Derivatives of Inner Functions and the Schwarz-Pick Lemma
Derivatives of Inner Functions and the
Schwarz-Pick Lemma
Fernando Pérez-González - U. de La Laguna
Workshop on Complex Analysis and Operator Theory,
Málaga, June 2016
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Joint research with
Jouni Rättyä
University of Eastern Finland- Joensuu
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Introduction
H(D) = algebra of all analytic functions in the open unit disc
D := {z : |z| < 1}.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Introduction
H(D) = algebra of all analytic functions in the open unit disc
D := {z : |z| < 1}.
Recall that Θ ∈ H(D) is called inner provided that Θ is bounded in
D, and it satisfies |Θ(e iθ )| = 1, a. e. e iθ ∈ T = ∂D .
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Introduction
H(D) = algebra of all analytic functions in the open unit disc
D := {z : |z| < 1}.
Recall that Θ ∈ H(D) is called inner provided that Θ is bounded in
D, and it satisfies |Θ(e iθ )| = 1, a. e. e iθ ∈ T = ∂D .
Θ = γBS
where |γ| = 1, B is a Blaschke product and S is a singular inner
function.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Introduction
For a given sequence {zn } in D satisfying the Blaschke condition
∞
X
(1 − |zn |2 ) < ∞ ,
n=1
(with the convention zn /|zn | = 0 for zn = 0), the Blaschke product
associated with {zn } is defined as
B(z) =
∞
Y
|zn | zn − z
.
zn 1 − zn z
n=1
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Introduction
The singular inner function S is defined by
Z
z +w
dσ(w ) ,
S(z) = exp
T z −w
where the measure σ on T is singular with respect to the Lebesgue
measure.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Introduction
The singular inner function S is defined by
Z
z +w
dσ(w ) ,
S(z) = exp
T z −w
where the measure σ on T is singular with respect to the Lebesgue
measure.
If the measure σ is atomic and consists of a point mass
concentrated in w ∈ T, then S is of the form
z +w
S(z) = exp γ
,
z −w
where 0 < γ < ∞ .
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Introduction
The singular inner function S is defined by
Z
z +w
dσ(w ) ,
S(z) = exp
T z −w
where the measure σ on T is singular with respect to the Lebesgue
measure.
If the measure σ is atomic and consists of a point mass
concentrated in w ∈ T, then S is of the form
z +w
S(z) = exp γ
,
z −w
where 0 < γ < ∞ .
Classical problem: given an inner function Θ and a certain space
X of analytic functions in D, when Θ0 ∈ X ?
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
For 0 < p ≤ ∞, the Hardy space is
H p = {f ∈ H(D) : kf kH p < ∞},
where
kf kH p := lı́m Mp (r , f ) < ∞,
r →1−
and
Mp (r , f ) :=
1
2π
Z
2π
|f (re iθ )|p dθ
0
is the integral mean of the function f .
p1
,
0 < p < ∞.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
Let Θ be an inner function with the canonical decomposition
Z it
∞
Y
e +z
|zn | zn − z
it
× exp −
dσ(e ) .
Θ(z) = γ
it
zn 1 − zn z
T e −z
n=1
Put
Z
∞
X
1 − |zn |2
dσ(e it )
fΘ (e ) =
+
.
iθ
it 2
|e iθ − zn |2
T |e − e |
iθ
n=1
Then,
Θ0
∈
H p (D),
0 < p ≤ ∞, if and only if, fΘ ∈ Lp (T).
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
J. Caughran and A. Shields, 1969 asked if there is a singular
1
inner function S such that S 0 ∈ H 2 (D).
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
J. Caughran and A. Shields, 1969 asked if there is a singular
1
inner function S such that S 0 ∈ H 2 (D).
M. Cullen (1971) : Let 0 < p < 21 and suppose that σSis a
positive singular measure
on T such that T \ supp σ = n In , arcs
P
1−2p
on satisfying that n |In |
< ∞. Let
Z it
e +z
it
dσ(e ) ,
S(z) = exp −
it
T e −z
then S 0 ∈ H p .
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
J. Caughran and A. Shields, 1969 asked if there is a singular
1
inner function S such that S 0 ∈ H 2 (D).
M. Cullen (1971) : Let 0 < p < 21 and suppose that σSis a
positive singular measure
on T such that T \ supp σ = n In , arcs
P
1−2p
on satisfying that n |In |
< ∞. Let
Z it
e +z
it
dσ(e ) ,
S(z) = exp −
it
T e −z
then S 0 ∈ H p .
Theorem (Ahern- Clark, 1974)
1
Let Θ be an inner function such that Θ0 ∈ H 2 . Then Θ is a
Blaschke product.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
Let E beSa closed subset of T so that its complement
T \ E = n In is a countable union of open intervals In , n ≥ 1. E is
called a Carleson set if |E | = 0 and
X
n
|In | log
1
< ∞.
|In |
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
Let E beSa closed subset of T so that its complement
T \ E = n In is a countable union of open intervals In , n ≥ 1. E is
called a Carleson set if |E | = 0 and
X
n
|In | log
1
< ∞.
|In |
Theorem (Ahern, 1979)
Let E ⊂ T closed and suppose |E | = 0 and is not a Carleson set.
Then there is a Blaschke sequence {zn } satisfying the following
properties:
(i) The set of accumulation points of {zn } is a subset of E ;
P
(ii) The sequence {zn } satisfies that n (1 − |zn |)α < ∞ for all
α > 21 ;
(iii) The Blaschke product B formed with {zn } is such that
B0 ∈
/ N.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
P
Theorem of Ahern shows that the condition (1 − |zn |)α < ∞,
α > 12 , is not sufficient to deduce that B 0 ∈ H p (D). Therefore,
more restrictive conditions are needed.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
P
Theorem of Ahern shows that the condition (1 − |zn |)α < ∞,
α > 12 , is not sufficient to deduce that B 0 ∈ H p (D). Therefore,
more restrictive conditions are needed.
A Blaschke sequence {zn } is said to be uniformly separated if there
is a δ > 0 such that
∞
Y
zj − zk 1 − zj zk > δ .
j6=k,j=1
The Blaschke product formed with a uniformly separated sequence
is called an interpolating Blaschke product. For this reason,
uniformly separated sequences are also called interpolating
sequences.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
Cohn (1983) proved that for interpolating sequence the two
conditions are equivalent.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
Cohn (1983) proved that for interpolating sequence the two
conditions are equivalent.
Theorem (Cohn, 1983)
Fix 0 < α < 12 . Let {zn } be an interpolating Blaschke sequence,
and let B be the corresponding
P Blaschke product. Then
B 0 ∈ H 1−α (D) if and only if (1 − |zn |)α < ∞ .
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Hardy spaces
Cohn (1983) proved that for interpolating sequence the two
conditions are equivalent.
Theorem (Cohn, 1983)
Fix 0 < α < 12 . Let {zn } be an interpolating Blaschke sequence,
and let B be the corresponding
P Blaschke product. Then
B 0 ∈ H 1−α (D) if and only if (1 − |zn |)α < ∞ .
Theorem (Girela-Peláez- Vukotić, 2007)
If theTzeros of a Blaschke product B lie in a Stolz domain, then
B 0 ∈ 0<p< 1 H p (D).
2
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Classical Bergman spaces
For 0 < p < ∞ y −1 < α < ∞, the classical weighted Bergman
spaces is
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Classical Bergman spaces
For 0 < p < ∞ y −1 < α < ∞, the classical weighted Bergman
spaces is
Apα = {f ∈ H(D) : kf kpAp < ∞}
α
where
kf kpAp :=
α
Z
D
|f (z)|p (1 − |z|2 )α dA(z) < ∞,
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Classical Bergman spaces
For 0 < p < ∞ y −1 < α < ∞, the classical weighted Bergman
spaces is
Apα = {f ∈ H(D) : kf kpAp < ∞}
α
where
kf kpAp :=
α
Z
|f (z)|p (1 − |z|2 )α dA(z) < ∞,
D
For us, a weight ω : D → (0, ∞) is integrable. It is radial if
ω(z) = ω(|z|) for all z ∈ D. The weighted Bergman space consists
of those f ∈ H(D) for which
Z
p
kf kAp :=
|f (z)|p ω(z) dA(z) < ∞,
ω
D
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Classical Bergman spaces
For the classical weighted Bergman spaces, the first results were
established by Ahern
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Classical Bergman spaces
For the classical weighted Bergman spaces, the first results were
established by Ahern
Theorem (Ahern, 1979)
Let Θ be an inner function, 1 ≤ p ≤ 2, α > −1.
(i) If α > p − 1 then Θ0 ∈ Apα ;
(ii) If p − 2 < α < p − 1 then Θ0 ∈ Apα if and only if Θ0 ∈ A1α−p+1 ;
(iii) If α ≤ p − 2 and p > 1, then Θ0 ∈ Apα if and only if Θ is a
finite Blaschke product.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Classical Bergman spaces
For the classical weighted Bergman spaces, the first results were
established by Ahern
Theorem (Ahern, 1979)
Let Θ be an inner function, 1 ≤ p ≤ 2, α > −1.
(i) If α > p − 1 then Θ0 ∈ Apα ;
(ii) If p − 2 < α < p − 1 then Θ0 ∈ Apα if and only if Θ0 ∈ A1α−p+1 ;
(iii) If α ≤ p − 2 and p > 1, then Θ0 ∈ Apα if and only if Θ is a
finite Blaschke product.
This result was generalized for the range 1 ≤ p < ∞ by Allan
Gluchoff (1987) using the notion of approximating Blaschke
product associated to an inner function introduced by Cohn in
1983.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Classical Bergman spaces
Lemma
If Θ is an inner function and BΘ is an approximating Blaschke
product of Θ one has that
1 − |Θ(z)| 1 − |BΘ (z)| .
Derivatives of Inner Functions and the Schwarz-Pick Lemma
State of the art
Classical Bergman spaces
Lemma
If Θ is an inner function and BΘ is an approximating Blaschke
product of Θ one has that
1 − |Θ(z)| 1 − |BΘ (z)| .
Theorem (Gluchoff, 1987)
Let Θ be an inner function and BΘ an approximating Blaschke
product with zero set {z
Pn }. If −1 < α < p − 1 where p ≥ 1, then
Θ0 ∈ Apα if and only if k (1 − |zk |)α−p+2 < ∞.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
We want to search the weights ω such that
Z
|Θ0 (z)|p ω(z) dA(z) < ∞ ,
D
for any inner function.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
We want to search the weights ω such that
Z
|Θ0 (z)|p ω(z) dA(z) < ∞ ,
D
for any inner function.
Theorem (Schwarz-Pick Lemma)
Let f : D → D be holomorphic. Then
|f 0 (z)| ≤
1 − |f (z)|
1 − |f (z)|2
.
,
2
1 − |z|
1 − |z|
z ∈ D.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
In other words, is there any essential loss of information when
integrating the function
QΘ (z) :=
1 − |Θ(z)|
,
1 − |z|
Schwarz-Pick quotient
instead of Θ0 ?
Z
0
p
Z
QΘ (z)p ω(z) dA(z)
Z 1 − |Θ(z)| p
ω(z) dA(z)
=
1 − |z|
D
|Θ (z)| ω(z) dA(z) D
for all inner functions Θ.
D
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
Weights
b if there exists C = C (ω) ≥ 1 such that
ω∈D
Z
ω
b (r ) =
1
ω(s) ds
r
satisfies the doubling property
1+r
ω
b (r ) ≤ C ω
b
,
2
0 ≤ r < 1.
b contains so-called regular and normal weights. Thus it contains
D
The standard power weights (1 − |z|2 )α ;
Rapidly increasing weights as (1 − |z|)−1 log
e
1−|z|
−β
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
There are several recent excellent papers by José Ángel Peláez
and Jouni Rättyä:
Memoirs of the AMS.
Advances in Math.
Math. Annalen.
Journal of Math. Pures et Appl.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
Lemma
Let ω be a radial weight. Then TFAE:
b
(i) ω ∈ D;
(ii) There exist C = C (ω) > 0 and β = β(ω) > 0 such that
ω
b (r ) ≤ C
1−r
1−t
β
ω
b (t),
0 ≤ r ≤ t < 1;
(iii) There exist C = C (ω) > 0 and γ = γ(ω) > 0 such that
Z t
1−t γ
ω(s) ds ≤ C ω
b (t), 0 ≤ t < 1;
1−s
0
(iv) The following asymptotic equality is valid
Z 1
1
x
s ω(s) ds ω
b 1−
, x ∈ [1, ∞) .
x
0
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
bp if
Write ω ∈ D
p
bp (ω) = sup (1 − r )
D
b (r )
0<r <1 ω
Z
0
r
ω(s)
ds < ∞.
(1 − s)p
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
bp if
Write ω ∈ D
p
bp (ω) = sup (1 − r )
D
b (r )
0<r <1 ω
Z
0
r
ω(s)
ds < ∞.
(1 − s)p
Condition (iii) yields
b=
D
[
p>0
bp .
D
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
bp if
Write ω ∈ D
p
bp (ω) = sup (1 − r )
D
b (r )
0<r <1 ω
Z
0
r
ω(s)
ds < ∞.
(1 − s)p
Condition (iii) yields
b=
D
[
bp .
D
p>0
Condition (ii) in Lemma shows that
ω
b (t) ω
b (r )
whenever
1−t 1−r,
0 ≤ r < t < 1.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
bp if
Write ω ∈ D
p
bp (ω) = sup (1 − r )
D
b (r )
0<r <1 ω
Z
0
r
ω(s)
ds < ∞.
(1 − s)p
Condition (iii) yields
b=
D
[
bp .
D
p>0
Condition (ii) in Lemma shows that
ω
b (t) ω
b (r )
whenever
1−t 1−r,
0 ≤ r < t < 1.
Proposition (Self-improving property)
bp , then ω ∈ D
bp−ε
Let 0 < p < ∞ and ω a radial weight. If ω ∈ D
p
for all ε ∈ (0, b
) and
Dp (ω)+1
bp (ω) ≤ D
bp−ε (ω) ≤
D
p
bp (ω))
p − ε(1 + D
bp (ω) .
D
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
Theorem
b Then the following assertions are
Let 0 < p < ∞ and ω ∈ D.
equivalent:
(i) kQΘ kLpω . kΘ0 kApω for all inner functions Θ;
(ii) kQΘn kLpω . kΘ0n kApω for all monomials Θn (z) = z n ;
(iii) µp,ω is a p-Carleson measure for Apω ;
bp .
(iv) ω ∈ D
dµp,ω (z) = (1 − |z|)
p−1
Z
0
|z|
ω(r )
dr
(1 − r )p
!
dA(z).
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
Corollary
bp . Then, for any all inner function Θ
Let 0 < p < ∞ and ω ∈ D
kΘ0 kApω kQΘ kLpω .
for all inner functions Θ.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
Theorem
b Then
Let 0 < p < ∞ and ω ∈ D.
Z
Z 1 − |Θ(z)| p
ω(z) dA(z)
|Θ0 (z)|p ω(z) dA(z) 1 − |z|
D
D
bp , i.e.,
for all inner functions Θ if and only if ω ∈ D
Z
(1 − r )p r ω(s)
sup
ds < ∞ .
p
b (r )
0<r <1 ω
0 (1 − s)
(3.1)
(3.2)
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
Theorem
b Then
Let 0 < p < ∞ and ω ∈ D.
Z
Z 1 − |Θ(z)| p
ω(z) dA(z)
|Θ0 (z)|p ω(z) dA(z) 1 − |z|
D
D
bp , i.e.,
for all inner functions Θ if and only if ω ∈ D
Z
(1 − r )p r ω(s)
sup
ds < ∞ .
p
b (r )
0<r <1 ω
0 (1 − s)
(3.1)
(3.2)
Let 0 < p < ∞, α > −1, we have
bp (α) < ∞ ⇔ kΘ0 k p kQΘ k p ,
p > α+1 ⇔ D
Aα
Lα
(Ahern, 1983).
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
Theorem
b Then
Let 0 < p < ∞ and ω ∈ D.
Z
Z 1 − |Θ(z)| p
ω(z) dA(z)
|Θ0 (z)|p ω(z) dA(z) 1 − |z|
D
D
bp , i.e.,
for all inner functions Θ if and only if ω ∈ D
Z
(1 − r )p r ω(s)
sup
ds < ∞ .
p
b (r )
0<r <1 ω
0 (1 − s)
(3.1)
(3.2)
Let 0 < p < ∞, α > −1, we have
bp (α) < ∞ ⇔ kΘ0 k p kQΘ k p ,
p > α+1 ⇔ D
Aα
Lα
(Ahern, 1983).
Using other techniques A. Baranov and R. Zaraouf (2016) have
recently rediscovered the same result for 1 < p < ∞.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Weighted Bergman Spaces
Theorem
b Then
Let 0 < p < ∞ and ω ∈ D.
Z
Z 1 − |Θ(z)| p
ω(z) dA(z)
|Θ0 (z)|p ω(z) dA(z) 1 − |z|
D
D
bp , i.e.,
for all inner functions Θ if and only if ω ∈ D
Z
(1 − r )p r ω(s)
sup
ds < ∞ .
p
b (r )
0<r <1 ω
0 (1 − s)
(3.1)
(3.2)
Let 0 < p < ∞, α > −1, we have
bp (α) < ∞ ⇔ kΘ0 k p kQΘ k p ,
p > α+1 ⇔ D
Aα
Lα
(Ahern, 1983).
Using other techniques A. Baranov and R. Zaraouf (2016) have
recently rediscovered the same result for 1 < p < ∞.
Estimate in (3.2) can be interpreted as a simplified Muckenhoupt
or Bekollé-Bonami condition.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
The inequality
Z
1 − |Θ(r ξ)| ≤ r
is valid for almost every ξ ∈ T.
1
Θ (sξ)ds ,
0
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
The inequality
Z
1 − |Θ(r ξ)| ≤ r
1
Θ (sξ)ds ,
0
is valid for almost every ξ ∈ T.
If we consider the linear average operator
R1 z |z| f s |z| ds
T (f )(z) :=
1 − |z|
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
The inequality
Z
1 − |Θ(r ξ)| ≤ r
1
Θ (sξ)ds ,
0
is valid for almost every ξ ∈ T.
If we consider the linear average operator
R1 z |z| f s |z| ds
T (f )(z) :=
1 − |z|
the right hand side of (3.1) is dominated by kT (Θ0 )kpLp .
ω
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
The inequality
Z
1 − |Θ(r ξ)| ≤ r
1
Θ (sξ)ds ,
0
is valid for almost every ξ ∈ T.
If we consider the linear average operator
R1 z |z| f s |z| ds
T (f )(z) :=
1 − |z|
the right hand side of (3.1) is dominated by kT (Θ0 )kpLp .
ω
Therefore, (3.1) is satisfied whenever T be bounded from the
Bergman space Apω to the Lebesgue space Lpω .
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
But we can improve the T operator a little bit.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
But we can improve the T operator a little bit. Take the lens
region
ζ 1
Γ(z) = ζ ∈ D : | arg z − arg ζ| <
1 − 2
z
with vertexes inside the unit disc.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
But we can improve the T operator a little bit. Take the lens
region
ζ 1
Γ(z) = ζ ∈ D : | arg z − arg ζ| <
1 − 2
z
with vertexes inside the unit disc.
And consider the maximal function,
N(f )(z) := sup |f (ζ)| ,
ζ∈Γ(z)
and the sublinear operator T N defined by
R1
z
N(f
)
s
|z|
|z| ds
T N (f )(z) :=
.
1 − |z|
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
Theorem (Main result extended)
b Then the following assertions are
Let 0 < p < ∞ and ω ∈ D.
equivalent:
(i) kQΘ kLpω . kΘ0 kApω for all inner functions Θ;
(ii) kQΘn kLpω . kΘ0n kApω for all monomials Θn (z) = z n ;
(iii) µp,ω is a p-Carleson measure for Apω ;
bp .
(iv) ω ∈ D
(v) T N : Apω → Lpω is bounded;
(vi) T : Apω → Lpω is bounded.
Moreover,
bp (ω) .
kT N kpAp →Lp kT kpAp →Lp kIdkpAp →Lp D
ω
ω
ω
ω
ω
ω
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
It is clear that
|T (f )(z)| ≤ T N (f )(z) for all z ∈ D.
T N (f ) dominates the maximal function N(f ) pointwise, and
therefore
N : Apω → Lpω is bounded whenever T N : Apω → Lpω is bounded.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
It is clear that
|T (f )(z)| ≤ T N (f )(z) for all z ∈ D.
T N (f ) dominates the maximal function N(f ) pointwise, and
therefore
N : Apω → Lpω is bounded whenever T N : Apω → Lpω is bounded.
From this and the main result follows that :
Corollary
If T : Apω → Lpω is bounded then N : Apω → Lpω is bounded.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Some integral operators
Tools used in the proof:
Carleson measures for Bergman spaces,
Hardy-Littlewood maximal theorem,
A classical result due to Muckenhoupt on double integrals,
Test functions,
bp .
Self-improving property for weight in D
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Hardy spaces and Schwarz-Pick
The facts
Apα → H p as α → −1+ , and
bp if and
The standard weight ω(z) = (1 − |z|)α belongs to D
only if p > α + 1,
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Hardy spaces and Schwarz-Pick
The facts
Apα → H p as α → −1+ , and
bp if and
The standard weight ω(z) = (1 − |z|)α belongs to D
only if p > α + 1,
Theorem
Let 0 < p < ∞. Then
kΘ0 kpH p
Z
sup
0≤r <1 0
for all inner function Θ.
2π
1 − |Θ(re iθ )|
1−r
p
dθ
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Further Observations
Write ω ∈ Dp , if
(1 − r )p−1
Dp (ω) = ess sup0<r <1
ω(r )
Z
0
r
ω(s)
ds < ∞.
(1 − s)p
bp the inclusion
If ω ∈ Dp , then Fubini’s theorem gives that ω ∈ D
being strict.
A reasoning similar to the one used in the proof of the main result
gives
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Further Observations
Write ω ∈ Dp , if
(1 − r )p−1
Dp (ω) = ess sup0<r <1
ω(r )
r
Z
0
ω(s)
ds < ∞.
(1 − s)p
bp the inclusion
If ω ∈ Dp , then Fubini’s theorem gives that ω ∈ D
being strict.
A reasoning similar to the one used in the proof of the main result
gives
Proposition
If ω ∈ Dp and 1 ≤ p < ∞ then,
Z
δ
1
1 − |Θ(re iθ )|
1−r
p
Z
ω(r ) dr ≤ C
1
|Θ0 (re iθ )|p ω(r ) dr , (6.1)
δ
for any δ ∈ (0, 1), for any inner function Θ and for almost all θ.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Further Observations
If p > 1, one can find a condition different from ω ∈ Dp that
ensures (6.1) by using a result by Muckenhoupt on Hardy
operators.
Proposition
Let 1 < p < ∞ and ω a radial weight. If
Z
sup
0<r <1 0
r
ω(s)
ds
(1 − s)p
Z
1
ω(s)
1
1−p
p−1
ds
< ∞,
(6.2)
r
then there exists a constant C = C (p, ω) > 0 such that (6.1) is
satisfied for almost all θ ∈ [0, 2π) and for all inner functions Θ.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Further Observations
If p > 1, one can find a condition different from ω ∈ Dp that
ensures (6.1) by using a result by Muckenhoupt on Hardy
operators.
Proposition
Let 1 < p < ∞ and ω a radial weight. If
Z
sup
0<r <1 0
r
ω(s)
ds
(1 − s)p
Z
1
ω(s)
1
1−p
p−1
ds
< ∞,
(6.2)
r
then there exists a constant C = C (p, ω) > 0 such that (6.1) is
satisfied for almost all θ ∈ [0, 2π) and for all inner functions Θ.
Write ω ∈ Mp if 1 < p < ∞ and ω satisfies (6.2).
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Further Observations
An immediate consequence of this Proposition is that
Corollary
If 1 < p < ∞ and ω ∈ Mp then
Z
0
p
Z
|Θ (z)| ω(z) dA(z) S(a)
S(a)
1 − |Θ(z)|
1 − |z|
p
ω(z) dA(z)
for all a ∈ D and any inner function Θ.
S(a) =
z ∈ D : 1 − |a| ≤ |z| < 1,
1 − |a|
| arg z − arg a| ≤
2
.
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Further Observations
The operator T is closely related to the Bergman projection
Z
f (ζ)
dA(ζ)
P(f )(z) =
2
D (1 − ζz)
b and ω
Write ω ∈ R, if ω ∈ D
b (r ) ω(r )(1 − r ) for all 0 ≤ r < 1.
Corollary
Let 1 ≤ p < ∞ and ω ∈ R. Then the following statements are
equivalent:
(i) T : Apω → Lpω is bounded;
(ii) P : Lpω → Apω is bounded;
bp .
(iii) ω ∈ D
Derivatives of Inner Functions and the Schwarz-Pick Lemma
Further Observations
Muchas gracias !!!!