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DEGENERATE POINCARE INEQUALITIES
Carlos Perez
University of the Basque Country&
BCAM, Basque Center for Applied Math.
Atelier d’Analyse Harmonique 2018CNRS-Paul Langevin Center
Aussois, March 30, 2018
Denegerate Poincare inequalities [email protected]
collaborators
most of the lecture will be on an almost finished work with:
1
Denegerate Poincare inequalities [email protected]
collaborators
most of the lecture will be on an almost finished work with:
Ezequiel Rela
1
Denegerate Poincare inequalities [email protected]
collaborators
most of the lecture will be on an almost finished work with:
Ezequiel Rela
And if I have time few results from a work in progress with
1
Denegerate Poincare inequalities [email protected]
collaborators
most of the lecture will be on an almost finished work with:
Ezequiel Rela
And if I have time few results from a work in progress with
Sheldy Ombrosi, Ezequiel Rela and Israel Rios-Rivera
1
Denegerate Poincare inequalities [email protected]
THE CLASSICAL POINCARE INEQUALITY
The simplest context: Rn with the metric associated to the cubes with thelebesgue measure.
2
Denegerate Poincare inequalities [email protected]
THE CLASSICAL POINCARE INEQUALITY
The simplest context: Rn with the metric associated to the cubes with thelebesgue measure.• (1,1) Poincare inequality:
1
|Q|
∫Q|f − fQ| ≤ cn
`(Q)
|Q|
∫Q|∇f |
fQ = 1|Q|
∫Q f = average of f over the cube Q &
2
Denegerate Poincare inequalities [email protected]
THE CLASSICAL POINCARE INEQUALITY
The simplest context: Rn with the metric associated to the cubes with thelebesgue measure.• (1,1) Poincare inequality:
1
|Q|
∫Q|f − fQ| ≤ cn
`(Q)
|Q|
∫Q|∇f |
fQ = 1|Q|
∫Q f = average of f over the cube Q & `(Q) = sidelength ofQ.
2
Denegerate Poincare inequalities [email protected]
THE CLASSICAL POINCARE INEQUALITY
The simplest context: Rn with the metric associated to the cubes with thelebesgue measure.• (1,1) Poincare inequality:
1
|Q|
∫Q|f − fQ| ≤ cn
`(Q)
|Q|
∫Q|∇f |
fQ = 1|Q|
∫Q f = average of f over the cube Q & `(Q) = sidelength ofQ.
• (p, p), p ≥ 1 Poincare inequality:(1
|Q|
∫Q|f − fQ|p dx
)1/p
≤ c `(Q)
(1
|Q|
∫Q|∇f |pdx
)1/p
2
Denegerate Poincare inequalities [email protected]
THE CLASSICAL POINCARE INEQUALITY
The simplest context: Rn with the metric associated to the cubes with thelebesgue measure.• (1,1) Poincare inequality:
1
|Q|
∫Q|f − fQ| ≤ cn
`(Q)
|Q|
∫Q|∇f |
fQ = 1|Q|
∫Q f = average of f over the cube Q & `(Q) = sidelength ofQ.
• (p, p), p ≥ 1 Poincare inequality:(1
|Q|
∫Q|f − fQ|p dx
)1/p
≤ c `(Q)
(1
|Q|
∫Q|∇f |pdx
)1/p
• Higher order case. Somewhat less-known result.
2
Denegerate Poincare inequalities [email protected]
THE CLASSICAL POINCARE INEQUALITY
The simplest context: Rn with the metric associated to the cubes with thelebesgue measure.• (1,1) Poincare inequality:
1
|Q|
∫Q|f − fQ| ≤ cn
`(Q)
|Q|
∫Q|∇f |
fQ = 1|Q|
∫Q f = average of f over the cube Q & `(Q) = sidelength ofQ.
• (p, p), p ≥ 1 Poincare inequality:(1
|Q|
∫Q|f − fQ|p dx
)1/p
≤ c `(Q)
(1
|Q|
∫Q|∇f |pdx
)1/p
• Higher order case. Somewhat less-known result.
1
|Q|
∫Q|f − P (Q, f)| ≤ cn
`(Q)m
|Q|
∫Q|∇mf |
2
Denegerate Poincare inequalities [email protected]
INTEGRAL REPRESENTATION FORMULA
The proof is based on the following classical formula:
|f(x)− fQ| ≤c
|Q|
∫Q
|∇f(z)||x− z|n−1
dz = c I1(|∇f |χQ
)(x).
3
Denegerate Poincare inequalities [email protected]
INTEGRAL REPRESENTATION FORMULA
The proof is based on the following classical formula:
|f(x)− fQ| ≤c
|Q|
∫Q
|∇f(z)||x− z|n−1
dz = c I1(|∇f |χQ
)(x).
• They are equivalent (Franchi-Lu-Wheeden)
3
Denegerate Poincare inequalities [email protected]
INTEGRAL REPRESENTATION FORMULA
The proof is based on the following classical formula:
|f(x)− fQ| ≤c
|Q|
∫Q
|∇f(z)||x− z|n−1
dz = c I1(|∇f |χQ
)(x).
• They are equivalent (Franchi-Lu-Wheeden)
Here
Iαf(x) = c∫Rn
f(y)
|x− y|n−αdy
0 < α < n.
3
Denegerate Poincare inequalities [email protected]
INTEGRAL REPRESENTATION FORMULA
The proof is based on the following classical formula:
|f(x)− fQ| ≤c
|Q|
∫Q
|∇f(z)||x− z|n−1
dz = c I1(|∇f |χQ
)(x).
• They are equivalent (Franchi-Lu-Wheeden)
Here
Iαf(x) = c∫Rn
f(y)
|x− y|n−αdy
0 < α < n.
These are called fractional integrals of order α or Riesz potentials.
3
Denegerate Poincare inequalities [email protected]
INTEGRAL REPRESENTATION FORMULA
The proof is based on the following classical formula:
|f(x)− fQ| ≤c
|Q|
∫Q
|∇f(z)||x− z|n−1
dz = c I1(|∇f |χQ
)(x).
• They are equivalent (Franchi-Lu-Wheeden)
Here
Iαf(x) = c∫Rn
f(y)
|x− y|n−αdy
0 < α < n.
These are called fractional integrals of order α or Riesz potentials.
The classical well–known estimates use these type of representation.
3
Denegerate Poincare inequalities [email protected]
INTEGRAL REPRESENTATION FORMULA
The proof is based on the following classical formula:
|f(x)− fQ| ≤c
|Q|
∫Q
|∇f(z)||x− z|n−1
dz = c I1(|∇f |χQ
)(x).
• They are equivalent (Franchi-Lu-Wheeden)
Here
Iαf(x) = c∫Rn
f(y)
|x− y|n−αdy
0 < α < n.
These are called fractional integrals of order α or Riesz potentials.
The classical well–known estimates use these type of representation.
It goes back to Sobolev.
3
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n.
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n. Observe that 1∗ = n
n−1 = n′
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n. Observe that 1∗ = n
n−1 = n′
(1
|Q|
∫Q|f − fQ|p
∗)1/p∗
≤ c `(Q)
(1
|Q|
∫Q|∇f |p
)1/p
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n. Observe that 1∗ = n
n−1 = n′
(1
|Q|
∫Q|f − fQ|p
∗)1/p∗
≤ c `(Q)
(1
|Q|
∫Q|∇f |p
)1/p
The SELF-IMPROVING property
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n. Observe that 1∗ = n
n−1 = n′
(1
|Q|
∫Q|f − fQ|p
∗)1/p∗
≤ c `(Q)
(1
|Q|
∫Q|∇f |p
)1/p
The SELF-IMPROVING property
Observations:
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n. Observe that 1∗ = n
n−1 = n′
(1
|Q|
∫Q|f − fQ|p
∗)1/p∗
≤ c `(Q)
(1
|Q|
∫Q|∇f |p
)1/p
The SELF-IMPROVING property
Observations:• These estimates are sharper than the corresponding (p, p) Poincare inequal-ities since p∗ > p by Jensen’s inequality.
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n. Observe that 1∗ = n
n−1 = n′
(1
|Q|
∫Q|f − fQ|p
∗)1/p∗
≤ c `(Q)
(1
|Q|
∫Q|∇f |p
)1/p
The SELF-IMPROVING property
Observations:• These estimates are sharper than the corresponding (p, p) Poincare inequal-ities since p∗ > p by Jensen’s inequality.• p∗ is optimal, that is, we cannot replace p∗ by a larger exponent.
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n. Observe that 1∗ = n
n−1 = n′
(1
|Q|
∫Q|f − fQ|p
∗)1/p∗
≤ c `(Q)
(1
|Q|
∫Q|∇f |p
)1/p
The SELF-IMPROVING property
Observations:• These estimates are sharper than the corresponding (p, p) Poincare inequal-ities since p∗ > p by Jensen’s inequality.• p∗ is optimal, that is, we cannot replace p∗ by a larger exponent.• p∗ is usually called the Sobolev exponent.
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n. Observe that 1∗ = n
n−1 = n′
(1
|Q|
∫Q|f − fQ|p
∗)1/p∗
≤ c `(Q)
(1
|Q|
∫Q|∇f |p
)1/p
The SELF-IMPROVING property
Observations:• These estimates are sharper than the corresponding (p, p) Poincare inequal-ities since p∗ > p by Jensen’s inequality.• p∗ is optimal, that is, we cannot replace p∗ by a larger exponent.• p∗ is usually called the Sobolev exponent.• One of the points of this talk is to show how to avoid such representationformulae.
4
Denegerate Poincare inequalities [email protected]
Sobolev inequalities
Sobolev inequalities can be seen as sharp versions of the (p, p) Poincareinequalities:
Let p∗ = pnn−p when 1 ≤ p < n. Observe that 1∗ = n
n−1 = n′
(1
|Q|
∫Q|f − fQ|p
∗)1/p∗
≤ c `(Q)
(1
|Q|
∫Q|∇f |p
)1/p
The SELF-IMPROVING property
Observations:• These estimates are sharper than the corresponding (p, p) Poincare inequal-ities since p∗ > p by Jensen’s inequality.• p∗ is optimal, that is, we cannot replace p∗ by a larger exponent.• p∗ is usually called the Sobolev exponent.• One of the points of this talk is to show how to avoid such representationformulae.•We will use Calderon-Zygmund theory instead.
4
Denegerate Poincare inequalities [email protected]
Elliptic TheoryThe case p > 1 is important in the theory of elliptic P.D.E. .
5
Denegerate Poincare inequalities [email protected]
Elliptic TheoryThe case p > 1 is important in the theory of elliptic P.D.E. .
•The elliptic Operator:
5
Denegerate Poincare inequalities [email protected]
Elliptic TheoryThe case p > 1 is important in the theory of elliptic P.D.E. .
•The elliptic Operator: Lu = div(A(x).∇u) = 0
5
Denegerate Poincare inequalities [email protected]
Elliptic TheoryThe case p > 1 is important in the theory of elliptic P.D.E. .
•The elliptic Operator: Lu = div(A(x).∇u) = 0
where λ|ξ|2 ≤ A(x)ξ.ξ ≤ Λ|ξ|2
5
Denegerate Poincare inequalities [email protected]
Elliptic TheoryThe case p > 1 is important in the theory of elliptic P.D.E. .
•The elliptic Operator: Lu = div(A(x).∇u) = 0
where λ|ξ|2 ≤ A(x)ξ.ξ ≤ Λ|ξ|2
• Goal: to prove local Holder continuity of the (weak) solutions of the equation.
5
Denegerate Poincare inequalities [email protected]
Elliptic TheoryThe case p > 1 is important in the theory of elliptic P.D.E. .
•The elliptic Operator: Lu = div(A(x).∇u) = 0
where λ|ξ|2 ≤ A(x)ξ.ξ ≤ Λ|ξ|2
• Goal: to prove local Holder continuity of the (weak) solutions of the equation.
• Classical theory: De Giorgi, Nash (local Holder continuity theory of solutions)
5
Denegerate Poincare inequalities [email protected]
Elliptic TheoryThe case p > 1 is important in the theory of elliptic P.D.E. .
•The elliptic Operator: Lu = div(A(x).∇u) = 0
where λ|ξ|2 ≤ A(x)ξ.ξ ≤ Λ|ξ|2
• Goal: to prove local Holder continuity of the (weak) solutions of the equation.
• Classical theory: De Giorgi, Nash (local Holder continuity theory of solutions)
• Moser (Harnack inequality from which Holder continuity of solutions can bederived). This became the standard machinery for these questions.
5
Denegerate Poincare inequalities [email protected]
Elliptic TheoryThe case p > 1 is important in the theory of elliptic P.D.E. .
•The elliptic Operator: Lu = div(A(x).∇u) = 0
where λ|ξ|2 ≤ A(x)ξ.ξ ≤ Λ|ξ|2
• Goal: to prove local Holder continuity of the (weak) solutions of the equation.
• Classical theory: De Giorgi, Nash (local Holder continuity theory of solutions)
• Moser (Harnack inequality from which Holder continuity of solutions can bederived). This became the standard machinery for these questions.
Key point besides the (2,2) PI,
5
Denegerate Poincare inequalities [email protected]
Elliptic TheoryThe case p > 1 is important in the theory of elliptic P.D.E. .
•The elliptic Operator: Lu = div(A(x).∇u) = 0
where λ|ξ|2 ≤ A(x)ξ.ξ ≤ Λ|ξ|2
• Goal: to prove local Holder continuity of the (weak) solutions of the equation.
• Classical theory: De Giorgi, Nash (local Holder continuity theory of solutions)
• Moser (Harnack inequality from which Holder continuity of solutions can bederived). This became the standard machinery for these questions.
Key point besides the (2,2) PI, the PS inequality:
(1
|Q|
∫Q|f − fQ|2
∗dx
)1/2∗
≤ c `(Q)
(1
|Q|
∫Q|∇f |2dx
)1/2
5
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
Late 60’s and early 70’s:
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
Late 60’s and early 70’s: Kruzkov, Murthy, Stampacchia, Trudinger.
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
Late 60’s and early 70’s: Kruzkov, Murthy, Stampacchia, Trudinger.
supQ
(1
|Q|
∫Qws dx
)1/s ( 1
|Q|
∫Qw−t dx
)1/t
<∞,1
s+
1
t<
2
n
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
Late 60’s and early 70’s: Kruzkov, Murthy, Stampacchia, Trudinger.
supQ
(1
|Q|
∫Qws dx
)1/s ( 1
|Q|
∫Qw−t dx
)1/t
<∞,1
s+
1
t<
2
n
• The relevant work is due Fabes-Kenig-Serapioni (1982),
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
Late 60’s and early 70’s: Kruzkov, Murthy, Stampacchia, Trudinger.
supQ
(1
|Q|
∫Qws dx
)1/s ( 1
|Q|
∫Qw−t dx
)1/t
<∞,1
s+
1
t<
2
n
• The relevant work is due Fabes-Kenig-Serapioni (1982),they removed the restriction in s, t,
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
Late 60’s and early 70’s: Kruzkov, Murthy, Stampacchia, Trudinger.
supQ
(1
|Q|
∫Qws dx
)1/s ( 1
|Q|
∫Qw−t dx
)1/t
<∞,1
s+
1
t<
2
n
• The relevant work is due Fabes-Kenig-Serapioni (1982),they removed the restriction in s, t,and consider the A2 condition instead:
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
Late 60’s and early 70’s: Kruzkov, Murthy, Stampacchia, Trudinger.
supQ
(1
|Q|
∫Qws dx
)1/s ( 1
|Q|
∫Qw−t dx
)1/t
<∞,1
s+
1
t<
2
n
• The relevant work is due Fabes-Kenig-Serapioni (1982),they removed the restriction in s, t,and consider the A2 condition instead:
[w]A2
= supQ
(1
|Q|
∫Qw dx
) (1
|Q|
∫Qw−1 dx
)
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
Late 60’s and early 70’s: Kruzkov, Murthy, Stampacchia, Trudinger.
supQ
(1
|Q|
∫Qws dx
)1/s ( 1
|Q|
∫Qw−t dx
)1/t
<∞,1
s+
1
t<
2
n
• The relevant work is due Fabes-Kenig-Serapioni (1982),they removed the restriction in s, t,and consider the A2 condition instead:
[w]A2
= supQ
(1
|Q|
∫Qw dx
) (1
|Q|
∫Qw−1 dx
)
• method of proof is based on the Moser iteration technique
6
Denegerate Poincare inequalities [email protected]
“Degenerate” elliptic equations
• “Degenerate” elliptic equations
λ |ξ|2w(x) ≤ A(x)ξ.ξ ≤ Λ |ξ|2w(x)
where w is a weight with some sort of singularity.
Late 60’s and early 70’s: Kruzkov, Murthy, Stampacchia, Trudinger.
supQ
(1
|Q|
∫Qws dx
)1/s ( 1
|Q|
∫Qw−t dx
)1/t
<∞,1
s+
1
t<
2
n
• The relevant work is due Fabes-Kenig-Serapioni (1982),they removed the restriction in s, t,and consider the A2 condition instead:
[w]A2
= supQ
(1
|Q|
∫Qw dx
) (1
|Q|
∫Qw−1 dx
)
• method of proof is based on the Moser iteration technique
6
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
(1
w(Q)
∫Q|f − fQ|2w
)12≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12,
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
(1
w(Q)
∫Q|f − fQ|2w
)12≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12,
and there is gain for some δ:
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
(1
w(Q)
∫Q|f − fQ|2w
)12≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12,
and there is gain for some δ:
(1
w(Q)
∫Q|f − fQ|2+δw
) 12+δ
≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
(1
w(Q)
∫Q|f − fQ|2w
)12≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12,
and there is gain for some δ:
(1
w(Q)
∫Q|f − fQ|2+δw
) 12+δ
≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12
• Due to Fabes-Kenig-Serapioni.
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
(1
w(Q)
∫Q|f − fQ|2w
)12≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12,
and there is gain for some δ:
(1
w(Q)
∫Q|f − fQ|2+δw
) 12+δ
≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12
• Due to Fabes-Kenig-Serapioni.• Method of proof is by fractional integrals.
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
(1
w(Q)
∫Q|f − fQ|2w
)12≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12,
and there is gain for some δ:
(1
w(Q)
∫Q|f − fQ|2+δw
) 12+δ
≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12
• Due to Fabes-Kenig-Serapioni.• Method of proof is by fractional integrals.Since there have been a lot of variants of these results:
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
(1
w(Q)
∫Q|f − fQ|2w
)12≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12,
and there is gain for some δ:
(1
w(Q)
∫Q|f − fQ|2+δw
) 12+δ
≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12
• Due to Fabes-Kenig-Serapioni.• Method of proof is by fractional integrals.Since there have been a lot of variants of these results:• Chanillo-Wheeden, Franchi-Lu-Wheeden, Chua-Wheeden
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
(1
w(Q)
∫Q|f − fQ|2w
)12≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12,
and there is gain for some δ:
(1
w(Q)
∫Q|f − fQ|2+δw
) 12+δ
≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12
• Due to Fabes-Kenig-Serapioni.• Method of proof is by fractional integrals.Since there have been a lot of variants of these results:• Chanillo-Wheeden, Franchi-Lu-Wheeden, Chua-WheedenWe will see again the gain again but it is not that precise anymore.
7
Denegerate Poincare inequalities [email protected]
Weighted Poincare and Poincare-Sobolev inequalties
If w ∈ A2
(1
w(Q)
∫Q|f − fQ|2w
)12≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12,
and there is gain for some δ:
(1
w(Q)
∫Q|f − fQ|2+δw
) 12+δ
≤ C(w)`(Q)
(1
w(Q)
∫Q|∇f |2w
)12
• Due to Fabes-Kenig-Serapioni.• Method of proof is by fractional integrals.Since there have been a lot of variants of these results:• Chanillo-Wheeden, Franchi-Lu-Wheeden, Chua-WheedenWe will see again the gain again but it is not that precise anymore.
• First example of this property is due to L. Saloff-Coste.
7
Denegerate Poincare inequalities [email protected]
GENERAL SITUATION
STARTING POINT:
1
|Q|
∫Q|f(y)− fQ| dy ≤ a(Q)
where a is a “functional”
8
Denegerate Poincare inequalities [email protected]
GENERAL SITUATION
STARTING POINT:
1
|Q|
∫Q|f(y)− fQ| dy ≤ a(Q)
where a is a “functional” a : Q → (0,∞)
8
Denegerate Poincare inequalities [email protected]
GENERAL SITUATION
STARTING POINT:
1
|Q|
∫Q|f(y)− fQ| dy ≤ a(Q)
where a is a “functional” a : Q → (0,∞)
where Q denotes the family of all cubes from Rn.
8
Denegerate Poincare inequalities [email protected]
GENERAL SITUATION
STARTING POINT:
1
|Q|
∫Q|f(y)− fQ| dy ≤ a(Q)
where a is a “functional” a : Q → (0,∞)
where Q denotes the family of all cubes from Rn.
Question: What kind of condition can we impose on a to get the self-improvingproperty?
8
Denegerate Poincare inequalities [email protected]
GENERAL SITUATION
STARTING POINT:
1
|Q|
∫Q|f(y)− fQ| dy ≤ a(Q)
where a is a “functional” a : Q → (0,∞)
where Q denotes the family of all cubes from Rn.
Question: What kind of condition can we impose on a to get the self-improvingproperty?
• There is the Lp self-improving (model example: Sobolev inequalities)
8
Denegerate Poincare inequalities [email protected]
GENERAL SITUATION
STARTING POINT:
1
|Q|
∫Q|f(y)− fQ| dy ≤ a(Q)
where a is a “functional” a : Q → (0,∞)
where Q denotes the family of all cubes from Rn.
Question: What kind of condition can we impose on a to get the self-improvingproperty?
• There is the Lp self-improving (model example: Sobolev inequalities)
• There is also exponential self-improving:
8
Denegerate Poincare inequalities [email protected]
GENERAL SITUATION
STARTING POINT:
1
|Q|
∫Q|f(y)− fQ| dy ≤ a(Q)
where a is a “functional” a : Q → (0,∞)
where Q denotes the family of all cubes from Rn.
Question: What kind of condition can we impose on a to get the self-improvingproperty?
• There is the Lp self-improving (model example: Sobolev inequalities)
• There is also exponential self-improving:
a lo John-Nirenberg or
8
Denegerate Poincare inequalities [email protected]
GENERAL SITUATION
STARTING POINT:
1
|Q|
∫Q|f(y)− fQ| dy ≤ a(Q)
where a is a “functional” a : Q → (0,∞)
where Q denotes the family of all cubes from Rn.
Question: What kind of condition can we impose on a to get the self-improvingproperty?
• There is the Lp self-improving (model example: Sobolev inequalities)
• There is also exponential self-improving:
a lo John-Nirenberg or
of Trudinger type
8
Denegerate Poincare inequalities [email protected]
model example
Our model example is associated to the fractional average
9
Denegerate Poincare inequalities [email protected]
model example
Our model example is associated to the fractional average
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
9
Denegerate Poincare inequalities [email protected]
model example
Our model example is associated to the fractional average
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
• They enjoy a Lp self-improving
9
Denegerate Poincare inequalities [email protected]
model example
Our model example is associated to the fractional average
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
• They enjoy a Lp self-improving• Motivated by the theory developed in the papers by Hajlasz, Heinonen andKoskela.
9
Denegerate Poincare inequalities [email protected]
model example
Our model example is associated to the fractional average
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
• They enjoy a Lp self-improving• Motivated by the theory developed in the papers by Hajlasz, Heinonen andKoskela.
• Other type of examples are given by
9
Denegerate Poincare inequalities [email protected]
model example
Our model example is associated to the fractional average
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
• They enjoy a Lp self-improving• Motivated by the theory developed in the papers by Hajlasz, Heinonen andKoskela.
• Other type of examples are given by
a(Q) = ν(Q)1p
9
Denegerate Poincare inequalities [email protected]
model example
Our model example is associated to the fractional average
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
• They enjoy a Lp self-improving• Motivated by the theory developed in the papers by Hajlasz, Heinonen andKoskela.
• Other type of examples are given by
a(Q) = ν(Q)1p
• related to the exponential self-improving property.
9
Denegerate Poincare inequalities [email protected]
the Dr conditionGOAL: find some conditions on a such if f satisfies
10
Denegerate Poincare inequalities [email protected]
the Dr conditionGOAL: find some conditions on a such if f satisfies
1
|Q|
∫Q|f − fQ| ≤ a(Q) Q ⊂ Rn
10
Denegerate Poincare inequalities [email protected]
the Dr conditionGOAL: find some conditions on a such if f satisfies
1
|Q|
∫Q|f − fQ| ≤ a(Q) Q ⊂ Rn
implies a Lr self–improving property of the form for some r > 1,(1
|Q|
∫Q|f(y)− fQ|r dy
)1/r
≤ c a(Q)
10
Denegerate Poincare inequalities [email protected]
the Dr conditionGOAL: find some conditions on a such if f satisfies
1
|Q|
∫Q|f − fQ| ≤ a(Q) Q ⊂ Rn
implies a Lr self–improving property of the form for some r > 1,(1
|Q|
∫Q|f(y)− fQ|r dy
)1/r
≤ c a(Q)
We impose a geometrical type condition which will be key in what follows:
10
Denegerate Poincare inequalities [email protected]
the Dr conditionGOAL: find some conditions on a such if f satisfies
1
|Q|
∫Q|f − fQ| ≤ a(Q) Q ⊂ Rn
implies a Lr self–improving property of the form for some r > 1,(1
|Q|
∫Q|f(y)− fQ|r dy
)1/r
≤ c a(Q)
We impose a geometrical type condition which will be key in what follows:
Let 0 < r <∞. We say that the functional a satisfies the Dr condition if thereexists a finite constant c such that for each cube Q and any family {Qi} ofpairwise disjoint dyadic subcubes of Q,∑
i
a(Qi)r|Qi| ≤ cr a(Q)r|Q|
10
Denegerate Poincare inequalities [email protected]
the Dr conditionGOAL: find some conditions on a such if f satisfies
1
|Q|
∫Q|f − fQ| ≤ a(Q) Q ⊂ Rn
implies a Lr self–improving property of the form for some r > 1,(1
|Q|
∫Q|f(y)− fQ|r dy
)1/r
≤ c a(Q)
We impose a geometrical type condition which will be key in what follows:
Let 0 < r <∞. We say that the functional a satisfies the Dr condition if thereexists a finite constant c such that for each cube Q and any family {Qi} ofpairwise disjoint dyadic subcubes of Q,∑
i
a(Qi)r|Qi| ≤ cr a(Q)r|Q|
• Resembles a little bit the Carleson condition.
10
Denegerate Poincare inequalities [email protected]
the Dr conditionGOAL: find some conditions on a such if f satisfies
1
|Q|
∫Q|f − fQ| ≤ a(Q) Q ⊂ Rn
implies a Lr self–improving property of the form for some r > 1,(1
|Q|
∫Q|f(y)− fQ|r dy
)1/r
≤ c a(Q)
We impose a geometrical type condition which will be key in what follows:
Let 0 < r <∞. We say that the functional a satisfies the Dr condition if thereexists a finite constant c such that for each cube Q and any family {Qi} ofpairwise disjoint dyadic subcubes of Q,∑
i
a(Qi)r|Qi| ≤ cr a(Q)r|Q|
• Resembles a little bit the Carleson condition.• r < s =⇒ Ds ⊂ Dr.
10
Denegerate Poincare inequalities [email protected]
the Dr conditionGOAL: find some conditions on a such if f satisfies
1
|Q|
∫Q|f − fQ| ≤ a(Q) Q ⊂ Rn
implies a Lr self–improving property of the form for some r > 1,(1
|Q|
∫Q|f(y)− fQ|r dy
)1/r
≤ c a(Q)
We impose a geometrical type condition which will be key in what follows:
Let 0 < r <∞. We say that the functional a satisfies the Dr condition if thereexists a finite constant c such that for each cube Q and any family {Qi} ofpairwise disjoint dyadic subcubes of Q,∑
i
a(Qi)r|Qi| ≤ cr a(Q)r|Q|
• Resembles a little bit the Carleson condition.• r < s =⇒ Ds ⊂ Dr.• Then we can define for a given a the optimal exponent
ra = sup{r : a ∈ Dr}.
10
Denegerate Poincare inequalities [email protected]
EXAMPLESRecall the fractional functional given by
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
,
11
Denegerate Poincare inequalities [email protected]
EXAMPLESRecall the fractional functional given by
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
,
Observe that with r = npn−αp we have
a(Qi)r|Qi| = ν(Qi)
r/p
11
Denegerate Poincare inequalities [email protected]
EXAMPLESRecall the fractional functional given by
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
,
Observe that with r = npn−αp we have
a(Qi)r|Qi| = ν(Qi)
r/p
and then if {Qi} is a family of disjoint dyadic subcubes of Q
11
Denegerate Poincare inequalities [email protected]
EXAMPLESRecall the fractional functional given by
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
,
Observe that with r = npn−αp we have
a(Qi)r|Qi| = ν(Qi)
r/p
and then if {Qi} is a family of disjoint dyadic subcubes of Q
∑i
a(Qi)r|Qi| =
∑i
ν(Qi)r/p ≤
∑i
ν(Qi)
r/p
11
Denegerate Poincare inequalities [email protected]
EXAMPLESRecall the fractional functional given by
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
,
Observe that with r = npn−αp we have
a(Qi)r|Qi| = ν(Qi)
r/p
and then if {Qi} is a family of disjoint dyadic subcubes of Q
∑i
a(Qi)r|Qi| =
∑i
ν(Qi)r/p ≤
∑i
ν(Qi)
r/p
≤ ν(Q)r/p = a(Q)r|Q|
which means that a ∈ Dr.
11
Denegerate Poincare inequalities [email protected]
EXAMPLESRecall the fractional functional given by
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
,
Observe that with r = npn−αp we have
a(Qi)r|Qi| = ν(Qi)
r/p
and then if {Qi} is a family of disjoint dyadic subcubes of Q
∑i
a(Qi)r|Qi| =
∑i
ν(Qi)r/p ≤
∑i
ν(Qi)
r/p
≤ ν(Q)r/p = a(Q)r|Q|
which means that a ∈ Dr.Some observations:• If α = 1, r = p∗, the Sobolev exponent.
11
Denegerate Poincare inequalities [email protected]
EXAMPLESRecall the fractional functional given by
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
,
Observe that with r = npn−αp we have
a(Qi)r|Qi| = ν(Qi)
r/p
and then if {Qi} is a family of disjoint dyadic subcubes of Q
∑i
a(Qi)r|Qi| =
∑i
ν(Qi)r/p ≤
∑i
ν(Qi)
r/p
≤ ν(Q)r/p = a(Q)r|Q|
which means that a ∈ Dr.Some observations:• If α = 1, r = p∗, the Sobolev exponent.
• If α = m = 1,2 · · · , r = mpn−pm, the Sobolev exponent related to higher
order PI.
11
Denegerate Poincare inequalities [email protected]
EXAMPLESRecall the fractional functional given by
a(Q) = `(Q)α(ν(Q)
|Q|
)1/p
,
Observe that with r = npn−αp we have
a(Qi)r|Qi| = ν(Qi)
r/p
and then if {Qi} is a family of disjoint dyadic subcubes of Q
∑i
a(Qi)r|Qi| =
∑i
ν(Qi)r/p ≤
∑i
ν(Qi)
r/p
≤ ν(Q)r/p = a(Q)r|Q|
which means that a ∈ Dr.Some observations:• If α = 1, r = p∗, the Sobolev exponent.
• If α = m = 1,2 · · · , r = mpn−pm, the Sobolev exponent related to higher
order PI.• If ν ≡ 1, a satisfies the Dr condition for every r > 1.
11
Denegerate Poincare inequalities [email protected]
Variants of the Dr condition
The (weighted) Dr(w) condition for some 0 < r < ∞: for each cube Q andfor any family {Qi} of pairwise disjoint cubes contained in Q,∑
i
a(Qi)rw(Qi) ≤ cr a(Q)rw(Q)
12
Denegerate Poincare inequalities [email protected]
Variants of the Dr condition
The (weighted) Dr(w) condition for some 0 < r < ∞: for each cube Q andfor any family {Qi} of pairwise disjoint cubes contained in Q,∑
i
a(Qi)rw(Qi) ≤ cr a(Q)rw(Q)
Theorem (Franchi, P, Wheeden, 1998)Let a ∈ Dr(w) for some r > 0 and let w ∈ A∞. Let f such that
1
|Q|
∫Q|f − fQ| ≤ a(Q),
12
Denegerate Poincare inequalities [email protected]
Variants of the Dr condition
The (weighted) Dr(w) condition for some 0 < r < ∞: for each cube Q andfor any family {Qi} of pairwise disjoint cubes contained in Q,∑
i
a(Qi)rw(Qi) ≤ cr a(Q)rw(Q)
Theorem (Franchi, P, Wheeden, 1998)Let a ∈ Dr(w) for some r > 0 and let w ∈ A∞. Let f such that
1
|Q|
∫Q|f − fQ| ≤ a(Q),
Then there exists a constant c such that
‖f − fQ‖Lr,∞(Q,w) ≤ c a(Q)
12
Denegerate Poincare inequalities [email protected]
Variants of the Dr condition
The (weighted) Dr(w) condition for some 0 < r < ∞: for each cube Q andfor any family {Qi} of pairwise disjoint cubes contained in Q,∑
i
a(Qi)rw(Qi) ≤ cr a(Q)rw(Q)
Theorem (Franchi, P, Wheeden, 1998)Let a ∈ Dr(w) for some r > 0 and let w ∈ A∞. Let f such that
1
|Q|
∫Q|f − fQ| ≤ a(Q),
Then there exists a constant c such that
‖f − fQ‖Lr,∞(Q,w) ≤ c a(Q)
• The proof combines Calderon–Zygmund theory with an appropriate variantof the good–λ inequality of Burkholder–Gundy.
12
Denegerate Poincare inequalities [email protected]
Variants of the Dr condition
The (weighted) Dr(w) condition for some 0 < r < ∞: for each cube Q andfor any family {Qi} of pairwise disjoint cubes contained in Q,∑
i
a(Qi)rw(Qi) ≤ cr a(Q)rw(Q)
Theorem (Franchi, P, Wheeden, 1998)Let a ∈ Dr(w) for some r > 0 and let w ∈ A∞. Let f such that
1
|Q|
∫Q|f − fQ| ≤ a(Q),
Then there exists a constant c such that
‖f − fQ‖Lr,∞(Q,w) ≤ c a(Q)
• The proof combines Calderon–Zygmund theory with an appropriate variantof the good–λ inequality of Burkholder–Gundy.• Can be extended to the context of a space of homogeneous type.
12
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
13
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
We start by using the L1 unweighted Poincare inequality
1
|Q||f(x)− fQ|dx ≤ cn`(Q)−
∫Q|∇f(x)|dx.
13
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
We start by using the L1 unweighted Poincare inequality
1
|Q||f(x)− fQ|dx ≤ cn`(Q)−
∫Q|∇f(x)|dx.
By the Ap condition, we obtain that
1
|Q||f(x)− fQ|dx ≤ cn [w]
1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
.
13
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
We start by using the L1 unweighted Poincare inequality
1
|Q||f(x)− fQ|dx ≤ cn`(Q)−
∫Q|∇f(x)|dx.
By the Ap condition, we obtain that
1
|Q||f(x)− fQ|dx ≤ cn [w]
1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
.
Hence appears naturally the weighted fractional integral
af(Q) := cn [w]1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
13
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
We start by using the L1 unweighted Poincare inequality
1
|Q||f(x)− fQ|dx ≤ cn`(Q)−
∫Q|∇f(x)|dx.
By the Ap condition, we obtain that
1
|Q||f(x)− fQ|dx ≤ cn [w]
1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
.
Hence appears naturally the weighted fractional integral
af(Q) := cn [w]1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
which satisfies trivially
13
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
We start by using the L1 unweighted Poincare inequality
1
|Q||f(x)− fQ|dx ≤ cn`(Q)−
∫Q|∇f(x)|dx.
By the Ap condition, we obtain that
1
|Q||f(x)− fQ|dx ≤ cn [w]
1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
.
Hence appears naturally the weighted fractional integral
af(Q) := cn [w]1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
which satisfies trivially a ∈ Dp(w)
13
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
We start by using the L1 unweighted Poincare inequality
1
|Q||f(x)− fQ|dx ≤ cn`(Q)−
∫Q|∇f(x)|dx.
By the Ap condition, we obtain that
1
|Q||f(x)− fQ|dx ≤ cn [w]
1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
.
Hence appears naturally the weighted fractional integral
af(Q) := cn [w]1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
which satisfies trivially a ∈ Dp(w)
and hence(1
w(Q)
∫Q|f(x)− fQ|pwdx
)1/p
≤ c `(Q)
(1
w(Q)
∫Q|∇f(x)|pwdx
)1/p
13
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
14
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
It is more interesting to get for some p∗ > p(1
w(Q)
∫Q|f(y)− fQ|p
∗w(y)dy
) 1p∗≤ c `(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1p
14
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
It is more interesting to get for some p∗ > p(1
w(Q)
∫Q|f(y)− fQ|p
∗w(y)dy
) 1p∗≤ c `(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1p
Hence the question reduces to understand the Dp∗(w) for a larger exponent.
Lemma Let w ∈ Ap, a ∈ Dp(n′+δ)(w) where δ depends on w.
14
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
It is more interesting to get for some p∗ > p(1
w(Q)
∫Q|f(y)− fQ|p
∗w(y)dy
) 1p∗≤ c `(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1p
Hence the question reduces to understand the Dp∗(w) for a larger exponent.
Lemma Let w ∈ Ap, a ∈ Dp(n′+δ)(w) where δ depends on w.
and hence
14
Denegerate Poincare inequalities [email protected]
How to recover the weighted Poincare-Sobolev inequality
It is more interesting to get for some p∗ > p(1
w(Q)
∫Q|f(y)− fQ|p
∗w(y)dy
) 1p∗≤ c `(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1p
Hence the question reduces to understand the Dp∗(w) for a larger exponent.
Lemma Let w ∈ Ap, a ∈ Dp(n′+δ)(w) where δ depends on w.
and hence(1
w(Q)
∫Q|f(y)− fQ|p(n′+δ)wdx
) 1p(n′+δ)
≤ c `(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1p
14
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators,
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
X21 +X2
2 =∂2
∂x2+ x2 ∂
2
∂y2.
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
X21 +X2
2 =∂2
∂x2+ x2 ∂
2
∂y2.
Nagel-Stein-Wainger proved that there is a metric dX
, the Carnot-Caratheodorymetric,
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
X21 +X2
2 =∂2
∂x2+ x2 ∂
2
∂y2.
Nagel-Stein-Wainger proved that there is a metric dX
, the Carnot-Caratheodorymetric, which is doubling with respect to the Lebesgue measure.
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
X21 +X2
2 =∂2
∂x2+ x2 ∂
2
∂y2.
Nagel-Stein-Wainger proved that there is a metric dX
, the Carnot-Caratheodorymetric, which is doubling with respect to the Lebesgue measure.Hence
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
X21 +X2
2 =∂2
∂x2+ x2 ∂
2
∂y2.
Nagel-Stein-Wainger proved that there is a metric dX
, the Carnot-Caratheodorymetric, which is doubling with respect to the Lebesgue measure.Hence
(Rn, dX, dx)
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
X21 +X2
2 =∂2
∂x2+ x2 ∂
2
∂y2.
Nagel-Stein-Wainger proved that there is a metric dX
, the Carnot-Caratheodorymetric, which is doubling with respect to the Lebesgue measure.Hence
(Rn, dX, dx)
becomes a space of homogeneous type.
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
X21 +X2
2 =∂2
∂x2+ x2 ∂
2
∂y2.
Nagel-Stein-Wainger proved that there is a metric dX
, the Carnot-Caratheodorymetric, which is doubling with respect to the Lebesgue measure.Hence
(Rn, dX, dx)
becomes a space of homogeneous type.•. There is a key PI:
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
X21 +X2
2 =∂2
∂x2+ x2 ∂
2
∂y2.
Nagel-Stein-Wainger proved that there is a metric dX
, the Carnot-Caratheodorymetric, which is doubling with respect to the Lebesgue measure.Hence
(Rn, dX, dx)
becomes a space of homogeneous type.•. There is a key PI:
1
|B|
∫B|f − fB| ≤ c
rB|B|
∫B|Xf |
15
Denegerate Poincare inequalities [email protected]
Other Interesting Poincare Inequalities
Subelliptic operators, from the theory of several complex variables.In R2:
(X1, X2) = (∂
∂x, x
∂
∂y)
associated to the Grushin operator:
X21 +X2
2 =∂2
∂x2+ x2 ∂
2
∂y2.
Nagel-Stein-Wainger proved that there is a metric dX
, the Carnot-Caratheodorymetric, which is doubling with respect to the Lebesgue measure.Hence
(Rn, dX, dx)
becomes a space of homogeneous type.•. There is a key PI:
1
|B|
∫B|f − fB| ≤ c
rB|B|
∫B|Xf |
• Jerison (1986)
15
Denegerate Poincare inequalities [email protected]
A NON–SMOOTH EXAMPLE
Consider the vector field
16
Denegerate Poincare inequalities [email protected]
A NON–SMOOTH EXAMPLE
Consider the vector field
Xα = (X1, X2) = (∂
∂x, |x|α
∂
∂y)
16
Denegerate Poincare inequalities [email protected]
A NON–SMOOTH EXAMPLE
Consider the vector field
Xα = (X1, X2) = (∂
∂x, |x|α
∂
∂y)
associated to the operator:
16
Denegerate Poincare inequalities [email protected]
A NON–SMOOTH EXAMPLE
Consider the vector field
Xα = (X1, X2) = (∂
∂x, |x|α
∂
∂y)
associated to the operator:
∂2
∂x2+ |x|2α
∂2
∂y2.
16
Denegerate Poincare inequalities [email protected]
A NON–SMOOTH EXAMPLE
Consider the vector field
Xα = (X1, X2) = (∂
∂x, |x|α
∂
∂y)
associated to the operator:
∂2
∂x2+ |x|2α
∂2
∂y2.
These type of non–smooth examples were considered by Franchi-Lanconelliin the mid 80’s.
16
Denegerate Poincare inequalities [email protected]
A NON–SMOOTH EXAMPLE
Consider the vector field
Xα = (X1, X2) = (∂
∂x, |x|α
∂
∂y)
associated to the operator:
∂2
∂x2+ |x|2α
∂2
∂y2.
These type of non–smooth examples were considered by Franchi-Lanconelliin the mid 80’s.As above there is a corresponding Carnot-Caratheodory metric d
Xsuch
(Rn, dX, dx)
becomes space of homogeneous type.
16
Denegerate Poincare inequalities [email protected]
A NON–SMOOTH EXAMPLE
Consider the vector field
Xα = (X1, X2) = (∂
∂x, |x|α
∂
∂y)
associated to the operator:
∂2
∂x2+ |x|2α
∂2
∂y2.
These type of non–smooth examples were considered by Franchi-Lanconelliin the mid 80’s.As above there is a corresponding Carnot-Caratheodory metric d
Xsuch
(Rn, dX, dx)
becomes space of homogeneous type.• There is another key PI:
16
Denegerate Poincare inequalities [email protected]
A NON–SMOOTH EXAMPLE
Consider the vector field
Xα = (X1, X2) = (∂
∂x, |x|α
∂
∂y)
associated to the operator:
∂2
∂x2+ |x|2α
∂2
∂y2.
These type of non–smooth examples were considered by Franchi-Lanconelliin the mid 80’s.As above there is a corresponding Carnot-Caratheodory metric d
Xsuch
(Rn, dX, dx)
becomes space of homogeneous type.• There is another key PI:
1
|B|
∫B|f − fB| ≤ c
rB|B|
∫B|Xαf |
16
Denegerate Poincare inequalities [email protected]
A NON–SMOOTH EXAMPLE
Consider the vector field
Xα = (X1, X2) = (∂
∂x, |x|α
∂
∂y)
associated to the operator:
∂2
∂x2+ |x|2α
∂2
∂y2.
These type of non–smooth examples were considered by Franchi-Lanconelliin the mid 80’s.As above there is a corresponding Carnot-Caratheodory metric d
Xsuch
(Rn, dX, dx)
becomes space of homogeneous type.• There is another key PI:
1
|B|
∫B|f − fB| ≤ c
rB|B|
∫B|Xαf |
• Franchi–Gutierrez–Wheeden (1994).
16
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then• the smooth case:
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then• the smooth case:(
1
|B|
∫B|f − fB|p
∗) 1p∗≤ c rB
(1
|B|
∫B|Xf |p
)1/p
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then• the smooth case:(
1
|B|
∫B|f − fB|p
∗) 1p∗≤ c rB
(1
|B|
∫B|Xf |p
)1/p
• Lu p > 1
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then• the smooth case:(
1
|B|
∫B|f − fB|p
∗) 1p∗≤ c rB
(1
|B|
∫B|Xf |p
)1/p
• Lu p > 1
• Franchi–Lu–Wheeden p = 1.
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then• the smooth case:(
1
|B|
∫B|f − fB|p
∗) 1p∗≤ c rB
(1
|B|
∫B|Xf |p
)1/p
• Lu p > 1
• Franchi–Lu–Wheeden p = 1.
• the non–smooth case:
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then• the smooth case:(
1
|B|
∫B|f − fB|p
∗) 1p∗≤ c rB
(1
|B|
∫B|Xf |p
)1/p
• Lu p > 1
• Franchi–Lu–Wheeden p = 1.
• the non–smooth case:
(1
|B|
∫B|f − fB|p
∗)1/p∗
≤ c rB
(1
|B|
∫B|Xαf |p
)1/p
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then• the smooth case:(
1
|B|
∫B|f − fB|p
∗) 1p∗≤ c rB
(1
|B|
∫B|Xf |p
)1/p
• Lu p > 1
• Franchi–Lu–Wheeden p = 1.
• the non–smooth case:
(1
|B|
∫B|f − fB|p
∗)1/p∗
≤ c rB
(1
|B|
∫B|Xαf |p
)1/p
Franchi–Gutierrez–Wheeden, p ≥ 1
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then• the smooth case:(
1
|B|
∫B|f − fB|p
∗) 1p∗≤ c rB
(1
|B|
∫B|Xf |p
)1/p
• Lu p > 1
• Franchi–Lu–Wheeden p = 1.
• the non–smooth case:
(1
|B|
∫B|f − fB|p
∗)1/p∗
≤ c rB
(1
|B|
∫B|Xαf |p
)1/p
Franchi–Gutierrez–Wheeden, p ≥ 1
• Each case has its own proof all of them based on a representation formula.
17
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev
Let p∗ = pDD−p where D = doubling order
Then• the smooth case:(
1
|B|
∫B|f − fB|p
∗) 1p∗≤ c rB
(1
|B|
∫B|Xf |p
)1/p
• Lu p > 1
• Franchi–Lu–Wheeden p = 1.
• the non–smooth case:
(1
|B|
∫B|f − fB|p
∗)1/p∗
≤ c rB
(1
|B|
∫B|Xαf |p
)1/p
Franchi–Gutierrez–Wheeden, p ≥ 1
• Each case has its own proof all of them based on a representation formula.•We avoid all these.
17
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
18
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
Dr(w) condition: for some 0 < r < ∞ namely for each ball B and for anyfamily {Bi} of pairwise disjoint balls contained in B,
18
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
Dr(w) condition: for some 0 < r < ∞ namely for each ball B and for anyfamily {Bi} of pairwise disjoint balls contained in B,∑
i
a(Bi)rwµ(Bi) ≤ cr a(B)rwµ(B)
18
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
Dr(w) condition: for some 0 < r < ∞ namely for each ball B and for anyfamily {Bi} of pairwise disjoint balls contained in B,∑
i
a(Bi)rwµ(Bi) ≤ cr a(B)rwµ(B)
Theorem (MacManus, P.) Let w ∈ A∞(µ) and suppose that
1
µ(B)
∫B|f − fB| dµ ≤ a(B)
18
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
Dr(w) condition: for some 0 < r < ∞ namely for each ball B and for anyfamily {Bi} of pairwise disjoint balls contained in B,∑
i
a(Bi)rwµ(Bi) ≤ cr a(B)rwµ(B)
Theorem (MacManus, P.) Let w ∈ A∞(µ) and suppose that
1
µ(B)
∫B|f − fB| dµ ≤ a(B)
Then, if δ > 0, there is a constant C independent of f and B such that
‖f − fB‖Lr,∞(B,w) ≤ C a((1 + δ)B)
18
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
Dr(w) condition: for some 0 < r < ∞ namely for each ball B and for anyfamily {Bi} of pairwise disjoint balls contained in B,∑
i
a(Bi)rwµ(Bi) ≤ cr a(B)rwµ(B)
Theorem (MacManus, P.) Let w ∈ A∞(µ) and suppose that
1
µ(B)
∫B|f − fB| dµ ≤ a(B)
Then, if δ > 0, there is a constant C independent of f and B such that
‖f − fB‖Lr,∞(B,w) ≤ C a((1 + δ)B)
• It is not so clean because of the factor (1 + δ). (lack of dyadic structure)
18
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
Dr(w) condition: for some 0 < r < ∞ namely for each ball B and for anyfamily {Bi} of pairwise disjoint balls contained in B,∑
i
a(Bi)rwµ(Bi) ≤ cr a(B)rwµ(B)
Theorem (MacManus, P.) Let w ∈ A∞(µ) and suppose that
1
µ(B)
∫B|f − fB| dµ ≤ a(B)
Then, if δ > 0, there is a constant C independent of f and B such that
‖f − fB‖Lr,∞(B,w) ≤ C a((1 + δ)B)
• It is not so clean because of the factor (1 + δ). (lack of dyadic structure)• Other situations: non homogeneous spaces
18
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
Dr(w) condition: for some 0 < r < ∞ namely for each ball B and for anyfamily {Bi} of pairwise disjoint balls contained in B,∑
i
a(Bi)rwµ(Bi) ≤ cr a(B)rwµ(B)
Theorem (MacManus, P.) Let w ∈ A∞(µ) and suppose that
1
µ(B)
∫B|f − fB| dµ ≤ a(B)
Then, if δ > 0, there is a constant C independent of f and B such that
‖f − fB‖Lr,∞(B,w) ≤ C a((1 + δ)B)
• It is not so clean because of the factor (1 + δ). (lack of dyadic structure)• Other situations: non homogeneous spaces(joint work with J. Orobitg)
18
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
Dr(w) condition: for some 0 < r < ∞ namely for each ball B and for anyfamily {Bi} of pairwise disjoint balls contained in B,∑
i
a(Bi)rwµ(Bi) ≤ cr a(B)rwµ(B)
Theorem (MacManus, P.) Let w ∈ A∞(µ) and suppose that
1
µ(B)
∫B|f − fB| dµ ≤ a(B)
Then, if δ > 0, there is a constant C independent of f and B such that
‖f − fB‖Lr,∞(B,w) ≤ C a((1 + δ)B)
• It is not so clean because of the factor (1 + δ). (lack of dyadic structure)• Other situations: non homogeneous spaces(joint work with J. Orobitg)•Weaker hypothesis: replace L1 norm by much weaker norms.
18
Denegerate Poincare inequalities [email protected]
THE SHARPEST RESULT IN THE GENERAL CONTEXT
Let (X, d, µ) be a metric space with a doubling measure µ
Dr(w) condition: for some 0 < r < ∞ namely for each ball B and for anyfamily {Bi} of pairwise disjoint balls contained in B,∑
i
a(Bi)rwµ(Bi) ≤ cr a(B)rwµ(B)
Theorem (MacManus, P.) Let w ∈ A∞(µ) and suppose that
1
µ(B)
∫B|f − fB| dµ ≤ a(B)
Then, if δ > 0, there is a constant C independent of f and B such that
‖f − fB‖Lr,∞(B,w) ≤ C a((1 + δ)B)
• It is not so clean because of the factor (1 + δ). (lack of dyadic structure)• Other situations: non homogeneous spaces(joint work with J. Orobitg)•Weaker hypothesis: replace L1 norm by much weaker norms.(joint work with A. Lerner)
18
Denegerate Poincare inequalities [email protected]
The small Dp condition
Definition 1 Let L > 1 and let Q be a cube. We will say that a family ofpairwise disjoint subcubes {Qi} of Q is L-small if
∑i
|Qi| ≤|Q|L
We will say {Qi} ∈ S(L)
19
Denegerate Poincare inequalities [email protected]
The small Dp condition
Definition 1 Let L > 1 and let Q be a cube. We will say that a family ofpairwise disjoint subcubes {Qi} of Q is L-small if
∑i
|Qi| ≤|Q|L
We will say {Qi} ∈ S(L)
Now, the correct notion of Dp condition in this context is the following.
19
Denegerate Poincare inequalities [email protected]
The small Dp condition
Definition 1 Let L > 1 and let Q be a cube. We will say that a family ofpairwise disjoint subcubes {Qi} of Q is L-small if
∑i
|Qi| ≤|Q|L
We will say {Qi} ∈ S(L)
Now, the correct notion of Dp condition in this context is the following.
Definition 1Let w be any weight and let s > 1. We say that the functional a satisfies theweighted SDs
p(w) condition for 0 < p < ∞ if there is a constant c such thatfor any cube Q and any family {Qi} of pairwise disjoint subcubes of Q suchthat {Qi} ∈ S(L), the following inequality holds:
∑i
a(Qi)pw(Qi) ≤ cp
(1
L
)psa(Q)pw(Q)
19
Denegerate Poincare inequalities [email protected]
main example
Let µ be any Radon measure and define
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
.
20
Denegerate Poincare inequalities [email protected]
main example
Let µ be any Radon measure and define
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
.
Let w be a weight, L > 1,1 ≤ p < n and let a ∈ SDnp (w).
20
Denegerate Poincare inequalities [email protected]
main example
Let µ be any Radon measure and define
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
.
Let w be a weight, L > 1,1 ≤ p < n and let a ∈ SDnp (w).
The proof is an easy consequence of Holder’s inequality. Let {Qi} ∈ S(L),then
20
Denegerate Poincare inequalities [email protected]
main example
Let µ be any Radon measure and define
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
.
Let w be a weight, L > 1,1 ≤ p < n and let a ∈ SDnp (w).
The proof is an easy consequence of Holder’s inequality. Let {Qi} ∈ S(L),then ∑
i
a(Qi)pw(Qi) =
∑i
`(Qi)pµ(Qi) =
∑i
|Qi|p/nµ(Qi)
20
Denegerate Poincare inequalities [email protected]
main example
Let µ be any Radon measure and define
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
.
Let w be a weight, L > 1,1 ≤ p < n and let a ∈ SDnp (w).
The proof is an easy consequence of Holder’s inequality. Let {Qi} ∈ S(L),then ∑
i
a(Qi)pw(Qi) =
∑i
`(Qi)pµ(Qi) =
∑i
|Qi|p/nµ(Qi)
≤
∑i
|Qi|
p/n∑i
µ(Qi)(n/p)′
1(n/p)′
20
Denegerate Poincare inequalities [email protected]
main example
Let µ be any Radon measure and define
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
.
Let w be a weight, L > 1,1 ≤ p < n and let a ∈ SDnp (w).
The proof is an easy consequence of Holder’s inequality. Let {Qi} ∈ S(L),then ∑
i
a(Qi)pw(Qi) =
∑i
`(Qi)pµ(Qi) =
∑i
|Qi|p/nµ(Qi)
≤
∑i
|Qi|
p/n∑i
µ(Qi)(n/p)′
1(n/p)′
≤(|Q|L
)p/nµ(Q) =
(1
L
)p/na(Q)pw(Q)
20
Denegerate Poincare inequalities [email protected]
A first result
Theorem. Let w be any weight. Consider a functional a satisfiyngSDs
p(w) with s > 1 and p ≥ 1. Suppose that
1
|Q|
∫Q|f − fQ| ≤ a(Q) (H)
for every cube Q. Then, there exists a dimensional constant cn such that forany cube Q
21
Denegerate Poincare inequalities [email protected]
A first result
Theorem. Let w be any weight. Consider a functional a satisfiyngSDs
p(w) with s > 1 and p ≥ 1. Suppose that
1
|Q|
∫Q|f − fQ| ≤ a(Q) (H)
for every cube Q. Then, there exists a dimensional constant cn such that forany cube QThen (
1
w(Q)
∫Q|f − fQ|pwdx
)1p
≤ s cn a(Q)
21
Denegerate Poincare inequalities [email protected]
A first result
Theorem. Let w be any weight. Consider a functional a satisfiyngSDs
p(w) with s > 1 and p ≥ 1. Suppose that
1
|Q|
∫Q|f − fQ| ≤ a(Q) (H)
for every cube Q. Then, there exists a dimensional constant cn such that forany cube QThen (
1
w(Q)
∫Q|f − fQ|pwdx
)1p
≤ s cn a(Q)
CorollaryLet (u, v) ∈ Ap. The the following Poincare (p, p) inequality holds(
1
u(Q)
∫Q|f − fQ|p u dx
)1/p
≤ cn[u, v]1pAp`(Q)
(1
u(Q)
∫Q|∇f |p v dx
)1/p
,
where cn is a dimensional constant.
21
Denegerate Poincare inequalities [email protected]
Two more corollaries
Corollary (A generalized John-Nirenberg)Let a be an increasing functional and suppose that f satisfies (H).
22
Denegerate Poincare inequalities [email protected]
Two more corollaries
Corollary (A generalized John-Nirenberg)Let a be an increasing functional and suppose that f satisfies (H).Then, ∥∥∥f − fQ∥∥∥expL(Q,w)
≤ cn[w]A∞ a(Q)
22
Denegerate Poincare inequalities [email protected]
Two more corollaries
Corollary (A generalized John-Nirenberg)Let a be an increasing functional and suppose that f satisfies (H).Then, ∥∥∥f − fQ∥∥∥expL(Q,w)
≤ cn[w]A∞ a(Q)
Corollary (The Keith-Zhong phenomenon)Let 1 < p0 and let (f, g) be a couple of functions satisfying
1
|Q|
∫Q|f − fQ| dx ≤ C[w]Ap0
`(Q)
(1
w(Q)
∫Qgp0 wdx
) 1p0
w ∈ Ap0
22
Denegerate Poincare inequalities [email protected]
Two more corollaries
Corollary (A generalized John-Nirenberg)Let a be an increasing functional and suppose that f satisfies (H).Then, ∥∥∥f − fQ∥∥∥expL(Q,w)
≤ cn[w]A∞ a(Q)
Corollary (The Keith-Zhong phenomenon)Let 1 < p0 and let (f, g) be a couple of functions satisfying
1
|Q|
∫Q|f − fQ| dx ≤ C[w]Ap0
`(Q)
(1
w(Q)
∫Qgp0 wdx
) 1p0
w ∈ Ap0
Then, for any 1 ≤ p < p0, the following estimate holds for any w ∈ Ap(1
w(Q)
∫Q|f − fQ|wdx
)1/p
≤ cC[w]Ap`(Q)
(1
w(Q)
∫Qgpwdx
)1/p
22
Denegerate Poincare inequalities [email protected]
Improving Poincare-Sobolev
As before let
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
23
Denegerate Poincare inequalities [email protected]
Improving Poincare-Sobolev
As before let
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
Lemma Let 1 ≤ q ≤ p < n, and let w ∈ Aq. If E > 1 we let p∗ be
1
p−
1
p∗=
1
nqE.
23
Denegerate Poincare inequalities [email protected]
Improving Poincare-Sobolev
As before let
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
Lemma Let 1 ≤ q ≤ p < n, and let w ∈ Aq. If E > 1 we let p∗ be
1
p−
1
p∗=
1
nqE.
Then, if {Qi} ∈ S(L), L > 1, the following inequality holds:
∑i
a(Qi)p∗w(Qi) ≤ [w]
p∗nqEAq
(1
L
) p∗nE′
a(Q)p∗w(Q)
23
Denegerate Poincare inequalities [email protected]
Improving Poincare-Sobolev
As before let
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
Lemma Let 1 ≤ q ≤ p < n, and let w ∈ Aq. If E > 1 we let p∗ be
1
p−
1
p∗=
1
nqE.
Then, if {Qi} ∈ S(L), L > 1, the following inequality holds:
∑i
a(Qi)p∗w(Qi) ≤ [w]
p∗nqEAq
(1
L
) p∗nE′
a(Q)p∗w(Q)
• The functional a “preserves smallness” with index nE′ and constant [w]1nqEAq
23
Denegerate Poincare inequalities [email protected]
Improving Poincare-Sobolev
As before let
a(Q) = `(Q)
(1
w(Q)µ(Q)
)1/p
Lemma Let 1 ≤ q ≤ p < n, and let w ∈ Aq. If E > 1 we let p∗ be
1
p−
1
p∗=
1
nqE.
Then, if {Qi} ∈ S(L), L > 1, the following inequality holds:
∑i
a(Qi)p∗w(Qi) ≤ [w]
p∗nqEAq
(1
L
) p∗nE′
a(Q)p∗w(Q)
• The functional a “preserves smallness” with index nE′ and constant [w]1nqEAq
• E can be seen as “error” it is the made.
23
Denegerate Poincare inequalities [email protected]
Consequence
Corollary Let 1 ≤ q ≤ p < n, and let w ∈ Aq. Let p∗ be defined by
1
p−
1
p∗=
1
n(q + log[w]Aq)
and suppose that f satisfies (H). Then
24
Denegerate Poincare inequalities [email protected]
Consequence
Corollary Let 1 ≤ q ≤ p < n, and let w ∈ Aq. Let p∗ be defined by
1
p−
1
p∗=
1
n(q + log[w]Aq)
and suppose that f satisfies (H). Then
(1
w(Q)
∫Q|f − fQ|p
∗wdx
) 1p∗≤ cn a(Q)
24
Denegerate Poincare inequalities [email protected]
Consequence
Corollary Let 1 ≤ q ≤ p < n, and let w ∈ Aq. Let p∗ be defined by
1
p−
1
p∗=
1
n(q + log[w]Aq)
and suppose that f satisfies (H). Then
(1
w(Q)
∫Q|f − fQ|p
∗wdx
) 1p∗≤ cn a(Q)
In particular(1
w(Q)
∫Q|f − fQ|p
∗wdx
) 1p∗≤ cn [w]
1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
24
Denegerate Poincare inequalities [email protected]
Consequence
Corollary Let 1 ≤ q ≤ p < n, and let w ∈ Aq. Let p∗ be defined by
1
p−
1
p∗=
1
n(q + log[w]Aq)
and suppose that f satisfies (H). Then
(1
w(Q)
∫Q|f − fQ|p
∗wdx
) 1p∗≤ cn a(Q)
In particular(1
w(Q)
∫Q|f − fQ|p
∗wdx
) 1p∗≤ cn [w]
1pAp`(Q)
(1
w(Q)
∫Q|∇f(x)|pw dx
)1/p
• Again, this is a very “clean” inequality.
24
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev via Good-λ
TheoremLet 1 ≤ q ≤ p < n an let w ∈ Aq. Let p∗ be defined by
1
p−
1
p∗=
1
nq
and suppose that f satisfies (H). Then
25
Denegerate Poincare inequalities [email protected]
Poincare-Sobolev via Good-λ
TheoremLet 1 ≤ q ≤ p < n an let w ∈ Aq. Let p∗ be defined by
1
p−
1
p∗=
1
nq
and suppose that f satisfies (H). Then
(1
w(Q)
∫Q|f − fQ|p
∗wdx
) 1p∗≤ c[w]
1nqAq
[w]2pAp`(Q)
(1
w(Q)
∫Q|∇f |pwdx
)1p
,
25
Denegerate Poincare inequalities [email protected]
Bloom BMO and Muckenhoupt-Wheeden
Let f be a locally integrable function and let w be a weight such that
‖f‖BMOw
= supQ
1
w(Q)
∫Q|f − fQ| <∞,
26
Denegerate Poincare inequalities [email protected]
Bloom BMO and Muckenhoupt-Wheeden
Let f be a locally integrable function and let w be a weight such that
‖f‖BMOw
= supQ
1
w(Q)
∫Q|f − fQ| <∞,
Theorema) A1 case: If w ∈ A1, there exists a constant c such that for any cube Qand any q > 1(
1
w(Q)
∫Q
(|f − fQ|
w
)qwdx
)1q
≤ c q [w]A1‖f‖BMOw
and hence for any cube Q∥∥∥∥f − fQw
∥∥∥∥expL(Q,w)
≤ c [w]A1‖f‖
BMOw
26
Denegerate Poincare inequalities [email protected]
Bloom BMO and Muckenhoupt-Wheeden
Let f be a locally integrable function and let w be a weight such that
‖f‖BMOw
= supQ
1
w(Q)
∫Q|f − fQ| <∞,
Theorema) A1 case: If w ∈ A1, there exists a constant c such that for any cube Qand any q > 1(
1
w(Q)
∫Q
(|f − fQ|
w
)qwdx
)1q
≤ c q [w]A1‖f‖BMOw
and hence for any cube Q∥∥∥∥f − fQw
∥∥∥∥expL(Q,w)
≤ c [w]A1‖f‖
BMOw
b) Ap case: If w ∈ Ap, 1 < p < ∞, there exists a constant c such that forany cube Q 1
w(Q)
∫Q
(|f − fQ|
w
)p′wdx
1p′
≤ c2np p′ [w]Ap‖f‖BMOw
26
Denegerate Poincare inequalities [email protected]
Using the sparse method
Theorema) A1 case: If w ∈ A1, there exists a constant c such that for any cube Q andq > 1(
1
w(Q)
∫Q
∣∣∣∣∣f(x)− fQw
∣∣∣∣∣q
w(x)dx
)1q
≤ cn‖f‖BMOw
qq′[w]1q′A1
[w]1qA∞
27
Denegerate Poincare inequalities [email protected]
Using the sparse method
Theorema) A1 case: If w ∈ A1, there exists a constant c such that for any cube Q andq > 1(
1
w(Q)
∫Q
∣∣∣∣∣f(x)− fQw
∣∣∣∣∣q
w(x)dx
)1q
≤ cn‖f‖BMOw
qq′[w]1q′A1
[w]1qA∞
b) Ap case: then, if 1 ≤ q ≤ p′
(1
w(Q)
∫Q
∣∣∣∣∣f(x)− fQw
∣∣∣∣∣q
w(x)dx
)1q
≤ cnp‖f‖BMOw
[w]1p′A∞[w]
1pAp
27
Denegerate Poincare inequalities [email protected]
Using the sparse method
Theorema) A1 case: If w ∈ A1, there exists a constant c such that for any cube Q andq > 1(
1
w(Q)
∫Q
∣∣∣∣∣f(x)− fQw
∣∣∣∣∣q
w(x)dx
)1q
≤ cn‖f‖BMOw
qq′[w]1q′A1
[w]1qA∞
b) Ap case: then, if 1 ≤ q ≤ p′
(1
w(Q)
∫Q
∣∣∣∣∣f(x)− fQw
∣∣∣∣∣q
w(x)dx
)1q
≤ cnp‖f‖BMOw
[w]1p′A∞[w]
1pAp
27
Denegerate Poincare inequalities [email protected]
Using the sparse method
Theorema) A1 case: If w ∈ A1, there exists a constant c such that for any cube Q andq > 1(
1
w(Q)
∫Q
∣∣∣∣∣f(x)− fQw
∣∣∣∣∣q
w(x)dx
)1q
≤ cn‖f‖BMOw
qq′[w]1q′A1
[w]1qA∞
b) Ap case: then, if 1 ≤ q ≤ p′
(1
w(Q)
∫Q
∣∣∣∣∣f(x)− fQw
∣∣∣∣∣q
w(x)dx
)1q
≤ cnp‖f‖BMOw
[w]1p′A∞[w]
1pAp
27
Denegerate Poincare inequalities [email protected]
Using the sparse method
Theorema) A1 case: If w ∈ A1, there exists a constant c such that for any cube Q andq > 1(
1
w(Q)
∫Q
∣∣∣∣∣f(x)− fQw
∣∣∣∣∣q
w(x)dx
)1q
≤ cn‖f‖BMOw
qq′[w]1q′A1
[w]1qA∞
b) Ap case: then, if 1 ≤ q ≤ p′
(1
w(Q)
∫Q
∣∣∣∣∣f(x)− fQw
∣∣∣∣∣q
w(x)dx
)1q
≤ cnp‖f‖BMOw
[w]1p′A∞[w]
1pAp
27