Measure Theory Marczewski Centennial Conference Bedle˛ vo ...measure07/slides/curbera.pdf ·...
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Extension versus representationThree examples
Measure TheoryMarczewski Centennial Conference
Bedlevo, September 2007
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Edward Marczewski
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Edward Marczewski
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Optimal domains for classical operatorsand vector measures:
a new look at old problems
Guillermo P. Curbera
Universidad de SevillaSpain
Bedlevo, September 2007
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Outline
1 Extension versus representationRepresentation theoremsExtension theorems
2 Three examplesThe Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Outline
1 Extension versus representationRepresentation theoremsExtension theorems
2 Three examplesThe Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Outline
1 Extension versus representationRepresentation theoremsExtension theorems
2 Three examplesThe Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
The Riesz representation theorem
Frigyes Riesz (1909)
Let Λ: C([0, 1]) → C be a positive linear operator. Then,there exists a finite Borel measure
ν : B0([0, 1]) → R
such that:
Λf =
∫f dν, f ∈ C([0, 1]).
Viewpoint: a representation theorem.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
The Riesz representation theorem
Frigyes Riesz (1909)
Let Λ: C([0, 1]) → C be a positive linear operator. Then,there exists a finite Borel measure
ν : B0([0, 1]) → R
such that:
Λf =
∫f dν, f ∈ C([0, 1]).
Viewpoint: a representation theorem.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
The Riesz representation theorem
Frigyes Riesz (1909)
Let Λ: C([0, 1]) → C be a positive linear operator. Then,there exists a finite Borel measure
ν : B0([0, 1]) → R
such that:
Λf =
∫f dν, f ∈ C([0, 1]).
Viewpoint:
a representation theorem.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
The Riesz representation theorem
Frigyes Riesz (1909)
Let Λ: C([0, 1]) → C be a positive linear operator. Then,there exists a finite Borel measure
ν : B0([0, 1]) → R
such that:
Λf =
∫f dν, f ∈ C([0, 1]).
Viewpoint: a representation theorem.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
The vector case
Bartle, Dunford, Schwartz (1955)
Let K be a compact Hausdorff space, X be a Banachspace, and T : C(K ) → X a weakly compact operator.Then, there exists a Borel (vector) measure
ν : B0(K ) → X
such that:
Tf =
∫f dν, f ∈ C(K ).
Viewpoint: a representation theorem.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
The vector case
Bartle, Dunford, Schwartz (1955)
Let K be a compact Hausdorff space, X be a Banachspace, and T : C(K ) → X a weakly compact operator.Then, there exists a Borel (vector) measure
ν : B0(K ) → X
such that:
Tf =
∫f dν, f ∈ C(K ).
Viewpoint: a representation theorem.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
The vector case
Bartle, Dunford, Schwartz (1955)
Let K be a compact Hausdorff space, X be a Banachspace, and T : C(K ) → X a weakly compact operator.Then, there exists a Borel (vector) measure
ν : B0(K ) → X
such that:
Tf =
∫f dν, f ∈ C(K ).
Viewpoint:
a representation theorem.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
The vector case
Bartle, Dunford, Schwartz (1955)
Let K be a compact Hausdorff space, X be a Banachspace, and T : C(K ) → X a weakly compact operator.Then, there exists a Borel (vector) measure
ν : B0(K ) → X
such that:
Tf =
∫f dν, f ∈ C(K ).
Viewpoint: a representation theorem.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Representation of operators I: Bochner
Let (Ω,Σ, µ) be a finite measure space and X be a Banachspace.
(Dunford, Pettis, Phillips (1940)Let T : L1(µ) → X be a weakly compact linear operator.Then, there exists g ∈ L∞(µ) such that
Tf =
∫f ·g dµ, f ∈ L1(µ).
Then A ∈ Σ 7→ ν(A) := T (χA) ∈ X , is a vector measurewith a Bochner integrable density with respect to µ.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Representation of operators I: Bochner
Let (Ω,Σ, µ) be a finite measure space and X be a Banachspace.
(Dunford, Pettis, Phillips (1940)Let T : L1(µ) → X be a weakly compact linear operator.Then, there exists g ∈ L∞(µ) such that
Tf =
∫f ·g dµ, f ∈ L1(µ).
Then A ∈ Σ 7→ ν(A) := T (χA) ∈ X , is a vector measurewith a Bochner integrable density with respect to µ.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Representation of operators I: Bochner
Let (Ω,Σ, µ) be a finite measure space and X be a Banachspace.
(Dunford, Pettis, Phillips (1940)Let T : L1(µ) → X be a weakly compact linear operator.Then, there exists g ∈ L∞(µ) such that
Tf =
∫f ·g dµ, f ∈ L1(µ).
Then A ∈ Σ 7→ ν(A) := T (χA) ∈ X , is a vector measurewith a Bochner integrable density with respect to µ.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Representation of operators II: BDS
Let (Ω,Σ) be a measure space, X be a Banach space, andν : Σ → X a vector measure.
The BDS–integral:Let f : Ω → R be a measurable function. It is integrable withrespect to ν if... (Lebesgue type integration).
L1(ν) suitably normed is a Banach space (of classes) ofintegrable functions.
Integration operator: f ∈ L1(ν) 7→ Iν(f ) =∫
f dν ∈ X .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Representation of operators II: BDS
Let (Ω,Σ) be a measure space, X be a Banach space, andν : Σ → X a vector measure.
The BDS–integral:Let f : Ω → R be a measurable function. It is integrable withrespect to ν if... (Lebesgue type integration).
L1(ν) suitably normed is a Banach space (of classes) ofintegrable functions.
Integration operator: f ∈ L1(ν) 7→ Iν(f ) =∫
f dν ∈ X .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Representation of operators II: BDS
Let (Ω,Σ) be a measure space, X be a Banach space, andν : Σ → X a vector measure.
The BDS–integral:Let f : Ω → R be a measurable function. It is integrable withrespect to ν if... (Lebesgue type integration).
L1(ν) suitably normed is a Banach space (of classes) ofintegrable functions.
Integration operator: f ∈ L1(ν) 7→ Iν(f ) =∫
f dν ∈ X .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–representation: Example
The Riemann–Liouville fractional integral of order α ∈ (0, 1):
Tα(f ) =1
Γ(α)
∫ 1
0
f (t)|x − t |α
dt , x ∈ [0, 1].
Tα : L∞([0, 1]) → Lp([0, 1]) continuous for all p ∈ [1,∞].
Tα is Bochner representable iff p < 1/α.
However, if νp(A) := Tα(χA) ∈ Lp([0, 1]), for A ∈ B0([0, 1]),then:
Tα(f ) = (BDS)−∫
f dνp, f ∈ L∞([0, 1]),
for all p ∈ [1,∞].
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–representation: Example
The Riemann–Liouville fractional integral of order α ∈ (0, 1):
Tα(f ) =1
Γ(α)
∫ 1
0
f (t)|x − t |α
dt , x ∈ [0, 1].
Tα : L∞([0, 1]) → Lp([0, 1]) continuous for all p ∈ [1,∞].
Tα is Bochner representable iff p < 1/α.
However, if νp(A) := Tα(χA) ∈ Lp([0, 1]), for A ∈ B0([0, 1]),then:
Tα(f ) = (BDS)−∫
f dνp, f ∈ L∞([0, 1]),
for all p ∈ [1,∞].
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–representation: Example
The Riemann–Liouville fractional integral of order α ∈ (0, 1):
Tα(f ) =1
Γ(α)
∫ 1
0
f (t)|x − t |α
dt , x ∈ [0, 1].
Tα : L∞([0, 1]) → Lp([0, 1]) continuous for all p ∈ [1,∞].
Tα is Bochner representable iff p < 1/α.
However, if νp(A) := Tα(χA) ∈ Lp([0, 1]), for A ∈ B0([0, 1]),then:
Tα(f ) = (BDS)−∫
f dνp, f ∈ L∞([0, 1]),
for all p ∈ [1,∞].
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–representation: Example
The Riemann–Liouville fractional integral of order α ∈ (0, 1):
Tα(f ) =1
Γ(α)
∫ 1
0
f (t)|x − t |α
dt , x ∈ [0, 1].
Tα : L∞([0, 1]) → Lp([0, 1]) continuous for all p ∈ [1,∞].
Tα is Bochner representable iff p < 1/α.
However, if νp(A) := Tα(χA) ∈ Lp([0, 1]), for A ∈ B0([0, 1]),then:
Tα(f ) = (BDS)−∫
f dνp, f ∈ L∞([0, 1]),
for all p ∈ [1,∞].
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–representation: Example
The Riemann–Liouville fractional integral of order α ∈ (0, 1):
Tα(f ) =1
Γ(α)
∫ 1
0
f (t)|x − t |α
dt , x ∈ [0, 1].
Tα : L∞([0, 1]) → Lp([0, 1]) continuous for all p ∈ [1,∞].
Tα is Bochner representable iff p < 1/α.
However, if νp(A) := Tα(χA) ∈ Lp([0, 1]), for A ∈ B0([0, 1]),then:
Tα(f ) = (BDS)−∫
f dνp, f ∈ L∞([0, 1]),
for all p ∈ [1,∞].
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Outline
1 Extension versus representationRepresentation theoremsExtension theorems
2 Three examplesThe Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–extension
The previous example has interesting features:
L∞([0, 1])Tα - Lp([0, 1])
Id?
L1(νp)Iνp
*
Moreover, the space L1(νp) is the largest Banach functionspace with order continuous norm for which such a factorizationexists (p 6= ∞).(Order continuous norm: fα ↓ 0 then ‖fα‖ ↓ 0.)
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–extension
The previous example has interesting features:
L∞([0, 1])Tα - Lp([0, 1])
Id?
L1(νp)Iνp
*
Moreover, the space L1(νp) is the largest Banach functionspace with order continuous norm for which such a factorizationexists (p 6= ∞).(Order continuous norm: fα ↓ 0 then ‖fα‖ ↓ 0.)
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–extension
The previous example has interesting features:
L∞([0, 1])Tα - Lp([0, 1])
Id?
L1(νp)Iνp
*
Moreover, the space L1(νp) is the largest Banach functionspace with order continuous norm for which such a factorizationexists (p 6= ∞).(Order continuous norm: fα ↓ 0 then ‖fα‖ ↓ 0.)
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–extension
The previous example has interesting features:
L∞([0, 1])Tα - Lp([0, 1])
Id?
L1(νp)Iνp
*
Moreover, the space L1(νp) is the largest Banach functionspace with order continuous norm for which such a factorizationexists (p 6= ∞).(Order continuous norm: fα ↓ 0 then ‖fα‖ ↓ 0.)
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–extension
More precisely, if there exists a Banach function space E withorder continuous norm, and an operator T : E → X whichextends Tα
L∞([0, 1])Tα - Lp([0, 1])
Id?
ET
*
then E ⊂ L1(νp) and the integration operator Iνp extends T .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
BDS–extension
More precisely, if there exists a Banach function space E withorder continuous norm, and an operator T : E → X whichextends Tα
L∞([0, 1])Tα - Lp([0, 1])
Id?
ET
*
then E ⊂ L1(νp) and the integration operator Iνp extends T .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Extension versus representation
The Riesz theorem:
C([0, 1])Λ
- C
Id?
L1(ν)Iν
*
The BDS theorem:
C(K )T
- X
Id?
L1(ν)Iν
*
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Extension versus representation
The Riesz theorem:
C([0, 1])Λ
- C
Id?
L1(ν)Iν
*
The BDS theorem:
C(K )T
- X
Id?
L1(ν)Iν
*
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
Extension versus representation
The Riesz theorem:
C([0, 1])Λ
- C
Id?
L1(ν)Iν
*
The BDS theorem:
C(K )T
- X
Id?
L1(ν)Iν
*
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
General situation
Theorem
T : E → X linearE Banach function spaceX Banach space
such thatTfn → Tf weakly in X
if fn ↑ f ∈ E
⇒
ν(A) := TχA
is σ–additiveE → L1(ν)Integration operator Iν
extends T
L1(ν) is the optimal domain for T (with order continuousnorm).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
General situation
Theorem
T : E → X linearE Banach function spaceX Banach space
such thatTfn → Tf weakly in X
if fn ↑ f ∈ E
⇒
ν(A) := TχA
is σ–additiveE → L1(ν)Integration operator Iν
extends T
L1(ν) is the optimal domain for T (with order continuousnorm).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
Representation theoremsExtension theorems
General situation
Theorem
T : E → X linearE Banach function spaceX Banach space
such thatTfn → Tf weakly in X
if fn ↑ f ∈ E
⇒
ν(A) := TχA
is σ–additiveE → L1(ν)Integration operator Iν
extends T
L1(ν) is the optimal domain for T (with order continuousnorm).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Outline
1 Extension versus representationRepresentation theoremsExtension theorems
2 Three examplesThe Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
The Volterra integral operator
Defined by
f 7−→ Vf (x) :=
∫ x
0f (t) dt , x ∈ [0, 1].
Let:
[V , L∞] =
f : V |f | ∈ L∞([0, 1])
[V , L1] =
f : V |f | ∈ L1([0, 1])
.
Then:
[V , L∞] = L1([0, 1]).
[V , L1] = L1((1− t)dt).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
The Volterra integral operator
Defined by
f 7−→ Vf (x) :=
∫ x
0f (t) dt , x ∈ [0, 1].
Let:
[V , L∞] =
f : V |f | ∈ L∞([0, 1])
[V , L1] =
f : V |f | ∈ L1([0, 1])
.
Then:
[V , L∞] = L1([0, 1]).
[V , L1] = L1((1− t)dt).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
The Volterra integral operator
Defined by
f 7−→ Vf (x) :=
∫ x
0f (t) dt , x ∈ [0, 1].
Let:
[V , L∞] =
f : V |f | ∈ L∞([0, 1])
[V , L1] =
f : V |f | ∈ L1([0, 1])
.
Then:
[V , L∞] = L1([0, 1]).
[V , L1] = L1((1− t)dt).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
The Volterra integral operator
Defined by
f 7−→ Vf (x) :=
∫ x
0f (t) dt , x ∈ [0, 1].
Let:
[V , L∞] =
f : V |f | ∈ L∞([0, 1])
[V , L1] =
f : V |f | ∈ L1([0, 1])
.
Then:
[V , L∞] = L1([0, 1]).
[V , L1] = L1((1− t)dt).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal Lp-domain
Let 1 < p < ∞.
QUESTION: What can we say about
[V , Lp] =
f : V |f | ∈ Lp([0, 1])?
ANSWER: Since Lp = (L1, L∞)1/p′,p, then
[V , Lp] =([V , L1], [V , L∞]
)1/p′,p
.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal Lp-domain
Let 1 < p < ∞.
QUESTION: What can we say about
[V , Lp] =
f : V |f | ∈ Lp([0, 1])?
ANSWER: Since Lp = (L1, L∞)1/p′,p, then
[V , Lp] =([V , L1], [V , L∞]
)1/p′,p
.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal Lp-domain
Let 1 < p < ∞.
QUESTION: What can we say about
[V , Lp] =
f : V |f | ∈ Lp([0, 1])?
ANSWER: Since Lp = (L1, L∞)1/p′,p, then
[V , Lp] =([V , L1], [V , L∞]
)1/p′,p
.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal X–domain
The result is valid more generally:
For X rearrangement invariant (i.e. interpolation) space on[0, 1] we define the optimal domain
[V , X ] =
f : V |f | ∈ X.
Since X = (L1, L∞)ρ, then
[V , X ] =([V , L1], [V , L∞]
)ρ.
For Volterra convolution operators (under certainconditions on φ):
f 7−→ Vφf (x) :=
∫ x
0f (t)φ(x − t) dt , x ∈ [0, 1].
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal X–domain
The result is valid more generally:
For X rearrangement invariant (i.e. interpolation) space on[0, 1] we define the optimal domain
[V , X ] =
f : V |f | ∈ X.
Since X = (L1, L∞)ρ, then
[V , X ] =([V , L1], [V , L∞]
)ρ.
For Volterra convolution operators (under certainconditions on φ):
f 7−→ Vφf (x) :=
∫ x
0f (t)φ(x − t) dt , x ∈ [0, 1].
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal X–domain
The result is valid more generally:
For X rearrangement invariant (i.e. interpolation) space on[0, 1] we define the optimal domain
[V , X ] =
f : V |f | ∈ X.
Since X = (L1, L∞)ρ, then
[V , X ] =([V , L1], [V , L∞]
)ρ.
For Volterra convolution operators (under certainconditions on φ):
f 7−→ Vφf (x) :=
∫ x
0f (t)φ(x − t) dt , x ∈ [0, 1].
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Outline
1 Extension versus representationRepresentation theoremsExtension theorems
2 Three examplesThe Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Sobolev’s classical inequality
Theorem (1938)
Let Ω ⊂ Rn be a bounded domain and let 1 ≤ p < n. Thereexist a constant C > 0 such that
‖u‖Lq(Ω) ≤ C ‖ |∇u| ‖Lp(Ω), u ∈ C10(Ω),
where q := npn−p .
Note: p < npn−p ⇒ ‖u‖p ≤ ‖u‖ np
n−p
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Sobolev’s classical inequality
Theorem (1938)
Let Ω ⊂ Rn be a bounded domain and let 1 ≤ p < n. Thereexist a constant C > 0 such that
‖u‖Lq(Ω) ≤ C ‖ |∇u| ‖Lp(Ω), u ∈ C10(Ω),
where q := npn−p .
Note: p < npn−p ⇒ ‖u‖p ≤ ‖u‖ np
n−p
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Sobolev’s classical inequality
Theorem (1938)
Let Ω ⊂ Rn be a bounded domain and let 1 ≤ p < n. Thereexist a constant C > 0 such that
‖u‖Lq(Ω) ≤ C ‖ |∇u| ‖Lp(Ω), u ∈ C10(Ω),
where q := npn−p .
Note: p < npn−p ⇒ ‖u‖p ≤ ‖u‖ np
n−p
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Refining Sobolev’s inequality
‖u‖Lq(Ω) ≤ C ‖ |∇u| ‖Lp(Ω), u ∈ C10(Ω)
For a fixed norm in the left hand side, finding a smaller norm inright hand side.
Optimal problem:
For a fixed norm in the left hand side, find the smallestnorm in right hand side.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Refining Sobolev’s inequality
‖u‖Lq(Ω) ≤ C
‖ |∇u| ‖Lp(Ω)
, u ∈ C10(Ω)
For a fixed norm in the left hand side,
finding a smaller norm inright hand side.
Optimal problem:
For a fixed norm in the left hand side, find the smallestnorm in right hand side.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Refining Sobolev’s inequality
‖u‖Lq(Ω) ≤ C ‖ |∇u| ‖Lp(Ω), u ∈ C10(Ω)
For a fixed norm in the left hand side, finding a smaller norm inright hand side.
Optimal problem:
For a fixed norm in the left hand side, find the smallestnorm in right hand side.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Refining Sobolev’s inequality
‖u‖Lq(Ω) ≤ C ‖ |∇u| ‖Lp(Ω), u ∈ C10(Ω)
For a fixed norm in the left hand side, finding a smaller norm inright hand side.
Optimal problem:
For a fixed norm in the left hand side, find the smallestnorm in right hand side.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Refining Sobolev’s inequality
‖u‖Lq(Ω) ≤ C ‖ |∇u| ‖Lp(Ω), u ∈ C10(Ω)
For a fixed norm in the left hand side, finding a smaller norm inright hand side.
Optimal problem:
For a fixed norm in the left hand side, find the smallestnorm in right hand side.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Refining Sobolev’s inequality
‖u‖Lq(Ω) ≤ C ‖ |∇u| ‖Lp(Ω), u ∈ C10(Ω)
For a fixed norm in the left hand side, finding a smaller norm inright hand side.
Optimal problem:
For a fixed norm in the left hand side, find the smallestnorm in right hand side.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
An example
Fix Lp–norm on the LHS (n′ < p < ∞).
Optimality within Lebesgue norms: the Lnp
n+p –norm isoptimal on the RHS, that is,
‖u‖Lp(Ω) ≤ C ‖ |∇u| ‖L
npn+p (Ω)
More general (rearrangement invariant, r.i.) norms: the
Lnp
n+p ,p–norm is optimal on the RHS, that is,
‖u‖p ≤ C ‖ |∇u| ‖ npn+p ,p,
sharper than the classical Sobolev’s inequality, since
‖ |∇u| ‖ npn+p ,p ≤ ‖ |∇u| ‖ np
n+p.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
An example
Fix Lp–norm on the LHS (n′ < p < ∞).
Optimality within Lebesgue norms: the Lnp
n+p –norm isoptimal on the RHS, that is,
‖u‖Lp(Ω) ≤ C ‖ |∇u| ‖L
npn+p (Ω)
More general (rearrangement invariant, r.i.) norms: the
Lnp
n+p ,p–norm is optimal on the RHS, that is,
‖u‖p ≤ C ‖ |∇u| ‖ npn+p ,p,
sharper than the classical Sobolev’s inequality, since
‖ |∇u| ‖ npn+p ,p ≤ ‖ |∇u| ‖ np
n+p.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
An example
Fix Lp–norm on the LHS (n′ < p < ∞).
Optimality within Lebesgue norms: the Lnp
n+p –norm isoptimal on the RHS, that is,
‖u‖Lp(Ω) ≤ C ‖ |∇u| ‖L
npn+p (Ω)
More general (rearrangement invariant, r.i.) norms: the
Lnp
n+p ,p–norm is optimal on the RHS, that is,
‖u‖p ≤ C ‖ |∇u| ‖ npn+p ,p,
sharper than the classical Sobolev’s inequality, since
‖ |∇u| ‖ npn+p ,p ≤ ‖ |∇u| ‖ np
n+p.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
An example
Fix Lp–norm on the LHS (n′ < p < ∞).
Optimality within Lebesgue norms: the Lnp
n+p –norm isoptimal on the RHS, that is,
‖u‖Lp(Ω) ≤ C ‖ |∇u| ‖L
npn+p (Ω)
More general (rearrangement invariant, r.i.) norms: the
Lnp
n+p ,p–norm is optimal on the RHS, that is,
‖u‖p ≤ C ‖ |∇u| ‖ npn+p ,p,
sharper than the classical Sobolev’s inequality, since
‖ |∇u| ‖ npn+p ,p ≤ ‖ |∇u| ‖ np
n+p.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
An example
Fix Lp–norm on the LHS (n′ < p < ∞).
Optimality within Lebesgue norms: the Lnp
n+p –norm isoptimal on the RHS, that is,
‖u‖Lp(Ω) ≤ C ‖ |∇u| ‖L
npn+p (Ω)
More general (rearrangement invariant, r.i.) norms: the
Lnp
n+p ,p–norm is optimal on the RHS, that is,
‖u‖p ≤ C ‖ |∇u| ‖ npn+p ,p,
sharper than the classical Sobolev’s inequality, since
‖ |∇u| ‖ npn+p ,p ≤ ‖ |∇u| ‖ np
n+p.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Reduction to one variable
Theorem (Edmunds, Kerman, Pick, 2000)
Let X , Y be r.i. spaces on [0,1]. TFAE:
‖u‖X(Ω) ≤ C‖ |∇u| ‖Y (Ω), for all u ∈ C10(Ω)
‖Tf‖X ≤ K‖f‖Y , for all f ∈ X
where T is the kernel operator associated with Sobolev’sinequality, defined for f : [0, 1] → R by
t ∈ [0, 1] 7−→ Tf (t) :=
∫ 1
tf (s)s
1n−1 ds
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Reduction to one variable
Theorem (Edmunds, Kerman, Pick, 2000)
Let X , Y be r.i. spaces on [0,1]. TFAE:
‖u‖X(Ω) ≤ C‖ |∇u| ‖Y (Ω), for all u ∈ C10(Ω)
‖Tf‖X ≤ K‖f‖Y , for all f ∈ X
where T is the kernel operator associated with Sobolev’sinequality, defined for f : [0, 1] → R by
t ∈ [0, 1] 7−→ Tf (t) :=
∫ 1
tf (s)s
1n−1 ds
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Reduction to one variable
Theorem (Edmunds, Kerman, Pick, 2000)
Let X , Y be r.i. spaces on [0,1]. TFAE:
‖u‖X(Ω) ≤ C‖ |∇u| ‖Y (Ω), for all u ∈ C10(Ω)
‖Tf‖X ≤ K‖f‖Y , for all f ∈ X
where T is the kernel operator associated with Sobolev’sinequality, defined for f : [0, 1] → R by
t ∈ [0, 1] 7−→ Tf (t) :=
∫ 1
tf (s)s
1n−1 ds
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimality & one-variable optimality
Given a r.i. space X on [0,1] (a r.i. space X (Ω) on Ω).
Which is the largest r.i. function space Y (Ω) such that
‖u‖X(Ω) ≤ C‖ |∇u| ‖Y (Ω)?
i.e. the weakest size condition on |∇u| so that u ∈ X (Ω)?
Which is the largest r.i. function space Y such that
T : Y → X continuously?
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimality & one-variable optimality
Given a r.i. space X on [0,1] (a r.i. space X (Ω) on Ω).
Which is the largest r.i. function space Y (Ω) such that
‖u‖X(Ω) ≤ C‖ |∇u| ‖Y (Ω)?
i.e. the weakest size condition on |∇u| so that u ∈ X (Ω)?
Which is the largest r.i. function space Y such that
T : Y → X continuously?
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimality & one-variable optimality
Given a r.i. space X on [0,1] (a r.i. space X (Ω) on Ω).
Which is the largest r.i. function space Y (Ω) such that
‖u‖X(Ω) ≤ C‖ |∇u| ‖Y (Ω)?
i.e. the weakest size condition on |∇u| so that u ∈ X (Ω)?
Which is the largest r.i. function space Y such that
T : Y → X continuously?
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimality & one-variable optimality
Given a r.i. space X on [0,1] (a r.i. space X (Ω) on Ω).
Which is the largest r.i. function space Y (Ω) such that
‖u‖X(Ω) ≤ C‖ |∇u| ‖Y (Ω)?
i.e. the weakest size condition on |∇u| so that u ∈ X (Ω)?
Which is the largest r.i. function space Y such that
T : Y → X continuously?
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal domains and vector measure
Given a r.i. function space X, which is the largest Banachfunction space Y such that
T : Y −→ X continuously?
The optimal domain [T , X ] =
f : T |f | ∈ X
.
Associated vector measure: νX (A) := T (χA).
L1(νX ) ⊆ [T , X ] ⊆ L1w (νX ).
If X has OC norm then [T , X ] = L1(νX ).
If X has the Fatou prop. then [T , X ] = L1w (νX ).
Examples. OC: Lp,q; Fatou: Lp,∞, Exp Lp.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal domains and vector measure
Given a r.i. function space X, which is the largest Banachfunction space Y such that
T : Y −→ X continuously?
The optimal domain [T , X ] =
f : T |f | ∈ X
.
Associated vector measure: νX (A) := T (χA).
L1(νX ) ⊆ [T , X ] ⊆ L1w (νX ).
If X has OC norm then [T , X ] = L1(νX ).
If X has the Fatou prop. then [T , X ] = L1w (νX ).
Examples. OC: Lp,q; Fatou: Lp,∞, Exp Lp.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal domains and vector measure
Given a r.i. function space X, which is the largest Banachfunction space Y such that
T : Y −→ X continuously?
The optimal domain [T , X ] =
f : T |f | ∈ X
.
Associated vector measure: νX (A) := T (χA).
L1(νX ) ⊆ [T , X ] ⊆ L1w (νX ).
If X has OC norm then [T , X ] = L1(νX ).
If X has the Fatou prop. then [T , X ] = L1w (νX ).
Examples. OC: Lp,q; Fatou: Lp,∞, Exp Lp.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal domains and vector measure
Given a r.i. function space X, which is the largest Banachfunction space Y such that
T : Y −→ X continuously?
The optimal domain [T , X ] =
f : T |f | ∈ X
.
Associated vector measure: νX (A) := T (χA).
L1(νX ) ⊆ [T , X ] ⊆ L1w (νX ).
If X has OC norm then [T , X ] = L1(νX ).
If X has the Fatou prop. then [T , X ] = L1w (νX ).
Examples. OC: Lp,q; Fatou: Lp,∞, Exp Lp.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal domains and vector measure
Given a r.i. function space X, which is the largest Banachfunction space Y such that
T : Y −→ X continuously?
The optimal domain [T , X ] =
f : T |f | ∈ X
.
Associated vector measure: νX (A) := T (χA).
L1(νX ) ⊆ [T , X ] ⊆ L1w (νX ).
If X has OC norm then [T , X ] = L1(νX ).
If X has the Fatou prop. then [T , X ] = L1w (νX ).
Examples. OC: Lp,q; Fatou: Lp,∞, Exp Lp.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal domains and vector measure
Given a r.i. function space X, which is the largest Banachfunction space Y such that
T : Y −→ X continuously?
The optimal domain [T , X ] =
f : T |f | ∈ X
.
Associated vector measure: νX (A) := T (χA).
L1(νX ) ⊆ [T , X ] ⊆ L1w (νX ).
If X has OC norm then [T , X ] = L1(νX ).
If X has the Fatou prop. then [T , X ] = L1w (νX ).
Examples. OC: Lp,q; Fatou: Lp,∞, Exp Lp.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal domains and vector measure
Given a r.i. function space X, which is the largest Banachfunction space Y such that
T : Y −→ X continuously?
The optimal domain [T , X ] =
f : T |f | ∈ X
.
Associated vector measure: νX (A) := T (χA).
L1(νX ) ⊆ [T , X ] ⊆ L1w (νX ).
If X has OC norm then [T , X ] = L1(νX ).
If X has the Fatou prop. then [T , X ] = L1w (νX ).
Examples. OC: Lp,q; Fatou: Lp,∞, Exp Lp.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Optimal domains and vector measure
Given a r.i. function space X, which is the largest Banachfunction space Y such that
T : Y −→ X continuously?
The optimal domain [T , X ] =
f : T |f | ∈ X
.
Associated vector measure: νX (A) := T (χA).
L1(νX ) ⊆ [T , X ] ⊆ L1w (νX ).
If X has OC norm then [T , X ] = L1(νX ).
If X has the Fatou prop. then [T , X ] = L1w (νX ).
Examples. OC: Lp,q; Fatou: Lp,∞, Exp Lp.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Compactness of the Sobolev imbedding
Theorem (Rellich Kondrachov, 1930, 1945)
Let 1 ≤ p < n, W 1,p0 (Ω) −→ Lq(Ω)
is compact for q < npn−p
is NOT compact at q = npn−p
Generalized Sobolev spaces:
For X (Ω) a r.i. space, the Sobolev space W 10 X (Ω) is the
closure of C10(Ω) for the norm:
‖u‖W 10 X(Ω) := ‖u‖X(Ω) + ‖ |∇u| ‖X(Ω).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Compactness of the Sobolev imbedding
Theorem (Rellich Kondrachov, 1930, 1945)
Let 1 ≤ p < n, W 1,p0 (Ω) −→ Lq(Ω)
is compact for q < npn−p
is NOT compact at q = npn−p
Generalized Sobolev spaces:
For X (Ω) a r.i. space, the Sobolev space W 10 X (Ω) is the
closure of C10(Ω) for the norm:
‖u‖W 10 X(Ω) := ‖u‖X(Ω) + ‖ |∇u| ‖X(Ω).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Compactness of the Sobolev imbedding
Theorem (Rellich Kondrachov, 1930, 1945)
Let 1 ≤ p < n, W 1,p0 (Ω) −→ Lq(Ω)
is compact for q < npn−p
is NOT compact at q = npn−p
Generalized Sobolev spaces:
For X (Ω) a r.i. space, the Sobolev space W 10 X (Ω) is the
closure of C10(Ω) for the norm:
‖u‖W 10 X(Ω) := ‖u‖X(Ω) + ‖ |∇u| ‖X(Ω).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Compactness of Sobolev imbedding
AIM:
Study the compactness/loss of noncompactnesphenomena of the optimal Sobolev’s imbedding
W 10 [T , X ](Ω) −→ X (Ω).
IDEA: Use the compactness/noncompactnes of theassociated kernel operator T : [T , X ] → X .
TOOL: The compactness of IνX: L1(νX ) → X is well
understood.
Note: we assume that [T , X ] is r.i., thus we denote [T , X ]ri .
The fundamental function of X , is ϕX (t) := ‖χ[0,t]‖X .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Compactness of Sobolev imbedding
AIM: Study the compactness/loss of noncompactnesphenomena of the optimal Sobolev’s imbedding
W 10 [T , X ](Ω) −→ X (Ω).
IDEA: Use the compactness/noncompactnes of theassociated kernel operator T : [T , X ] → X .
TOOL: The compactness of IνX: L1(νX ) → X is well
understood.
Note: we assume that [T , X ] is r.i., thus we denote [T , X ]ri .
The fundamental function of X , is ϕX (t) := ‖χ[0,t]‖X .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Compactness of Sobolev imbedding
AIM: Study the compactness/loss of noncompactnesphenomena of the optimal Sobolev’s imbedding
W 10 [T , X ](Ω) −→ X (Ω).
IDEA: Use the compactness/noncompactnes of theassociated kernel operator T : [T , X ] → X .
TOOL: The compactness of IνX: L1(νX ) → X is well
understood.
Note: we assume that [T , X ] is r.i., thus we denote [T , X ]ri .
The fundamental function of X , is ϕX (t) := ‖χ[0,t]‖X .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Compactness of Sobolev imbedding
AIM: Study the compactness/loss of noncompactnesphenomena of the optimal Sobolev’s imbedding
W 10 [T , X ](Ω) −→ X (Ω).
IDEA: Use the compactness/noncompactnes of theassociated kernel operator T : [T , X ] → X .
TOOL: The compactness of IνX: L1(νX ) → X is well
understood.
Note: we assume that [T , X ] is r.i., thus we denote [T , X ]ri .
The fundamental function of X , is ϕX (t) := ‖χ[0,t]‖X .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Compactness of Sobolev imbedding
AIM: Study the compactness/loss of noncompactnesphenomena of the optimal Sobolev’s imbedding
W 10 [T , X ](Ω) −→ X (Ω).
IDEA: Use the compactness/noncompactnes of theassociated kernel operator T : [T , X ] → X .
TOOL: The compactness of IνX: L1(νX ) → X is well
understood.
Note: we assume that [T , X ] is r.i., thus we denote [T , X ]ri .
The fundamental function of X , is ϕX (t) := ‖χ[0,t]‖X .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Compactness of Sobolev imbedding
AIM: Study the compactness/loss of noncompactnesphenomena of the optimal Sobolev’s imbedding
W 10 [T , X ](Ω) −→ X (Ω).
IDEA: Use the compactness/noncompactnes of theassociated kernel operator T : [T , X ] → X .
TOOL: The compactness of IνX: L1(νX ) → X is well
understood.
Note: we assume that [T , X ] is r.i., thus we denote [T , X ]ri .
The fundamental function of X , is ϕX (t) := ‖χ[0,t]‖X .
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Generalized Rellich-Kondrachov theorem
Theorem
Let X be a r.i. space and W 10 [T , X ]ri(Ω) −→ X (Ω) be the
optimal Sobolev’s imbedding.
a) NONCOMPACTNESS
Condition: t−1/n′ϕX (t) decreasing.
Lp,q([0, 1]), p ≥ n′, 1 ≤ q ≤ ∞; Exp Lp.spaces “smaller" than Ln′,∞([0, 1]).
b) COMPACTNESS
Condition: t−1/n′ϕX (t) → 0.
Lp,q([0, 1]), p < n′, 1 ≤ q ≤ ∞; Lp logL, p < n′.spaces “larger" than Ln′,∞([0, 1]).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Generalized Rellich-Kondrachov theorem
Theorem
Let X be a r.i. space and W 10 [T , X ]ri(Ω) −→ X (Ω) be the
optimal Sobolev’s imbedding.
a) NONCOMPACTNESS
Condition: t−1/n′ϕX (t) decreasing.
Lp,q([0, 1]), p ≥ n′, 1 ≤ q ≤ ∞; Exp Lp.spaces “smaller" than Ln′,∞([0, 1]).
b) COMPACTNESS
Condition: t−1/n′ϕX (t) → 0.
Lp,q([0, 1]), p < n′, 1 ≤ q ≤ ∞; Lp logL, p < n′.spaces “larger" than Ln′,∞([0, 1]).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Generalized Rellich-Kondrachov theorem
Theorem
Let X be a r.i. space and W 10 [T , X ]ri(Ω) −→ X (Ω) be the
optimal Sobolev’s imbedding.
a) NONCOMPACTNESS
Condition: t−1/n′ϕX (t) decreasing.
Lp,q([0, 1]), p ≥ n′, 1 ≤ q ≤ ∞; Exp Lp.spaces “smaller" than Ln′,∞([0, 1]).
b) COMPACTNESS
Condition: t−1/n′ϕX (t) → 0.
Lp,q([0, 1]), p < n′, 1 ≤ q ≤ ∞; Lp logL, p < n′.spaces “larger" than Ln′,∞([0, 1]).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Outline
1 Extension versus representationRepresentation theoremsExtension theorems
2 Three examplesThe Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Extension of the Hausforff-Young inequality
Theorem (Hausdorff-Young inequality)
For 1 ≤ p ≤ 2 the Fourier transform F maps Lp(T) into `p′(Z),where 1/p′ + 1/p = 1, and
‖f‖p′ ≤ ‖f‖p, f ∈ Lp(T).
QUESTION: Is the Hausdorff-Young inequality optimal?
That is, keeping the range space `p′(Z) fixed, is it possible tocontinuously extend
F : Lp(T) → `p′(Z)
to a Banach function space over T larger than Lp(T)?
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Extension of the Hausforff-Young inequality
Theorem (Hausdorff-Young inequality)
For 1 ≤ p ≤ 2 the Fourier transform F maps Lp(T) into `p′(Z),where 1/p′ + 1/p = 1, and
‖f‖p′ ≤ ‖f‖p, f ∈ Lp(T).
QUESTION: Is the Hausdorff-Young inequality optimal?
That is, keeping the range space `p′(Z) fixed, is it possible tocontinuously extend
F : Lp(T) → `p′(Z)
to a Banach function space over T larger than Lp(T)?
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Extension of the Hausforff-Young inequality
Theorem (Hausdorff-Young inequality)
For 1 ≤ p ≤ 2 the Fourier transform F maps Lp(T) into `p′(Z),where 1/p′ + 1/p = 1, and
‖f‖p′ ≤ ‖f‖p, f ∈ Lp(T).
QUESTION: Is the Hausdorff-Young inequality optimal?
That is, keeping the range space `p′(Z) fixed, is it possible tocontinuously extend
F : Lp(T) → `p′(Z)
to a Banach function space over T larger than Lp(T)?
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Extension of the Hausforff-Young inequality
Theorem (Hausdorff-Young inequality)
For 1 ≤ p ≤ 2 the Fourier transform F maps Lp(T) into `p′(Z),where 1/p′ + 1/p = 1, and
‖f‖p′ ≤ ‖f‖p, f ∈ Lp(T).
QUESTION: Is the Hausdorff-Young inequality optimal?
That is, keeping the range space `p′(Z) fixed, is it possible tocontinuously extend
F : Lp(T) → `p′(Z)
to a Banach function space over T larger than Lp(T)?
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Theorem (Mockenhaupt-Ricker)
For 1 < p < 2 the optimal domain [F , `p′ ] satisfies
(a) F maps [F , `p′ ] into `p′ continuously.
(b) Lp(T) ( [F , `p′ ] ( L1(T).
(c) [F , `p′ ] =
f ∈ L1(T) : ˆfχA ∈ `p′(Z),∀A ∈ B0(T)
.
NOTES:
The optimal domain [F , `p′ ] = L1(νp′), where νp′ is thevector measure associated to the Fourier transform
A ∈ B0(T) 7→ νp′(A) := F (χA) ∈ `p′(Z).
(c) solves a question of R. E. Edwards, 1967.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Theorem (Mockenhaupt-Ricker)
For 1 < p < 2 the optimal domain [F , `p′ ] satisfies
(a) F maps [F , `p′ ] into `p′ continuously.
(b) Lp(T) ( [F , `p′ ] ( L1(T).
(c) [F , `p′ ] =
f ∈ L1(T) : ˆfχA ∈ `p′(Z),∀A ∈ B0(T)
.
NOTES:
The optimal domain [F , `p′ ] = L1(νp′), where νp′ is thevector measure associated to the Fourier transform
A ∈ B0(T) 7→ νp′(A) := F (χA) ∈ `p′(Z).
(c) solves a question of R. E. Edwards, 1967.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Theorem (Mockenhaupt-Ricker)
For 1 < p < 2 the optimal domain [F , `p′ ] satisfies
(a) F maps [F , `p′ ] into `p′ continuously.
(b) Lp(T) ( [F , `p′ ] ( L1(T).
(c) [F , `p′ ] =
f ∈ L1(T) : ˆfχA ∈ `p′(Z),∀A ∈ B0(T)
.
NOTES:
The optimal domain [F , `p′ ] = L1(νp′), where νp′ is thevector measure associated to the Fourier transform
A ∈ B0(T) 7→ νp′(A) := F (χA) ∈ `p′(Z).
(c) solves a question of R. E. Edwards, 1967.
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Some references
G. Curbera, Volterra convolution operators with values in rearrangement invariant spaces,J. London Math. Soc. (1999).
G. Curbera & O. Delgado, Optimal domains for L0–valued operators via stochastic Measures,Positivity (to appear).
G. Curbera & W. Ricker, Optimal domains for kernel operators via interpolation,Math. Nachr. (2002).
G. Curbera & W. Ricker, Optimal domain for the kernel operator associated with Sobolev’s inequality,Studia Math. (2003).
G. Curbera & W. Ricker, Banach lattices with the Fatou property and optimal domains of kernel operators,Indag. Math. (2006).
G. Curbera & W. Ricker, Compactness properties of Sobolev imbeddings for rearrangement invariant norms,Trans. Amer. Math. Soc. (2007).
G. Curbera & W. Ricker, Can optimal rearrangement invariant Sobolev imbeddings be further extended?,Indiana Univ. Math. J. (2007).
O. Delgado, Optimal domains for kernel operators on [0,∞)× [0,∞),Studia Math. (2006).
O. Delgado & J. Soria, Optimal domain for the Hardy operator,J. Funct. Anal. (2007).
G. Mockenhaupt & W. Ricker, Optimal extension of the Hausdorff-Young inequality,Crelle’s J. (to appear).
Guillermo P. Curbera Optimal domains and vector measures
Extension versus representationThree examples
The Volterra operatorSobolev’s inequalityThe Hausdorff-Young inequality
Vector measure meeting
Katholische Universität Eichstätt-Ingolstadt (Germany)
Third Meeting on
Vector Measures, Integration and Applications.
Sept. 24–27, 2008.
Guillermo P. Curbera Optimal domains and vector measures