PaataIvanisvili KentStateUniversity 2017 …pivanisv/bellman.pdfPaata Ivanisvili Kent State...
Transcript of PaataIvanisvili KentStateUniversity 2017 …pivanisv/bellman.pdfPaata Ivanisvili Kent State...
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Sharp estimates in harmonic analysis
Paata Ivanisvili
Kent State University
2017
University of California, Irvine
Paata Ivanisvili Kent State University Sharp estimates
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Bellman function for extremal problems in BMO.(joint with N. Osipov, D. Stolyarov, P. Zatitskiy, V. Vasyunin)
Paata Ivanisvili Kent State University Sharp estimates
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Surface of minimal area k1 + k2 = 0.
Paata Ivanisvili Kent State University Sharp estimates
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Minimal concave function k1k2 = 0
Paata Ivanisvili Kent State University Sharp estimates
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Find the minimal concave function
B(x , y) is minimal concave function on Ω with B|∂Ω = f .
ProblemGiven a wire find B .
Paata Ivanisvili Kent State University Sharp estimates
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Projection
Paata Ivanisvili Kent State University Sharp estimates
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Foliation
This we call foliation.
Problem (equivalent)
Given a wire find foliation.
Paata Ivanisvili Kent State University Sharp estimates
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Foliation
This we call foliation.
Problem (equivalent)
Given a wire find foliation.
Paata Ivanisvili Kent State University Sharp estimates
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Concave envelope
Paata Ivanisvili Kent State University Sharp estimates
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Foliation
Paata Ivanisvili Kent State University Sharp estimates
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Angle and Cup
Demonstrate
Paata Ivanisvili Kent State University Sharp estimates
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Angle and Cup
DemonstratePaata Ivanisvili Kent State University Sharp estimates
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Boundary value problem: homogeneous Monge–Ampère
B(x , y) is minimal concave function on Ω with B|∂Ω = f .
What if B(x , y) is minimal concave function on Ω \ D?
Paata Ivanisvili Kent State University Sharp estimates
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Boundary value problem: homogeneous Monge–Ampère
B(x , y) is minimal concave function on Ω with B|∂Ω = f .What if B(x , y) is minimal concave function on Ω \ D?
Paata Ivanisvili Kent State University Sharp estimates
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Erasing a hole
Will not be minimal!
Paata Ivanisvili Kent State University Sharp estimates
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Erasing a hole
Will not be minimal!
Paata Ivanisvili Kent State University Sharp estimates
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Erasing a hole: Ω \ D
HessB ≤ 0 on Ω \ DB|∂Ω = f .B is minimal among all such solutions.det(Hess B) = 0 on Ω \ D;
Paata Ivanisvili Kent State University Sharp estimates
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Erasing a hole: Ω \ D
HessB ≤ 0 on Ω \ D
B|∂Ω = f .B is minimal among all such solutions.det(Hess B) = 0 on Ω \ D;
Paata Ivanisvili Kent State University Sharp estimates
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Erasing a hole: Ω \ D
HessB ≤ 0 on Ω \ DB|∂Ω = f .
B is minimal among all such solutions.det(Hess B) = 0 on Ω \ D;
Paata Ivanisvili Kent State University Sharp estimates
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Erasing a hole: Ω \ D
HessB ≤ 0 on Ω \ DB|∂Ω = f .B is minimal among all such solutions.
det(Hess B) = 0 on Ω \ D;
Paata Ivanisvili Kent State University Sharp estimates
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Erasing a hole: Ω \ D
HessB ≤ 0 on Ω \ DB|∂Ω = f .B is minimal among all such solutions.det(Hess B) = 0 on Ω \ D;
Paata Ivanisvili Kent State University Sharp estimates
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Evolution of the surface
Paata Ivanisvili Kent State University Sharp estimates
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Annulus and parabolic strips
Ω \D
Paata Ivanisvili Kent State University Sharp estimates
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Left tangent lines 2D
x1
x2
x2 = x21
x2 = x21 + ε
2
Paata Ivanisvili Kent State University Sharp estimates
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Left tangent lines 3D
Paata Ivanisvili Kent State University Sharp estimates
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Right tangent lines 2D
x1
x2
x2 = x21
x2 = x21 + ε
2
Paata Ivanisvili Kent State University Sharp estimates
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Right tangent lines 3D
Paata Ivanisvili Kent State University Sharp estimates
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Cup
x1
x2
x2 = x21
x2 = x21 + ε
2
A0
B0
U1 U2
Ωcup
ΩRΩL
Paata Ivanisvili Kent State University Sharp estimates
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Cup
Paata Ivanisvili Kent State University Sharp estimates
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Angle
x1
x2
V
x2 = x21
x2 = x21 + ε
2
Paata Ivanisvili Kent State University Sharp estimates
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Angle
Paata Ivanisvili Kent State University Sharp estimates
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Angle and cup
x1
x2
U−
U+
A
B
V
Paata Ivanisvili Kent State University Sharp estimates
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Angle and cup
Paata Ivanisvili Kent State University Sharp estimates
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Angle and cup: evolution
Paata Ivanisvili Kent State University Sharp estimates
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Two cups
Paata Ivanisvili Kent State University Sharp estimates
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Two cups
Paata Ivanisvili Kent State University Sharp estimates
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Two cups: evolution
Paata Ivanisvili Kent State University Sharp estimates
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John–Nirenberg inequality and BMO space
supI⊆[0,1]
1|I |
∣∣∣∣{t ∈ I : ∣∣∣∣ϕ(t)− 1|I |∫Iϕ
∣∣∣∣ ≥ λ}∣∣∣∣ ≤ c1e−c2λ/‖ϕ‖BMO2([0,1])
where
‖ϕ‖2BMO2([0,1])def= sup
I⊆[0,1]
1|I |
∫I
∣∣∣∣ϕ− 1|I |∫Iϕ
∣∣∣∣2 .QuestionFind sharp constants c1, c2 > 0.
Paata Ivanisvili Kent State University Sharp estimates
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John–Nirenberg inequality and BMO space
supI⊆[0,1]
1|I |
∣∣∣∣{t ∈ I : ∣∣∣∣ϕ(t)− 1|I |∫Iϕ
∣∣∣∣ ≥ λ}∣∣∣∣ ≤ c1e−c2λ/‖ϕ‖BMO2([0,1])where
‖ϕ‖2BMO2([0,1])def= sup
I⊆[0,1]
1|I |
∫I
∣∣∣∣ϕ− 1|I |∫Iϕ
∣∣∣∣2 .
QuestionFind sharp constants c1, c2 > 0.
Paata Ivanisvili Kent State University Sharp estimates
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John–Nirenberg inequality and BMO space
supI⊆[0,1]
1|I |
∣∣∣∣{t ∈ I : ∣∣∣∣ϕ(t)− 1|I |∫Iϕ
∣∣∣∣ ≥ λ}∣∣∣∣ ≤ c1e−c2λ/‖ϕ‖BMO2([0,1])where
‖ϕ‖2BMO2([0,1])def= sup
I⊆[0,1]
1|I |
∫I
∣∣∣∣ϕ− 1|I |∫Iϕ
∣∣∣∣2 .QuestionFind sharp constants c1, c2 > 0.
Paata Ivanisvili Kent State University Sharp estimates
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Extremal problems on BMO: I & II
Bε(x1, x2)def= sup
ϕ : ‖ϕ‖BMO≤ε
{∫ 10
f (ϕ(t))dt :
∫ 10ϕ = x1,
∫ 10ϕ2 = x2,
}
Example: f (t) = 1(−∞,−λ)∪(λ,∞)(t) - John–Nirenberg inequality.
Main results (IOSVZ):1. Bε is minimal concave function on Ω0 \ Ωε with B|∂Ω0 = f
where Ωε = {(x1, x2) ⊂ R2 : x2 ≥ x21 + ε2}2. For each ε we find Bε (stationary algorithm).3. We described evolution of Bε, ε ≥ 0 (dynamical approach).
The torsion of (t, t2, f (t)) changes sign finitely many times.
Paata Ivanisvili Kent State University Sharp estimates
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Extremal problems on BMO: I & II
Bε(x1, x2)def= sup
ϕ : ‖ϕ‖BMO≤ε
{∫ 10
f (ϕ(t))dt :
∫ 10ϕ = x1,
∫ 10ϕ2 = x2,
}
Example: f (t) = 1(−∞,−λ)∪(λ,∞)(t) - John–Nirenberg inequality.
Main results (IOSVZ):1. Bε is minimal concave function on Ω0 \ Ωε with B|∂Ω0 = f
where Ωε = {(x1, x2) ⊂ R2 : x2 ≥ x21 + ε2}2. For each ε we find Bε (stationary algorithm).3. We described evolution of Bε, ε ≥ 0 (dynamical approach).
The torsion of (t, t2, f (t)) changes sign finitely many times.
Paata Ivanisvili Kent State University Sharp estimates
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Extremal problems on BMO: I & II
Bε(x1, x2)def= sup
ϕ : ‖ϕ‖BMO≤ε
{∫ 10
f (ϕ(t))dt :
∫ 10ϕ = x1,
∫ 10ϕ2 = x2,
}
Example: f (t) = 1(−∞,−λ)∪(λ,∞)(t) - John–Nirenberg inequality.
Main results (IOSVZ):1. Bε is minimal concave function on Ω0 \ Ωε with B|∂Ω0 = f
where Ωε = {(x1, x2) ⊂ R2 : x2 ≥ x21 + ε2}2. For each ε we find Bε (stationary algorithm).3. We described evolution of Bε, ε ≥ 0 (dynamical approach).
The torsion of (t, t2, f (t)) changes sign finitely many times.
Paata Ivanisvili Kent State University Sharp estimates
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Extremal problems on BMO: I & II
Bε(x1, x2)def= sup
ϕ : ‖ϕ‖BMO≤ε
{∫ 10
f (ϕ(t))dt :
∫ 10ϕ = x1,
∫ 10ϕ2 = x2,
}
Example: f (t) = 1(−∞,−λ)∪(λ,∞)(t) - John–Nirenberg inequality.
Main results (IOSVZ):1. Bε is minimal concave function on Ω0 \ Ωε with B|∂Ω0 = f
where Ωε = {(x1, x2) ⊂ R2 : x2 ≥ x21 + ε2}2. For each ε we find Bε (stationary algorithm).3. We described evolution of Bε, ε ≥ 0 (dynamical approach).
The torsion of (t, t2, f (t)) changes sign finitely many times.
Paata Ivanisvili Kent State University Sharp estimates
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Other applications
Theorem (Ivanisvili–Jaye–Nazarov)
For any p > 1 and n ≥ 1 there exists A(p, n) > 1 such that∫Rn
(supB3x
1|B|
∫B|f (y)|dy
)pdx ≥ A(p, n)
∫Rn|f (x)|pdx , ∀f ∈ Lp
NO for centered maximal function p > nn−2 . f (x) = min{|x |n−1, 1}.
Theorem (P.I.)
supεI∈{−1,1}
∫ 10
(∑I∈D
εI 〈f , hI 〉hI)2
+ τ2f 2
p/2 dx ≤ C ∫ 10|f |pdx
Settles an open problem of Boros–Janakiraman–Volberg
Paata Ivanisvili Kent State University Sharp estimates
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Other applications
Theorem (Ivanisvili–Jaye–Nazarov)
For any p > 1 and n ≥ 1 there exists A(p, n) > 1 such that∫Rn
(supB3x
1|B|
∫B|f (y)|dy
)pdx ≥ A(p, n)
∫Rn|f (x)|pdx , ∀f ∈ Lp
NO for centered maximal function p > nn−2 .
f (x) = min{|x |n−1, 1}.
Theorem (P.I.)
supεI∈{−1,1}
∫ 10
(∑I∈D
εI 〈f , hI 〉hI)2
+ τ2f 2
p/2 dx ≤ C ∫ 10|f |pdx
Settles an open problem of Boros–Janakiraman–Volberg
Paata Ivanisvili Kent State University Sharp estimates
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Other applications
Theorem (Ivanisvili–Jaye–Nazarov)
For any p > 1 and n ≥ 1 there exists A(p, n) > 1 such that∫Rn
(supB3x
1|B|
∫B|f (y)|dy
)pdx ≥ A(p, n)
∫Rn|f (x)|pdx , ∀f ∈ Lp
NO for centered maximal function p > nn−2 . f (x) = min{|x |n−1, 1}.
Theorem (P.I.)
supεI∈{−1,1}
∫ 10
(∑I∈D
εI 〈f , hI 〉hI)2
+ τ2f 2
p/2 dx ≤ C ∫ 10|f |pdx
Settles an open problem of Boros–Janakiraman–Volberg
Paata Ivanisvili Kent State University Sharp estimates
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Other applications
Theorem (Ivanisvili–Jaye–Nazarov)
For any p > 1 and n ≥ 1 there exists A(p, n) > 1 such that∫Rn
(supB3x
1|B|
∫B|f (y)|dy
)pdx ≥ A(p, n)
∫Rn|f (x)|pdx , ∀f ∈ Lp
NO for centered maximal function p > nn−2 . f (x) = min{|x |n−1, 1}.
Theorem (P.I.)
supεI∈{−1,1}
∫ 10
(∑I∈D
εI 〈f , hI 〉hI)2
+ τ2f 2
p/2 dx ≤ C ∫ 10|f |pdx
Settles an open problem of Boros–Janakiraman–Volberg
Paata Ivanisvili Kent State University Sharp estimates
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The end
Thank you
Paata Ivanisvili Kent State University Sharp estimates
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