Factorization of operators on Banach (function) spaces Emiel Lorist - - PowerPoint PPT Presentation

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

Euclidean structures

• Project started in the early 2000s
• Understand the role R-boundedness from an

abstract operator-theoretic viewpoint

• Connected to completely bounded maps
• Project on hold since Nigel passed away
• Studied and improved/extended parts of the

manuscript in 2016

• Joined the project in 2018, rewrote and modernized

the manuscript

• Euclidean structures
• Part I: Representation of operator families
• n 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

Euclidean structures

• Project started in the early 2000s
• Understand the role R-boundedness from an

abstract operator-theoretic viewpoint

• Connected to completely bounded maps
• Project on hold since Nigel passed away
• Studied and improved/extended parts of the

manuscript in 2016

• Joined the project in 2018, rewrote and modernized

the manuscript

• Euclidean structures
• Part I: Representation of operator families
• n 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

Euclidean structures

• Project started in the early 2000s
• Understand the role R-boundedness from an

abstract operator-theoretic viewpoint

• Connected to completely bounded maps
• Project on hold since Nigel passed away
• Studied and improved/extended parts of the

manuscript in 2016

• Joined the project in 2018, rewrote and modernized

the manuscript

• Euclidean structures
• Part I: Representation of operator families
• n 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

Euclidean structures

• Project started in the early 2000s
• Understand the role R-boundedness from an

abstract operator-theoretic viewpoint

• Connected to completely bounded maps
• Project on hold since Nigel passed away
• Studied and improved/extended parts of the

manuscript in 2016

• Joined the project in 2018, rewrote and modernized

the manuscript

• Euclidean structures
• Part I: Representation of operator families
• n 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

R-boundedness

Definition

Let X be a Banach space and Γ ⊆ L(X). Then we say that Γ is R-bounded if for any finite sequences (Tk)n

k=1 in Γ and (xk)n k=1 in X

• E
• n
• k=1

εkTkxk

• 2

X

1/2 ≤ C

• E
• n
• k=1

εkxk

• 2

X

1/2 , where (εk)n

k=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

• 21/2 is a norm on X n.
• A Euclidean structure is such a norm with a left and right ideal property.

2 / 12

R-boundedness

Definition

Let X be a Banach space and Γ ⊆ L(X). Then we say that Γ is R-bounded if for any finite sequences (Tk)n

k=1 in Γ and (xk)n k=1 in X

• E
• n
• k=1

εkTkxk

• 2

X

1/2 ≤ C

• E
• n
• k=1

εkxk

• 2

X

1/2 , where (εk)n

k=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

• 21/2 is a norm on X n.
• A Euclidean structure is such a norm with a left and right ideal property.

2 / 12

R-boundedness

Definition

Let X be a Banach space and Γ ⊆ L(X). Then we say that Γ is R-bounded if for any finite sequences (Tk)n

k=1 in Γ and (xk)n k=1 in X

• E
• n
• k=1

εkTkxk

• 2

X

1/2 ≤ C

• E
• n
• k=1

εkxk

• 2

X

1/2 , where (εk)n

k=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

• 21/2 is a norm on X n.
• A Euclidean structure is such a norm with a left and right ideal property.

2 / 12

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 = xX, x ∈ X, Axα ≤ Axα, x ∈ X n, A ∈ Mm,n(C), (Tx1, · · · , Txn)α ≤ C Txα, x ∈ X n, T ∈ L(X), A Euclidean structure induces a norm on the finite rank operators from ℓ2 to X.

3 / 12

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 = xX, x ∈ X, Axα ≤ Axα, x ∈ X n, A ∈ Mm,n(C), (Tx1, · · · , Txn)α ≤ C Txα, x ∈ X n, T ∈ L(X), A Euclidean structure induces a norm on the finite rank operators from ℓ2 to X. Non-example:

• On a Banach space X:

xR :=

• E
• n
• k=1

εkxk

• 2

X

1

2 ,

x ∈ X n, is not a Euclidean structure. It fails the right-ideal property.

3 / 12

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 = xX, x ∈ X, Axα ≤ Axα, x ∈ X n, A ∈ Mm,n(C), (Tx1, · · · , Txn)α ≤ C Txα, x ∈ X n, T ∈ L(X), A Euclidean structure induces a norm on the finite rank operators from ℓ2 to X. Examples:

• On any Banach space X: The Gaussian structure

xγ :=

• E
• n
• k=1

γkxk

• 2

X

1

2 ,

x ∈ X n, where (γk)n

k=1 is a sequence of independent normalized Gaussians.

• On a Banach lattice X: The ℓ2-structure

xℓ2 :=

• n
• k=1

|xk|21/2

• X,

x ∈ X n.

3 / 12

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 = xX, x ∈ X, Axα ≤ Axα, x ∈ X n, A ∈ Mm,n(C), (Tx1, · · · , Txn)α ≤ C Txα, x ∈ X n, T ∈ L(X), A Euclidean structure induces a norm on the finite rank operators from ℓ2 to X. Examples:

• On any Banach space X: The Gaussian structure

xγ :=

• E
• n
• k=1

γkxk

• 2

X

1

2 ,

x ∈ X n, where (γk)n

k=1 is a sequence of independent normalized Gaussians.

• On a Banach lattice X: The ℓ2-structure

xℓ2 :=

• n
• k=1

|xk|21/2

• X,

x ∈ X n.

3 / 12

α-boundedness

Definition

Let X be a Banach space, α an Euclidean structure and Γ ⊆ L(X). Then we say that Γ is α-bounded if for any T = diag(T1, · · · , Tn) with T1, · · · , Tn ∈ Γ Txα ≤ Cxα, x ∈ X n.

• α-boundedness implies uniform boundedness
• On a Banach space X with finite cotype

xγ =

• E
• n
• k=1

γkxk

• 2

X

1

2 ≃

• E
• n
• k=1

εkxk

• 2

X

1

2 = xR,

so γ-boundedness is equivalent to R-boundedness

• On a Banach lattice X with finite cotype

xℓ2 =

• n
• k=1

|xk|21/2

• X ≃
• E
• n
• k=1

εkxk

• 2

X

1

2 = xR,

so ℓ2-boundedness is equivalent to R-boundedness.

4 / 12

α-boundedness

Definition

Let X be a Banach space, α an Euclidean structure and Γ ⊆ L(X). Then we say that Γ is α-bounded if for any T = diag(T1, · · · , Tn) with T1, · · · , Tn ∈ Γ Txα ≤ Cxα, x ∈ X n.

• α-boundedness implies uniform boundedness
• On a Banach space X with finite cotype

xγ =

• E
• n
• k=1

γkxk

• 2

X

1

2 ≃

• E
• n
• k=1

εkxk

• 2

X

1

2 = xR,

so γ-boundedness is equivalent to R-boundedness

• On a Banach lattice X with finite cotype

xℓ2 =

• n
• k=1

|xk|21/2

• X ≃
• E
• n
• k=1

εkxk

• 2

X

1

2 = xR,

so ℓ2-boundedness is equivalent to R-boundedness.

4 / 12

α-boundedness

Definition

Let X be a Banach space, α an Euclidean structure and Γ ⊆ L(X). Then we say that Γ is α-bounded if for any T = diag(T1, · · · , Tn) with T1, · · · , Tn ∈ Γ Txα ≤ Cxα, x ∈ X n.

• α-boundedness implies uniform boundedness
• On a Banach space X with finite cotype

xγ =

• E
• n
• k=1

γkxk

• 2

X

1

2 ≃

• E
• n
• k=1

εkxk

• 2

X

1

2 = xR,

so γ-boundedness is equivalent to R-boundedness

• On a Banach lattice X with finite cotype

xℓ2 =

• n
• k=1

|xk|21/2

• X ≃
• E
• n
• k=1

εkxk

• 2

X

1

2 = xR,

so ℓ2-boundedness is equivalent to R-boundedness.

4 / 12

Factorization through a Hilbert space

Theorem (Kwapie´ n ’72 and Maurey ’74)

Let X and Y be a Banach spaces and T ∈ L(X, Y ). If X has type 2 and Y cotype 2, then there is a Hilbert space H and operators S ∈ L(X, H) and U ∈ L(H, Y ) s.t. T = US.

Corollary (Kwapie´ n ’72)

Let X be a Banach space. X has type 2 and cotype 2 if and only if X is isomorphic to a Hilbert space.

5 / 12

Factorization through a Hilbert space

Theorem (Kwapie´ n ’72 and Maurey ’74)

Let X and Y be a Banach spaces and T ∈ L(X, Y ). If X has type 2 and Y cotype 2, then there is a Hilbert space H and operators S ∈ L(X, H) and U ∈ L(H, Y ) s.t. T = US.

Corollary (Kwapie´ n ’72)

Let X be a Banach space. X has type 2 and cotype 2 if and only if X is isomorphic to a Hilbert space.

5 / 12

Factorization through a Hilbert space

Theorem (Kwapie´ n ’72 and Maurey ’74)

Let X and Y be a Banach spaces and T ∈ L(X, Y ). If X has type 2 and Y cotype 2, then there is a Hilbert space H and operators S ∈ L(X, H) and U ∈ L(H, Y ) s.t. T = US.

Corollary (Kwapie´ n ’72)

Let X be a Banach space. X has type 2 and cotype 2 if and only if X is isomorphic to 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, Y cotype 2 and Γ1 is γ-bounded, then there is a Hilbert space H, a

• T ∈ L(X, H) for every T ∈ Γ1

and U ∈ L(H, Y ) s.t. the following diagram commutes:

X Y H

• T

T U

5 / 12

Factorization through a Hilbert space

Theorem (Kwapie´ n ’72 and Maurey ’74)

Let X and Y be a Banach spaces and T ∈ L(X, Y ). If X has type 2 and Y cotype 2, then there is a Hilbert space H and operators S ∈ L(X, H) and U ∈ L(H, Y ) s.t. T = US.

Corollary (Kwapie´ n ’72)

Let X be a Banach space. X has type 2 and cotype 2 if and only if X is isomorphic to 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, Y cotype 2 and Γ1 and Γ2 are γ-bounded, then there is a Hilbert space H, a

• T ∈ 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

5 / 12

Factorization through weighted L2

For a measure space (S, Σ, µ) and a weight w : S → R+ let L2(S, w) be the space of all 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 ℓ2-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 g0L2(S,w) ≤ C g0X, g1X ≤ C g1L2(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

|Tkfk|2 1

2

Then we have Tf 2

ℓ2 = g12 X ≤ C 2 n

• k=1
• S

|Tkfk|2w dµ ≤ C 4

n

• k=1
• S

|fk|2w dµ ≤ C 6g02

X = f 2 ℓ2

6 / 12

Factorization through weighted L2

For a measure space (S, Σ, µ) and a weight w : S → R+ let L2(S, w) be the space of all 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 ℓ2-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 g0L2(S,w) ≤ C g0X, g1X ≤ C g1L2(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

|Tkfk|2 1

2

Then we have Tf 2

ℓ2 = g12 X ≤ C 2 n

• k=1
• S

|Tkfk|2w dµ ≤ C 4

n

• k=1
• S

|fk|2w dµ ≤ C 6g02

X = f 2 ℓ2

6 / 12

Factorization through weighted L2

For a measure space (S, Σ, µ) and a weight w : S → R+ let L2(S, w) be the space of all 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 ℓ2-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 g0L2(S,w) ≤ C g0X, g1X ≤ C g1L2(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

|Tkfk|2 1

2

Then we have Tf 2

ℓ2 = g12 X ≤ C 2 n

• k=1
• S

|Tkfk|2w dµ ≤ C 4

n

• k=1
• S

|fk|2w dµ ≤ C 6g02

X = f 2 ℓ2

6 / 12

Factorization through weighted L2

For a measure space (S, Σ, µ) and a weight w : S → R+ let L2(S, w) be the space of all 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 ℓ2-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 g0L2(S,w) ≤ C g0X, g1X ≤ C g1L2(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

|Tkfk|2 1

2

Then we have Tf 2

ℓ2 = g12 X ≤ C 2 n

• k=1
• S

|Tkfk|2w dµ ≤ C 4

n

• k=1
• S

|fk|2w dµ ≤ C 6g02

X = f 2 ℓ2

6 / 12

Application: Harmonic analysis

Let p ∈ [1, ∞). For f ∈ Lp(R) define the Hilbert transform Hf (x) := p. v. 1 π

• R

1 x − y f (y) dy, x ∈ R, = F −1 ξ → −i sgn(ξ)Ff (ξ)

• (x).

For f ∈ Lp(Rd) define the Hardy–Littlewood maximal operator Mf (x) := sup

r>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 on

the Bochner space Lp(Rd; X).

• These have numerous applications in both harmonic analysis and PDE.

7 / 12

Application: Harmonic analysis

Let p ∈ [1, ∞). For f ∈ Lp(R) define the Hilbert transform Hf (x) := p. v. 1 π

• R

1 x − y f (y) dy, x ∈ R, = F −1 ξ → −i sgn(ξ)Ff (ξ)

• (x).

For f ∈ Lp(Rd) define the Hardy–Littlewood maximal operator Mf (x) := sup

r>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 on

the Bochner space Lp(Rd; X).

• These have numerous applications in both harmonic analysis and PDE.

7 / 12

Application: Harmonic analysis

Let p ∈ [1, ∞). For f ∈ Lp(R) define the Hilbert transform Hf (x) := p. v. 1 π

• R

1 x − y f (y) dy, x ∈ R, = F −1 ξ → −i sgn(ξ)Ff (ξ)

• (x).

For f ∈ Lp(Rd) define the Hardy–Littlewood maximal operator Mf (x) := sup

r>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 on

the Bochner space Lp(Rd; X).

• These have numerous applications in both harmonic analysis and PDE.

7 / 12

Application: Harmonic analysis

Let p ∈ [1, ∞). For f ∈ Lp(R) define the Hilbert transform Hf (x) := p. v. 1 π

• R

1 x − y f (y) dy, x ∈ R, = F −1 ξ → −i sgn(ξ)Ff (ξ)

• (x).

For f ∈ Lp(Rd) define the Hardy–Littlewood maximal operator Mf (x) := sup

r>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 on

the Bochner space Lp(Rd; X).

• These have numerous applications in both harmonic analysis and PDE.

7 / 12

Application: Harmonic analysis

Let p ∈ [1, ∞) and let X be a Banach space. For f ∈ Lp(R; X) define the vector-valued Hilbert transform

• Hf (x) := p. v.
• R

1 x − y f (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 lattice Hardy–Littlewood maximal operator

• Mf (x) := sup

r>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, ∞) (Calder´

• n–Benedek–Panzone ’62).
• Boundedness of

M is independent of p ∈ (1, ∞) and d ∈ N (Garc´ ıa-Cuerva–Macias–Torrea ’93).

• Boundedness of

H and M depends on the geometry of X.

8 / 12

Application: Harmonic analysis

Let p ∈ [1, ∞) and let X be a Banach space. For f ∈ Lp(R; X) define the vector-valued Hilbert transform

• Hf (x) := p. v.
• R

1 x − y f (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 lattice Hardy–Littlewood maximal operator

• Mf (x) := sup

r>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, ∞) (Calder´

• n–Benedek–Panzone ’62).
• Boundedness of

M is independent of p ∈ (1, ∞) and d ∈ N (Garc´ ıa-Cuerva–Macias–Torrea ’93).

• Boundedness of

H and M depends on the geometry of X.

8 / 12

Application: Harmonic analysis

Let p ∈ [1, ∞) and let X be a Banach space. For f ∈ Lp(R; X) define the vector-valued Hilbert transform

• Hf (x) := p. v.
• R

1 x − y f (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 lattice Hardy–Littlewood maximal operator

• Mf (x) := sup

r>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, ∞) (Calder´

• n–Benedek–Panzone ’62).
• Boundedness of

M is independent of p ∈ (1, ∞) and d ∈ N (Garc´ ıa-Cuerva–Macias–Torrea ’93).

• Boundedness of

H and M depends on the geometry of X.

8 / 12

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)n

k=0 in Lp(Ω; X) has

Unconditional Martingale Differences (dfk)n

k=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.

9 / 12

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)n

k=0 in Lp(Ω; X) has

Unconditional Martingale Differences (dfk)n

k=1.

• That is, if for some p ∈ (1, ∞) and all signs ǫk = ±1 we have
• n
• k=1

ǫkdfn

• 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)n

k=0 in Lp(Ω; X) has

Unconditional Martingale Differences (dfk)n

k=1.

• Reflexive Lebesgue, Lorentz, (Musielak)-Orlicz, Sobolev, Besov spaces and

Schatten 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)n

k=0 in Lp(Ω; X) has

Unconditional Martingale Differences (dfk)n

k=1.

• Reflexive Lebesgue, Lorentz, (Musielak)-Orlicz, Sobolev, Besov spaces and

Schatten 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 MLp(R;X)→Lp(R;X) ≤ C H2

Lp(R;X)→Lp(R;X)

• Apply factorization theorem with Γ = {

H} and g1 = Mg0 to transfer the question 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) g0L2(R×S,w) ≤ C g0Lp(R;X), Mg0Lp(R;X) ≤ C Mg0L2(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 bounded
• n L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R, w(·, s)) for s ∈ S, M is as well.
• Thus

Mg0Lp(R;X) g0Lp(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 MLp(R;X)→Lp(R;X) ≤ C H2

Lp(R;X)→Lp(R;X)

• Apply factorization theorem with Γ = {

H} and g1 = Mg0 to transfer the question 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) g0L2(R×S,w) ≤ C g0Lp(R;X), Mg0Lp(R;X) ≤ C Mg0L2(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 bounded
• n L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R, w(·, s)) for s ∈ S, M is as well.
• Thus

Mg0Lp(R;X) g0Lp(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 MLp(R;X)→Lp(R;X) ≤ C H2

Lp(R;X)→Lp(R;X)

• Apply factorization theorem with Γ = {

H} and g1 = Mg0 to transfer the question 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) g0L2(R×S,w) ≤ C g0Lp(R;X), Mg0Lp(R;X) ≤ C Mg0L2(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 bounded
• n L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R, w(·, s)) for s ∈ S, M is as well.
• Thus

Mg0Lp(R;X) g0Lp(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 MLp(R;X)→Lp(R;X) ≤ C H2

Lp(R;X)→Lp(R;X)

• Apply factorization theorem with Γ = {

H} and g1 = Mg0 to transfer the question 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) g0L2(R×S,w) ≤ C g0Lp(R;X), Mg0Lp(R;X) ≤ C Mg0L2(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 bounded
• n L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R, w(·, s)) for s ∈ S, M is as well.
• Thus

Mg0Lp(R;X) g0Lp(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 MLp(R;X)→Lp(R;X) ≤ C H2

Lp(R;X)→Lp(R;X)

• Apply factorization theorem with Γ = {

H} and g1 = Mg0 to transfer the question 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) g0L2(R×S,w) ≤ C g0Lp(R;X), Mg0Lp(R;X) ≤ C Mg0L2(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 bounded
• n L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R, w(·, s)) for s ∈ S, M is as well.
• Thus

Mg0Lp(R;X) g0Lp(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 MLp(R;X)→Lp(R;X) ≤ C H2

Lp(R;X)→Lp(R;X)

• Apply factorization theorem with Γ = {

H} and g1 = Mg0 to transfer the question 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) g0L2(R×S,w) ≤ C g0Lp(R;X), Mg0Lp(R;X) ≤ C Mg0L2(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 bounded
• n L2(R, v) (Muckenhoupt ’72,...).
• As H is bounded on L2(R, w(·, s)) for s ∈ S, M is as well.
• Thus

Mg0Lp(R;X) g0Lp(R;X)

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Application: Banach space geometry

Let X be a Banach space and p ∈ (1, ∞). Let (fk)n

k=0 be a finite martingale in

Lp(Ω; X) with difference sequence (dfk)n

k=1. Then X has the UMD property if and

• nly if for all signs ǫk = ±1 we have
• n
• k=1

ǫkdfn

• 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)n

k=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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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

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

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

Thank you for your attention!

12 / 12