Inequalities and transport

Inequalities and transport

Jean DOLBEAULT

Ceremade, Universit´e Paris IX-Dauphine, Place de Lattre de Tassigny,

75775 Paris C´edex 16, France

E-mail: dolbeaul@ceremade.dauphine.fr

http://www.ceremade.dauphine.fr/

edolbeaul/

Optimal transport

[Cordero-Erausquin, Gangbo, Houdr´e, Nazaret, Villani], [Agueh, Ghoussoub, Kang]

• Sobolev inequality: kfk_L2^∗ ≤ S k∇fk_L2

• (Standard) logarithmic Sobolev inequality

• Logarithmic Sobolev inequality in W^1,p(IR^N)

Sobolev inequalities

kf|_L2^∗ ≤ S k∇fk_L2

N ≥ 3. Optimal function: f(x) = (σ + |x|²)^−(N−2)/2. A proof based on mass transportation:

inf

1 2λ²

Z

IR^N |∇f|² dx :

Z

IR^N |f|^2∗ dx = 1

= ⁿ⁽ⁿ⁻²⁾

2(n−1) sup

R

IR^N |g|^2∗(1−1n)dy − ^λ₂^{2 R}_IRN |y|² |g|^2∗dy : ^R

IR^N |g|^2∗dy = 1

3

Mass transportation: basic results

µ and ν two Borel probability measures on IR^N. T : IR^N → IR^N T#µ = ν ⇐⇒ ν(A) = µ(T⁻¹(A)) for any Borel measurable set A.

Theorem 1 (Brenier, McCann) ∃T = ∇φ such that T#µ = ν and φ is convex.

µ = F(x) dx, ν = G(x) dx, ^R_IRN F(x) dx = ^R

IR^N G(y) dy = 1

∀ b ∈ C(IR^N,IR⁺)

Z

IR^N b(y)G(y)dy =

Z

IR^N b(∇φ(x))F(x) dx Under technical assumptions: φ ∈ C², supp(F) or supp(G) is convex... [Caffarelli] φ solves the Monge-Amp`ere equation

G(∇φ) det Hess(φ) = F

A proof of the Sobolev inequality

G(∇φ)⁻¹ⁿ = ( det Hess(φ))¹ⁿF⁻ⁿ¹ ≤ 1

n ∆φ F⁻¹ⁿ

Z

G(y)¹⁻ⁿ¹dy ≤ 1 n

Z

G(∇φ(x))¹⁻¹ⁿ( det Hess(φ))¹ⁿ∆φ dx

= 1 n

Z

F¹⁻¹ⁿ∆φ dx = −1 n

Z

∇(F¹⁻¹ⁿ) · ∇φ dx

by the arithmetic-geometric inequality. F = |f|^2∗, G = |g|^2∗

Z

|g|^2∗(1−1n)dy ≤ −2(n − 1) n(n − 2)

Z

(f

n

n−2)∇f · ∇φ dx

n(n − 2) 2(n − 1)

Z

|g|^2∗(1−1n)dy ≤ 2 λ²

Z

|∇f|²dx + λ² 2

Z

|f|^2∗|∇φ|²dx by Young’s inequality. Use: ^R F |∇φ|² dx = ^R G|y|²dy

5

A proof of the standard logarithmic Sobolev inequality

G(y) = e^−|y|2/2, F(x) = f(x) e^−|x|2/2, ∇φ#F dx = G dy.

e^{−|∇φ|2/2} det Hess(φ) = f(x) e^−|x|2/2 θ(x) = φ(x) − ¹₂ |x|²

f(x) e^−|x|2/2 = det (Id + Hess(θ))e^{−|x+∇θ(x)|2/2}

log f − |x|²/2 = −|x + ∇θ(x)|²/2 + log ^hdet (Id + Hess(θ))ⁱ

≤ −|x + ∇θ(x)|²/2 + ∆θ

(use log(1 + t) ≤ t). Let dµ(x) = (2π)^−n/2e^−|x|2/2dx.

log f ≤ −1

2 |∇θ|² − x · ∇θ + ∆θ

Z

f log f dµ ≤ −1 ^Z

q

f ∇θ + ∇f

√

2

dµ + 1 ^Z |∇f|²

dµ ≤ 1 ^Z |∇f|² dµ

Logarithmic Sobolev inequality in

G(y) = c_p,ne⁻

p

p−1 |y|^p/(p−1)

=: f_∞(y), F(x) = f(x) c_p,n e⁻

p

p−1 |x|^p/(p−1)

∇φ#F dx = G dy, dµ(x) = f_∞^p (x) dx f(x)e⁻

p

p−1 |x|^p/(p−1)

= det (Hess(φ))e⁻

p

p−1 |x+∇θ(x)|^p/(p−1)

f^p(x) = f_∞^p (∇φ) det (Id + Hess(φ))

Z

f^p log f^p dµ =

Z

f^p logf_∞^p dµ +

Z

(∆φ − n)f^pdµ

Z

∆φ f^pdµ = −p

Z

f^p−1∇f·∇φ dµ ≤ λ^−q q

Z

|f|^p |∇φ|^p/(p−1) + λ^p p

Z

|∇f|^pdµ

using Young’s inequality: X = f^p−1∇φ, Y = ∇f

Z

X · Y dµ ≤ λ^−q

q kXk^q_q + λ^p

p kY k^p_p

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