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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