factorization of operators on banach (function) spaces · functional calculus maximal regularity of...
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![Page 1: Factorization of operators on Banach (function) spaces · Functional calculus Maximal regularity of PDE’s Stochastic integration in Banach spaces E P n k=1 " kx k 2 1=2 is a norm](https://reader033.vdocuments.us/reader033/viewer/2022060314/5f0b81197e708231d430d85f/html5/thumbnails/1.jpg)
Factorization of operators on Banach (function) spaces
Emiel Lorist
Delft University of Technology, The Netherlands
Madrid, Spain
September 9, 2019
Joint work with Nigel Kalton and Lutz Weis
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Euclidean structures• Project started in the early 2000s
• Understand the role R-boundedness from anabstract operator-theoretic viewpoint
• Connected to completely bounded maps
• Project on hold since Nigel passed away
• Studied and improved/extended parts of themanuscript in 2016
• Joined the project in 2018, rewrote and modernizedthe manuscript
• Euclidean structures• Part I: Representation of operator families
on a Hilbert space• Part II: Factorization of operator families• Part III: Vector-valued function spaces• Part IV: Sectorial operators and H∞-calculus• Part V: Counterexamples
• Project to be finished this fall
Nigel Kalton
Lutz Weis
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Euclidean structures• Project started in the early 2000s
• Understand the role R-boundedness from anabstract operator-theoretic viewpoint
• Connected to completely bounded maps
• Project on hold since Nigel passed away
• Studied and improved/extended parts of themanuscript in 2016
• Joined the project in 2018, rewrote and modernizedthe manuscript
• Euclidean structures• Part I: Representation of operator families
on a Hilbert space• Part II: Factorization of operator families• Part III: Vector-valued function spaces• Part IV: Sectorial operators and H∞-calculus• Part V: Counterexamples
• Project to be finished this fall
Nigel Kalton
Lutz Weis
1 / 12
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Euclidean structures• Project started in the early 2000s
• Understand the role R-boundedness from anabstract operator-theoretic viewpoint
• Connected to completely bounded maps
• Project on hold since Nigel passed away
• Studied and improved/extended parts of themanuscript in 2016
• Joined the project in 2018, rewrote and modernizedthe manuscript
• Euclidean structures• Part I: Representation of operator families
on a Hilbert space• Part II: Factorization of operator families• Part III: Vector-valued function spaces• Part IV: Sectorial operators and H∞-calculus• Part V: Counterexamples
• Project to be finished this fall
Nigel Kalton
Lutz Weis
1 / 12
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Euclidean structures• Project started in the early 2000s
• Understand the role R-boundedness from anabstract operator-theoretic viewpoint
• Connected to completely bounded maps
• Project on hold since Nigel passed away
• Studied and improved/extended parts of themanuscript in 2016
• Joined the project in 2018, rewrote and modernizedthe manuscript
• Euclidean structures• Part I: Representation of operator families
on a Hilbert space• Part II: Factorization of operator families• Part III: Vector-valued function spaces• Part IV: Sectorial operators and H∞-calculus• Part V: Counterexamples
• Project to be finished this fall
Nigel Kalton
Lutz Weis
1 / 12
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R-boundedness
Definition
Let X be a Banach space and Γ ⊆ L(X ). Then we say that Γ is R-bounded if forany finite sequences (Tk)nk=1 in Γ and (xk)nk=1 in X(
E∥∥∥ n∑k=1
εkTkxk
∥∥∥2
X
)1/2
≤ C(E∥∥∥ n∑k=1
εkxk
∥∥∥2
X
)1/2
,
where (εk)nk=1 is a sequence of independent Rademacher variables.
• R-boundedness is a strengthening of uniform boundedness.• Equivalent to uniform boundedness on Hilbert spaces.
• R-boundedness plays a (key) role in e.g.• Schauder multipliers• Operator-valued Fourier multiplier theory• Functional calculus• Maximal regularity of PDE’s• Stochastic integration in Banach spaces
•(E∥∥∑n
k=1 εkxk∥∥2)1/2
is a norm on X n.• A Euclidean structure is such a norm with a left and right ideal property.
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R-boundedness
Definition
Let X be a Banach space and Γ ⊆ L(X ). Then we say that Γ is R-bounded if forany finite sequences (Tk)nk=1 in Γ and (xk)nk=1 in X(
E∥∥∥ n∑k=1
εkTkxk
∥∥∥2
X
)1/2
≤ C(E∥∥∥ n∑k=1
εkxk
∥∥∥2
X
)1/2
,
where (εk)nk=1 is a sequence of independent Rademacher variables.
• R-boundedness is a strengthening of uniform boundedness.• Equivalent to uniform boundedness on Hilbert spaces.
• R-boundedness plays a (key) role in e.g.• Schauder multipliers• Operator-valued Fourier multiplier theory• Functional calculus• Maximal regularity of PDE’s• Stochastic integration in Banach spaces
•(E∥∥∑n
k=1 εkxk∥∥2)1/2
is a norm on X n.• A Euclidean structure is such a norm with a left and right ideal property.
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R-boundedness
Definition
Let X be a Banach space and Γ ⊆ L(X ). Then we say that Γ is R-bounded if forany finite sequences (Tk)nk=1 in Γ and (xk)nk=1 in X(
E∥∥∥ n∑k=1
εkTkxk
∥∥∥2
X
)1/2
≤ C(E∥∥∥ n∑k=1
εkxk
∥∥∥2
X
)1/2
,
where (εk)nk=1 is a sequence of independent Rademacher variables.
• R-boundedness is a strengthening of uniform boundedness.• Equivalent to uniform boundedness on Hilbert spaces.
• R-boundedness plays a (key) role in e.g.• Schauder multipliers• Operator-valued Fourier multiplier theory• Functional calculus• Maximal regularity of PDE’s• Stochastic integration in Banach spaces
•(E∥∥∑n
k=1 εkxk∥∥2)1/2
is a norm on X n.• A Euclidean structure is such a norm with a left and right ideal property.
2 / 12
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Euclidean structures
Definition
Let X be a Banach space. A Euclidean structure α is a family of norms ‖·‖α on X n
for all n ∈ N such that
‖(x)‖α = ‖x‖X , x ∈ X ,
‖Ax‖α ≤ ‖A‖‖x‖α, x ∈ X n, A ∈ Mm,n(C),
‖(Tx1, · · · ,Txn)‖α ≤ C ‖T‖‖x‖α, x ∈ X n, T ∈ L(X ),
A Euclidean structure induces a norm on the finite rank operators from `2 to X .
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Euclidean structures
Definition
Let X be a Banach space. A Euclidean structure α is a family of norms ‖·‖α on X n
for all n ∈ N such that
‖(x)‖α = ‖x‖X , x ∈ X ,
‖Ax‖α ≤ ‖A‖‖x‖α, x ∈ X n, A ∈ Mm,n(C),
‖(Tx1, · · · ,Txn)‖α ≤ C ‖T‖‖x‖α, x ∈ X n, T ∈ L(X ),
A Euclidean structure induces a norm on the finite rank operators from `2 to X .
Non-example:
• On a Banach space X :
‖x‖R :=(E∥∥ n∑k=1
εkxk∥∥2
X
) 12, x ∈ X n,
is not a Euclidean structure. It fails the right-ideal property.
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Euclidean structures
Definition
Let X be a Banach space. A Euclidean structure α is a family of norms ‖·‖α on X n
for all n ∈ N such that
‖(x)‖α = ‖x‖X , x ∈ X ,
‖Ax‖α ≤ ‖A‖‖x‖α, x ∈ X n, A ∈ Mm,n(C),
‖(Tx1, · · · ,Txn)‖α ≤ C ‖T‖‖x‖α, x ∈ X n, T ∈ L(X ),
A Euclidean structure induces a norm on the finite rank operators from `2 to X .
Examples:
• On any Banach space X : The Gaussian structure
‖x‖γ :=(E∥∥ n∑k=1
γkxk∥∥2
X
) 12, x ∈ X n,
where (γk)nk=1 is a sequence of independent normalized Gaussians.
• On a Banach lattice X : The `2-structure
‖x‖`2 :=∥∥∥( n∑
k=1
|xk |2)1/2
∥∥∥X, x ∈ X n.
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Euclidean structures
Definition
Let X be a Banach space. A Euclidean structure α is a family of norms ‖·‖α on X n
for all n ∈ N such that
‖(x)‖α = ‖x‖X , x ∈ X ,
‖Ax‖α ≤ ‖A‖‖x‖α, x ∈ X n, A ∈ Mm,n(C),
‖(Tx1, · · · ,Txn)‖α ≤ C ‖T‖‖x‖α, x ∈ X n, T ∈ L(X ),
A Euclidean structure induces a norm on the finite rank operators from `2 to X .
Examples:
• On any Banach space X : The Gaussian structure
‖x‖γ :=(E∥∥ n∑k=1
γkxk∥∥2
X
) 12, x ∈ X n,
where (γk)nk=1 is a sequence of independent normalized Gaussians.
• On a Banach lattice X : The `2-structure
‖x‖`2 :=∥∥∥( n∑
k=1
|xk |2)1/2
∥∥∥X, x ∈ X n.
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α-boundedness
Definition
Let X be a Banach space, α an Euclidean structure and Γ ⊆ L(X ). Then we say thatΓ is α-bounded if for any T = diag(T1, · · · ,Tn) with T1, · · · ,Tn ∈ Γ
‖Tx‖α ≤ C‖x‖α, x ∈ X n.
• α-boundedness implies uniform boundedness
• On a Banach space X with finite cotype
‖x‖γ =(E∥∥ n∑k=1
γkxk∥∥2
X
) 12 '
(E∥∥ n∑k=1
εkxk∥∥2
X
) 12
= ‖x‖R,
so γ-boundedness is equivalent to R-boundedness
• On a Banach lattice X with finite cotype
‖x‖`2 =∥∥( n∑
k=1
|xk |2)1/2∥∥
X'(E∥∥ n∑k=1
εkxk∥∥2
X
) 12
= ‖x‖R,
so `2-boundedness is equivalent to R-boundedness.
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α-boundedness
Definition
Let X be a Banach space, α an Euclidean structure and Γ ⊆ L(X ). Then we say thatΓ is α-bounded if for any T = diag(T1, · · · ,Tn) with T1, · · · ,Tn ∈ Γ
‖Tx‖α ≤ C‖x‖α, x ∈ X n.
• α-boundedness implies uniform boundedness
• On a Banach space X with finite cotype
‖x‖γ =(E∥∥ n∑k=1
γkxk∥∥2
X
) 12 '
(E∥∥ n∑k=1
εkxk∥∥2
X
) 12
= ‖x‖R,
so γ-boundedness is equivalent to R-boundedness
• On a Banach lattice X with finite cotype
‖x‖`2 =∥∥( n∑
k=1
|xk |2)1/2∥∥
X'(E∥∥ n∑k=1
εkxk∥∥2
X
) 12
= ‖x‖R,
so `2-boundedness is equivalent to R-boundedness.
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α-boundedness
Definition
Let X be a Banach space, α an Euclidean structure and Γ ⊆ L(X ). Then we say thatΓ is α-bounded if for any T = diag(T1, · · · ,Tn) with T1, · · · ,Tn ∈ Γ
‖Tx‖α ≤ C‖x‖α, x ∈ X n.
• α-boundedness implies uniform boundedness
• On a Banach space X with finite cotype
‖x‖γ =(E∥∥ n∑k=1
γkxk∥∥2
X
) 12 '
(E∥∥ n∑k=1
εkxk∥∥2
X
) 12
= ‖x‖R,
so γ-boundedness is equivalent to R-boundedness
• On a Banach lattice X with finite cotype
‖x‖`2 =∥∥( n∑
k=1
|xk |2)1/2∥∥
X'(E∥∥ n∑k=1
εkxk∥∥2
X
) 12
= ‖x‖R,
so `2-boundedness is equivalent to R-boundedness.
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Factorization through a Hilbert space
Theorem (Kwapien ’72 and Maurey ’74)
Let X and Y be a Banach spaces and T ∈ L(X ,Y ). If X has type 2 and Y cotype2, then there is a Hilbert space H and operators S ∈ L(X ,H) and U ∈ L(H,Y ) s.t.
T = US .
Corollary (Kwapien ’72)
Let X be a Banach space. X has type 2 and cotype 2 if and only if X is isomorphicto a Hilbert space.
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Factorization through a Hilbert space
Theorem (Kwapien ’72 and Maurey ’74)
Let X and Y be a Banach spaces and T ∈ L(X ,Y ). If X has type 2 and Y cotype2, then there is a Hilbert space H and operators S ∈ L(X ,H) and U ∈ L(H,Y ) s.t.
T = US .
Corollary (Kwapien ’72)
Let X be a Banach space. X has type 2 and cotype 2 if and only if X is isomorphicto a Hilbert space.
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Factorization through a Hilbert space
Theorem (Kwapien ’72 and Maurey ’74)
Let X and Y be a Banach spaces and T ∈ L(X ,Y ). If X has type 2 and Y cotype2, then there is a Hilbert space H and operators S ∈ L(X ,H) and U ∈ L(H,Y ) s.t.
T = US .
Corollary (Kwapien ’72)
Let X be a Banach space. X has type 2 and cotype 2 if and only if X is isomorphicto a Hilbert space.
Theorem (Kalton–L.–Weis ’19)
Let X and Y be a Banach spaces, Γ1 ⊆ L(X ,Y ) . If X has type 2, Ycotype 2 and Γ1 is γ-bounded, then there is a Hilbert space H, aT ∈ L(X ,H) for every T ∈ Γ1 and U ∈ L(H,Y ) s.t.the following diagram commutes:
X Y
H
T
T
U
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Factorization through a Hilbert space
Theorem (Kwapien ’72 and Maurey ’74)
Let X and Y be a Banach spaces and T ∈ L(X ,Y ). If X has type 2 and Y cotype2, then there is a Hilbert space H and operators S ∈ L(X ,H) and U ∈ L(H,Y ) s.t.
T = US .
Corollary (Kwapien ’72)
Let X be a Banach space. X has type 2 and cotype 2 if and only if X is isomorphicto a Hilbert space.
Theorem (Kalton–L.–Weis ’19)
Let X and Y be a Banach spaces, Γ1 ⊆ L(X ,Y ) and Γ2 ⊆ L(Y ). If X has type 2, Ycotype 2 and Γ1 and Γ2 are γ-bounded, then there is a Hilbert space H, aT ∈ L(X ,H) for every T ∈ Γ1, a S ∈ L(H) for every S ∈ Γ2 and U ∈ L(H,Y ) s.t.the following diagram commutes:
X Y Y
H H
T
T S
U
S
U
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Factorization through weighted L2
For a measure space (S ,Σ, µ) and a weight w : S → R+ let L2(S ,w) be the space ofall measurable f : S → C such that
‖f ‖L2(S,w) :=(∫
S
|f |2w dµ)1/2
<∞
Theorem (Kalton–L.–Weis ’19)
Let X be an order-continuous Banach function space over (S , µ) and let Γ ⊆ L(X ).Γ is `2-bounded if and only if for any g0, g1 ∈ X there is a weight w : S → R+ s.t.
‖Tf ‖L2(S,w) ≤ C ‖f ‖L2(S,w), f ∈ X ∩ L2(S ,w), T ∈ Γ
‖g0‖L2(S,w) ≤ C ‖g0‖X ,
‖g1‖X ≤ C ‖g1‖L2(S,w).
The if statement is trivial. Indeed for f1, · · · , fn ∈ X and T1, · · · ,Tn ∈ Γ set
g0 :=( n∑k=1
|fk |2) 1
2 , g1 :=( n∑k=1
|Tk fk |2) 1
2
Then we have
‖Tf ‖2`2 = ‖g1‖2
X ≤ C 2n∑
k=1
∫S
|Tk fk |2w dµ ≤ C 4n∑
k=1
∫S
|fk |2w dµ ≤ C 6‖g0‖2X = ‖f ‖2
`2
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Factorization through weighted L2
For a measure space (S ,Σ, µ) and a weight w : S → R+ let L2(S ,w) be the space ofall measurable f : S → C such that
‖f ‖L2(S,w) :=(∫
S
|f |2w dµ)1/2
<∞
Theorem (Kalton–L.–Weis ’19)
Let X be an order-continuous Banach function space over (S , µ) and let Γ ⊆ L(X ).Γ is `2-bounded if and only if for any g0, g1 ∈ X there is a weight w : S → R+ s.t.
‖Tf ‖L2(S,w) ≤ C ‖f ‖L2(S,w), f ∈ X ∩ L2(S ,w), T ∈ Γ
‖g0‖L2(S,w) ≤ C ‖g0‖X ,
‖g1‖X ≤ C ‖g1‖L2(S,w).
The if statement is trivial. Indeed for f1, · · · , fn ∈ X and T1, · · · ,Tn ∈ Γ set
g0 :=( n∑k=1
|fk |2) 1
2 , g1 :=( n∑k=1
|Tk fk |2) 1
2
Then we have
‖Tf ‖2`2 = ‖g1‖2
X ≤ C 2n∑
k=1
∫S
|Tk fk |2w dµ ≤ C 4n∑
k=1
∫S
|fk |2w dµ ≤ C 6‖g0‖2X = ‖f ‖2
`2
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Factorization through weighted L2
For a measure space (S ,Σ, µ) and a weight w : S → R+ let L2(S ,w) be the space ofall measurable f : S → C such that
‖f ‖L2(S,w) :=(∫
S
|f |2w dµ)1/2
<∞
Theorem (Kalton–L.–Weis ’19)
Let X be an order-continuous Banach function space over (S , µ) and let Γ ⊆ L(X ).Γ is `2-bounded if and only if for any g0, g1 ∈ X there is a weight w : S → R+ s.t.
‖Tf ‖L2(S,w) ≤ C ‖f ‖L2(S,w), f ∈ X ∩ L2(S ,w), T ∈ Γ
‖g0‖L2(S,w) ≤ C ‖g0‖X ,
‖g1‖X ≤ C ‖g1‖L2(S,w).
The if statement is trivial. Indeed for f1, · · · , fn ∈ X and T1, · · · ,Tn ∈ Γ set
g0 :=( n∑k=1
|fk |2) 1
2 , g1 :=( n∑k=1
|Tk fk |2) 1
2
Then we have
‖Tf ‖2`2 = ‖g1‖2
X ≤ C 2n∑
k=1
∫S
|Tk fk |2w dµ ≤ C 4n∑
k=1
∫S
|fk |2w dµ ≤ C 6‖g0‖2X = ‖f ‖2
`2
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Factorization through weighted L2
For a measure space (S ,Σ, µ) and a weight w : S → R+ let L2(S ,w) be the space ofall measurable f : S → C such that
‖f ‖L2(S,w) :=(∫
S
|f |2w dµ)1/2
<∞
Theorem (Kalton–L.–Weis ’19)
Let X be an order-continuous Banach function space over (S , µ) and let Γ ⊆ L(X ).Γ is `2-bounded if and only if for any g0, g1 ∈ X there is a weight w : S → R+ s.t.
‖Tf ‖L2(S,w) ≤ C ‖f ‖L2(S,w), f ∈ X ∩ L2(S ,w), T ∈ Γ
‖g0‖L2(S,w) ≤ C ‖g0‖X ,
‖g1‖X ≤ C ‖g1‖L2(S,w).
The if statement is trivial. Indeed for f1, · · · , fn ∈ X and T1, · · · ,Tn ∈ Γ set
g0 :=( n∑k=1
|fk |2) 1
2 , g1 :=( n∑k=1
|Tk fk |2) 1
2
Then we have
‖Tf ‖2`2 = ‖g1‖2
X ≤ C 2n∑
k=1
∫S
|Tk fk |2w dµ ≤ C 4n∑
k=1
∫S
|fk |2w dµ ≤ C 6‖g0‖2X = ‖f ‖2
`2
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Application: Harmonic analysis
Let p ∈ [1,∞). For f ∈ Lp(R) define the Hilbert transform
Hf (x) := p. v.1
π
∫R
1
x − yf (y) dy , x ∈ R,
= F−1(ξ 7→ −i sgn(ξ)F f (ξ))(x).
For f ∈ Lp(Rd) define the Hardy–Littlewood maximal operator
Mf (x) := supr>0
1
|B(x , r)|
∫Rd
|f (y)| dy , x ∈ Rd .
• Very important operators in harmonic analysis.
• Both H and M are bounded on Lp for p ∈ (1,∞).
• We are interested in the tensor extensions of H and M onthe Bochner space Lp(Rd ;X ).
• These have numerous applications in both harmonic analysis and PDE.
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Application: Harmonic analysis
Let p ∈ [1,∞). For f ∈ Lp(R) define the Hilbert transform
Hf (x) := p. v.1
π
∫R
1
x − yf (y) dy , x ∈ R,
= F−1(ξ 7→ −i sgn(ξ)F f (ξ))(x).
For f ∈ Lp(Rd) define the Hardy–Littlewood maximal operator
Mf (x) := supr>0
1
|B(x , r)|
∫Rd
|f (y)| dy , x ∈ Rd .
• Very important operators in harmonic analysis.
• Both H and M are bounded on Lp for p ∈ (1,∞).
• We are interested in the tensor extensions of H and M onthe Bochner space Lp(Rd ;X ).
• These have numerous applications in both harmonic analysis and PDE.
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Application: Harmonic analysis
Let p ∈ [1,∞). For f ∈ Lp(R) define the Hilbert transform
Hf (x) := p. v.1
π
∫R
1
x − yf (y) dy , x ∈ R,
= F−1(ξ 7→ −i sgn(ξ)F f (ξ))(x).
For f ∈ Lp(Rd) define the Hardy–Littlewood maximal operator
Mf (x) := supr>0
1
|B(x , r)|
∫Rd
|f (y)| dy , x ∈ Rd .
• Very important operators in harmonic analysis.
• Both H and M are bounded on Lp for p ∈ (1,∞).
• We are interested in the tensor extensions of H and M onthe Bochner space Lp(Rd ;X ).
• These have numerous applications in both harmonic analysis and PDE.
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Application: Harmonic analysis
Let p ∈ [1,∞). For f ∈ Lp(R) define the Hilbert transform
Hf (x) := p. v.1
π
∫R
1
x − yf (y) dy , x ∈ R,
= F−1(ξ 7→ −i sgn(ξ)F f (ξ))(x).
For f ∈ Lp(Rd) define the Hardy–Littlewood maximal operator
Mf (x) := supr>0
1
|B(x , r)|
∫Rd
|f (y)| dy , x ∈ Rd .
• Very important operators in harmonic analysis.
• Both H and M are bounded on Lp for p ∈ (1,∞).
• We are interested in the tensor extensions of H and M onthe Bochner space Lp(Rd ;X ).
• These have numerous applications in both harmonic analysis and PDE.
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Application: Harmonic analysis
Let p ∈ [1,∞) and let X be a Banach space. For f ∈ Lp(R;X ) define thevector-valued Hilbert transform
Hf (x) := p. v.
∫R
1
x − yf (y) dy , x ∈ R
where the integral is interpreted in the Bochner sense.
Let X be a Banach function space. For f ∈ Lp(Rd ;X ) define the latticeHardy–Littlewood maximal operator
Mf (x) := supr>0
1
|B(x , r)|
∫Rd
|f (y)| dy , x ∈ Rd ,
where the supremum is taken in the lattice sense.
• Boundedness of H is independent of p ∈ (1,∞)(Calderon–Benedek–Panzone ’62).
• Boundedness of M is independent of p ∈ (1,∞) and d ∈ N(Garcıa-Cuerva–Macias–Torrea ’93).
• Boundedness of H and M depends on the geometry of X .
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Application: Harmonic analysis
Let p ∈ [1,∞) and let X be a Banach space. For f ∈ Lp(R;X ) define thevector-valued Hilbert transform
Hf (x) := p. v.
∫R
1
x − yf (y) dy , x ∈ R
where the integral is interpreted in the Bochner sense.
Let X be a Banach function space. For f ∈ Lp(Rd ;X ) define the latticeHardy–Littlewood maximal operator
Mf (x) := supr>0
1
|B(x , r)|
∫Rd
|f (y)| dy , x ∈ Rd ,
where the supremum is taken in the lattice sense.
• Boundedness of H is independent of p ∈ (1,∞)(Calderon–Benedek–Panzone ’62).
• Boundedness of M is independent of p ∈ (1,∞) and d ∈ N(Garcıa-Cuerva–Macias–Torrea ’93).
• Boundedness of H and M depends on the geometry of X .
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Application: Harmonic analysis
Let p ∈ [1,∞) and let X be a Banach space. For f ∈ Lp(R;X ) define thevector-valued Hilbert transform
Hf (x) := p. v.
∫R
1
x − yf (y) dy , x ∈ R
where the integral is interpreted in the Bochner sense.
Let X be a Banach function space. For f ∈ Lp(Rd ;X ) define the latticeHardy–Littlewood maximal operator
Mf (x) := supr>0
1
|B(x , r)|
∫Rd
|f (y)| dy , x ∈ Rd ,
where the supremum is taken in the lattice sense.
• Boundedness of H is independent of p ∈ (1,∞)(Calderon–Benedek–Panzone ’62).
• Boundedness of M is independent of p ∈ (1,∞) and d ∈ N(Garcıa-Cuerva–Macias–Torrea ’93).
• Boundedness of H and M depends on the geometry of X .
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Theorem (Burkholder ’83 and Bourgain ’83)
Let X be a Banach space and p ∈ (1,∞). The following are equivalent:
(i) X has the UMD property.
(ii) H is bounded on Lp(R;X ).
• X has the UMD property if any finite martingale (fk)nk=0 in Lp(Ω;X ) hasUnconditional Martingale Differences (dfk)nk=1.
Theorem (Bourgain ’84 and Rubio de Francia ’86)
Let X be a Banach function space and p ∈ (1,∞). The following is equivalent to (i)and (ii)
(iii) M is bounded on Lp(Rd ;X ) and on Lp(Rd ;X ∗).
• (iii)⇒(ii) is “easy” and quantitative.
• (ii)⇒(iii) is very involved and technical.
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Theorem (Burkholder ’83 and Bourgain ’83)
Let X be a Banach space and p ∈ (1,∞). The following are equivalent:
(i) X has the UMD property.
(ii) H is bounded on Lp(R;X ).
• X has the UMD property if any finite martingale (fk)nk=0 in Lp(Ω;X ) hasUnconditional Martingale Differences (dfk)nk=1.
• That is, if for some p ∈ (1,∞) and all signs εk = ±1 we have∥∥∥ n∑k=1
εkdfn∥∥∥Lp(Ω;X )
.∥∥∥ n∑k=1
dfn∥∥∥Lp(Ω;X )
,
Theorem (Bourgain ’84 and Rubio de Francia ’86)
Let X be a Banach function space and p ∈ (1,∞). The following is equivalent to (i)and (ii)
(iii) M is bounded on Lp(Rd ;X ) and on Lp(Rd ;X ∗).
• (iii)⇒(ii) is “easy” and quantitative.
• (ii)⇒(iii) is very involved and technical.
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Theorem (Burkholder ’83 and Bourgain ’83)
Let X be a Banach space and p ∈ (1,∞). The following are equivalent:
(i) X has the UMD property.
(ii) H is bounded on Lp(R;X ).
• X has the UMD property if any finite martingale (fk)nk=0 in Lp(Ω;X ) hasUnconditional Martingale Differences (dfk)nk=1.
• Reflexive Lebesgue, Lorentz, (Musielak)-Orlicz, Sobolev, Besov spaces andSchatten classes have the UMD property.
• L1 and L∞ do not have the UMD property.
• Linear constant dependence is an open problem
Theorem (Bourgain ’84 and Rubio de Francia ’86)
Let X be a Banach function space and p ∈ (1,∞). The following is equivalent to (i)and (ii)
(iii) M is bounded on Lp(Rd ;X ) and on Lp(Rd ;X ∗).
• (iii)⇒(ii) is “easy” and quantitative.
• (ii)⇒(iii) is very involved and technical.
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Theorem (Burkholder ’83 and Bourgain ’83)
Let X be a Banach space and p ∈ (1,∞). The following are equivalent:
(i) X has the UMD property.
(ii) H is bounded on Lp(R;X ).
• X has the UMD property if any finite martingale (fk)nk=0 in Lp(Ω;X ) hasUnconditional Martingale Differences (dfk)nk=1.
• Reflexive Lebesgue, Lorentz, (Musielak)-Orlicz, Sobolev, Besov spaces andSchatten classes have the UMD property.
• L1 and L∞ do not have the UMD property.
• Linear constant dependence is an open problem
Theorem (Bourgain ’84 and Rubio de Francia ’86)
Let X be a Banach function space and p ∈ (1,∞). The following is equivalent to (i)and (ii)
(iii) M is bounded on Lp(Rd ;X ) and on Lp(Rd ;X ∗).
• (iii)⇒(ii) is “easy” and quantitative.
• (ii)⇒(iii) is very involved and technical.
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Application: Harmonic analysis
Theorem (Kalton–L.–Weis ’19)
Let X be a Banach function space on (S , µ) and p ∈ (1,∞). If H is bounded on
Lp(R;X ), then M is bounded on Lp(R;X ) with
‖M‖Lp(R;X )→Lp(R;X ) ≤ C ‖H‖2Lp(R;X )→Lp(R;X )
• Apply factorization theorem with Γ = H and g1 = Mg0 to transfer thequestion from Lp(R;X ) to L2(R× S ,w):For any g0 ∈ Lp(R;X ) there is a weight w : R× S → R+ such that
‖Hf ‖L2(R×S,w) ≤ C ‖f ‖L2(R×S,w), f ∈ Lp(R;X ) ∩ L2(R× S ,w)
‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),
‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).
• By Fubini it suffices to show the boundedness of M on L2(R,w(·, s)) for s ∈ S .
• For a weight v : R→ R+, M is bounded on L2(R, v) if and only if H is boundedon L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R,w(·, s)) for s ∈ S , M is as well.
• Thus ‖Mg0‖Lp(R;X ) . ‖g0‖Lp(R;X )
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Application: Harmonic analysis
Theorem (Kalton–L.–Weis ’19)
Let X be a Banach function space on (S , µ) and p ∈ (1,∞). If H is bounded on
Lp(R;X ), then M is bounded on Lp(R;X ) with
‖M‖Lp(R;X )→Lp(R;X ) ≤ C ‖H‖2Lp(R;X )→Lp(R;X )
• Apply factorization theorem with Γ = H and g1 = Mg0 to transfer thequestion from Lp(R;X ) to L2(R× S ,w):For any g0 ∈ Lp(R;X ) there is a weight w : R× S → R+ such that
‖Hf ‖L2(R×S,w) ≤ C ‖f ‖L2(R×S,w), f ∈ Lp(R;X ) ∩ L2(R× S ,w)
‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),
‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).
• By Fubini it suffices to show the boundedness of M on L2(R,w(·, s)) for s ∈ S .
• For a weight v : R→ R+, M is bounded on L2(R, v) if and only if H is boundedon L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R,w(·, s)) for s ∈ S , M is as well.
• Thus ‖Mg0‖Lp(R;X ) . ‖g0‖Lp(R;X )
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Application: Harmonic analysis
Theorem (Kalton–L.–Weis ’19)
Let X be a Banach function space on (S , µ) and p ∈ (1,∞). If H is bounded on
Lp(R;X ), then M is bounded on Lp(R;X ) with
‖M‖Lp(R;X )→Lp(R;X ) ≤ C ‖H‖2Lp(R;X )→Lp(R;X )
• Apply factorization theorem with Γ = H and g1 = Mg0 to transfer thequestion from Lp(R;X ) to L2(R× S ,w):For any g0 ∈ Lp(R;X ) there is a weight w : R× S → R+ such that
‖Hf ‖L2(R×S,w) ≤ C ‖f ‖L2(R×S,w), f ∈ Lp(R;X ) ∩ L2(R× S ,w)
‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),
‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).
• By Fubini it suffices to show the boundedness of M on L2(R,w(·, s)) for s ∈ S .
• For a weight v : R→ R+, M is bounded on L2(R, v) if and only if H is boundedon L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R,w(·, s)) for s ∈ S , M is as well.
• Thus ‖Mg0‖Lp(R;X ) . ‖g0‖Lp(R;X )
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Application: Harmonic analysis
Theorem (Kalton–L.–Weis ’19)
Let X be a Banach function space on (S , µ) and p ∈ (1,∞). If H is bounded on
Lp(R;X ), then M is bounded on Lp(R;X ) with
‖M‖Lp(R;X )→Lp(R;X ) ≤ C ‖H‖2Lp(R;X )→Lp(R;X )
• Apply factorization theorem with Γ = H and g1 = Mg0 to transfer thequestion from Lp(R;X ) to L2(R× S ,w):For any g0 ∈ Lp(R;X ) there is a weight w : R× S → R+ such that
‖Hf ‖L2(R×S,w) ≤ C ‖f ‖L2(R×S,w), f ∈ Lp(R;X ) ∩ L2(R× S ,w)
‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),
‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).
• By Fubini it suffices to show the boundedness of M on L2(R,w(·, s)) for s ∈ S .
• For a weight v : R→ R+, M is bounded on L2(R, v) if and only if H is boundedon L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R,w(·, s)) for s ∈ S , M is as well.
• Thus ‖Mg0‖Lp(R;X ) . ‖g0‖Lp(R;X )
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Application: Harmonic analysis
Theorem (Kalton–L.–Weis ’19)
Let X be a Banach function space on (S , µ) and p ∈ (1,∞). If H is bounded on
Lp(R;X ), then M is bounded on Lp(R;X ) with
‖M‖Lp(R;X )→Lp(R;X ) ≤ C ‖H‖2Lp(R;X )→Lp(R;X )
• Apply factorization theorem with Γ = H and g1 = Mg0 to transfer thequestion from Lp(R;X ) to L2(R× S ,w):For any g0 ∈ Lp(R;X ) there is a weight w : R× S → R+ such that
‖Hf ‖L2(R×S,w) ≤ C ‖f ‖L2(R×S,w), f ∈ Lp(R;X ) ∩ L2(R× S ,w)
‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),
‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).
• By Fubini it suffices to show the boundedness of M on L2(R,w(·, s)) for s ∈ S .
• For a weight v : R→ R+, M is bounded on L2(R, v) if and only if H is boundedon L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R,w(·, s)) for s ∈ S , M is as well.
• Thus ‖Mg0‖Lp(R;X ) . ‖g0‖Lp(R;X )
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Application: Harmonic analysis
Theorem (Kalton–L.–Weis ’19)
Let X be a Banach function space on (S , µ) and p ∈ (1,∞). If H is bounded on
Lp(R;X ), then M is bounded on Lp(R;X ) with
‖M‖Lp(R;X )→Lp(R;X ) ≤ C ‖H‖2Lp(R;X )→Lp(R;X )
• Apply factorization theorem with Γ = H and g1 = Mg0 to transfer thequestion from Lp(R;X ) to L2(R× S ,w):For any g0 ∈ Lp(R;X ) there is a weight w : R× S → R+ such that
‖Hf ‖L2(R×S,w) ≤ C ‖f ‖L2(R×S,w), f ∈ Lp(R;X ) ∩ L2(R× S ,w)
‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),
‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).
• By Fubini it suffices to show the boundedness of M on L2(R,w(·, s)) for s ∈ S .
• For a weight v : R→ R+, M is bounded on L2(R, v) if and only if H is boundedon L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R,w(·, s)) for s ∈ S , M is as well.
• Thus ‖Mg0‖Lp(R;X ) . ‖g0‖Lp(R;X )
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Application: Banach space geometryLet X be a Banach space and p ∈ (1,∞). Let (fk)nk=0 be a finite martingale inLp(Ω;X ) with difference sequence (dfk)nk=1. Then X has the UMD property if andonly if for all signs εk = ±1 we have∥∥∥ n∑
k=1
εkdfn∥∥∥Lp(Ω;X )
(1)
.∥∥∥ n∑k=1
dfn∥∥∥Lp(Ω;X )
,
if and only if∥∥∥ n∑k=1
dfn∥∥∥Lp(Ω;X )
(2)
.∥∥∥ n∑k=1
εkdfn∥∥∥Lp(Ω×Ω′;X )
(3)
.∥∥∥ n∑k=1
dfn∥∥∥Lp(Ω;X )
,
where (ε)nk=1 is a Rademacher sequence on Ω′.
• (2) does not imply (1).
• It is an open question whether (3) implies (1)
Theorem (Kalton–L.–Weis ’19)
Let X be a Banach function space. Then (3) implies (1).
• Proof similar to previous slide.
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![Page 42: Factorization of operators on Banach (function) spaces · Functional calculus Maximal regularity of PDE’s Stochastic integration in Banach spaces E P n k=1 " kx k 2 1=2 is a norm](https://reader033.vdocuments.us/reader033/viewer/2022060314/5f0b81197e708231d430d85f/html5/thumbnails/42.jpg)
Outlook
• In the Euclidean structures manuscript:
• More factorization theorems.
• Representation of an α-bounded family of operators on a Hilbert space.
• Applications to interpolation, function spaces, functional calculus.
• More applications of factorization through weighted L2:
• Show the necessity of UMD for the R-boundedness of certain operators
• Potentially many more applications!
• To appear on arXiv before Christmas!
12 / 12
![Page 43: Factorization of operators on Banach (function) spaces · Functional calculus Maximal regularity of PDE’s Stochastic integration in Banach spaces E P n k=1 " kx k 2 1=2 is a norm](https://reader033.vdocuments.us/reader033/viewer/2022060314/5f0b81197e708231d430d85f/html5/thumbnails/43.jpg)
Outlook
• In the Euclidean structures manuscript:
• More factorization theorems.
• Representation of an α-bounded family of operators on a Hilbert space.
• Applications to interpolation, function spaces, functional calculus.
• More applications of factorization through weighted L2:
• Show the necessity of UMD for the R-boundedness of certain operators
• Potentially many more applications!
• To appear on arXiv before Christmas!
12 / 12
![Page 44: Factorization of operators on Banach (function) spaces · Functional calculus Maximal regularity of PDE’s Stochastic integration in Banach spaces E P n k=1 " kx k 2 1=2 is a norm](https://reader033.vdocuments.us/reader033/viewer/2022060314/5f0b81197e708231d430d85f/html5/thumbnails/44.jpg)
Outlook
• In the Euclidean structures manuscript:
• More factorization theorems.
• Representation of an α-bounded family of operators on a Hilbert space.
• Applications to interpolation, function spaces, functional calculus.
• More applications of factorization through weighted L2:
• Show the necessity of UMD for the R-boundedness of certain operators
• Potentially many more applications!
• To appear on arXiv before Christmas!
12 / 12
![Page 45: Factorization of operators on Banach (function) spaces · Functional calculus Maximal regularity of PDE’s Stochastic integration in Banach spaces E P n k=1 " kx k 2 1=2 is a norm](https://reader033.vdocuments.us/reader033/viewer/2022060314/5f0b81197e708231d430d85f/html5/thumbnails/45.jpg)
Thank you for your attention!
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