Young’s inequality

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Lyapunov functionals for the heat equation and sharp inequalities

Giuseppe Toscani

Department of Mathematics University of Pavia, Italy

Frontiers of Mathematics and Applications III Santander, august 13-17, 2012

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Outlines Inequalities and Lyapunov functionals Young’s inequality

Outline

1 Inequalities and Lyapunov functionals Introduction

Entropy power inequality

2 Young’s inequality

A new proof of H¨older inequality

Lyapunov functionals and Young’s inequality

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Outline

1 Inequalities and Lyapunov functionals Introduction

Entropy power inequality

2 Young’s inequality

A new proof of H¨older inequality

Lyapunov functionals and Young’s inequality

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Outlines Inequalities and Lyapunov functionals Young’s inequality Introduction

Young’s inequality

Beckner, (1975); Brascamp and Lieb (1976)

kf ∗gk_r ≤(A_pA_qA_r⁰)ⁿkfk_pkgk_q.

f ∈L^p(Rⁿ), g ∈L^q(Rⁿ),1<p,q,r <∞ and 1/p+ 1/q = 1 + 1/r. The constant A_m

A_m = m^1/m m^01/m⁰

!1/2

primes always denote dual exponents,1/m+ 1/m⁰= 1.

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Young’s inequality

Beckner, (1975); Brascamp and Lieb (1976)

kf ∗gk_r ≤(A_pA_qA_r⁰)ⁿkfk_pkgk_q.

f ∈L^p(Rⁿ), g ∈L^q(Rⁿ),1<p,q,r <∞ and 1/p+ 1/q = 1 + 1/r. The constant A_m

A_m = m^1/m m^01/m⁰

!1/2

primes always denote dual exponents,1/m+ 1/m⁰= 1.

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Method of proof

The best constants in Young’s inequality were found by Beckner using tensorisation argumentsand rearrangements of functions.

Brascamp and Lieb derived them from a more general inequality, which is nowadays known as theBrascamp-Lieb inequality.

The expression of thebest constant, in the case in which both f andg are probability density functions, is obtained by noticing that Young’s inequality is saturated by Gaussian densities.

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Method of proof

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Method of proof

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Reverse Young’s inequality

Brascamp and Lieb noticed that the sharp form of Young inequality also holds in the so-calledreverse case

kf ∗gk_r ≥(ApAqA_r⁰)ⁿkfk_pkgk_q.

where now0<p,q,r <1 while, as in Young’s inequality, 1/p+ 1/q = 1 + 1/r. In this case, however, the dual exponents p⁰,q⁰,r⁰ are negative, and the constantAm

A_m = m^1/m

|m⁰|^1/|m⁰^|

!1/2

.

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Reverse Young’s inequality

Brascamp and Lieb noticed that the sharp form of Young inequality also holds in the so-calledreverse case

kf ∗gk_r ≥(ApAqA_r⁰)ⁿkfk_pkgk_q.

where now0<p,q,r <1 while, as in Young’s inequality, 1/p+ 1/q = 1 + 1/r. In this case, however, the dual exponents p⁰,q⁰,r⁰ are negative, and the constantAm

A_m = m^1/m

|m⁰|^1/|m⁰^|

!1/2

.

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Other proofs

The proof of this sharp reverse Young inequality was subsequentlysimplified byBarthe, (1998).

The original proof in Brascamp-Lieb was rather complicated, and usedtensorisation, Schwarz symmetrisation,

Brunn-Minkowskiand some not so intuitive phenomenon for the measure in high dimension.

The proof of the main result of Barthe relies on a

parametrization of functions which was suggested by Brunn’s proof of the Brunn-Minkowski inequality.

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Other proofs

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Other proofs

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Other proofs II

A different proof in Bennett, Bez (2009).

Bennett-Bez gave asemigroup proof of the optimal Young inequality identifying a super-solutionof the heat equation.

We will start here from aLyapunov functional, by proving its monotonicity. Fundamental tooltheinvarianceof the functional with respect to the scaling dilation

f(x)→f_a(x) = 1 aⁿf x

aⁿ

, a>0.

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Other proofs II

aⁿ

, a>0.

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Other proofs II

aⁿ

, a>0.

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Connections with other inequalities

Lieb enlightened connections of the sharp form of Young inequality with other inequalities.

By letting p,q,r →1 in the sharp form of Young’s inequality reduces to another well-known inequality in information theory, known asShannon’s entropy power inequality

e^2H(f^∗g) ≥e^2H(f⁾+ e^2H(g⁾,

In Shannon’s inequality f andg are probability density functions, and

H(f) =− Z

R

f(x) logf(x)dx

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Outlines Inequalities and Lyapunov functionals Young’s inequality Entropy power inequality

Connections with other inequalities

e^2H(f^∗g) ≥e^2H(f⁾+ e^2H(g⁾,

H(f) =− Z

R

f(x) logf(x)dx

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Connections with other inequalities

e^2H(f^∗g) ≥e^2H(f⁾+ e^2H(g⁾,

H(f) =− Z

R

f(x) logf(x)dx

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Entropy power inequality

The first rigorous proof of entropy power inequality was given byStam (1959), and generalized byBlackman (1965) to n-dimensional random vectors.

The proof is based on an identity which couples Fisher’s information with Shannon’s entropy functional

The original proofs of Blachman and Stam make a substantial use of the solution to the heat equation

∂f(x,t)

∂t = ∆f(x,t),

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Entropy power inequality

∂f(x,t)

∂t = ∆f(x,t),

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Entropy power inequality

∂f(x,t)

∂t = ∆f(x,t),

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A Lyapunov functional

The proof of Stam is based on the following argument. Let f(x,t) andg(x,t) be two solutions of the heat equation with different coefficients of diffusion, corresponding to the initial data f(x) (respectivelyg(x)).

Consider the evolution in time of the functional Θ_f_,g(t) defined by

Θ_f_,g(t) = e^2H(f^(t))+ e^2H(g^(t)) e^2H(f^(t)∗g^(t)) .

Evaluating the time derivative of Θ_f_,g(t), shows that Θ_f_,g(t) is increasing in time, and converges towards the constant value Θf,g(+∞) = 1.

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A Lyapunov functional

The proof of Stam is based on the following argument. Let f(x,t) andg(x,t) be two solutions of the heat equation with different coefficients of diffusion, corresponding to the initial data f(x) (respectivelyg(x)).

Consider the evolution in time of the functional Θ_f_,g(t) defined by

Θ_f_,g(t) = e^2H(f^(t))+ e^2H(g^(t)) e^2H(f^(t)∗g^(t)) .

Evaluating the time derivative of Θ_f_,g(t), shows that Θ_f_,g(t) is increasing in time, and converges towards the constant value Θf,g(+∞) = 1.

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H¨ older’s inequality

H¨older’s inequalityfor integrals states that, ifp,q>1 are such that 1/p+ 1/q= 1

Z

R

|f(x)g(x)|dx ≤ Z

R

|f(x)|^pdx

1/pZ

R

|g(x)|^qdx 1/q

.

Moreover, there isequality if and onlyf andg are such that there exist positive real numbersa andb such that

af^p(x) =bg^q(x)almost everywhere.

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Outlines Inequalities and Lyapunov functionals Young’s inequality A new proof of H¨older inequality

H¨ older’s inequality

H¨older’s inequalityfor integrals states that, ifp,q>1 are such that 1/p+ 1/q= 1

Z

R

|f(x)g(x)|dx ≤ Z

R

|f(x)|^pdx

1/pZ

R

|g(x)|^qdx 1/q

.

Moreover, there isequality if and onlyf andg are such that there exist positive real numbersa andb such that

af^p(x) =bg^q(x)almost everywhere.

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Classical proof

H¨older’s inequality can be proven in many ways, for example resorting to Young’s inequality for constants, which states that, if 1/p+ 1/q= 1

cd ≤ c^p p +d^q

q ,

for all nonnegativec andd

Equality is achieved if and only if c^p =d^q.

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Classical proof

H¨older’s inequality can be proven in many ways, for example resorting to Young’s inequality for constants, which states that, if 1/p+ 1/q= 1

cd ≤ c^p p +d^q

q ,

for all nonnegativec andd

Equality is achieved if and only if c^p =d^q.

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A new proof

A different way to achieveH¨older’s inequalityis the following.

Let Φu,v(t) be the functional

Φu,v(t) = Z

R

u(x,t)^1/pv(x,t)^1/qdx,

where 1/p+ 1/q = 1.

u(x,t) andv(x,t), t>0, aresolutions to the heat equation corresponding to the initial values u(x)∈L¹(R) (respectively v(x)∈L¹(R)).

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A new proof

A different way to achieveH¨older’s inequalityis the following.

Let Φu,v(t) be the functional

Φu,v(t) = Z

R

u(x,t)^1/pv(x,t)^1/qdx,

where 1/p+ 1/q = 1.

u(x,t) andv(x,t), t>0, aresolutions to the heat equation corresponding to the initial values u(x)∈L¹(R) (respectively v(x)∈L¹(R)).

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A new proof II

The functionalφu,v(t) is increasing in timefrom

Φu,v(t = 0) = Z

R

u(x)^1/pv(x)^1/qdx,

to

t→∞lim Φu,v(t) = Z

R

u(x)dx Z

R

v(x)dx.

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A new proof II

The functionalφu,v(t) is increasing in timefrom

Φu,v(t = 0) = Z

R

u(x)^1/pv(x)^1/qdx,

to

R

u(x)dx Z

R

v(x)dx.

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A new proof III

Let us evaluate the time derivative of Φ(t)

Φ⁰_u,v(t) = Z

R

h

(u(x,t)^1/p)_tv(x,t)^1/q+u(x,t)^1/p(v(x,t)^1/q)_t i

dx=

Z

R

1

pu^1/p−1v^1/qu_xx+ 1

qu^1/pv^1/q−1v_xx

dx = Z

R

1

pu^−1/qv^1/qu_xx+1

qu^1/pv^−1/pv_xx

dx.

Integrating by parts we get

Φ⁰_u,v(t) = 1 pq

Z

R

u^1/p(x,t)v^1/q(x,t)

u_x(x,t)

u(x,t) −v_x(x,t) v(x,t)

2

dx ≥0

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A new proof III

Let us evaluate the time derivative of Φ(t)

Φ⁰_u,v(t) = Z

R

h

(u(x,t)^1/p)_tv(x,t)^1/q+u(x,t)^1/p(v(x,t)^1/q)_t i

dx=

Z

R

1

pu^1/p−1v^1/qu_xx+ 1

qu^1/pv^1/q−1v_xx

dx = Z

R

1

pu^−1/qv^1/qu_xx+1

qu^1/pv^−1/pv_xx

dx.

Integrating by parts we get

Φ⁰_u,v(t) = 1 pq

Z

R

u^1/p(x,t)v^1/q(x,t)

u_x(x,t)

u(x,t) −v_x(x,t) v(x,t)

2

dx ≥0

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Consequences

The functional Φu,v(t) is increasing in time. Note that the time derivative of the functional is equal to zeroif and only if, for every t>0

ux(x,t)

u(x,t) −vx(x,t) v(x,t) = 0.

for all x

This condition can be rewritten as

d

dx logu(x,t) v(v,t) = 0.

Φ⁰(t) = 0 if and only ifu(x,t) =c v(x,t)for some positive constant c.

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Consequences

ux(x,t)

u(x,t) −vx(x,t) v(x,t) = 0.

for all x

d

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Consequences

ux(x,t)

u(x,t) −vx(x,t) v(x,t) = 0.

for all x

d

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Consequences II

The functional Φ(t) is monotone increasing, and it will reach its eventual maximum value as time t→ ∞.

The computation of the limit value uses in a substantial way thescaling invariance of Φ.

The value of Φ_u,v(t) does not change if we scale u(x,t) and v(x,t) according to

u(x,t)→U(x,t) =√

1 + 2tu(x√

1 + 2t,t) v(x,t)→V(x,t) =√

1 + 2tv(x√

1 + 2t,t).

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Consequences II

u(x,t)→U(x,t) =√

1 + 2tu(x√

1 + 2t,t) v(x,t)→V(x,t) =√

1 + 2tv(x√

1 + 2t,t).

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Consequences II

u(x,t)→U(x,t) =√

1 + 2tu(x√

1 + 2t,t) v(x,t)→V(x,t) =√

1 + 2tv(x√

1 + 2t,t).

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Consequences III

It is well-known that

t→∞lim U(x,t) =M₁(x) Z

R

u(x)dx lim

t→∞V(x,t) =M₁(x) Z

R

v(x)dx,

M₁(x) is the Gaussian density inRof variance equal to 1.

The monotonicity of the functional Φ(t) implies the inequality

Z

R

u(x)^1/pv(x)^1/qdx ≤ Z

R

u(x)dx

1/pZ

R

v(x)dx 1/q

,

withequality if and only ifu(x) =cv(x)for some constantc.

Settingf =u^1/p andg =v^1/q proves bothH¨older inequalityand the equality cases.

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Consequences III

R

u(x)dx lim

t→∞V(x,t) =M₁(x) Z

R

v(x)dx,

Z

R

u(x)dx

1/pZ

R

v(x)dx 1/q

,

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Consequences III

R

u(x)dx lim

t→∞V(x,t) =M₁(x) Z

R

v(x)dx,

Z

R

u(x)dx

1/pZ

R

v(x)dx 1/q

,

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Consequences III

R

u(x)dx lim

t→∞V(x,t) =M₁(x) Z

R

v(x)dx,

Z

R

u(x)dx

1/pZ

R

v(x)dx 1/q

,

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Comments

Despite its apparent complexity, this way ofproof is based on a solid physical argument, namely the monotonicity in time of a Lyapunov functional of the solution to the heat equation.

This gives a clear indication that many inequalitiesreflect the physical principle of thetendency of a system to move towards the state of maximum entropy.

This idea also applies to proveYoung’s inequality.

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Comments

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Comments

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Outlines Inequalities and Lyapunov functionals Young’s inequality Lyapunov functionals and Young’s inequality

The Lyapunov functional related to Young’s inequality

The key functional to study is

Ψ_u,v(t) = Z

R

u(x,t)^1/p∗v(x,t)^1/qr

dx 1/r

.

As in Young’s inequality, 1/p+ 1/q = 1 + 1/r u(x,t) andv(x,t) are solutions of the heat equation corresponding to the initial datau(x) (respectivelyv(x)).

However, these solutions correspond to two different heat equations, with different coefficients of diffusions, say α and β.

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The Lyapunov functional related to Young’s inequality

Ψ_u,v(t) = Z

R

dx 1/r

.

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The Lyapunov functional related to Young’s inequality

Ψ_u,v(t) = Z

R

dx 1/r

.

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Main result

Theorem

Let 1/p+ 1/q= 1 + 1/r, and u(x,t)and v(x,t), t>0, are solutions to the heat equation corresponding to the initial values u(x)∈L¹(R) (respectively v(x)∈L¹(R)). Then, if p,q,r >1, and the diffusion coefficients are given byα=q⁰/p (respectivelyβ=p⁰/q), or by a multiple of them,Ψ_u,v(t)is increasing in timefrom

Ψ_u,v(t = 0) = Z

R

u(x)^1/p∗v(x)^1/qr

dx 1/r

, to the limit value

t→∞lim Ψ_u,v(t) = (A_pA_qA_r0)^1/2 Z

R

u(x)dx

1/pZ

R

v(x)dx 1/q

.

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Proof

Compute the time derivative of the functional Ψ_u,v(t). To shorten, let us denote

h(x,t) =u(x,t)^1/p∗v(x,t)^1/q. Computations give

1 r

d dt

Z

R

h^r(x,t)dx =−(α+β)(r−1) Z

R

h^r−2(x,t) (h_x(x,t))²dx+

αp p⁰

Z

R

h^r−1(x,t)A(u^1/p,v^1/q)(x,t)dx+

βq q⁰

Z

R

h^r−1(x,t)B(u^1/p,v^1/q)(x,t)dx.

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Proof

Compute the time derivative of the functional Ψ_u,v(t). To shorten, let us denote

h(x,t) =u(x,t)^1/p∗v(x,t)^1/q. Computations give

1 r

d dt

Z

R

h^r(x,t)dx =−(α+β)(r−1) Z

R

h^r−2(x,t) (h_x(x,t))²dx+

αp p⁰

Z

R

h^r−1(x,t)A(u^1/p,v^1/q)(x,t)dx+

βq q⁰

Z

R

h^r−1(x,t)B(u^1/p,v^1/q)(x,t)dx.

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Proof

AandB are defined by

A(f,g)(x,t) = Z

R

(f_x(x−y,t))²

f(x−y) g(y,t)dy,

B(f,g)(x,t) = Z

R

f(x−y)(g_y(y,t))² g(y) dy. Since

dΨ_u,v(t)

dt = Ψ_u,v(t)^1−r d dt

Z

R

h^r(x,t)dx,

thesign of the time derivativeof the functional Ψ_u,v(t) depends of the sign of the expressionon the right-hand side of the previous equality.

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Proof

AandB are defined by

A(f,g)(x,t) = Z

R

(f_x(x−y,t))²

f(x−y) g(y,t)dy,

B(f,g)(x,t) = Z

R

f(x−y)(g_y(y,t))² g(y) dy. Since

dΨ_u,v(t)

dt = Ψ_u,v(t)^1−r d dt

Z

R

h^r(x,t)dx,

thesign of the time derivativeof the functional Ψ_u,v(t) depends of the sign of the expressionon the right-hand side of the previous equality.

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Key result

Lemma

Let f(x)and g(x)be probability density functions such that both A(f,g)and B(f,g)are well defined. Then, for all positive constants a,b and r>0

a²+b²+ 2abr Z

R

(f ∗g)^r−2((f ∗g)_x)² dx≤

a² Z

R

(f ∗g)^r−1A(f,g)dx+b² Z

R

(f ∗g)^r−1B(f,g)dx.

Moreover, there isequality if and only if, for any positive constant c and constants m₁,m₂,f and g are Gaussian densities, f(x) =M_ca(x−m₁) and g(x) =M_cb(x−m₂).

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Proof II

If we choose a²=αp/p⁰,b²=βq/q⁰, the coefficient of the term on the left-hand side

a²+b²+ 2abr=αp p⁰+βq

q⁰+ 2p αβ

rpq p⁰q⁰r. Let us introduce, for any givenr>1 the function

Γ(α, β) = (α+β)(r−1)−

αp p⁰+βq

q⁰+ 2p αβ

rpq p⁰q⁰r

.

The function Γ is jointly convex, and possesses the half-line of extremals

β= p q⁰·p⁰

qα.

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Proof II

q⁰+ 2p αβ

Γ(α, β) = (α+β)(r−1)−

αp p⁰+βq

q⁰+ 2p αβ

rpq p⁰q⁰r

.

β= p q⁰·p⁰

qα.

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Proof II

q⁰+ 2p αβ

Γ(α, β) = (α+β)(r−1)−

αp p⁰+βq

q⁰+ 2p αβ

rpq p⁰q⁰r

.

β= p

q⁰·p⁰ qα.

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Proof III

Along the half-line the functional Φ_u,v(t) isincreasing with respect tot.

Scalingu(x,t) andv(x,t), we conclude that the functional will keep its maximum value as time goes to infinity, and

R

u(x)dx 1/pZ

R

v(x)dx 1/q

C(p,q,r) The constant

C(p,q,r) = Z

R

M_q0/p(x)^1/p∗M_p0/q(x)^1/q r

dx.

1/r

=(ApAqA_r⁰)^1/2. Easy togeneralizeto dimensionn>1.

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Proof III

R

u(x)dx 1/pZ

R

v(x)dx 1/q

C(p,q,r) = Z

R

M_q0/p(x)^1/p∗M_p0/q(x)^1/q r

dx.

1/r

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Proof III

R

u(x)dx 1/pZ

R

v(x)dx 1/q

C(p,q,r) = Z

R

M_q0/p(x)^1/p∗M_p0/q(x)^1/q r

dx.

1/r

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Proof III

R

u(x)dx 1/pZ

R

v(x)dx 1/q

C(p,q,r) = Z

R

M_q0/p(x)^1/p∗M_p0/q(x)^1/q r

dx.

1/r

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References

G. Toscani, Heat equation and the sharp Young’s inequality.

Preprinthttp://arxiv.org/abs/1204.2086

G. Toscani, Lyapunov functionals for the heat equation and sharp inequalities. In press onAcc. Pel. Pericolanti (2012)

G. Toscani, An information-theoretic proof of Nash’s inequality, Preprinthttp://arxiv.org/abs/1206.2130

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References

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