Sharp estimates in harmonic analysis
Paata Ivanisvili
Kent State University
2017
University of California, Irvine
Paata Ivanisvili Kent State University Sharp estimates
Bellman function for extremal problems in BMO.(joint with N. Osipov, D. Stolyarov, P. Zatitskiy, V. Vasyunin)
Paata Ivanisvili Kent State University Sharp estimates
Surface of minimal area k1 + k2 = 0.
Paata Ivanisvili Kent State University Sharp estimates
Minimal concave function k1k2 = 0
Paata Ivanisvili Kent State University Sharp estimates
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
Projection
Paata Ivanisvili Kent State University Sharp estimates
Foliation
This we call foliation.
Problem (equivalent)
Given a wire find foliation.
Paata Ivanisvili Kent State University Sharp estimates
Foliation
This we call foliation.
Problem (equivalent)
Given a wire find foliation.
Paata Ivanisvili Kent State University Sharp estimates
Concave envelope
Paata Ivanisvili Kent State University Sharp estimates
Foliation
Paata Ivanisvili Kent State University Sharp estimates
Angle and Cup
Demonstrate
Paata Ivanisvili Kent State University Sharp estimates
Angle and Cup
DemonstratePaata Ivanisvili Kent State University Sharp estimates
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
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
Erasing a hole
Will not be minimal!
Paata Ivanisvili Kent State University Sharp estimates
Erasing a hole
Will not be minimal!
Paata Ivanisvili Kent State University Sharp estimates
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
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
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
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
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
Evolution of the surface
Paata Ivanisvili Kent State University Sharp estimates
Annulus and parabolic strips
Ω \D
Paata Ivanisvili Kent State University Sharp estimates
Left tangent lines 2D
x1
x2
x2 = x21
x2 = x21 + ε
2
Paata Ivanisvili Kent State University Sharp estimates
Left tangent lines 3D
Paata Ivanisvili Kent State University Sharp estimates
Right tangent lines 2D
x1
x2
x2 = x21
x2 = x21 + ε
2
Paata Ivanisvili Kent State University Sharp estimates
Right tangent lines 3D
Paata Ivanisvili Kent State University Sharp estimates
Cup
x1
x2
x2 = x21
x2 = x21 + ε
2
A0
B0
U1 U2
Ωcup
ΩRΩL
Paata Ivanisvili Kent State University Sharp estimates
Cup
Paata Ivanisvili Kent State University Sharp estimates
Angle
x1
x2
V
x2 = x21
x2 = x21 + ε
2
Paata Ivanisvili Kent State University Sharp estimates
Angle
Paata Ivanisvili Kent State University Sharp estimates
Angle and cup
x1
x2
U−
U+
A
B
V
Paata Ivanisvili Kent State University Sharp estimates
Angle and cup
Paata Ivanisvili Kent State University Sharp estimates
Angle and cup: evolution
Paata Ivanisvili Kent State University Sharp estimates
Two cups
Paata Ivanisvili Kent State University Sharp estimates
Two cups
Paata Ivanisvili Kent State University Sharp estimates
Two cups: evolution
Paata Ivanisvili Kent State University Sharp estimates
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
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
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
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
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
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
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
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
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
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
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
The end
Thank you
Paata Ivanisvili Kent State University Sharp estimates
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