a| 2 |1 − a · z|2 n+1 ,
where Jϕa(z) denotes the real Jacobian of ϕa at z. For example, in dimension
n = 1, ϕa(z) = a−z
1−az, 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 2log 1 +|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
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− ca|2 t2ρ2 a + |Qaz| 2 t2ρ a < 1 , where ca = (1− t2) a 1− t2|a|2, ρa = 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 ∈ L 1(dλ n)∩ 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 Rγ,t
: H (Bn)→ H (Bn) given in [38] by Rγ,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 Rγ,t is denoted Rγ,t, then Proposition 1.14 of [38] yields
Rγ,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, Rγ,t is approximately differentiation of order t.
From Theorem 6.1 and Theorem 6.4 of [38] we have that the derivatives Rγ,mf (z)
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)∈ Lp(dλn) for some m > n p, m∈ N, 1− |z|2 m∇mf (z)∈ Lp(dλn) for all m > n p, m∈ N, 1− |z|2 mRγ,mf (z)∈ Lp(dλn) for some m > n p, m + n + γ /∈ −N, 1− |z|2 mRγ,mf (z)∈ Lp(dλn) for all m > n p, m + n + γ /∈ −N. Moreover, with σ (z) = 1 − |z|2, we have for 1 < p < ∞,
C−1 σm1Rγ,m1f Lp(dλ n) (11) ≤ m2−1 k=0 ∇kf (0) + Bn 1− |z|2 m2∇m2f (z) p dλn(z) 1 p ≤ C σm1Rγ,m1f Lp(dλ n)
for all m1, m2 > np, 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 = np + 1 and setting
Bp = Bp(Bn) = f ∈ H (Bn) : σmR0,mf Lp(dλ
n) <∞ . (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
A2α: f, g α = Bn f (z) g (z)dνα(z) , f, g∈ A2α, where dνα(z) = 1− |z| 2 α dz. Recall that Kα w(z) = Kα(z, w) = (1− w · z)−n−1−α
is the reproducing kernel for A2
α (Theorem 2.7 in [38]):
f (w) = f, Kw αα =
Bn
f (z) Kα
This formula continues to hold as well for f ∈ Ap
α, 1 < p < ∞, since the
polynomials are dense in Apα.
Corollary 6.5 of [38] states that Rγ,n+1+αp is a bounded invertible operator from
Bp onto Apα, provided that neither n + γ nor n + γ +n+1+αp is a negative integer.
It turns out to be convenient to take γ = α −n+1+αp 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 Rαt = Rα−t,t.
Note that the operators Rαt and their inverses (Rαt)−1=Rα−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 Rα
n+1+α p
is a bounded invertible operator from Bp onto Apα, and that Rαn+1+α
p
is a bounded invertible operator from Bp onto Apα, 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
and Bp using ·, · α as follows:
f, g α,p= Rαn+1+α p f,R α n+1+α p g α = Bn Rαn+1+α p f (z)R α n+1+α p g (z)dνα(z) = Bn 1− |z|2 n+1+α p Rαn+1+α p f (z) 1− |z| 2 n+1+αp Rα n+1+α p g (z) dλn(z) . With Kα
w(z) the reproducing kernel for A2α, we have that the kernel
kα,pw (z) = Rαn+1+α p −1 Rαn+1+α p −1 Kwα(z) (13)
satisfies the following reproducing formula for Bp:
f (w) = f, kwα,p α,p= Bn Rαn+1+α p f (z)Rα n+1+α p kwα,p(z)dνα(z) , f ∈ Bp. (14)
Thus we have the following theorem.
Theorem 4 Let 1 < p < ∞ and α > −1. Then the dual space of Bp can be
identified with Bp under the pairing ·, · α,p, and the reproducing kernel k α,p w for
From (13) and (10) we have Rαn+1+α p kwα,p(z) = Rαn+1+α p −1 Kwα(z) (15) = Rα−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 kα,p
w is comparable to 1 + log 1−|w|1 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 kα,p
w (z) be the reproducing kernel for
Bp relative to the pairing ·, · α,p given in Theorem 4 above. Let {zj}∞j=1 be a
sequence in the unit ball Bn. Then the following conditions are equivalent.
1. {zj}∞j=1 interpolates Bp: The map f → f (zj) kzα,pj B p ∞ j=1
takes Bp boundedly into and onto p.
(16) 2. The following norm equivalence holds:
∞ j=1 aj kα,p zj kzα,pj B p B p ≈ ∞ j=1 |aj|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 kα,pzj −p Bp δzj is a Bp-Carleson measure.
Theorem 6 Let 1 < p < ∞, α > −1 and kα,pw (z) be the reproducing kernel for Bp relative to the pairing ·, · α,p given in Theorem 4 above. Let {zj}∞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. {zj}∞j=1 interpolates MBp:
The map f → {f (zj)}∞j=1 takes MBp boundedly into and onto
∞. (19)
2. kα,p zj
n
j=1 is an unconditional basic sequence in Bp: ∞ j=1 bjkα,pzj Bp ≤ C ∞ j=1 ajkzα,pj Bp , whenever |bj| ≤ |aj| . (20) 3. {zj}∞j=1 = cαj ∞ j=1 where {αj} ∞
j=1 is a sequence in a Bergman tree Tn
sat-isfying
β (zi, 0)≤ Cβ (zi, zj) , i = j and (21)
∞
j=1
(1 + d (αj, o))1−pδαj satisfies the tree condition (1).
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
kwα,p B p ≈ 1 + log 1 1− |w|2 1 p ≈ (1 + β (0, w))p1 . (22)
Proof. Using (15) and m = n+1+α p > n p , we compute that kwα,p Bp = Bn 1− |z|2 n+1+α p Rαn+1+α p kα,pw (z) p dλn(z) 1 p = Bn 1− |z|2 1− w · z n+1+α dλn(z) 1 p = Bn 1− |z|2 α |1 − w · z|n+1+αdz 1 p ≈ 1 + log 1 1− |w|2 1 p
by Theorem 1.12 of [38]. Note also that 1 + log 1
1−|w|2 ≈ d (α) if w ∈ Kα.
Thus we can restate condition (18) in the equivalent form,
β (zi, 0)≤ Cβ (zi, zj) and (23) ∞ j=1 1 + log 1 1− |zj| 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
→ Bp
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 ∈ MBp such that fk MBp ≤ C, fk(zj) = δ
j
k and k|fk(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 + 1
n−1 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).
2.2
Multiplier space necessity
We begin with the straightforward necessity implications; (19) implies (20), (16) implies (17), and (17) implies (18). For the most part, we follow Böe [12], who in turn generalized the Hilbert space arguments in Marshall and Sundberg [23]. First, we have that condition (20) follows from (19) using that the reproduc-ing kernels are eigenfunctions of adjoints of multiplier operators: Mϕ∗ kzα,pj =
ϕ (zj)kzα,pj . Indeed, for all f ∈ Bp,
f, Mϕ∗ kzα,pj = ϕf, kα,pzj = ϕ (zj) f (zj) = ϕ (zj) f, kα,pzj = f, ϕ (zj)k
α,p zj .
Now if we choose ϕ ∈ MBp so that bj = ϕ (zj)aj, then
∞ j=1 bjkα,pzj Bp = Mϕ∗ ∞ j=1 ajkzα,pj Bp ≤ Mϕ ∞ j=1 ajkα,pzj Bp .
We note in passing that the identity M∗
ϕ(kzα,p) = ϕ (z)kzα,p shows that multipliers
are bounded: Mϕ op= Mϕ op∗ = sup f ∈Bp Mϕ∗f f ≥ supz Mϕ∗kα,pz kα,pz = sup z |ϕ (z)| = ϕ H ∞.
Next we prove the equivalence of (16) and (17), the arguments being short and essentially reversible. First, if the map T f = f (zj)
kα,pzj B p ∞ j=1 in (16) maps Bp into p, then we have
∞ j=1 aj kα,p zj kα,pzj B p B p = sup f Bp=1 f, ∞ j=1 aj kα,p zj kzα,pj B p α,p = sup f Bp=1 ∞ j=1 f (zj) kzα,pj B p aj ≤ sup f Bp=1 ∞ j=1 f (zj) kα,pzj B p p 1p {aj}∞j=1 p ≤ C {aj}∞j=1 p .
If the map T is also onto, then its adjoint T∗, given by
T∗ {aj}∞j=1 = ∞ j=1 aj kα,p zj kα,pzj B p , satisfies T∗ {aj}∞j=1 Bp ≥ c {aj} ∞ j=1 p ,
which is the opposite inequality in (17), and completes the proof that (16) implies (17). Conversely, if the inequality in (17) holds, then
∞ j=1 f (zj) kα,pzj B p p 1p (24) = sup {aj}∞j=1 p =1 ∞ j=1 f (zj) kα,pzj B p aj = sup {aj}∞j=1 p =1 ∞ j=1 f, ajk α,p zj kzα,pj B p α,p ≤ sup {aj}∞j=1 p =1 f Bp ∞ j=1 aj kα,p zj kα,pzj B p B p ≤ C f Bp,
and thus the map T in (16) is into. If the reverse inequalityz in (17) also holds, then T∗ {aj}∞j=1 Bp = ∞ j=1 aj kα,p zj kzα,pj B p B p ≥ c {aj}∞j=1 p ,
which shows that T is also onto.
Note We have shown in particular that the inequality in (17) implies that the map T in (16) is into. This will be used below.
The implication (17) implies (18) will now follow if we show that (16) implies (18). The Carleson embedding in (18) is a restatement that the map T in (16) is into. Indeed, the left side of (24) is f
Lp ∞ j=1 k α,p zj −pB p δzj
, and thus shows that the Carleson embedding in (18) holds. To obtain the separation condition, fix i and use that T is onto to obtain f ∈ Bp satisfying f (zi) = 1and f (zj) = 0 for
i = j. It now follows from the open mapping theorem and the Hölder estimate |f (z) − f (w)| ≤ C f Bpβ (z, w) 1 p , z, w ∈ B n, (25) that f Bp ≤ C T f p = C |f (z i)| kziα,p B p = C|f (zi)− f (zj)| kα,pzi Bp ≤ C f Bp β (zi, zj) 1 p β (zi, 0) 1 p , for all i = j.
2.2.1 The necessity of separation and Carleson
Now we turn to proving the more difficult necessity implication (20) implies (18). First we dispose of the easy part - namely that the separation condition in (18) follows from (20). Indeed, by (22), (20) and (25) we have
(1 + β (0, zi)) 1 p ≈ kα,p zi Bp ≤ C k α,p zi − k α,p zj Bp = C sup f Bp=1 f, kα,pzi − kzα,pj α,p = C sup f Bp=1|f (z i)− f (zj)| ≤ Cβ (zi, zj) 1 p .
It remains to prove that the Carleson embedding follows from (20). For this, we show that (20) implies (17) for both 1 < p < 1 + n−11 and p = 2, and also that (20) implies the inequality of (17) for p > 2. The note above then yields that the map T in (16) is into, which we showed above is a restatement of the Carleson embedding.
2.2.2 The case 1 < p < 1 + n−11
Here we prove the implication (20) implies (17) for the special case 1 < p < 1 + n−11 . Given 1 < p < 1 + n−11 , we make the choice −1 < α < ∞ to satisfy
p = n + 1 + α
n + α , (26)
which accounts for our restriction 1 < p < 1 + 1
n−1. Note that p = n + 1 + α, so that n + 1 + α p = n + α, n + 1 + α p = 1.
Thus in this case we have Rα
n+1+α p = Rα n+α and Rαn+1+α p = Rα 1 where α is as in (26), so that f, g α,p= Rαn+αf,Rα1g A2 α = Bn 1− |z|2 n+αRαn+αf (z) 1− |z|2 Rα 1g (z)dλn(z) .
The point of the choice of p in (26) is that Rα1k
α,p w (z) =
1 1− w · z
has positive real part in the ball. Let {zj}∞j=1 be a sequence in the ball Bn. We
will need the following two results.
Lemma 8 (Lemma 3.1 in [12]) If {fj}∞j=1 is an unconditional basic sequence of
positive functions in Lq(dµ), 1 < q < ∞, then ∞ j=1 |ajfj| Lq(dµ) ≈ Cq sup j≥1 |a jfj| Lq(dµ) ≈ Cq ∞ j=1 |aj|q fj qLq(dµ) 1 q .
Proof. For convenience we sketch Böe’s proof, which we will need to adapt to holomorphic trees later anyway. Since the fn are positive and unconditional in
Lq(dµ)
, we have, letting {rj(t)}∞j=1 denote the Rademacher functions, ∞ j=1 |ajfj| Lq(dµ) ≤ ∞ j=1 |aj| fj Lq(dµ) ≤ C ∞ j=1 rj(t) ajfj Lq(dµ)
for all t ∈ [0, 1] (since |aj| = |rj(t) aj|). Now average the qth power of this
inequality over t ∈ [0, 1], and use Khinchine’s inequality to obtain
∞ j=1 |ajfj| q Lq(dµ) ≤ Cq 1 0 ∞ j=1 rj(t) ajfj q Lq(dµ) dt = Cq 1 0 ∞ j=1 rj(t) ajfj q dtdµ ≤ Cqq {ajfj}∞j=1 q 2dµ. Since {ajfj}∞j=1 2 ≤ {ajfj} ∞ j=1 1 2 1 {ajfj} ∞ j=1 1 2
∞, we have by the
Cauchy-Schwartz inequality, ∞ j=1 |ajfj| q Lq(dµ) ≤ Cqq {ajfj}∞j=1 q 1dµ 1 2 {ajfj}∞j=1 q ∞dµ 1 2 ,
which yields the inequality
∞ j=1 |ajfj| Lq(dµ) ≤ Cq sup j≥1|a jfj| Lq(dµ) .
Thus the expressions
∞ j=1 |ajfj| r 1 r Lq(dµ)
are all comparable for 1 < r < ∞, and the choice r = q yields the final equivalence in the lemma.
Lemma 9 For −1 < α < ∞, 1 < q < ∞ and F ∈ H (Bn) with Im F (0) = 0,
Bn |F (z)|qdνα(z) 1 q ≈ Bn |Re F (z)|qdνα(z) 1 q . (27)
Proof. The Korànyi-Vagi theorem (Theorem 6.3.1 in [29]) shows the equivalence of the left and right hand sides in (27) when the measure dνα(z)on the ball Bnis
replaced by surface measure dσ (z) on the sphere ∂Bn(and F is say a polynomial).
Note that dσ corresponds to limα→−1dνα. This immediately yields (27) by an
Now suppose that (20) holds. Since p = n + 1 + α, we have from (15) that Rα1k α,p w (z) = 1 1− w · z. (28)
We now compute that
∞ j=1 aj kα,p zj kα,pzj B p B p = 1− |z|2 Rα1 ∞ j=1 aj kα,p zj kzα,pj B p Lp(dλ n) = 1− |z|2 ∞ j=1 aj Rα 1kα,pzj kzα,pj B p Lp(dλ n) = ∞ j=1 aj kzα,pj −1 Bp 1 1− zj · z Lp(dν α)
since p = n + 1 + α. Now by the lemmas above, and using p = n + 1 + α and fj = kzα,pj −1 Bp Re 1 1− zj · z > 0, we continue with ∞ j=1 aj kzα,pj −1 Bp 1 1− zj · z Lp(dν α) ≈ ∞ j=1 aj kzα,pj −1 Bp Re 1 1− zj · z Lp(dν α) ≈ ∞ j=1 |aj| p fj p Lp(dν α) 1 p ≈ ∞ j=1 |aj|p 1 p , since fj Lp(dνα)= k α,p zj −1 Bp Re 1 1− zj · z Lp(dν α) ≈ kα,pzj −1 Bp 1 1− zj · z Lp(dν α) = kα,pzj −1 Bp 1− |z| 2 R1kzα,pj Lp(dλ n) = kα,pzj −1 Bp k α,p zj Bp = 1
upon using the second lemma above once more. This completes the proof of condition (17) in the case 1 < p < 1 + n−11 .
2.2.3 The case p≥ 2
Here we show that (20) implies the inequality in (17) for p > 2, and also that (20) implies (17) for p = 2. First we claim that the unconditional basis condition (20) and Khinchine’s inequality yield the inequality
∞ j=1 aj kα,p zj kzα,pj B p B p ≤ C ∞ j=1 |aj|p 1 p
for p ≥ 2, and with equality in the case p = 2. To see this, we compute using first (20) and then Khinchine, that for any m > pn, we have
∞ j=1 aj kα,pzj kzα,pj B p p Bp ≈ 1 0 ∞ j=1 aj kzα,pj kα,pzj B p rj(t) p Bp dt = 1 0 Bn ∞ j=1 aj 1− |z|2 mRαmkzα,pj (z) kzα,pj B p rj(t) p dλn(z) dt ≈ Bn ∞ j=1 aj 1− |z|2 mRα mkzα,pj (z) kα,pzj B p 2 p 2 dλn(z) .
Since p2 ≤ 1, we continue with
∞ j=1 aj kα,p zj kzα,pj B p p Bp ≤ C Bn ∞ j=1 aj 1− |z|2 mRα mkzα,pj (z) kα,pzj B p p dλn(z) = ∞ j=1 |aj|p kα,pzj −p Bp Bn 1− |z|2 mRαmkzα,pj (z) p dλn(z) = ∞ j=1 |aj|p ,
which is the inequality in (17). In the case p = 2 we have equality, and so then (17).
References
[1] J. Ajohu dqg J. MfCduwk|, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics 44 (2002), AMS, Providence, RI. [2] J. Ajohu dqg J. E. McCduwk|, Complete Nevanlinna-Pick kernels, J.
