on k-monotonicity - lth · interpolation of weighted lp-spacesa theorem of hardy, littlewood and...

Interpolation of weighted Lp -spaces A theorem of Hardy, Littlewood and Polya Generlizations of HLP

On K-monotonicity

Gunnar Sparr

Lund 2010-10-04

Lund 2010-10-04 1 / 21


Interpolation spaces

Banach spaces A,B, . . .Banach couples ~A = {A0,A1}, ~B = {B0,B1} . . .Notation: T : ~A −→ ~B ⇐⇒ T : A0 → B0, T : A1 → B1 boundedly

Definition

A,B interpolation spaces with respect to ~A, ~B iff

[γInt] : T : ~A→ ~B with norms ≤ 1 =⇒ T : A→ B with norm ≤ γ

For γ = 1: Exact interpolation spaces, [Ex-Int]For non-specified γ: [Int]

If ~A = ~B, A = B, we say ”A interpolation space with respect to ~A”

Lund 2010-10-04 2 / 21


Case ~A = {L1, L∞}

Theorem (Calderon)

A exact interpolation space w.r.t. ~A iff

f ∈ A,

∫ t

0g∗(s) ds ≤

∫ t

0f ∗(s) ds for t > 0 implies g ∈ A, ‖g‖A ≤ ‖f ‖A

Theorem (Mitjagin)

A exact interpolation space w.r.t. ~A iff

if Tf (x) = ε(x)f (γ(x)), with |ε(x)| = 1 all x , γ measure preserving

then ‖T‖A ≤ 1

Lund 2010-10-04 3 / 21


Relation between Calderon and Mitjagin

Consider n-dimensional case ~A = {`n1, `n∞}‖T‖ ≤ 1 on ~A ⇐⇒

∑rows |tij | ≤ 1,

∑columns |tij | ≤ 1

Subclass:

Definition

D = set of Doubly stochastic matrices:matrices T = (tij) with tij ≥ 0,

∑i tij = 1,

∑j tij = 1

Theorem (Birkhoff)

D = convex hull of permutation matrices P

Leads to Mitjagin theorem

Lund 2010-10-04 4 / 21


Theorem HLP

Definition

f ∗ = (f ∗1 , . . . , f∗n ), f ∗1 ≥ . . . ≥ f ∗n , denotes the non-increasing

rearrangement of f = (f1, . . . , fn)

Theorem (Hardy-Littlewood-Polya, HLP)

Let f = (f1, . . . , fn), g = (g1, . . . , gn) with fi , gi ≥ 0, i = 1, . . . , n. Theng = Tf with T ∈ D iff

k∑1

g∗i ≤k∑1

f ∗i , k = 1, . . . , n − 1,

n∑1

g∗i =n∑1

f ∗i

Leads to Calderon theorem

Lund 2010-10-04 5 / 21


General case ~A, ~B

Definition (Peetre K -functional)

For f ∈ A0 + A1, define

K (t, f ; ~A) = inf ||f0||0 + t||f1||1, f = f0 + f1, fi ∈ Ai , i = 0, 1

Example (~A = {L1, L∞})Here

K (t, f ; L1, L∞) =

∫ t

0f ∗(s) ds

Condition on A in Calderon theorem:

f ∈ A, K (t, g) ≤ K (t, f ), t > 0⇐⇒ g ∈ A, ‖g‖A ≤ ‖f ‖A

Lund 2010-10-04 6 / 21


Inspires to

Definition (K-monotonicity)

Notation: g ≤ f [K ]⇐⇒ K (t, g ; ~B) ≤ K (t, f ; ~A), t > 0

A,B are γK -monotonic with respect to ~A, ~B iff

[γKmon] : f ∈ A, g ≤ f [K ] =⇒ g ∈ B, ‖g‖B ≤ γ‖f ‖A

For γ = 1: Exactly K-monotonic, [Ex-Kmon]For non-specified γ: [Kmon]

Calderon theorem inspires to

Definition (Calderon property)

~A, ~B Exactly Calderon iff [Ex-Kmon] ⇐⇒ [Ex-Int]~A, ~B Calderon iff [Kmon] ⇐⇒ [Int]

Lund 2010-10-04 7 / 21


Observation: If T : ~A→ ~B with norm ≤ 1, then

K (t,Tf ; ~B) ≤ inff=f0+f1 ‖Tf0‖B0 + t‖Tf1‖B1 ≤inff=f0+f1 ‖f0‖A0 + t‖f1‖A1 = K (t, f ; ~A)

Theorem

[Ex-Kmon] =⇒ [Ex-Int]

Proof.

‖Tf ‖i ≤ ‖f ‖i , i = 0, 1 =⇒ K (t,Tf ; ~B) ≤ K (t, f ; ~A)

=⇒[Kmon]‖Tf ‖B ≤ ‖f ‖A

To verify Calderon property, crucial problem is to prove [Int] =⇒ [Kmon]Would follow from:Given f , g with g ≤ f [K ], construct T with norm ≤ γ such that g = Tf

Lund 2010-10-04 8 / 21


Weighted Lp-spaces

Consider Lp,α with norm (∫|f α|p dµ)1/p

In particular finite dimensional spaces `np,α, norm (∑n

1 |fiαi |p)1/p

Let ~̀n~p,α = (`np0,α, `np1)

Definition

Functional M(t, f ; ~̀n~p,α) =∑n

1 min(|fiαi |p0 , t|fi |p1)

Order relation:

g ≤ f [M]⇐⇒ M(t, g) ≤ M(t, f ), t > 0

A,B are γM-monotonic with respect to ~A, ~B iff

[γMmon] : f ∈ A, g ≤ f [M] =⇒ g ∈ B, ‖g‖B ≤ ‖f ‖A

For γ = 1: Exactly M-monotonic, [Ex-Mmon]For non-specified γ: [Mmon]

Lund 2010-10-04 9 / 21


Properties of K and M (without proof):

K=M for p0 = p1 = 1

Theorem:g ≤ f [M] =⇒ g ≤ f [K ]

g ≤ f [K ] =⇒ g ≤ γ~pf [M] for some γ~p

(best constant known)

[Ex-Kmon] =⇒ [Ex-Mmon] =⇒ [γpKmon]

[Kmon] ⇐⇒ [Mmon]

Lemma (Main lemma)

g ≤ f [M] =⇒ there exists T : ~̀n~p,α → ~̀n~p,β with norm ≤ 1,

such that Tf = g

Consequence:

Theorem

{~̀n~p,α, ~̀n~p,β} is Calderon

Lund 2010-10-04 10 / 21


Results on Calderon property

{L1, L∞} Calderon 1966, Mitjagin 1965

{Lp, L∞} and {L1, Lq} Lorentz-Shimogaki 1971

{Lαp , Lβp}, 1 ≤ p <∞ Sedaev-Semenov 1971 and Sedaev 1973

{Lαp , Lβq}, 1 ≤ p, q <∞ Sparr 1978, Cwikel 1976

{Lαp , Lβq}, 0 < p, q <∞, with restrictions on measures, Sparr 1978

{~Aθ0,q0 , ~Aθ1,q1} for every ~A, 0 < θ0, θ1 < 1, 1 ≤ q0, q1 ≤ ∞ Cwikel

Example {L1,W11 } showing that not every couple is Calderon, Cwikel

1976

. . .

Lund 2010-10-04 11 / 21


Definitions (1)

Definition

S = set of column stochastic matrices, i.e.Θ = (θij) with θij ≥ 0,

∑i θij = 1 (column sums)

For vectors ~x = (x0i , x

1i ) with x0

i , x1i > 0, order relation ”inclination”

is defined by~x � ~y iff x1/x0 ≤ y1/y0

Notation: ΘX = Y ⇐⇒∑

j θij ~xj = ~yi , where xj , yi ∈ R2

Lund 2010-10-04 12 / 21


Definitions (2)

Definition

To X = {~x1, . . . ,~xn} is associated a parallelotope ωX with lower border γX

ω = {∑j

εj~xj | 0 ≤ εj ≤ 1}

Lund 2010-10-04 13 / 21


Matrix lemma 1

Lemma (1)

Suppose∑

j∈J ~xj =∑

i∈I ~yi with I , J finite. The following conditions areequivalent:

Y = ΘX with Θ ∈ SωY ⊆ ωX

γX ≤ γY

Lund 2010-10-04 14 / 21


Proof

Non-trivial implication: γX ≤ γY =⇒ Y = ΘX with Θ ∈ S.Proof by induction. Split into two parallelotopes

Lund 2010-10-04 15 / 21


Proof, contd

Induction assumption for the two parallelotopes:~y1 = θ11~x1 + λ1~z

. . .

~yk = θk1~x1 + λk~z~yk+1 = θk+1,2~x2 + . . .+ θk+1,n~xn

. . .

~ym = θm2~x2 + . . .+ θmn~xn

~z = µ2~x2 + . . .+ µn~xn

with all column sums = 1

Insert ~z into the first group.Yields Y = ΘX with all Θ-column sums = 1

Lund 2010-10-04 16 / 21


Special case: HLP

Theorem HLP corresponds to the special case ~xi = (xi , 1), ~yj = (yj , 1)

(∑k

1 y∗i , k) ≤ (∑k

1 x∗i , k)

Lund 2010-10-04 17 / 21


Matrix lemma 2

Lemma (2)

The following conditions are equivalent:

(i) ΘX ≤ Y with Θ ∈ S(ii) γX ≤ γY(iii)

∑j∈J

min(x0j , tx1

j ) ≤∑i∈I

min(y0i , ty

1i ), t > 0

Here ΘX ≤ Y means componentwise inequalityAlso true for I , J non-finite, with appropriate conditions on S

Lund 2010-10-04 18 / 21


Proof: (ii)⇔ (iii) consquence of convexity and Legendre transform

Lγ(t)def= inf

s(tγ(s)− s) = |OM| = |ON|+ |NM| =∑

j≥kx0j + t

∑j<k

x1j =

∑j

min(x0j , tx1

j )

Increasing order with respect to �

Lund 2010-10-04 19 / 21


Proof of Main Lemma

Proof.

Matrix lemma 2:∑j min(x0

j , tx1j ) ≤

∑i min(y0

i , ty1i ), t > 0⇒ there exists Θ ∈

S such that ΘX ≤ YApply to

n∑1

min(|gi |p0βp0i , t|gi |p1) ≤

n∑1

min(|fi |p0αp0i , t|fi |

p1)

LetT = diag (g)ΘTdiag (1/f )

Tf = g?Tf = diag (g) ΘT diag (1/f )f︸︷︷︸

1︸︷︷︸1

= g

where 1T = [1 1 . . . 1]Lund 2010-10-04 20 / 21


Proof of Main Lemma, contd

T : `np → `np, norm ≤ 1?Convexity of xp implies

|(Tφ)j |p = |gj |p|ΘTdiag (1/f )φ|p = |gj |p|∑i

θij1

fiφi |p ≤ |gj |p

∑i

θij|φi |p

|fi |p∑j

|(Tφ)j |p ≤∑j

|gj |p∑i

θij |φifi|p =

∑i

(∑j

θij |gj |p/|fi |p︸︷︷︸≤1

)|φi |p

T : `np,α → `np,α,norm ≤ 1?Analogously

Lund 2010-10-04 21 / 21

on k-monotonicity - lth · interpolation of weighted lp-spacesa theorem of hardy, littlewood and...

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