A SHARP NORM ESTIMATE OF
THE BERGMAN PROJECTION ON Lp SPACES
KEHE ZHU
A BSTRACT. We show that the norm of the Bergman projection on Lp of
the unit ball in Cn is comparable to csc(π/p) for 1 < p < ∞.
1. I NTRODUCTION
Throughout the paper we fix a positive integer n and let B denote the
open unit ball in Cn . For −1 < α < ∞ let
dvα (z) = (α + 1)(1 − |z|2 )α dv(z),
where dv is the normalized volume measure on B.
For 0 < p < ∞ let
Apα = H(B) ∩ Lp (B, dvα )
denote the weighted Bergman space on B with standard radial weights.
Here H(B) is the space of all holomorphic functions in B. It is easy to
see that Apα is closed in Lp (B, dvα ). We will use k kp,α for the norm in
Lp (B, dvα ).
We use Pα to denote the orthogonal projection from L2 (B, dvα ) onto A2α .
It is well known that Pα is an integral operator on L2 (B, dvα ),
Z
Pα f (z) = Kα (z, w)f (w) dvα (w),
B
where the integral kernel is given by
1
.
(1 − hz, wi)n+1+α
It is also well known that, for 1 < p < ∞, the Bergman projection Pα maps
Lp (B, dvα ) boundedly onto Apα . See Section 7.1 of [2] for example.
The purpose of this paper is to give a sharp estimate of the norm of Pα
on Lp (B, dvα ). Our main result is the following theorem.
Kα (z, w) =
Theorem. For any −1 < α < ∞ there exists a constant C > 0, depending on α and n but not on p, such that the norm of the operator
Pα : Lp (B, dvα ) → Apα
1991 Mathematics Subject Classification. 32A36 and 32A25.
1
2
KEHE ZHU
satisfies the estimate
C −1 csc
π
π
≤ kPα kp ≤ C csc
p
p
for all 1 < p < ∞.
It is easy to see that the quantity csc(π/p) is comparable to p as p → ∞
and comparable 1/(p − 1) as p → 1.
This work was done while I was visiting the University of Marseille I in
France. I wish to thank the Centre de Mathematiques et d’Informatique at
Marseille I, and Professor Hassan Youssfi in particular, for a very nice visit.
2. P RELIMINARIES
The proof of our main result still depends on the two traditional tools,
namely, Forelli-Rudin type estimates for certain integrals on the ball and
Schur’s test for the boundedness of integral operators on Lp spaces. But
three new ingredients are necessary here. First, we need a more precise
version of the Forelli-Rudin estimates, namely, how the estimates depend
on various parameters. Second, we need to find the right test function for
Schur’s lemma in order to control the parameters in the Forelli-Rudin estimates. And finally, we need to show that our estimates are sharp in a certain
sense.
Lemma 1. For any T > 0 there exists a constant C > 0, depending on n
and T but not on t, such that
Z
dσ(ζ)
CΓ(t)
≤
n+t
(1 − |z|2 )t
S |1 − hz, ζi|
for all z ∈ B and 0 < t < T , where S is the unit sphere in Cn and σ is the
normalized Lebesgue measure on S.
Proof. By the proof of Proposition 1.4.10 in [2],
Z
∞
dσ(ζ)
Γ(n) X
Γ2 (k + λ)
(1)
=
|z|2k ,
n+t
2 (λ)
|1
−
hz,
ζi|
Γ
Γ(k
+
1)Γ(k
+
n)
S
k=0
where λ = (n + t)/2. Also,
∞
(2)
X Γ(k + t)
1
=
|z|2k .
(1 − |z|2 )t
Γ(k
+
1)Γ(t)
k=0
As t goes from 0 to T , the parameter λ goes from n/2 to (n + T )/2, so the
quotient Γ(n)/Γ2 (λ) is bounded below away from 0 and bounded above
away from infinity. We now use Stirling’s formula to compare (1) and (2).
NORM OF BERGMAN PROJECTION
3
For each non-negative k let
ak = ak (t) =
Γ2 (k + λ)
.
Γ(k + n)Γ(k + t)
It is obvious that if k > 0 then ak is positive and bounded in t ∈ (0, T );
this is also true when k = 0, because the gamma function is bounded away
from 0 on the interval (0, ∞). We need to show that there exists a positive
constant C (depending only on n and T ) such that ak ≤ C for all k ≥ 0 and
0 < t < T.
By Stirling’s formula, there exist positive constants C1 and M such that
C1−1 ≤
Γ(x)
≤ C1
1
xx− 2 e−x
for all x ≥ M . It then follows easily that there exists a constant C2 > 0,
depending only on n and T , such that
k+n k+t p
(k + n)(k + t)
λ−n
λ−t
ak ≤ C2 1 +
1+
.
k+n
k+t
k+λ
As k → +∞, the right hand side above approaches C2 uniformly for t ∈
(0, T ) (when n is fixed), because
y
x
lim 1 +
= ex ,
y→∞
y
and the convergence is uniform for x in any finite interval. This proves the
desired estimate.
Lemma 2. Given any T > 0 and A > −1, there exists a constant C > 0,
depending on n, T , and A, but not on t and α, such that
Z
(1 − |w|2 )α dv(w)
CΓ(α + 1)Γ(t)
≤
n+1+α+t
(1 − |z|2 )t
B |1 − hz, wi|
for all −1 < α < A, 0 < t < T , and z ∈ B.
Proof. Let I denote the integral concerned. By the proof of Proposition
1.4.10 in [2],
∞
(3)
Γ(n + 1)Γ(α + 1) X
Γ2 (k + λ)
I=
|z|2k ,
2
Γ (λ)
Γ(k + 1)Γ(n + 1 + α + k)
k=0
where λ = (n + 1 + α + t)/2. The desired result then follows from (2), (3),
and Stirling’s formula. The details are exactly the same as in the proof of
Lemma 1.
4
KEHE ZHU
Lemma 3. Suppose µ is a positive measure on a space X and H(x, y) is
a positive kernel on X. If there exists a constant C > 0 and a positive
function h(x) on X such that
Z
H(x, y)h(y)q dµ(y) ≤ Ch(x)q
X
for all x in X, and
Z
H(x, y)h(x)p dµ(x) ≤ Ch(y)p
X
for all y in X, then the integral operator
Z
T f (x) =
H(x, y)f (y) dµ(y)
X
is bounded on Lp (X, µ) with norm not exceeding C. Here 1 < p < ∞ and
1/p + 1/q = 1.
Proof. See [3] for example.
3. P ROOF OF THE M AIN R ESULT
We now prove the main result of the paper.
Theorem 4. For any −1 < α < ∞ there exists a constant C > 0, depending on α and n but not on p, such that the norm of the operator
Pα : Lp (B, dvα ) → Apα
satisfies the estimate
C −1 csc
π
π
≤ kPα kp ≤ C csc
p
p
for all 1 < p < ∞.
Proof. Fix 1 < p < ∞ and let q be the conjugate exponent,
1 1
+ = 1.
p q
Consider the function
h(z) = (1 − |z|2 )−(α+1)/(pq)
on B and the operator
Z
T f (z) =
B
f (w) dvα (w)
|1 − hz, wi|n+1+α
NORM OF BERGMAN PROJECTION
5
on Lp (B, dvα ). By Lemma 2, with T = α + 1 and A = α, there exists a
constant C > 0, independent of p, such that
α+1
Z
Z
(1 − |w|2 ) q −1 dv(w)
h(w)q dvα (w)
= (α + 1)
n+1+α
n+1+ α+1
−1+ α+1
q
p
B |1 − hz, wi|
B |1 − hz, wi|
C(α + 1)Γ α+1
Γ α+1
q
p
≤
(1 − |z|2 )(α+1)/p
α+1
α+1
= C(α + 1)Γ
Γ
h(z)q .
q
p
Similarly,
Z
B
h(z)p dvα (z)
≤ C(α + 1)Γ
|1 − hz, wi|n+1+α
α+1
p
α+1
Γ
h(w)p .
q
It follows from Lemma 3 that the norm of the operator T on Lp (B, dvα ),
and hence the norm of Pα on Lp (B, dvα ), does not exceed
α+1
α+1
Γ
.
C(α + 1)Γ
q
p
If α = 0, a well-known property of the gamma function gives
1
π
1
Γ
=
;
Γ
p
q
sin πp
see [1].
If α 6= 0, we can find a constant C1 > 0, independent of p but dependent
on α, such that
α+1
α+1
C1
Γ
Γ
≤
.
q
p
sin πp
In fact, because of the symmetry of the sine function and the conjugacy
between p and q, we only need to consider the case in which p is very large.
In this case, the factor Γ((α + 1)/q) is bounded from above and from below,
and
α+1
1
Γ
∼p∼
,
p
sin πp
because
xΓ(x) = Γ(x + 1) ∼ 1
when x is a small positive number.
To prove that the norm estimate
C
kPα kp ≤
sin πp
6
KEHE ZHU
is sharp, we only need to consider the case when p > 2; the case when
1 < p < 2 then follows from duality and the symmetry of the function
sin(π/p). Note again that for p > 2 the constant sin(π/p) is comparable to
1/p.
So we assume that p > 2 and consider the function
f (z) = log(1 − z1 ) − log(1 − z1 )
= 2i arg(1 − z1 ).
It is clear that |f (z)| ≤ 2π for all z ∈ B, so that the norm of f in Lp (B, dvα )
does not exceed 2π. On the other hand, it is easy to see that
Pα f (z) = log(1 − z1 );
see Theorem 7.1.4(b) in [2].
It is well known that every function g in Apα satisfies the pointwise estimate
(4)
|g(z)| ≤
kgkp,α
,
(1 − |z|2 )(n+1+α)/p
z ∈ B.
In fact, for any fixed z in B, a change of variables shows that the function g
and G have the same norm in Apα , where
p1
(1 − |z|2 )n+1+α
,
w ∈ B,
G(w) = g ◦ ϕz (w)
(1 − hw, zi)2(n+1+α)
and ϕz is the involutive automorphism of B that interchanges the orgin and
the point z; see Section 2.2 in [2]. The obvious estimate |G(0)| ≤ kGkp,α
then leads to (4).
Let g = Pα f and z = (r, 0, · · · , 0) in (4), where 0 < r < 1. We obtain
kPα f kp,α ≥ (1 − r2 )
≥ (1 − r)
1
1−r
1
log
.
1−r
n+1+α
p
n+1+α
p
log
In particular, if r = 1 − e−p , then
kPα f kp,α ≥ pe−(n+1+α) .
This shows that
kPα f kp,α
p
≥
,
kf kp,α
2πen+1+α
so the norm of Pα on Lp (B, dvα ) is at least p/(2πen+1+α ). This completes
the proof of the theorem.
NORM OF BERGMAN PROJECTION
7
Note that the main result can be restated as follows: there exists a constant C > 0 such that
p2
p2
≤ kPα kp ≤ C
C −1
p−1
p−1
for every p ∈ (1, ∞). The quotient p2 /(p − 1) can also be replaced by pq
or p + q, where 1/p + 1/q = 1.
4. R ELATED Q UESTIONS
Our main result shows how fast the norm of the Bergman projection
Pα on Lp (B, dvα ) grows as p increases to infinity or as p decreases to 1,
when α is fixed. A related question is to determine how the norm of Pα
on Lp (B, dvα ) depends on α when p is fixed. In particular, we are interested in estimates of this norm when p is fixed and when α approaches −1.
We conjecture that the norm of Pα on Lp (B, dvα ) remains bounded if p is
fixed in (1, ∞) and when α approaches −1. A direct proof of this, such as
the one in the previous section, will give a proof for the boundedness of the
Cauchy-Szëgo projection on Lp spaces of the unit sphere when 1 < p < ∞.
It is of course well known that the Cauchy-Szëgo projection Q is bounded
on Lp spaces of the unit sphere when 1 < p < ∞. However, we are not
aware of any estimates for the norm of Q on Lp .
Another natural problem is to find sharp norm estimates for the Bergman
projection on Lp spaces of other domains, such as strongly pseudo-convex
domains in Cn .
R EFERENCES
[1] J.B. Conway, Functions of One Complex Variable, Springer-Verlag, New York, 1978.
[2] W. Rudin, Function Theory in the Unit Ball of Cn , Springer-Verlag, New York, 1980.
[3] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.
D EPARTMENT OF M ATHEMATICS , SUNY, A LBANY, NY 12222, USA
E-mail address: kzhu@math.albany.edu