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Mini-course on the Dirichlet space
Thomas Ransford
Universite Laval, Quebec
Focus Program on Analytic Functions Spaces and ApplicationsFields Institute, Toronto, July 2021
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Chapter 1
Introduction
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What is the Dirichlet space?
The Dirichlet space DD is the set of f holomorphic in D whose Dirichlet integral is finite:
D(f ) :=1
π
∫D|f ′(z)|2 dA(z) <∞.
If f (z) =∑
k≥0 akzk , then D(f ) =
∑k≥0 k |ak |2.
Consequently D ⊂ H2.
D is a Hilbert space with respect to the norm ‖ · ‖D given by
‖f ‖2D := ‖f ‖2
H2 +D(f ) =∑k≥0
(k + 1)|ak |2.
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What is the Dirichlet space?
The Dirichlet space DD is the set of f holomorphic in D whose Dirichlet integral is finite:
D(f ) :=1
π
∫D|f ′(z)|2 dA(z) <∞.
If f (z) =∑
k≥0 akzk , then D(f ) =
∑k≥0 k |ak |2.
Consequently D ⊂ H2.
D is a Hilbert space with respect to the norm ‖ · ‖D given by
‖f ‖2D := ‖f ‖2
H2 +D(f ) =∑k≥0
(k + 1)|ak |2.
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History and motivation
Very very brief history of D:
Beurling (1930’s–1940’s)
Carleson (1950’s–1960’s)
. . .
Some reasons for studying D:Potential theory, energy, capacity
Geometric interpretation, Mobius invariance
Weighted shifts, invariant subspaces
Borderline case, still many open problems
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History and motivation
Very very brief history of D:Beurling (1930’s–1940’s)
Carleson (1950’s–1960’s)
. . .
Some reasons for studying D:Potential theory, energy, capacity
Geometric interpretation, Mobius invariance
Weighted shifts, invariant subspaces
Borderline case, still many open problems
![Page 7: Mini-course on the Dirichlet space · 2021. 7. 15. · What is the Dirichlet space? The Dirichlet space D Dis the set of f holomorphic in D whose Dirichlet integral is nite: D(f)](https://reader035.vdocuments.mx/reader035/viewer/2022070221/613778780ad5d2067648a3ee/html5/thumbnails/7.jpg)
History and motivation
Very very brief history of D:Beurling (1930’s–1940’s)
Carleson (1950’s–1960’s)
. . .
Some reasons for studying D:Potential theory, energy, capacity
Geometric interpretation, Mobius invariance
Weighted shifts, invariant subspaces
Borderline case, still many open problems
![Page 8: Mini-course on the Dirichlet space · 2021. 7. 15. · What is the Dirichlet space? The Dirichlet space D Dis the set of f holomorphic in D whose Dirichlet integral is nite: D(f)](https://reader035.vdocuments.mx/reader035/viewer/2022070221/613778780ad5d2067648a3ee/html5/thumbnails/8.jpg)
History and motivation
Very very brief history of D:Beurling (1930’s–1940’s)
Carleson (1950’s–1960’s)
. . .
Some reasons for studying D:Potential theory, energy, capacity
Geometric interpretation, Mobius invariance
Weighted shifts, invariant subspaces
Borderline case, still many open problems
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What to study?
Some topics of interest:
Boundary behavior
Zeros
Multipliers
Reproducing kernel
Interpolation
Conformal invariance
Shift-invariant subspaces
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Where to find out more about D?
Survey articles:
W. Ross, The classical Dirichlet space, Recent advances inoperator-related function theory, 171–197, Contemp. Math., 393,Amer. Math. Soc., Providence, RI, 2006.
N. Arcozzi, R. Rochberg, E. Sawyer, B. Wick, The Dirichlet space: asurvey, New York J. Math. 17A (2011), 45–86.
Monographs:
O. El-Fallah, K. Kellay, J. Mashreghi, T. Ransford, A primer on theDirichlet space, Cambridge University Press, Cambridge, 2014
N. Arcozzi, R. Rochberg, E. Sawyer, B. Wick, The Dirichlet spaceand related function spaces, Amer. Math. Soc., Providence RI, 2019.
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Chapter 2
Capacity
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Energy
Let µ be a finite positive Borel measure on T.
Energy of µ
I (µ) :=
∫T
∫T
log2
|λ− ζ|dµ(λ) dµ(ζ).
May have I (µ) = +∞.
Formula for I (µ) in terms of Fourier coefficients of µ:
I (µ) =∑k≥1
|µ(k)|2
k+ µ(T)2 log 2.
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Capacity of compact sets
Capacity of compact F ⊂ T
c(F ) := 1/ inf{I (µ) : µ is a probability measure on F}.
Elementary properties:
F1 ⊂ F2 ⇒ c(F1) ≤ c(F2)
Fn ↓ F ⇒ c(Fn) ↓ c(F )
c(F1 ∪ F2) ≤ c(F1) + c(F2)
Examples:
c(F ) ≤ 1/ log(2/ diam(F ))
c(F ) = 0 if F is finite or countable
c(F ) ≥ 1/ log(2πe/|F |). In particular c(F ) = 0⇒ |F | = 0.
c(F ) > 0 if F is the (circular) middle-third Cantor set.
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Capacity of compact sets
Capacity of compact F ⊂ T
c(F ) := 1/ inf{I (µ) : µ is a probability measure on F}.
Elementary properties:
F1 ⊂ F2 ⇒ c(F1) ≤ c(F2)
Fn ↓ F ⇒ c(Fn) ↓ c(F )
c(F1 ∪ F2) ≤ c(F1) + c(F2)
Examples:
c(F ) ≤ 1/ log(2/ diam(F ))
c(F ) = 0 if F is finite or countable
c(F ) ≥ 1/ log(2πe/|F |). In particular c(F ) = 0⇒ |F | = 0.
c(F ) > 0 if F is the (circular) middle-third Cantor set.
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Capacity of compact sets
Capacity of compact F ⊂ T
c(F ) := 1/ inf{I (µ) : µ is a probability measure on F}.
Elementary properties:
F1 ⊂ F2 ⇒ c(F1) ≤ c(F2)
Fn ↓ F ⇒ c(Fn) ↓ c(F )
c(F1 ∪ F2) ≤ c(F1) + c(F2)
Examples:
c(F ) ≤ 1/ log(2/ diam(F ))
c(F ) = 0 if F is finite or countable
c(F ) ≥ 1/ log(2πe/|F |). In particular c(F ) = 0⇒ |F | = 0.
c(F ) > 0 if F is the (circular) middle-third Cantor set.
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Capacity of general sets
Inner capacity of E ⊂ T
c(E ) := sup{c(F ) : compact F ⊂ E}
Outer capacity of E ⊂ T
c∗(E ) := inf{c(U) : open U ⊃ E}
c∗(∪nEn) ≤∑
n c∗(En) (not true for c(·)).
c∗(E ) = c(E ) if E is Borel (Choquet’s capacitability theorem)
A property holds q.e. if it holds outside an E with c∗(E ) = 0.
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Equilibrium measures
Let F be a compact subset of T. Recall that
c(F ) := 1/ inf{I (µ) : µ is a probability measure on F}.
Measure µ attaining the inf is called an equilibrium measure for F .
Proposition
If c(F ) > 0, then F admits a unique equilibrium measure.
Fundamental theorem of potential theory (Frostman, 1935)
Let µ be the equilibrium measure for F , and Vµ be its potential, i.e.
Vµ(z) :=
∫T
log2
|z − ζ|dµ(ζ).
Then Vµ ≤ 1/c(F ) on T, and Vµ = 1/c(F ) q.e. on F .
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Equilibrium measures
Let F be a compact subset of T. Recall that
c(F ) := 1/ inf{I (µ) : µ is a probability measure on F}.
Measure µ attaining the inf is called an equilibrium measure for F .
Proposition
If c(F ) > 0, then F admits a unique equilibrium measure.
Fundamental theorem of potential theory (Frostman, 1935)
Let µ be the equilibrium measure for F , and Vµ be its potential, i.e.
Vµ(z) :=
∫T
log2
|z − ζ|dµ(ζ).
Then Vµ ≤ 1/c(F ) on T, and Vµ = 1/c(F ) q.e. on F .
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Chapter 3
Boundary behavior
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Preliminary remarks
Every f ∈ D has non-tangential limits a.e. on T (as f ∈ H2).
There exists f ∈ D such that limr→1− |f (r)| =∞.
Example: Consider
f (z) :=∑k≥2
zk
k log k.
Then
D(f ) =∑k≥2
k1
(k log k)2=∑k≥2
1
k(log k)2<∞,
but
lim infr→1−
f (r) ≥∑k≥2
1
k log k=∞.
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Preliminary remarks
Every f ∈ D has non-tangential limits a.e. on T (as f ∈ H2).
There exists f ∈ D such that limr→1− |f (r)| =∞.
Example: Consider
f (z) :=∑k≥2
zk
k log k.
Then
D(f ) =∑k≥2
k1
(k log k)2=∑k≥2
1
k(log k)2<∞,
but
lim infr→1−
f (r) ≥∑k≥2
1
k log k=∞.
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Beurling’s theorem
Theorem (Beurling, 1940)
If f ∈ D then f has non-tangential limits q.e. on T.
Remarks:
Beurling actually proved his result just for radial limits
Beurling’s theorem is sharp in the following sense:
Theorem (Carleson, 1952)
Given compact E ⊂ T of capacity zero, there exists f ∈ D such thatlimr→1− |f (rζ)| =∞ for all ζ ∈ E .
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Beurling’s theorem
Theorem (Beurling, 1940)
If f ∈ D then f has non-tangential limits q.e. on T.
Remarks:
Beurling actually proved his result just for radial limits
Beurling’s theorem is sharp in the following sense:
Theorem (Carleson, 1952)
Given compact E ⊂ T of capacity zero, there exists f ∈ D such thatlimr→1− |f (rζ)| =∞ for all ζ ∈ E .
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Capacitary weak-type and strong-type inequalities
Notation: Let f ∈ D. For ζ ∈ T, we write f ∗(ζ) := limr→1− f (rζ).Also A,B denote absolute positive constants.
Weak-type inequality (Beurling, 1940)
c(|f ∗| > t) ≤ A‖f ‖2D/t
2 (t > 0).
Corollary
|{|f ∗| > t}| ≤ Ae−Bt2/‖f ‖2
D (t > 0).
Strong-type inequality (Hansson, 1979)∫ ∞0
c(|f ∗| > t) t dt ≤ A‖f ‖2D.
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Capacitary weak-type and strong-type inequalities
Notation: Let f ∈ D. For ζ ∈ T, we write f ∗(ζ) := limr→1− f (rζ).Also A,B denote absolute positive constants.
Weak-type inequality (Beurling, 1940)
c(|f ∗| > t) ≤ A‖f ‖2D/t
2 (t > 0).
Corollary
|{|f ∗| > t}| ≤ Ae−Bt2/‖f ‖2
D (t > 0).
Strong-type inequality (Hansson, 1979)∫ ∞0
c(|f ∗| > t) t dt ≤ A‖f ‖2D.
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Douglas’ formula
Theorem (Douglas, 1931)
If f ∈ H2, then
D(f ) =1
4π2
∫T
∫T
∣∣∣ f ∗(λ)− f ∗(ζ)
λ− ζ
∣∣∣2 |dλ| |dζ|.
Corollary
If f ∈ D, then f has oricyclic limits a.e. in T.
non-tangential approach region oricyclic approach region
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Douglas’ formula
Theorem (Douglas, 1931)
If f ∈ H2, then
D(f ) =1
4π2
∫T
∫T
∣∣∣ f ∗(λ)− f ∗(ζ)
λ− ζ
∣∣∣2 |dλ| |dζ|.Corollary
If f ∈ D, then f has oricyclic limits a.e. in T.
non-tangential approach region oricyclic approach region
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Exponential approach region
Theorem (Nagel–Rudin–Shapiro, 1982)
If f ∈ D then, for a.e. ζ ∈ D, we have f (z)→ f ∗(ζ) as z → ζ in theexponential approach region
|z − ζ| < κ(
log1
1− |z |
)−1.
Remarks:
Approach region is ‘widest possible’.
This is an a.e. result (not q.e.).
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Carleson’s formula
Notation: Let f ∈ H2 with canonical factorization f = BSO.Let (an) be the zeros of B, and σ be the singular measure of S .
Theorem (Carleson, 1960)
D(f ) =
∫T
∫T
(|f ∗(λ)|2 − |f ∗(ζ)|2)(log |f ∗(λ)| − log |f ∗(ζ)|)|λ− ζ|2
|dλ|2π
|dζ|2π
+
∫T
(∑n
1− |an|2
|ζ − an|2+
∫T
2
|λ− ζ|2dσ(λ)
)|f ∗(ζ)|2 |dζ|
2π.
Corollary 1
If f belongs to D then so does its outer factor.
Corollary 2
The only inner functions in D are finite Blaschke products.
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Carleson’s formula
Notation: Let f ∈ H2 with canonical factorization f = BSO.Let (an) be the zeros of B, and σ be the singular measure of S .
Theorem (Carleson, 1960)
D(f ) =
∫T
∫T
(|f ∗(λ)|2 − |f ∗(ζ)|2)(log |f ∗(λ)| − log |f ∗(ζ)|)|λ− ζ|2
|dλ|2π
|dζ|2π
+
∫T
(∑n
1− |an|2
|ζ − an|2+
∫T
2
|λ− ζ|2dσ(λ)
)|f ∗(ζ)|2 |dζ|
2π.
Corollary 1
If f belongs to D then so does its outer factor.
Corollary 2
The only inner functions in D are finite Blaschke products.
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Carleson’s formula
Notation: Let f ∈ H2 with canonical factorization f = BSO.Let (an) be the zeros of B, and σ be the singular measure of S .
Theorem (Carleson, 1960)
D(f ) =
∫T
∫T
(|f ∗(λ)|2 − |f ∗(ζ)|2)(log |f ∗(λ)| − log |f ∗(ζ)|)|λ− ζ|2
|dλ|2π
|dζ|2π
+
∫T
(∑n
1− |an|2
|ζ − an|2+
∫T
2
|λ− ζ|2dσ(λ)
)|f ∗(ζ)|2 |dζ|
2π.
Corollary 1
If f belongs to D then so does its outer factor.
Corollary 2
The only inner functions in D are finite Blaschke products.
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Some further developments
Chang–Marshall theorem (1985):
sup{∫
Texp(|f ∗(e iθ)|2) dθ : f (0) = 0, D(f ) ≤ 1
}<∞.
Trade-off between approach regions and exceptional sets.Borichev (1994), Twomey (2002)
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Chapter 4
Zeros
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Preliminary remarks
A sequence (zn) in D (possibly with repetitions) is:
a zero set for D if ∃f ∈ D vanishing on (zn) but f 6≡ 0;
a uniqueness set for D if it is not a zero set.
Proposition
If (zn) is a zero set for D, then ∃f ∈ D vanishing precisely on (zn).
It is well known that (zn) is a zero set for the Hardy space H2 iff∑n
(1− |zn|) <∞.
What about the Dirichlet space?
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Preliminary remarks
A sequence (zn) in D (possibly with repetitions) is:
a zero set for D if ∃f ∈ D vanishing on (zn) but f 6≡ 0;
a uniqueness set for D if it is not a zero set.
Proposition
If (zn) is a zero set for D, then ∃f ∈ D vanishing precisely on (zn).
It is well known that (zn) is a zero set for the Hardy space H2 iff∑n
(1− |zn|) <∞.
What about the Dirichlet space?
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The three cases
Case I (obvious)∑n(1− |zn|) =∞ ⇒ (zn) is a uniqueness set for D.
Case II (Shapiro–Shields, 1962)∑n 1/| log(1− |zn|)| <∞ ⇒ (zn) is a zero set for D.
Case III (Nagel–Rudin–Shapiro, 1982)
If (zn) satisfies neither condition, then there exist a zero set (z ′n) and auniqueness set (z ′′n ) with |zn| = |z ′n| = |z ′′n | for all n.
Thus, in Case III, the arguments of (zn) matter. Back to this later.
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The three cases
Case I (obvious)∑n(1− |zn|) =∞ ⇒ (zn) is a uniqueness set for D.
Case II (Shapiro–Shields, 1962)∑n 1/| log(1− |zn|)| <∞ ⇒ (zn) is a zero set for D.
Case III (Nagel–Rudin–Shapiro, 1982)
If (zn) satisfies neither condition, then there exist a zero set (z ′n) and auniqueness set (z ′′n ) with |zn| = |z ′n| = |z ′′n | for all n.
Thus, in Case III, the arguments of (zn) matter. Back to this later.
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The three cases
Case I (obvious)∑n(1− |zn|) =∞ ⇒ (zn) is a uniqueness set for D.
Case II (Shapiro–Shields, 1962)∑n 1/| log(1− |zn|)| <∞ ⇒ (zn) is a zero set for D.
Case III (Nagel–Rudin–Shapiro, 1982)
If (zn) satisfies neither condition, then there exist a zero set (z ′n) and auniqueness set (z ′′n ) with |zn| = |z ′n| = |z ′′n | for all n.
Thus, in Case III, the arguments of (zn) matter. Back to this later.
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Boundary zero sets
Let E be a closed subset of T. It is called a Carleson set if∫T
log( 2
dist(ζ,E )
)|dζ| <∞.
Theorem (Carleson 1952)
If E is a Carleson set, then ∃f ∈ A1(D) with f −1(0) = E .
Theorem (Carleson 1952, Brown–Cohn 1985)
If c(E ) = 0, then ∃f ∈ D ∩ A(D) with f −1(0) = E .
Neither result implies the other.
Clearly, if |E | > 0, then E is a boundary uniqueness set for D. Butthere also exist closed uniqueness sets E with |E | = 0.
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Boundary zero sets
Let E be a closed subset of T. It is called a Carleson set if∫T
log( 2
dist(ζ,E )
)|dζ| <∞.
Theorem (Carleson 1952)
If E is a Carleson set, then ∃f ∈ A1(D) with f −1(0) = E .
Theorem (Carleson 1952, Brown–Cohn 1985)
If c(E ) = 0, then ∃f ∈ D ∩ A(D) with f −1(0) = E .
Neither result implies the other.
Clearly, if |E | > 0, then E is a boundary uniqueness set for D. Butthere also exist closed uniqueness sets E with |E | = 0.
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Boundary zero sets
Let E be a closed subset of T. It is called a Carleson set if∫T
log( 2
dist(ζ,E )
)|dζ| <∞.
Theorem (Carleson 1952)
If E is a Carleson set, then ∃f ∈ A1(D) with f −1(0) = E .
Theorem (Carleson 1952, Brown–Cohn 1985)
If c(E ) = 0, then ∃f ∈ D ∩ A(D) with f −1(0) = E .
Neither result implies the other.
Clearly, if |E | > 0, then E is a boundary uniqueness set for D. Butthere also exist closed uniqueness sets E with |E | = 0.
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Boundary zero sets
Let E be a closed subset of T. It is called a Carleson set if∫T
log( 2
dist(ζ,E )
)|dζ| <∞.
Theorem (Carleson 1952)
If E is a Carleson set, then ∃f ∈ A1(D) with f −1(0) = E .
Theorem (Carleson 1952, Brown–Cohn 1985)
If c(E ) = 0, then ∃f ∈ D ∩ A(D) with f −1(0) = E .
Neither result implies the other.
Clearly, if |E | > 0, then E is a boundary uniqueness set for D. Butthere also exist closed uniqueness sets E with |E | = 0.
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Arguments of zero sets
We return to zero sets within D, now considering their arguments.
Theorem (Caughran, 1970)
Let (e iθn) be a sequence in T. The following are equivalent:
(rneiθn) is a zero set for D whenever
∑n(1− rn) <∞.
E := {e iθn : n ≥ 1} is a Carleson set.
Example of a Blaschke sequence that is a uniqueness set for D
zn :=(
1− 1
n(log n)2
)e i/ log n
There is still no satisfactory complete characterization of zero sets.
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Arguments of zero sets
We return to zero sets within D, now considering their arguments.
Theorem (Caughran, 1970)
Let (e iθn) be a sequence in T. The following are equivalent:
(rneiθn) is a zero set for D whenever
∑n(1− rn) <∞.
E := {e iθn : n ≥ 1} is a Carleson set.
Example of a Blaschke sequence that is a uniqueness set for D
zn :=(
1− 1
n(log n)2
)e i/ log n
There is still no satisfactory complete characterization of zero sets.
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Arguments of zero sets
We return to zero sets within D, now considering their arguments.
Theorem (Caughran, 1970)
Let (e iθn) be a sequence in T. The following are equivalent:
(rneiθn) is a zero set for D whenever
∑n(1− rn) <∞.
E := {e iθn : n ≥ 1} is a Carleson set.
Example of a Blaschke sequence that is a uniqueness set for D
zn :=(
1− 1
n(log n)2
)e i/ log n
There is still no satisfactory complete characterization of zero sets.
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Some further developments
Carleson sets as zero sets for A∞(D)Taylor–Williams (1970)
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Chapter 5
Multipliers
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Preliminary remarks
Proposition
D is not an algebra.
Proof: Suppose D is an algebra.
By closed graph theorem, D isomorphic to a Banach algebra.
f 7→ f (z) is a character, so |f (z)| ≤ spectral radius of f .
Therefore every f ∈ D is bounded. Contradiction.
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Preliminary remarks
Proposition
D is not an algebra.
Proof: Suppose D is an algebra.
By closed graph theorem, D isomorphic to a Banach algebra.
f 7→ f (z) is a character, so |f (z)| ≤ spectral radius of f .
Therefore every f ∈ D is bounded. Contradiction.
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Preliminary remarks
Proposition
D is not an algebra.
Proof: Suppose D is an algebra.
By closed graph theorem, D isomorphic to a Banach algebra.
f 7→ f (z) is a character, so |f (z)| ≤ spectral radius of f .
Therefore every f ∈ D is bounded. Contradiction.
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Preliminary remarks
Proposition
D is not an algebra.
Proof: Suppose D is an algebra.
By closed graph theorem, D isomorphic to a Banach algebra.
f 7→ f (z) is a character, so |f (z)| ≤ spectral radius of f .
Therefore every f ∈ D is bounded. Contradiction.
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Preliminary remarks
Proposition
D is not an algebra.
Proof: Suppose D is an algebra.
By closed graph theorem, D isomorphic to a Banach algebra.
f 7→ f (z) is a character, so |f (z)| ≤ spectral radius of f .
Therefore every f ∈ D is bounded. Contradiction.
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Multipliers
Definition
A multiplier for D is a function φ such that φf ∈ D for all f ∈ D. The setof multipliers is an algebra, denoted by M(D).
Remark: In the case of Hardy spaces, M(H2) = H∞.
When is φ a multiplier of D?
Necessary condition: φ ∈ D ∩ H∞
Sufficient condition: φ′ ∈ H∞
To completely characterize multipliers, we introduce a new notion.
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Multipliers
Definition
A multiplier for D is a function φ such that φf ∈ D for all f ∈ D. The setof multipliers is an algebra, denoted by M(D).
Remark: In the case of Hardy spaces, M(H2) = H∞.
When is φ a multiplier of D?
Necessary condition: φ ∈ D ∩ H∞
Sufficient condition: φ′ ∈ H∞
To completely characterize multipliers, we introduce a new notion.
![Page 55: Mini-course on the Dirichlet space · 2021. 7. 15. · What is the Dirichlet space? The Dirichlet space D Dis the set of f holomorphic in D whose Dirichlet integral is nite: D(f)](https://reader035.vdocuments.mx/reader035/viewer/2022070221/613778780ad5d2067648a3ee/html5/thumbnails/55.jpg)
Multipliers
Definition
A multiplier for D is a function φ such that φf ∈ D for all f ∈ D. The setof multipliers is an algebra, denoted by M(D).
Remark: In the case of Hardy spaces, M(H2) = H∞.
When is φ a multiplier of D?
Necessary condition: φ ∈ D ∩ H∞
Sufficient condition: φ′ ∈ H∞
To completely characterize multipliers, we introduce a new notion.
![Page 56: Mini-course on the Dirichlet space · 2021. 7. 15. · What is the Dirichlet space? The Dirichlet space D Dis the set of f holomorphic in D whose Dirichlet integral is nite: D(f)](https://reader035.vdocuments.mx/reader035/viewer/2022070221/613778780ad5d2067648a3ee/html5/thumbnails/56.jpg)
Multipliers
Definition
A multiplier for D is a function φ such that φf ∈ D for all f ∈ D. The setof multipliers is an algebra, denoted by M(D).
Remark: In the case of Hardy spaces, M(H2) = H∞.
When is φ a multiplier of D?
Necessary condition: φ ∈ D ∩ H∞
Sufficient condition: φ′ ∈ H∞
To completely characterize multipliers, we introduce a new notion.
![Page 57: Mini-course on the Dirichlet space · 2021. 7. 15. · What is the Dirichlet space? The Dirichlet space D Dis the set of f holomorphic in D whose Dirichlet integral is nite: D(f)](https://reader035.vdocuments.mx/reader035/viewer/2022070221/613778780ad5d2067648a3ee/html5/thumbnails/57.jpg)
Carleson measures
Definition
A measure µ on D is a Carleson measure for D if ∃C such that∫D|f |2 dµ ≤ C‖f ‖2
D (f ∈ D).
With this notion in hand, it is quite easy to characterize multipliers:
Proposition
φ ∈M(D) iff both φ ∈ H∞ and |φ′|2 dA is a Carleson measure for D.
Begs a new question: how to characterize Carleson measures?
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Carleson measures
Definition
A measure µ on D is a Carleson measure for D if ∃C such that∫D|f |2 dµ ≤ C‖f ‖2
D (f ∈ D).
With this notion in hand, it is quite easy to characterize multipliers:
Proposition
φ ∈M(D) iff both φ ∈ H∞ and |φ′|2 dA is a Carleson measure for D.
Begs a new question: how to characterize Carleson measures?
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Carleson measures
Definition
A measure µ on D is a Carleson measure for D if ∃C such that∫D|f |2 dµ ≤ C‖f ‖2
D (f ∈ D).
With this notion in hand, it is quite easy to characterize multipliers:
Proposition
φ ∈M(D) iff both φ ∈ H∞ and |φ′|2 dA is a Carleson measure for D.
Begs a new question: how to characterize Carleson measures?
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Characterization of Carleson measures
Let µ be a finite positive measure on D.S(I ) := {re iθ : 1− |I | < r < 1, e iθ ∈ I}.Carleson (1962): µ is Carleson for H2 iff µ(S(I )) = O(|I |).
When is µ a Carleson measure for D?
Theorem (Wynn, 2011)
The condition µ(S(I )) = O(ψ(|I |)) is:
necessary if ψ(x) := 1/ log(1/x);
sufficient if ψ(x) := 1/ log(1/x)(log log(1/x))α with α > 1.
Theorem (Stegenga, 1980)
µ is a Carleson measure for D iff there is a constant A such that, for everyfinite set of disjoint closed subarcs I1, . . . , In of T,
µ(∪nj=1S(Ij)
)≤ Ac
(∪nj=1Ij
).
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Characterization of Carleson measures
Let µ be a finite positive measure on D.S(I ) := {re iθ : 1− |I | < r < 1, e iθ ∈ I}.Carleson (1962): µ is Carleson for H2 iff µ(S(I )) = O(|I |).
When is µ a Carleson measure for D?
Theorem (Wynn, 2011)
The condition µ(S(I )) = O(ψ(|I |)) is:
necessary if ψ(x) := 1/ log(1/x);
sufficient if ψ(x) := 1/ log(1/x)(log log(1/x))α with α > 1.
Theorem (Stegenga, 1980)
µ is a Carleson measure for D iff there is a constant A such that, for everyfinite set of disjoint closed subarcs I1, . . . , In of T,
µ(∪nj=1S(Ij)
)≤ Ac
(∪nj=1Ij
).
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Characterization of Carleson measures
Let µ be a finite positive measure on D.S(I ) := {re iθ : 1− |I | < r < 1, e iθ ∈ I}.Carleson (1962): µ is Carleson for H2 iff µ(S(I )) = O(|I |).
When is µ a Carleson measure for D?
Theorem (Wynn, 2011)
The condition µ(S(I )) = O(ψ(|I |)) is:
necessary if ψ(x) := 1/ log(1/x);
sufficient if ψ(x) := 1/ log(1/x)(log log(1/x))α with α > 1.
Theorem (Stegenga, 1980)
µ is a Carleson measure for D iff there is a constant A such that, for everyfinite set of disjoint closed subarcs I1, . . . , In of T,
µ(∪nj=1S(Ij)
)≤ Ac
(∪nj=1Ij
).
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Multipliers and reproducing kernels
If f ∈ D and w ∈ D, then f (w) = 〈f , kw 〉D, where
kw (z) :=1
wzlog( 1
1− wz
)(w , z ∈ D).
The function kw is called the reproducing kernel for w .
Proposition
Let φ ∈M(D) and define Mφ : D → D by Mφ(f ) := φf . Then
M∗φ(kw ) = φ(w)kw (w ∈ D).
Proof: For all f ∈ D, we have
〈f ,M∗φ(kw )〉D = 〈φf , kw 〉D = φ(w)f (w) = φ(w)〈f , kw 〉D = 〈f , φ(w)kw 〉D.
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Multipliers and reproducing kernels
If f ∈ D and w ∈ D, then f (w) = 〈f , kw 〉D, where
kw (z) :=1
wzlog( 1
1− wz
)(w , z ∈ D).
The function kw is called the reproducing kernel for w .
Proposition
Let φ ∈M(D) and define Mφ : D → D by Mφ(f ) := φf . Then
M∗φ(kw ) = φ(w)kw (w ∈ D).
Proof: For all f ∈ D, we have
〈f ,M∗φ(kw )〉D = 〈φf , kw 〉D = φ(w)f (w) = φ(w)〈f , kw 〉D = 〈f , φ(w)kw 〉D.
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Multipliers and reproducing kernels
If f ∈ D and w ∈ D, then f (w) = 〈f , kw 〉D, where
kw (z) :=1
wzlog( 1
1− wz
)(w , z ∈ D).
The function kw is called the reproducing kernel for w .
Proposition
Let φ ∈M(D) and define Mφ : D → D by Mφ(f ) := φf . Then
M∗φ(kw ) = φ(w)kw (w ∈ D).
Proof: For all f ∈ D, we have
〈f ,M∗φ(kw )〉D = 〈φf , kw 〉D = φ(w)f (w) = φ(w)〈f , kw 〉D = 〈f , φ(w)kw 〉D.
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Pick interpolation
Problem: Given z1, . . . , zn ∈ D and w1, . . . ,wn ∈ D, does there existφ ∈M(D) with ‖Mφ‖ ≤ 1 such that φ(zj) = wj for all j?
Theorem (Agler, 1988)
φ exists iff the matrix (1− w iwj)〈kzi , kzj 〉D is positive semi-definite.
Necessity is a simple consequence of the preceding proposition. Thesame argument works for any RKHS.
Sufficiency is a property of the Dirichlet kernel (‘Pick property’).
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Pick interpolation
Problem: Given z1, . . . , zn ∈ D and w1, . . . ,wn ∈ D, does there existφ ∈M(D) with ‖Mφ‖ ≤ 1 such that φ(zj) = wj for all j?
Theorem (Agler, 1988)
φ exists iff the matrix (1− w iwj)〈kzi , kzj 〉D is positive semi-definite.
Necessity is a simple consequence of the preceding proposition. Thesame argument works for any RKHS.
Sufficiency is a property of the Dirichlet kernel (‘Pick property’).
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Pick interpolation
Problem: Given z1, . . . , zn ∈ D and w1, . . . ,wn ∈ D, does there existφ ∈M(D) with ‖Mφ‖ ≤ 1 such that φ(zj) = wj for all j?
Theorem (Agler, 1988)
φ exists iff the matrix (1− w iwj)〈kzi , kzj 〉D is positive semi-definite.
Necessity is a simple consequence of the preceding proposition. Thesame argument works for any RKHS.
Sufficiency is a property of the Dirichlet kernel (‘Pick property’).
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Interpolating sequences
A sequence (zn)n≥1 in D is an interpolating sequence for M(D) if{(φ(z1), φ(z2), φ(z3), . . . ) : φ ∈M(D)
}= `∞.
Theorem (Marshall–Sundberg (1990’s), Bishop (1990’s), Bøe (2005))
The following are equivalent:
(zn)n≥1 is an interpolating sequence for M(D);∑n
δzn‖kzn‖2
is a D-Carleson measure and supn,mn 6=m
|〈kzn , kzm〉D|‖kzn‖D‖kzm‖D
< 1.
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Interpolating sequences
A sequence (zn)n≥1 in D is an interpolating sequence for M(D) if{(φ(z1), φ(z2), φ(z3), . . . ) : φ ∈M(D)
}= `∞.
Theorem (Marshall–Sundberg (1990’s), Bishop (1990’s), Bøe (2005))
The following are equivalent:
(zn)n≥1 is an interpolating sequence for M(D);∑n
δzn‖kzn‖2
is a D-Carleson measure and supn,mn 6=m
|〈kzn , kzm〉D|‖kzn‖D‖kzm‖D
< 1.
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Factorization theorems
We say f is cyclic for D if M(D)f = D.
Clearly f cyclic ⇒ f (z) 6= 0 for all z ∈ D. The converse is false.
f is cyclic for H2 iff f is an outer function (Beurling).
‘Inner-outer’ factorization (Jury–Martin, 2019)
If f ∈ D, then f = φg , where φ ∈M(D) and g is cyclic in D.
Smirnov factorization (Aleman–Hartz–McCarthy–Richter, 2017)
If f ∈ D, then f = φ1/φ2, where φ1, φ2 ∈M(D) and φ2 is cyclic in D.
Corollary
Given f ∈ D, there exists φ ∈M(D) with the same zero set.Consequently, the union of two zero sets is again one.
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Factorization theorems
We say f is cyclic for D if M(D)f = D.
Clearly f cyclic ⇒ f (z) 6= 0 for all z ∈ D. The converse is false.
f is cyclic for H2 iff f is an outer function (Beurling).
‘Inner-outer’ factorization (Jury–Martin, 2019)
If f ∈ D, then f = φg , where φ ∈M(D) and g is cyclic in D.
Smirnov factorization (Aleman–Hartz–McCarthy–Richter, 2017)
If f ∈ D, then f = φ1/φ2, where φ1, φ2 ∈M(D) and φ2 is cyclic in D.
Corollary
Given f ∈ D, there exists φ ∈M(D) with the same zero set.Consequently, the union of two zero sets is again one.
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Factorization theorems
We say f is cyclic for D if M(D)f = D.
Clearly f cyclic ⇒ f (z) 6= 0 for all z ∈ D. The converse is false.
f is cyclic for H2 iff f is an outer function (Beurling).
‘Inner-outer’ factorization (Jury–Martin, 2019)
If f ∈ D, then f = φg , where φ ∈M(D) and g is cyclic in D.
Smirnov factorization (Aleman–Hartz–McCarthy–Richter, 2017)
If f ∈ D, then f = φ1/φ2, where φ1, φ2 ∈M(D) and φ2 is cyclic in D.
Corollary
Given f ∈ D, there exists φ ∈M(D) with the same zero set.Consequently, the union of two zero sets is again one.
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Factorization theorems
We say f is cyclic for D if M(D)f = D.
Clearly f cyclic ⇒ f (z) 6= 0 for all z ∈ D. The converse is false.
f is cyclic for H2 iff f is an outer function (Beurling).
‘Inner-outer’ factorization (Jury–Martin, 2019)
If f ∈ D, then f = φg , where φ ∈M(D) and g is cyclic in D.
Smirnov factorization (Aleman–Hartz–McCarthy–Richter, 2017)
If f ∈ D, then f = φ1/φ2, where φ1, φ2 ∈M(D) and φ2 is cyclic in D.
Corollary
Given f ∈ D, there exists φ ∈M(D) with the same zero set.Consequently, the union of two zero sets is again one.
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Some further developments
Further characterizations of multipliers and Carleson measures for DArcozzi–Rochberg–Sawyer (2002)
Reverse Carleson measuresFricain–Hartmann–Ross (2017)
Corona problem for M(D)Tolokonnikov (1991), Xiao (1998), Trent (2004)
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Chapter 6
Conformal invariance
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Preliminary remarks
Let φ : D→ C and f : φ(D)→ C be holomorphic functions.Write nφ(w) for the number of solutions z of φ(z) = w .
Change-of-variable formula
D(f ◦ φ) =1
π
∫φ(D)|f ′(w)|2nφ(w) dA(w).
Corollary 1
If φ is injective, then D(φ) = (area of φ(D))/π.
Corollary 2
If f ∈ D and φ ∈ aut(D), then f ◦ φ ∈ D and D(f ◦ φ) = D(f ).
This last property more-or-less characterizes D.
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Preliminary remarks
Let φ : D→ C and f : φ(D)→ C be holomorphic functions.Write nφ(w) for the number of solutions z of φ(z) = w .
Change-of-variable formula
D(f ◦ φ) =1
π
∫φ(D)|f ′(w)|2nφ(w) dA(w).
Corollary 1
If φ is injective, then D(φ) = (area of φ(D))/π.
Corollary 2
If f ∈ D and φ ∈ aut(D), then f ◦ φ ∈ D and D(f ◦ φ) = D(f ).
This last property more-or-less characterizes D.
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Preliminary remarks
Let φ : D→ C and f : φ(D)→ C be holomorphic functions.Write nφ(w) for the number of solutions z of φ(z) = w .
Change-of-variable formula
D(f ◦ φ) =1
π
∫φ(D)|f ′(w)|2nφ(w) dA(w).
Corollary 1
If φ is injective, then D(φ) = (area of φ(D))/π.
Corollary 2
If f ∈ D and φ ∈ aut(D), then f ◦ φ ∈ D and D(f ◦ φ) = D(f ).
This last property more-or-less characterizes D.
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Characterization of D via Mobius invariance
Notation:
H := a vector space of holomorphic functions on D〈·, ·〉 := a semi-inner product on H and E(f ) := 〈f , f 〉.
Theorem (Arazy–Fisher 1985, slightly modified)
Assume:
if f ∈ H and φ ∈ aut(D), then f ◦ φ ∈ H and E(f ◦ φ) = E(f );
‖f ‖2 := |f (0)|2 + E(f ) defines a Hilbert-space norm on H;
convergence in this norm implies pointwise convergence on D;
H contains a non-constant function.
Then H = D and E(·) ≡ aD(·) some constant a > 0.
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Composition operators
Given holomorphic φ : D→ D, define Cφ : Hol(D)→ Hol(D) by
Cφ(f ) := f ◦ φ.
If φ ∈ aut(D) then Cφ : D → D. For which other φ is this true?
If φ(z) :=∑
k≥1 2−kz4k , then φ : D→ D, but Cφ(D) 6⊂ D as φ /∈ D.
Theorem (MacCluer–Shapiro, 1986)
Cφ : D → D ⇐⇒∫S(I )
nφ dA = O(|I |2).
Corollary (El-Fallah–Kellay–Shabankhah–Youssfi, 2011)
Conditions for Cφ : D → D:
necessary: D(φk) = O(k) as k →∞.
sufficient: D(φk) = O(1) as k →∞.
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Composition operators
Given holomorphic φ : D→ D, define Cφ : Hol(D)→ Hol(D) by
Cφ(f ) := f ◦ φ.
If φ ∈ aut(D) then Cφ : D → D. For which other φ is this true?
If φ(z) :=∑
k≥1 2−kz4k , then φ : D→ D, but Cφ(D) 6⊂ D as φ /∈ D.
Theorem (MacCluer–Shapiro, 1986)
Cφ : D → D ⇐⇒∫S(I )
nφ dA = O(|I |2).
Corollary (El-Fallah–Kellay–Shabankhah–Youssfi, 2011)
Conditions for Cφ : D → D:
necessary: D(φk) = O(k) as k →∞.
sufficient: D(φk) = O(1) as k →∞.
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Composition operators
Given holomorphic φ : D→ D, define Cφ : Hol(D)→ Hol(D) by
Cφ(f ) := f ◦ φ.
If φ ∈ aut(D) then Cφ : D → D. For which other φ is this true?
If φ(z) :=∑
k≥1 2−kz4k , then φ : D→ D, but Cφ(D) 6⊂ D as φ /∈ D.
Theorem (MacCluer–Shapiro, 1986)
Cφ : D → D ⇐⇒∫S(I )
nφ dA = O(|I |2).
Corollary (El-Fallah–Kellay–Shabankhah–Youssfi, 2011)
Conditions for Cφ : D → D:
necessary: D(φk) = O(k) as k →∞.
sufficient: D(φk) = O(1) as k →∞.
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Composition operators
Given holomorphic φ : D→ D, define Cφ : Hol(D)→ Hol(D) by
Cφ(f ) := f ◦ φ.
If φ ∈ aut(D) then Cφ : D → D. For which other φ is this true?
If φ(z) :=∑
k≥1 2−kz4k , then φ : D→ D, but Cφ(D) 6⊂ D as φ /∈ D.
Theorem (MacCluer–Shapiro, 1986)
Cφ : D → D ⇐⇒∫S(I )
nφ dA = O(|I |2).
Corollary (El-Fallah–Kellay–Shabankhah–Youssfi, 2011)
Conditions for Cφ : D → D:
necessary: D(φk) = O(k) as k →∞.
sufficient: D(φk) = O(1) as k →∞.
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Weighted composition operators
Theorem (Mashreghi–J. Ransford–T. Ransford, 2018)
Let T : D → Hol(D) be a linear map. The following are equivalent:
T maps nowhere-vanishing functions to nowhere-vanishing functions.
∃ holomorphic functions φ : D→ D and ψ : D→ C \ {0} such that
Tf = ψ.(f ◦ φ) (f ∈ D).
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Some further developments
Compact composition operators on DMacCluer, Shapiro (1986)
Composition operators in Schatten classesLefevre, Li, Queffelec, Rodrıguez-Piazza (2013)
Geometry of φ(D) when Cφ is Hilbert–SchmidtGallardo-Gutierrez, Gonzalez (2003)
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Chapter 7
Weighted Dirichlet spaces
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The Dα spaces
Definition
For −1 < α ≤ 1, write Dα for the set of holomorphic f on D with
Dα(f ) :=1
π
∫D|f ′(z)|2(1− |z |2)α dA(z) <∞.
Properties:
Dα(∑
k akzk) �
∑k k
1−α|ak |2
D0 = D and D1∼= H2
If 0 < α < 1, then Dα is ‘akin’ to D (using Riesz capacity cα).
If −1 < α < 0, then Dα is a subalgebra of the disk algebra.
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The Dα spaces
Definition
For −1 < α ≤ 1, write Dα for the set of holomorphic f on D with
Dα(f ) :=1
π
∫D|f ′(z)|2(1− |z |2)α dA(z) <∞.
Properties:
Dα(∑
k akzk) �
∑k k
1−α|ak |2
D0 = D and D1∼= H2
If 0 < α < 1, then Dα is ‘akin’ to D (using Riesz capacity cα).
If −1 < α < 0, then Dα is a subalgebra of the disk algebra.
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The Dµ spaces
Given a finite positive measure µ on T, write Pµ for its Poisson integral:
Pµ(z) :=
∫T
1− |z |2
|ζ − z |2dµ(ζ) (z ∈ D).
Definition (Richter, 1991)
Given µ, we denote by Dµ the set of holomorphic f on D such that
Dµ(f ) :=1
π
∫D|f ′(z)|2Pµ(z) dA(z) <∞.
If µ = dθ/2π, then Dµ = D, the classical Dirichlet space.
If µ = δζ , then Dµ is the local Dirichlet space at ζ, denoted Dζ .
Note: Can recover Dµ(f ) from Dζ(f ) using Fubini’s theorem:
Dµ(f ) =
∫TDζ(f ) dµ(ζ).
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The Dµ spaces
Given a finite positive measure µ on T, write Pµ for its Poisson integral:
Pµ(z) :=
∫T
1− |z |2
|ζ − z |2dµ(ζ) (z ∈ D).
Definition (Richter, 1991)
Given µ, we denote by Dµ the set of holomorphic f on D such that
Dµ(f ) :=1
π
∫D|f ′(z)|2Pµ(z) dA(z) <∞.
If µ = dθ/2π, then Dµ = D, the classical Dirichlet space.
If µ = δζ , then Dµ is the local Dirichlet space at ζ, denoted Dζ .
Note: Can recover Dµ(f ) from Dζ(f ) using Fubini’s theorem:
Dµ(f ) =
∫TDζ(f ) dµ(ζ).
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The Dµ spaces
Given a finite positive measure µ on T, write Pµ for its Poisson integral:
Pµ(z) :=
∫T
1− |z |2
|ζ − z |2dµ(ζ) (z ∈ D).
Definition (Richter, 1991)
Given µ, we denote by Dµ the set of holomorphic f on D such that
Dµ(f ) :=1
π
∫D|f ′(z)|2Pµ(z) dA(z) <∞.
If µ = dθ/2π, then Dµ = D, the classical Dirichlet space.
If µ = δζ , then Dµ is the local Dirichlet space at ζ, denoted Dζ .
Note: Can recover Dµ(f ) from Dζ(f ) using Fubini’s theorem:
Dµ(f ) =
∫TDζ(f ) dµ(ζ).
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The Dµ spaces
Given a finite positive measure µ on T, write Pµ for its Poisson integral:
Pµ(z) :=
∫T
1− |z |2
|ζ − z |2dµ(ζ) (z ∈ D).
Definition (Richter, 1991)
Given µ, we denote by Dµ the set of holomorphic f on D such that
Dµ(f ) :=1
π
∫D|f ′(z)|2Pµ(z) dA(z) <∞.
If µ = dθ/2π, then Dµ = D, the classical Dirichlet space.
If µ = δζ , then Dµ is the local Dirichlet space at ζ, denoted Dζ .
Note: Can recover Dµ(f ) from Dζ(f ) using Fubini’s theorem:
Dµ(f ) =
∫TDζ(f ) dµ(ζ).
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Properties of Dµ (Richter–Sundberg, 1991)
Dµ ⊂ H2 and is Hilbert space w.r.t. ‖f ‖2Dµ
:= ‖f ‖2H2 +Dµ(f ).
Douglas formula: if f ∈ Dµ, then f ∗ exists µ-a.e. and
Dµ(f ) =
∫T
∫T
|f ∗(λ)− f ∗(ζ)|2
|λ− ζ|2|dλ|2π
dµ(ζ).
Special case: f ∈ Dζ ⇐⇒ f (z) = a + (z − ζ)g(z) where g ∈ H2,
and then Dζ(f ) = ‖g‖2H2 .
Carleson formula for Dµ(f ).
Polynomials are dense in Dµ.
Dµ(fr ) ≤ 4Dµ(f ) (where fr (z) := f (rz)).Can replace 4 by 1 (Sarason 1997, using de Branges–Rovnyak spaces).
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Properties of Dµ (Richter–Sundberg, 1991)
Dµ ⊂ H2 and is Hilbert space w.r.t. ‖f ‖2Dµ
:= ‖f ‖2H2 +Dµ(f ).
Douglas formula: if f ∈ Dµ, then f ∗ exists µ-a.e. and
Dµ(f ) =
∫T
∫T
|f ∗(λ)− f ∗(ζ)|2
|λ− ζ|2|dλ|2π
dµ(ζ).
Special case: f ∈ Dζ ⇐⇒ f (z) = a + (z − ζ)g(z) where g ∈ H2,
and then Dζ(f ) = ‖g‖2H2 .
Carleson formula for Dµ(f ).
Polynomials are dense in Dµ.
Dµ(fr ) ≤ 4Dµ(f ) (where fr (z) := f (rz)).Can replace 4 by 1 (Sarason 1997, using de Branges–Rovnyak spaces).
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Properties of Dµ (Richter–Sundberg, 1991)
Dµ ⊂ H2 and is Hilbert space w.r.t. ‖f ‖2Dµ
:= ‖f ‖2H2 +Dµ(f ).
Douglas formula: if f ∈ Dµ, then f ∗ exists µ-a.e. and
Dµ(f ) =
∫T
∫T
|f ∗(λ)− f ∗(ζ)|2
|λ− ζ|2|dλ|2π
dµ(ζ).
Special case: f ∈ Dζ ⇐⇒ f (z) = a + (z − ζ)g(z) where g ∈ H2,
and then Dζ(f ) = ‖g‖2H2 .
Carleson formula for Dµ(f ).
Polynomials are dense in Dµ.
Dµ(fr ) ≤ 4Dµ(f ) (where fr (z) := f (rz)).Can replace 4 by 1 (Sarason 1997, using de Branges–Rovnyak spaces).
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Properties of Dµ (Richter–Sundberg, 1991)
Dµ ⊂ H2 and is Hilbert space w.r.t. ‖f ‖2Dµ
:= ‖f ‖2H2 +Dµ(f ).
Douglas formula: if f ∈ Dµ, then f ∗ exists µ-a.e. and
Dµ(f ) =
∫T
∫T
|f ∗(λ)− f ∗(ζ)|2
|λ− ζ|2|dλ|2π
dµ(ζ).
Special case: f ∈ Dζ ⇐⇒ f (z) = a + (z − ζ)g(z) where g ∈ H2,
and then Dζ(f ) = ‖g‖2H2 .
Carleson formula for Dµ(f ).
Polynomials are dense in Dµ.
Dµ(fr ) ≤ 4Dµ(f ) (where fr (z) := f (rz)).Can replace 4 by 1 (Sarason 1997, using de Branges–Rovnyak spaces).
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Properties of Dµ (Richter–Sundberg, 1991)
Dµ ⊂ H2 and is Hilbert space w.r.t. ‖f ‖2Dµ
:= ‖f ‖2H2 +Dµ(f ).
Douglas formula: if f ∈ Dµ, then f ∗ exists µ-a.e. and
Dµ(f ) =
∫T
∫T
|f ∗(λ)− f ∗(ζ)|2
|λ− ζ|2|dλ|2π
dµ(ζ).
Special case: f ∈ Dζ ⇐⇒ f (z) = a + (z − ζ)g(z) where g ∈ H2,
and then Dζ(f ) = ‖g‖2H2 .
Carleson formula for Dµ(f ).
Polynomials are dense in Dµ.
Dµ(fr ) ≤ 4Dµ(f ) (where fr (z) := f (rz)).Can replace 4 by 1 (Sarason 1997, using de Branges–Rovnyak spaces).
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Properties of Dµ (Richter–Sundberg, 1991)
Dµ ⊂ H2 and is Hilbert space w.r.t. ‖f ‖2Dµ
:= ‖f ‖2H2 +Dµ(f ).
Douglas formula: if f ∈ Dµ, then f ∗ exists µ-a.e. and
Dµ(f ) =
∫T
∫T
|f ∗(λ)− f ∗(ζ)|2
|λ− ζ|2|dλ|2π
dµ(ζ).
Special case: f ∈ Dζ ⇐⇒ f (z) = a + (z − ζ)g(z) where g ∈ H2,
and then Dζ(f ) = ‖g‖2H2 .
Carleson formula for Dµ(f ).
Polynomials are dense in Dµ.
Dµ(fr ) ≤ 4Dµ(f ) (where fr (z) := f (rz)).Can replace 4 by 1 (Sarason 1997, using de Branges–Rovnyak spaces).
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Some further developments
Capacities for Dµ.Chacon (2011), Guillot (2012)
Estimates for reproducing kernel and capacities in Dµ.El-Fallah, Elmadani, Kellay (2019)
Superharmonic weightsAleman (1993)
Dµ has the complete Pick propertyShimorin (2002)
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Chapter 8
Shift-invariant subspaces
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Preliminary remarks
Notation:
T a bounded operator on a Hilbert space HLat(T ,H) := the lattice of closed T -invariant subspaces of H.
Mz := the shift operator (multiplication by z).
Theorem (Beurling, 1948)
If M∈ Lat(Mz ,H2) \ {0}, then M = θH2 where θ is inner.
Analogue for Lat(Mz ,D)?
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Preliminary remarks
Notation:
T a bounded operator on a Hilbert space HLat(T ,H) := the lattice of closed T -invariant subspaces of H.
Mz := the shift operator (multiplication by z).
Theorem (Beurling, 1948)
If M∈ Lat(Mz ,H2) \ {0}, then M = θH2 where θ is inner.
Analogue for Lat(Mz ,D)?
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Preliminary remarks
Notation:
T a bounded operator on a Hilbert space HLat(T ,H) := the lattice of closed T -invariant subspaces of H.
Mz := the shift operator (multiplication by z).
Theorem (Beurling, 1948)
If M∈ Lat(Mz ,H2) \ {0}, then M = θH2 where θ is inner.
Analogue for Lat(Mz ,D)?
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The shift operator on Dµ
Write (T ,H) := (Mz ,D). Clearly:
(1) ‖T 2f ‖2 − 2‖Tf ‖2 + ‖f ‖2 = 0 for all f ∈ H.
(2) ∩n≥0Tn(H) = {0}.
(3) dim(H T (H)) = 1.
It turns out that the same properties hold if (T ,H) := (Mz ,Dµ).Conversely:
Theorem (Richter, 1991)
Let T be an operator on a Hilbert space H satisfying (1),(2),(3). Thenthere exists a unique finite measure µ on T such that (T ,H) is unitarilyequivalent to (Mz ,Dµ).
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The shift operator on Dµ
Write (T ,H) := (Mz ,D). Clearly:
(1) ‖T 2f ‖2 − 2‖Tf ‖2 + ‖f ‖2 = 0 for all f ∈ H.
(2) ∩n≥0Tn(H) = {0}.
(3) dim(H T (H)) = 1.
It turns out that the same properties hold if (T ,H) := (Mz ,Dµ).Conversely:
Theorem (Richter, 1991)
Let T be an operator on a Hilbert space H satisfying (1),(2),(3). Thenthere exists a unique finite measure µ on T such that (T ,H) is unitarilyequivalent to (Mz ,Dµ).
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The shift operator on Dµ
Write (T ,H) := (Mz ,D). Clearly:
(1) ‖T 2f ‖2 − 2‖Tf ‖2 + ‖f ‖2 = 0 for all f ∈ H.
(2) ∩n≥0Tn(H) = {0}.
(3) dim(H T (H)) = 1.
It turns out that the same properties hold if (T ,H) := (Mz ,Dµ).Conversely:
Theorem (Richter, 1991)
Let T be an operator on a Hilbert space H satisfying (1),(2),(3). Thenthere exists a unique finite measure µ on T such that (T ,H) is unitarilyequivalent to (Mz ,Dµ).
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Invariant subspaces of (Mz ,D)
Let M∈ Lat(Mz ,D).
Clearly (Mz ,M) satisfies properties (1),(2).
If M 6= {0}, then (3) also holds (Richter–Shields 1988).
Leads to:
Theorem (Richter 1991, Richter–Sundberg 1992)
Let M∈ Lat(Mz ,D) and let φ ∈MMz(M) with φ 6≡ 0. Then:
φ is a multiplier for D.
M = φDµ where dµ := |φ∗|2 dθ.
Corollary
M is cyclic (i.e. singly generated as an invariant subspace).
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Invariant subspaces of (Mz ,D)
Let M∈ Lat(Mz ,D).
Clearly (Mz ,M) satisfies properties (1),(2).
If M 6= {0}, then (3) also holds (Richter–Shields 1988).
Leads to:
Theorem (Richter 1991, Richter–Sundberg 1992)
Let M∈ Lat(Mz ,D) and let φ ∈MMz(M) with φ 6≡ 0. Then:
φ is a multiplier for D.
M = φDµ where dµ := |φ∗|2 dθ.
Corollary
M is cyclic (i.e. singly generated as an invariant subspace).
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Invariant subspaces of (Mz ,D)
Let M∈ Lat(Mz ,D).
Clearly (Mz ,M) satisfies properties (1),(2).
If M 6= {0}, then (3) also holds (Richter–Shields 1988).
Leads to:
Theorem (Richter 1991, Richter–Sundberg 1992)
Let M∈ Lat(Mz ,D) and let φ ∈MMz(M) with φ 6≡ 0. Then:
φ is a multiplier for D.
M = φDµ where dµ := |φ∗|2 dθ.
Corollary
M is cyclic (i.e. singly generated as an invariant subspace).
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Cyclic invariant subspaces
Problem: Given f ∈ D, identify [f ]D, the closed invariant subspace of Dgenerated by f .
Theorem (Richter–Sundberg 1992)
Let f ∈ D have inner-outer factorization f = fi fo . Then
[f ]D = fi [fo ]D ∩ D = [fo ]D ∩ fiH2.
It remains to identify [fo ]D. We might expect that [fo ]D = D. However,another phenomenon intervenes, that of boundary zeros.
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Cyclic invariant subspaces
Problem: Given f ∈ D, identify [f ]D, the closed invariant subspace of Dgenerated by f .
Theorem (Richter–Sundberg 1992)
Let f ∈ D have inner-outer factorization f = fi fo . Then
[f ]D = fi [fo ]D ∩ D = [fo ]D ∩ fiH2.
It remains to identify [fo ]D. We might expect that [fo ]D = D. However,another phenomenon intervenes, that of boundary zeros.
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Cyclic invariant subspaces
Problem: Given f ∈ D, identify [f ]D, the closed invariant subspace of Dgenerated by f .
Theorem (Richter–Sundberg 1992)
Let f ∈ D have inner-outer factorization f = fi fo . Then
[f ]D = fi [fo ]D ∩ D = [fo ]D ∩ fiH2.
It remains to identify [fo ]D. We might expect that [fo ]D = D. However,another phenomenon intervenes, that of boundary zeros.
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Cyclic invariant subspaces and boundary zeros
Notation: Given E ⊂ T, write DE := {h ∈ D : h∗ = 0 q.e. on E}.
Theorem (Carleson 1952)
DE is closed in D. Hence DE ∈ Lat(Mz ,D).
Corollary
Let f ∈ D and let E := {f ∗ = 0}. Then [f ]D ⊂ DE .
Open problem
Let f ∈ D be outer and let E := {f ∗ = 0}. Then do we have [f ]D = DE?In particular, if c(E ) = 0, then do we have [f ]D = D?
Special case where c(E ) = 0 is a celebrated conjecture of Brown–Shields
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Cyclic invariant subspaces and boundary zeros
Notation: Given E ⊂ T, write DE := {h ∈ D : h∗ = 0 q.e. on E}.
Theorem (Carleson 1952)
DE is closed in D. Hence DE ∈ Lat(Mz ,D).
Corollary
Let f ∈ D and let E := {f ∗ = 0}. Then [f ]D ⊂ DE .
Open problem
Let f ∈ D be outer and let E := {f ∗ = 0}. Then do we have [f ]D = DE?In particular, if c(E ) = 0, then do we have [f ]D = D?
Special case where c(E ) = 0 is a celebrated conjecture of Brown–Shields
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Cyclic invariant subspaces and boundary zeros
Notation: Given E ⊂ T, write DE := {h ∈ D : h∗ = 0 q.e. on E}.
Theorem (Carleson 1952)
DE is closed in D. Hence DE ∈ Lat(Mz ,D).
Corollary
Let f ∈ D and let E := {f ∗ = 0}. Then [f ]D ⊂ DE .
Open problem
Let f ∈ D be outer and let E := {f ∗ = 0}. Then do we have [f ]D = DE?In particular, if c(E ) = 0, then do we have [f ]D = D?
Special case where c(E ) = 0 is a celebrated conjecture of Brown–Shields
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Cyclic invariant subspaces and boundary zeros
Notation: Given E ⊂ T, write DE := {h ∈ D : h∗ = 0 q.e. on E}.
Theorem (Carleson 1952)
DE is closed in D. Hence DE ∈ Lat(Mz ,D).
Corollary
Let f ∈ D and let E := {f ∗ = 0}. Then [f ]D ⊂ DE .
Open problem
Let f ∈ D be outer and let E := {f ∗ = 0}. Then do we have [f ]D = DE?In particular, if c(E ) = 0, then do we have [f ]D = D?
Special case where c(E ) = 0 is a celebrated conjecture of Brown–Shields
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Brown–Shields conjecture
f ∈ D is cyclic for D if [f ]D = D. Necessary conditions for cyclicity:
f is outer;
E := {f ∗ = 0} is of capacity zero.
Conjecture (Brown–Shields, 1984)
These conditions are also sufficient.
Partial results:
Theorem (Hedenmalm–Shields, 1990)
If f ∈ D ∩ A(D) is outer and if E := {f = 0} is countable, then f is cyclic.
Theorem (El-Fallah–Kellay–Ransford, 2009)
If f ∈ D ∩ A(D) is outer and if E := {f = 0} satisfies, for some ε > 0,
|Et | = O(tε) (t → 0+) and
∫ 1
0dt/|Et | =∞,
then f is cyclic.
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Brown–Shields conjecture
f ∈ D is cyclic for D if [f ]D = D. Necessary conditions for cyclicity:
f is outer;
E := {f ∗ = 0} is of capacity zero.
Conjecture (Brown–Shields, 1984)
These conditions are also sufficient.
Partial results:
Theorem (Hedenmalm–Shields, 1990)
If f ∈ D ∩ A(D) is outer and if E := {f = 0} is countable, then f is cyclic.
Theorem (El-Fallah–Kellay–Ransford, 2009)
If f ∈ D ∩ A(D) is outer and if E := {f = 0} satisfies, for some ε > 0,
|Et | = O(tε) (t → 0+) and
∫ 1
0dt/|Et | =∞,
then f is cyclic.
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Brown–Shields conjecture
f ∈ D is cyclic for D if [f ]D = D. Necessary conditions for cyclicity:
f is outer;
E := {f ∗ = 0} is of capacity zero.
Conjecture (Brown–Shields, 1984)
These conditions are also sufficient.
Partial results:
Theorem (Hedenmalm–Shields, 1990)
If f ∈ D ∩ A(D) is outer and if E := {f = 0} is countable, then f is cyclic.
Theorem (El-Fallah–Kellay–Ransford, 2009)
If f ∈ D ∩ A(D) is outer and if E := {f = 0} satisfies, for some ε > 0,
|Et | = O(tε) (t → 0+) and
∫ 1
0dt/|Et | =∞,
then f is cyclic.
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Some further developments
Shift-invariant subspaces and cyclicity in DµRichter, Sundberg (1992)Guillot (2012)El-Fallah, Elmadani, Kellay (2016)
Optimal polynomial approximantsCatherine Beneteau and co-authors (2015 onwards)
Cyclicity in Dirichlet spaces on the bi-diskKnese–Kosinski–Ransford–Sola (2019)
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Conclusion: a shameless advertisement