Majorization, Interpolation and noncommutative Khintchine inequalities

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Khintchine inequalities

Léonard Cadilhac

To cite this version:

Léonard Cadilhac. Majorization, Interpolation and noncommutative Khintchine inequalities. Studia Mathematica, Instytut Matematyczny - Polska Akademii Nauk, In press. �hal-02485755�

KHINTCHINE INEQUALITY

LÉONARD CADILHAC

Abstract. Let 0 < p < q ≤ ∞ and α ∈ (0,∞]. We give a characterization of quasi-Banach interpolation spaces for the couple(Lp(0, α), Lq(0, α))in terms of two monotonicity properties, extending known results which mainly dealt with Banach spaces. This enables us to recover recent results of Cwikel and Nilsson on sequence spaces and to solve a conjecture of Levitina, Sukochev and Zanin in the setting of function spaces. We apply the results obtained to characterize symmetric spaces in which the standard forms of the noncommutative Khintchine inequalities hold.

1. Introduction

This paper is motivated by two dierent problems, one in the classical theory of L_p- spaces and in particular symmetric spaces and the other in noncommutative harmonic analysis. The two have become closely related during the last decade, since noncom- mutative symmetric spaces have been found to be a nice setting in which to generalize classical theorems and formulate new results. Examples linked to the present paper can be found in [24], [17], [19], [9], [10] and [20].

Question 1: Interpolation of L_p-spaces and right-majorization. Characterisa- tions and sucient conditions garanteeing that a symmetric space is an interpolation space for a couple of L_p-spaces have been investigated in the past decades. Sucient conditions can be formulated in terms of convexity, concavity or Boyd indices and char- acterisations rely on the computation of the K-functional for the couple (L_p, L_q), see [16] for a survey. In [20], Levitina, Sukochev and Zanin conjecture a new characterisa- tion of sequence spaces E for which there exists p <2 such that E is an interpolation space for the couple (`<sup>p</sup>, `²). It can be stated in terms of right-majorization and right- monotonicity. Let us dene these notions as well as their left counterparts. For functions (or sequences)f andg which admit nonincreasing rearrangementsf^∗ andg^∗,

• we say that f right-majorizesg and writef . g if: ∀t >0,R∞

t f^∗≥R∞ t g^∗,

• we say that f left-majorizesg and writef g if: ∀t >0,Rt

0f^∗ ≥Rt 0g^∗. For any p, q∈(0,∞), a symmetric quasi-Banach spaceE is said to be:

• right-q-monotone if there exists C > 0 such that for all f ∈ E and g ∈ L0,

|f|^q.|g|^q⇒g∈E,kgk_E ≤Ckfk_E.

• left-p-monotone if there exists C > 0 such that for all f ∈ E and g ∈ L0,

|f|^p |g|^p⇒g∈E,kgk_E ≤Ckfk_E. For a denition of L₀ see Subsection 2.2.

Conjecture 1.1 (Levitina, Sukochev, Zanin). Let E be a quasi-Banach symmetric se- quence space. Then E is right-2-monotone if and only if there existsp∈(0,2] such that E is an interpolation space between `<sup>p</sup> and`².

1

Cwikel and Nilsson proved the conjecture for Banach spaces with the Fatou property in [7]. We recover their result and prove that the conjecture holds for function spaces.

Theorem 1.2. Let E be a quasi-Banach symmetric function space and q∈(0,∞). The two following properties are equivalent:

(1) E is right-q-monotone,

(2) there exists p≤q such that E is an interpolation space for the couple (Lp, Lq).

The rst ingredient to prove this theorem is a characterisation of interpolation spaces for the couple(L_p, L_q)given in Theorem 4.1 i.e. a symmetric spaceEis an interpolation space for the couple (L_p, L_q) if and only if it is left-p-monotone and right-q-monotone.

Although we formulate the result dierently, it is an generalisation to the quasi-Banach case of a known fact (see Theorem 7.2 in [16]). The second is to remark that for any quasi-Banach symmetric spaceE, there existsp >0such thatEis an interpolation space for the couple(L_p, L∞). This can be obtained directly from the literature by considering the Boyd indices ofE,αE andβE (see Subsection 2.4).

In section 3, we extend part of the Lorentz-Shimogaki theorem to the quasi-Banach setting. This was, to the best of our knowledge, not a direct consequence of previous works despite the fact that very similar results can be found, see [6], [27].

In section 4, the characterisation, which is our main theorem, is proven as well as its partial extension to sequence spaces. We also give an application to p-convexications.

In section 5, we relate left-p-monotonicity to p-convexity and right-q-monotonicity to q-concavity. A direct consequence of this, combined with the main theorem is to recover and generalize known sucient conditions for a symmetric spaceEto be an interpolation space for the couple (L_p, L_q). In the following theorem, we compile some consequences of Theorem 3.1, Theorem 4.1, Section 5 and Subsection 2.4 in that direction.

Theorem 1.3. Assume that E is a symmetric space and p, q∈(0,∞) with p < q. For E to be an interpolation space for the couple (L_p, L_q), it suces that E verify one of the conditions among:

P1. E is left-p-monotone, P2. p <1/βE,

P3. E has the Fatou property and isp-convex,

P4. E is an interpolation space for the couple (Lp, L∞), and one of the conditions among:

Q1. E is right-q-monotone, Q2. q >1/αE,

Q3. E has the Fatou property and isq-concave.

Theorems of this form have already appeared in the literature, for example in [9, Theorem 3.2], [1, Theorem 1] or in the survey [16]. The version presented here is quite general, except for the fact that we do not consider the hypothesis of being separable which could be used instead of the Fatou property.

Question 2: Noncommutative Khintchine inequalities in symmetric spaces.

Noncommutative Khintchine inequalities have been introduced in [22] forL_p-spaces and have since been a crucial tool, in particular for the development of noncommutative harmonic analysis. They have been further studied by many dierent authors and in the general context of symmetric spaces ([23], [24], [19], [9]). The aim of Khintchine

inequalities is, given a specic sequence of random variables (ξi)i∈N in a (noncommu- tative) probability spaceA (independent Rademacher variables, free Haar unitaries i.e.

freely independent variables which are uniformly distributed on the unit circle ofC ...) and a measure space of coecients Mto provide computable expressions for norms of elements of the form:

Gx:=

∞

X

i=1

xi⊗ξi,

dened inM⊗A, wherex= (x_i)i∈Nis a nite sequence of elements of M.

Before going into more details, let us introduce a notation that will be used in the remainder of the text. For quantities A and B, we write A(x) . B(y) if there exists a constant c independent of x and y such that A(x) ≤ cB(y). Additionally, we write A(x)≈B(y) if A(x).B(y) and B(y).A(x).

The original Khintchine inequalities considered Rademacher variables and M com- mutative. In this case for any p∈(0,∞),

kGxk_p≈

∞

X

i=0

|x_i|²

!1/2 p

.

In the noncommutative context however, the formulation of noncommutative Khintchine inequalities in Lp depends of whether p ≤2 or p ≥2. This is due to the fact that two dierent square functions can be dened:

Sc(x) =

∞

X

i=1

x^∗_ixi

!1/2

andSr(x) =

∞

X

i=1

xix^∗_i

!1/2

.

With these notations, the noncommutative Khintchine inequalities state that if p ∈ (2,∞):

kGxk_p ≈max

kS_c(x)k_p,kS_r(x)k_p , and if p≤2:

kGxk_p ≈infn

kS_c(z)k_p+kS_r(y)k_p :y+z=xo .

It is therefore natural to try and characterize the symmetric spaces in which one of these (quasi-)norm equivalences hold. More precisely, denote by S(M) the space of nite sequences of nitely supported elements of Mand dene the following properties:

• Kh∩(E,M): for anyx∈S(M),

kGxk_E ≈max (kS_c(x)k_E,kS_r(x)k_E) =:kxk_R

E∩C_E,

• KhΣ(E,M): for any x∈S(M),

kGxk_E ≈inf{kS_c(z)k_E+kS_r(y)k_E :y+z=x}=:kxk_R

E+C_E.

IfB(`²)⊗L_∞(0,1)embeds (by a unital trace preserving homomorphism) in M, we char- acterize the symmetric spaces having properties Kh∩(E,M) and Kh_Σ(E,M) in terms of monotonicity and interpolation properties if the sequence(ξi)i∈Nis constituted of free Haar unitaries or independent Rademacher variables.

Theorem 1.4. Let E be a quasi-Banach symmetric space with the Fatou property. As- sume that M=B(`<sup>2</sup>)⊗L<sub>∞</sub>(0,1)and that (ξ<sub>i</sub>)i∈N is a sequence of free Haar unitaries in A. Denote by E:B(`²)→ B(`²) the conditional expectation onto the diagonal. Then the following properties are equivalent:

(i) Kh∩(E,M),

(ii) E is left-2-monotone,

(iii) E is an interpolation space for the couple (L2, L∞), (iv) ∀x∈ M⁺,

(E ⊗Id)(x²)^1/2

E .kxk_E. The following properties are also equivalent:

(i) KhΣ(E,M),

(ii) E is right-2-monotone,

(iii) there existsp <2 such that E is an interpolation space for the couple (Lp, L2), (iv) ∀x∈ M⁺,kxk_E .

(E ⊗Id)(x²)^1/2 E.

A similar statement holds for Rademacher variables. In this caseKh∩(L∞,M)never holds but this problem can be dealt with thanks to works of Astashkin [2]. We are very grateful to D. Zanin for pointing out this reference to us.

In the theorem above, the most dicult implications to prove are (ii) ⇒ (i) (or (iii) ⇒ (i)) and for these we will refer to [10] where the case of Kh∩ is handled, [26]

whereKhΣis proven ifE is anLp-space and [5] which allows to interpolate the previous result and obtain Kh_Σ for a general symmetric space E. The equivalence between (ii) and(iii)is of purely commutative nature. It is where the two problems we are interested in intersect and is obtained in sections 3 and 4.

In section 6, we prove that (i) ⇒ (iv) ⇒ (ii). In particular, (iv) ⇒ (ii) is an application of the following well-known theorem (see [13]):

Theorem 1.5 (Schur-Horn). Let N ∈ N. Let a, b ∈ R^N+ be non-increasing sequences such that:

∀k≤N,

k

X

i=1

a_i ≥

k

X

i=1

b_i and

N

X

i=1

a_i=

N

X

i=1

b_i. Then,

• b belongs to the convex hull of

(a_σ(1), . . . , a_σ(N)) :σ ∈S_n ⊂R^N,

• there exists a Hermitian matrix in M ∈Mn(C) such that the eigenvalues of M are given by a and the diagonal ofM is given by b.

2. Preliminaries

2.1. Interpolation. For a detailed exposition of interpolation theory, see [3]. We simply recall here the main denitions and properties that will be used later on. Note that interpolation theory is often dened in the context of Banach spaces but translates well to the quasi-Banach setting ([3], section 2.9). We start with the denition of an interpolation space.

Denition 2.1. Let(A, B) be a compatible couple of quasi-Banach spaces. We say that a quasi-Banach space E is an interpolation space for this couple if A∩B ⊂E ⊂A+B with constant C >0 if for every bounded operator T :A+B → A+B such that T|_A (resp.T|_B) is a contraction from A toA (resp. B toB),T is bounded fromE toE with norm less than C. IfC = 1, we say that E is an exact interpolation space.

Both for explicit constructions of interpolation spaces with the real method and for the general theory of interpolation, theK-functional is a fundamental tool. It is dened as follows:

Denition 2.2. Let(A, B)be a compatible couple of quasi-Banach spaces andx∈A+B. For all t >0 dene the K-functional of x by:

K_t(x, A, B) = inf{kyk_A+tkzk_B :y∈A, z ∈B, y+z=x}.

It enables to state a simple sucient condition for a space to be an interpolation space.

Denition 2.3. Let(A, B) be a compatible couple of quasi-Banach spaces. We say that a quasi-Banach spaceE isK-monotone for (A, B) with constantC >0 ifA∩B⊂E⊂ A+B and for all x∈E and y∈A+B, verifying

∀t >0, K_t(x, A, B)≥K_t(y, A, B), then y∈E and

kyk_E ≤Ckxk_E.

The following fact is well-known. We prove it here in the context of quasi-Banach spaces for completion.

Proposition 2.4. If E isK-monotone with constantC for the couple (A, B) then E is an interpolation space between A andB with the same constant.

Proof. Let T : A+B → A+B be a bounded operator such that its restriction to A (resp. B) is a bounded operator with norm1onA(resp. B). Letx∈E and lety=T x, y ∈ A+B. Let t > 0,A+tB is an exact interpolation space for the couple (A, B) so kxk_A+tB ≥ kyk_A+tB. This means that for all t >0, Kt(x, A, B) ≥Kt(y, A, B). Hence, by K-monotonicity of E,y ∈E and kyk_E ≤Ckxk_E. So T denes a bounded operator

of norm less thanC on E.

If reciprocally, every interpolation space of a couple (A, B) is K-monotone then we say that(A, B)is an Calderón couple.

2.2. Symmetric spaces. We start by introducing some notations. Measures will be, if not mentioned otherwise, denoted byν. If (Ω, ν) is a measure space, denote byL₀(Ω, ν) or simply L0(Ω) (if no confusion can occur) the set of measurable functions f on Ω such that ν({|f|> t}) is nite for some t∈R. To anyf ∈L₀(Ω), we associate a non- increasing rearrangement which is a function inL0(0, ν(Ω)), denoted by f^∗ or µ(f) and dened for t∈(0, ν(Ω))by:

f^∗(t) =µ_t(f) = inf{kf1_Ak_∞:ν(Ω\A)≥t}.

A quasi-Banach symmetric space on Ω is a nonzero subspace of L0(Ω) which is re- arrangement invariant (the quasi-norm of a function only depends on its distribution) and equipped with an increasing quasi-norm. A symmetric space E is said to have the Fatou property if for every increasing net (f_n)n∈I of elements ofE such that(kf_nk_E)n∈I

is bounded andfn↑f a.e., thenf ∈Eandkf_nk_E ↑ kfk_E. An introduction to symmetric spaces can be found in [18].

For the remainder of the paper, x α∈(0,∞]. Denote simply by L_p,p ∈(0,∞]the spaceLp(0, α)where(0, α)is equiped with the Lebesgue measure. A quasi-Banach sym- metric space on(0, α) will be called a symmetric function space. Similarly, a symmetric

sequence space is a quasi-Banach symmetric space on N endowed with the counting measure. Note that ifν(Ω)≤α, we can dene a space E(Ω)by:

E(Ω) ={f ∈L₀(Ω) :f^∗∈E} and ∀f ∈E(Ω),kfk_E(Ω)=kf^∗k_E. We end this subsection by a technical lemma that will be useful later on.

Lemma 2.5 (Hölder type inequality for positive operators). LetT be a positive operator on L0. Then, for anya, b∈L⁺₀ and s, s⁰>0such that1 =s⁻¹+s⁰⁻¹,

T(ab)≤T(a^s)^1/sT(b^s0)^1/s0. Proof. Recall that for anyx, y∈R⁺,

xy= inf

λ>0

λ^sx^s

s +λ^−s0y^s0 s⁰ . Hence,

T(a^s)^1/sT(b^s0)^1/s0 = inf

λ>0

λ^sT(a^s)

s +λ^−s0T(b^s0) s⁰ = inf

λ>0T λ^sa^s

s +λ^−s0b^s0 s⁰

!

≥T(ab).

2.3. Interpolation of Lp-spaces. The notion of left-p-monotonicity is almost equiva- lent toK-monotonicity for the couple(Lp, L∞). Indeed, theK-functional of the couple (L_p, L∞)takes the following form (see [12]):

(1) K_t(x, L_p, L∞)≈

Z t^p 0

(x^∗)^{p 1/p}

.

More generally, the K-functional of the couple (L_p, L_q), 0 < p < q < ∞ has been computed up to constants depending on p and q and can be found in [12]. Let r = (p⁻¹−q⁻¹)⁻¹. Then, for allf inLp+Lq and t >0,

(2) Kt(f, Lp, Lq)≈ Z t^r

0

(f^∗)^p 1/p

+t Z ∞

t^r

(f^∗)^q 1/q

.

Combined with the following theorem, this allows, in many cases, to give a precise description of the interpolation spaces for the couple(L_p, L_q) ([21],[6],[27]).

Theorem 2.6. Let α ∈ (0,∞]. The following couples of quasi-Banach spaces are Calderón couples:

• (Lorentz-Shimogaki) p≥1, (L_p(0, α), L∞(0, α)) and (L₁(0, α), L_p(0, α)),

• (Sparr) p, q≥1, (Lp(Ω), Lq(Ω))for any σ-nite measure space Ω.

• (Sparr, Cwikel) p, q∈(0,∞), (L_p(0, α), L_q(0, α)),

• (Cwikel) p∈(0,∞], (`<sup>p</sup>, `^∞).

The denition of left-p-monotonicity and right-q-monotonicity of a spaceE, contrary toK-monotonicity for a couple(A, B), does not impose a condition of the typeA∩B ⊂ E ⊂A+B. In the following lemma, we show that this condition is automatically veried.

Lemma 2.7. Let ∞ > q ≥ p > 0 and C > 0. Let E be a quasi-Banach symmetric function space. Then:

(1) ifE is left-p-monotone and right-q-monotone,L_p∩L_q⊂E ⊂L_p+L_q,

(2) ifE is left-p-monotone,Lp∩L∞⊂E⊂Lp+L∞.

Proof. We only prove (1) since (2) can be obtained by similar arguments. Let us prove the rst inclusion. Since E is a symmetric function space, it contains characteristic functions of sets of nite measure (see [8]). Let f ∈ (L_p ∩L_q)⁺. Decompose f into f1 =f1_{{f >1}} andf2 =f1_{f≤1}. Set,

h₁ = kf₁k_q

ν({f >1})^1/q1_{{f >1}} and h₂ =1_(0,kf2k^p

p).

Since h^q₁ is the mean of f₁^q (on the support off₁), h^q₁. f₁^q. Hence, f₁ ∈E. Now let us show that h^p₂ f₂^p. Ift <kf₂k^p_p,

Z t 0

h^p₂ = Z t

0

1≥ Z t

0

(f₂^∗)^p, and if t≥ kf₂k^p_p,

Z t 0

h^p₂ =kf₂k^p_p= Z α

0

(f₂^∗)^p≥ Z t

0

(f₂^∗)^p. Consequently,f2 ∈E. So f belongs toE.

Let us now prove the second inclusion. Letf /∈L_p+L_q. Decomposef intof₁ and f₂ as above. By assumption, f1∈/ Lp+Lq or f2 ∈/Lp+Lq.

case 1: f₁ ∈/ L_p+L_q. Thenf₁ is not in L_p, this means thatf₁^p a1[0,min(1,α)] for all a >0, hencekf₁k_E =∞,f₁ ∈/E.

case 2: f2 ∈/ Lp+Lq. Similarly,f2∈/ Lqsof₂^q.a1[0,min(1,α)]for alla >0sof2 ∈/E.

2.4. Boyd indices. Boyd indices play an important role in describing the boundedness properties of classical operators on symmetric spaces ([4], [16]). They intervene in our study as a handy tool to connect some properties of quasi-Banach symmetric spaces to interpolation. LetE be a symmetric function space on(0, α).

Denition 2.8. Let t >0. We denote byD_t the dilation operator on E given by:

D_t : E → E

f 7→ [s7→f(s/t)] ,

where by convention f(s) = 0 ifs≥α. The Boyd indices ofE are given by:

α_E = lim

t→0

logkD_tk_E→E logt = sup

t<1

logkD_tk_E→E logt , and

β_E = lim

t→∞

logkD_tk_E→E logt = inf

t>1

logkD_tk_E→E logt .

Remark 2.9. Sinceα_E is dened as a supremum, ifα_E = 0thenkD_tk_E→E = 1for all t∈(0,1].

Remark 2.10. SinceE is a quasi-Banach space,β_E <∞.

Proof. By denition, there exists a constantC >0such that for anyf, g∈E,kf+gk_E ≤ C(kfk_E +kgk_E). Consequently, kD₂k_E→E ≤2C and β_E ≤ ^{log 2C}_{log 2} .

For any p, q∈(0,∞), consider the map:

H^(p) : E → E

f 7→ h

s7→ ¹_sR_s

0(f^∗)^p1/pi , and:

H_(q) : E → E

f 7→ h

s7→ ¹_sR∞

s (f^∗)^q1/qi . The following lemma is a particular case of [25, Theorem 2].

Lemma 2.11. Let E be a symmetric function space such that p < _β¹

E and q > _α¹

E. ThenH^(p) andH_(q) are bounded onE.

Remark 2.12. In [25], contrary to the denition we took, symmetric spaces are auto- matically assumed to verify some version of the Fatou property. However, the proof of Lemma 2.11 given in [25] does not use this hypothesis so we do not reproduce it here.

As a consequence of Lemma 2.11, we obtain an interpolation result which can already be found in S. Dirksen's PhD thesis [8].

Proposition 2.13. Assume that p < _β¹

E. Then, E is an interpolation space for the couple(Lp, L∞).

Proof. By Lemma 2.11, H^(p) is well-dened and bounded from E to E, in particular E ⊂Lp+L∞. Letf ∈E andg∈Lp+L∞ and assume that f^p g^p. This means that H^(p)(f)≥H^(p)(g)≥g^∗. SinceH^(p)(f)belongs toE this means thatgbelongs toE and that:

kgk_E ≤

H^(p)(f)

E .kfk_E.

Hence, E isK-monotone for the couple (Lp, L∞). This concludes the proof by Proposi-

tion 2.4.

Combining the previous proof and Remark 2.10, we can obtain the following.

Corollary 2.14. There existsp >0 such thatE is is left-p-monotone.

Recall also the following classical result [25, Theorem 3]. Once again, the proof given in [25] applies without modication to our context since it only uses the boundedness of H^(p) and H_(q) combined with (2).

Proposition 2.15. Assume that 0< p < _β¹

E ≤ _α¹

E < q≤ ∞. Then, E is an interpola- tion space for the couple (L_p, L_q).

The next lemma will be needed in the last section of the paper. We follow [2, Lemma 7.2] but we provide a more precise statement.

Lemma 2.16. Suppose thatαE = 0. Let ϕ∈L∞(0,1)consider the map:

T_ϕ : E → E((0,1)×(0, α)) f 7→ [(s, t)→ϕ(s)f(t)] . Then,|||T_ϕ|||=kϕk_∞.

Proof. It is clear that |||T_ϕ||| ≤ kϕk_∞. Let a <kϕk_∞. This means that ν({|ϕ|> a}) = b >0. Remark that for any g∈E andt >0,

νn

T_a1|ϕ|>a(g) > to

=ν({|ϕ|> a} × {|ag|> t}) =bν({|ag|> t}). And since b≤1,

ν(|aD_b(g)|> t) =bν({|ag|> t}). Hence, µ(T_a1|ϕ|>a(g)) =aµ(D_b(g)).

Since α_E = 0,kD_bk_E→E = 1. Let ε >0, there exists x∈E such that kxk_E = 1 and kD_b(x)k_E ≥1−ε. Hence,

|||T_ϕ||| ≥ kT_ϕ(x)k_E ≥

Ta1|ϕ|>a(x)

E =akD_b(f)k_E ≥a(1−ε).

Since this is true for anya <kϕk_∞ andε >0, we can conclude that|||T_ϕ||| ≥ kϕk_∞. 3. Interpolation spaces for the couple (Lp, L∞)

Recall thatα∈(0,∞]is xed and that for anyp∈[0,∞]we write Lp =Lp(0, α). We are interested in the following extension of the Lorentz-Shimogaki theorem to p <1: Theorem 3.1. Let p <1 and E be a symmetric function space. Then E is an interpo- lation space for the couple (L_p, L∞) if and only if E is left-p-monotone.

We will deduce the theorem above from the following proposition:

Proposition 3.2. Letf, g∈Lp+L∞andd >0. Suppose thatf andgare non-negative, non-increasing, left-continuous, take values in D = {dⁿ}_n∈Z∪ {0} and f^p g^p. Then there exists an operator T :Lp+L∞→Lp+L∞ and a positive function h such that:

• h≤2^1/pf

• T h^∗=g,

• the restrictions of T on L_p and L∞ are contractions.

Proof. Dene h := 2^1/pf12f^p≥g^p. Let t > 0, let us show that Rt

0h^p ≥Rt

0g^p. Note that g^p >2f^p ⇔g^p <2(g^p−f^p).

Z t 0

h^p− Z t

0

g^p = Z t

0

1_2f^p≥g^p(2f^p−g^p)−1_2f^p_<g^pg^p

≥ Z t

0

1_2f^p≥g^p2(f^p−g^p)−1_2f^p_<g^p2(g^p−f^p)

= 2 Z t

0

f^p−g^p

≥0.

Denote X:=kgk_p∈(0,∞]. Dene:

H:t7→Rt

0 h^p andG:t7→Rt 0g^p.

Letx∈[0, X), note thatG⁻¹(x) is well-dened sinceGis continuous and increasing on [0, β], the support of g. Dene also H⁻¹(x) = minH⁻¹({x}) (H is continuous). Note that h(H⁻¹(x))6= 0. Indeed, by denition of t_x := H⁻¹(x), for all ε > 0, Rtx

tx−εh 6= 0 so there exists an increasing sequence (t_n) converging to t_x and such that h(t_n) 6= 0.

This means that 2f(tn)≥ g(tn). Using the fact that f is left-continuous and g is non- increasing, we obtain 2f(t_x) ≥ g(t_x). And since we showed that H ≥ G, H⁻¹(x) ≤ G⁻¹(x), henceg^p(H⁻¹(x))≥g^p(G⁻¹(x)). In conclusion:

(3) h^p(H⁻¹(x))≥g^p(G⁻¹(x)).

Consider now h^∗ and H∗ : t 7→ Rt

0(h^∗)^p. By construction, h restricted to {h > 0} is non-increasing andh^∗ is non-increasing too. Sinceh andh^∗ also have a countable range, they are of the form P

i∈N

ai1Ii where theIiare intervals. Lettin the support of hso that tis not an end point of one of those intervals then since h is non-increasing:

Z

h>h(t)

h <

Z t 0

h <

Z

h≥h(t)

h.

Similarly, ift⁰ is in the support ofh^∗ and not at an end point, Z

h^∗>h^∗(t⁰)

h^∗<

Z _t⁰

0

h^∗<

Z

h^∗≥h^∗(t⁰)

h^∗.

Sinceh and h^∗ have the same distribution, this means that for all s >0, R

h>sh=R

h^∗>sh^∗ and R

h≥sh=R

h^∗≥sh^∗.

By the previous inequalities, we deduce that except for a countable number of tand t⁰, if H(t) =H∗(t⁰) thenh(t) =h^∗(t⁰). SinceH∗ is strictly increasing onH_∗⁻¹((0, X)), this implies by (3) that for allx∈(0, X), except maybe a countable number,

(4) h^∗(H_∗⁻¹(x)) =h(H⁻¹(x))≥g(G⁻¹(x)).

Let φ = H_∗⁻¹◦G on (0, β). Remark that since h^∗ and g have range in 2^1/pD and D respectively,H∗ andGare piecewise ane on every compact of(0, β). SinceH∗◦φ=G, by derivating, for allt∈(0, α), except maybe a countable number,

(5) φ⁰(t)(h^∗◦φ)^p(t) =g^p(t).

Hence deneT :L_p+L∞→L_p+L∞by T(u) = (φ^01/p·u◦φ)◦1_(0,β). By the change of variables formula, T is a contraction on Lp. Indeed, one can apply the formula on any compact of(0, β)and obtain thep-norm as a limit. Lett∈(0, α)and write t=G⁻¹(x), then by (4):

φ⁰(t) = (H_∗⁻¹◦G)⁰(t) = g^p(t)

(h^∗)^p(H∗⁻¹(G(t)) =

g(G⁻¹(x)) h^∗(H∗⁻¹(x))

p

≤1,

except for a countable number of values of t. SoT is also a contraction onL∞. Finally,

by (5), T h^∗=g.

Proof of Theorem 3.1. The reverse implication is a consequence of Proposition 2.4 and Lemma 2.7. So let us focus on the direct implication. Let E be a interpolation space for the couple (L_p, L∞)with constant C. Let f ∈E and g∈L_p+L∞ such that for all t >0,Kt(f, Lp, L∞)≥Kt(g, Lp, L∞). By (1), we can replace this condition byf^p g^p. Letd∈(1,∞) and denef2 =d^{blogd(f∗)c+1} andg2=d^blogd(g∗)c. We havef^∗ ≤f2≤df^∗, g^∗ ≤dg₂≤dg^∗, hencef₂^p g^p₂. Furthermore,f₂ andg₂take values inD={dⁿ}_n∈Z∪{0}.

Takeh andT given by Proposition 3.2. Since h≤2^1/pf2, h∈E so by the denition of an interpolation space,T(h^∗) =g₂ ∈E and nally g∈E. Furthermore:

kgk_E =kg^∗k_E ≤dkg₂k_E ≤dCkhk_E

≤dC2^1/pkf₂k_E ≤d²C2^1/pkf^∗k_E =d²C2^1/pkfk_E.

This is true for anyd∈(1,∞) so,kgk_E ≤C2^1/pkfk_E. 4. Interpolation spaces for the couple (L_p, L_q)

4.1. Function spaces. In this section, we prove the main theorem of this paper, a characterization of interpolation spaces between Lp and Lq. The proof uses a strategy similar as [7]. Let us rst state the result precisely.

Theorem 4.1. Let0< p≤q <∞ andE be a quasi-Banach symmetric function space.

Then E is left-p-monotone and right-q-monotone if and only if E is an interpolation space between Lp andLq.

Proof of the reverse implication. Suppose thatEis an interpolation for the pair(Lp, Lq). Then by reiteration, E is an interpolation space for the pair(L_p, L∞) and by Theorem 3.1, E is left-p-monotone. Let f ∈ E and g ∈Lp+Lq such that f^q. g^q. Assume that f and gare positive by taking if necessary their modules. We will show that g∈E and kgk_E .kfk_E. Recall that (Lp, Lq) is a Calderón couple (Theorem 2.6), so it is enough to show that for allt >0,

K_t(g, L_p, L_q).K_t(f, L_p, L_q).

Recall also that, for all h∈L_p+L_q and t >0 ([12]):

Kt(h, Lp, Lq)≈ Z t^r

0

(h^∗)^{p 1/p}

+t Z ∞

t^r

(h^∗)^q 1/q

,

wherer= (p⁻¹−q⁻¹)⁻¹. Let t >0. First, since f^q. g^q, t

Z ∞ t^r

(g^∗)^q 1/q

≤t Z ∞

t^α

(f^∗)^q 1/q

≤K_t(f, L_p, L_q).

We now have to estimate the other term i.e to prove that:

Z t^r 0

(g^∗)^p 1/p

.Kt(f, Lp, Lq).

Suppose now that f and g are bounded so that f^q and g^q belong to L1. Using again the fact that f^q. g^q, it is easy to see that there exists g⁰ ≥ g such that kg⁰k_q = kfk_q and g^0q f^q (for example by letting g⁰ = g+h where h is a function taking only the valueskfk_∞+ 1and 0andkg⁰k_q =kfk_q). Then by [11] (Theorem 4.7, (1)), there exists a unital, positive, integral preserving operator T such that T((g^0∗)^q) = (f^∗)^q. Hence T((g^∗)^q)≤(f^∗)^q. Denee=1_(0,t^r₎ and write,

Z t^r 0

(g^∗)^p 1/p

= Z ∞

0

e(g^∗)^p 1/p

= Z ∞

0

T(e(g^∗)^p) 1/p

.

Now, apply Lemma 2.5 with s=q/p and s⁰ =r/p to obtain:

Z t^r 0

(g^∗)^{p 1/p}

≤ Z ∞

0

T(e)^p/rT((g^∗)^q)^p/q 1/p

. Z t^r

0

(f^∗)^pT(e)^p/r 1/p

+ Z ∞

t^r

(f^∗)^pT(e)^p/r 1/p

To estimate the left summand, we use the fact that kT ek_∞ ≤1 and for the right sum- mand, we apply Hölder's inequality.

≤ Z t^r

0

(f^∗)^p 1/p

+ Z ∞

t^r

(f^∗)^q

1/qZ ∞ t^r

T(1_(0,t^r₎) 1/r

.Kt(f, Lp, Lq) where we used that R∞

t^r T(1_(0,t^r₎)1/r

≤ R∞

0 T(1_(0,t^r₎)1/r

= t. To conclude for un- bounded functions, it suces to approximateg^∗ and f^∗. A way to do it is to apply the previous inequality tog^∗(.−ε) and f^∗(.−ε) and letεgo to 0. For the direct implication, the following lemma is the key argument. It enables us to relate left-p-monotonicity and right-q-monotonicity to K-monotonicity for the couple (L_p, L_q).

Lemma 4.2. Let 0 < p ≤ q < ∞, r > 0 and f, g ∈ (L_p +L_q)⁺ be non-increasing right-continuous functions such that for allt >0:

Z t^r 0

f^{p 1/p}

+t Z ∞

t^r

f^q 1/q

≥ Z t^r

0

g^{p 1/p}

+t Z ∞

t^r

g^q 1/q

,

then there exist nonegative functions h and l such that h+l =g, h is non-increasing, f^ph^p andf^q. l^q. More precisely, for allt >0,

Z ∞ t

f^q≥ Z ∞

t

l^q. Proof. Let A := {t ∈ R⁺ : Rt

0f^p ≥ Rt

0g^p} and B := {t ∈ R⁺ : R∞

t f^q ≥ R∞ t g^q}. A and B are closed sets and A∪B = (0,∞), denote A^c = R⁺\A. For t ∈ R⁺, set a(t) = minA∩[t,∞)(a(t) =∞ if A∩[t,∞) =∅) and for convenience, g(∞) = 0.

Deneh:t7→g(a(t)) andl=g−h,g andl are clearly non-negative.

Note that for t∈A^c, ifa(t) 6=∞ then f(a(t))≥g(a(t)). Indeed, assume by contra- diction thatf(a(t))< g(a(t)). By left-continuity off and g there existss∈(t, a(t))(in particular, s /∈A) such thatf < g on(s, a(t)). Hence:

Z a(t) 0

f^p= Z s

0

f^p+ Z a(t)

s

f^p <

Z s 0

g^p+ Z a(t)

s

g^p = Z a(t)

0

g^p,

which contradicts the fact that a(t)∈A. This implies that f(t)≥h(t)for any t∈A^c. Let us now check that the decomposition veries what we claimed. Let t ∈R⁺ and let b(t) = maxA∩[0, t](remark that 0∈A so b is well-dened). Clearly (b(t), t)⊂A^c and b(t)∈A. Hence,

Z t 0

h^p = Z b(t)

0

h^p+ Z t

b(t)

h^p ≤ Z b(t)

0

g^p+ Z t

b(t)

f^p≤ Z t

0

f^p.

Finally, set c(t) = infA^c ∩[t,∞) (c(t) = ∞ if A^c ∩[t,∞) = ∅). Note that since B∪A=R⁺, we always havec(t)∈B and remark thatl is supported in A^c. Hence,

Z ∞ t

l^p ≤ Z ∞

c(t)

g^p ≤ Z ∞

c(t)

f^p ≤ Z ∞

t

f^p, and since lis positive:

Z ∞ t

(l^∗)^q ≤ Z ∞

t

l^q≤ Z ∞

t

f^q= Z ∞

t

(f^∗)^q.

We are now ready to conclude the proof of the main theorem.

End of the proof of Theorem 4.1. By Proposition 2.4, it suces to prove that E is K- monotone. We already know from Lemma 2.7 thatLp∩L_q⊂E ⊂Lp+Lq. Assume that f 6= 0. Suppose that for all t >0,K_t(f, L_p, L_q) ≥K_t(g, L_p, L_q). By the formula (2) for theK-functional of the couple (Lp, Lq), there exists a constant c >0 independent off and gsuch that for all t >0:

Z t^α 0

(cf^∗)^p 1/p

+t Z ∞

t^α

(cf^∗)^q 1/q

≥ Z t^α

0

(g^∗)^p 1/p

+t Z ∞

t^α

(g^∗)^q 1/q

,

So by Lemma 4.2 applied to cf^∗ and g^∗, there exists two functions h and l such that h+l=g^∗,(cf)^p h^p and(cf)^q. l^q. Hence, by the assumptions onE,h, l∈E sog∈E and

kgk_E =kg^∗k_E .khk_E +klk_E ≤2cKkfk_E.

We can now obtain the theorem stated in the introduction as a corollary.

Proof of Theorem 1.2. (1)⇒(2). By corollary 2.14 there existspwhich can be chosen to be less thanq such thatE is left-p-monotone. Hence by Theorem 4.1,E ∈Int(L_p, L_q).

(2)⇒(1). This is true by Theorem 4.1.

Corollary 4.3. LetE be a symmetric quasi-Banach function space. Assume thatα_E 6=

0. Then, for any q∈(1/α_E,∞),E is right-q-monotone.

Proof. By Remark 2.10 and Proposition 2.15,E ∈Int(Lp, Lq)for some p≤q. Remark 4.4. Let us dene, as a convention, that every symmetric spaceE is right-∞- monotone. Then, Theorem 4.1 holds for q=∞ and generalises Theorem 3.1.

4.2. An application to convexications. Recall that if E is a symmetric function space, ther-convexication of E is given by:

E^(r)={f ∈L₀(0, α) :|f|^r∈E}, with kfk_E(r) =k|f|^rk_E. Lemma 4.5. LetE be a symmetric function space. Let p, q, r∈(0,∞). Then:

• E is left-p-monotone if and only if E^(r) is left-rp-monotone,

• E is right-q-monotone if and only if E^(r) is right-rq-monotone.

Proof. Let f ∈E^(r) and g∈ L0(0, α) such that |f|^rp |g|^rp. By denition |f|^r ∈E so by left-p-monotonicity of E, |g|^r ∈E and k|g|^rk_E .k|f|^rk_E. This exactly means that g∈E^(r) andkgk_E(r) .kfk_E(r).

The rest of the statement is checked similarly.

We can now deduce the following corollary, which extends a result of Montgomery- Smith to quasi-Banach spaces.

Corollary 4.6. Let p < q ∈ (0,∞] and r ∈ (0,∞). Let E be a symmetric function space. Then the following are equivalent:

(1) E is an interpolation space for the couple(Lp, Lq), (2) E^(r) is an interpolation space for the couple (L_rp, L_rq).

Proof. If q6= 0, this is immediate by Lemma 4.5 and Theorem 4.1. Ifq = 0, this is also

direct using Lemma 4.5 and Theorem 3.1.

4.3. Sequence spaces. We will now prove the main theorem in the context of symmetric sequence spaces rather than symmetric function spaces. We will use the fact that `^∞ can be embedded into L∞ by:

u7→

∞

X

i=1

ui1[i−1,i).

Hence, we will identify `^∞ with the subalgebra of L∞ of functions a.e. constant on intervals of the form [i, i+ 1),i∈N. Denote byE the conditional expectation fromL∞

to`<sup>∞</sup>. The notions of left and right majorization can be extended without modications to sequences and coincide with the usual notion. Let uand v in`^∞:

uv⇔ ∀n∈N, Pn i=1

u^∗_i ≥ Pⁿ

i=1

v^∗_i and u . v⇔ ∀n∈N,

∞

P

i=n

u^∗_i ≥

∞

P

i=n

v^∗_i. Let us now state the result of this subsection:

Theorem 4.7. Let 0< p≤q <∞ andE be a quasi-Banach symmetric sequence space.

If E is left-p-monotone and right-q-monotone then E is an interpolation space between

`p and`q.

Proof. By Proposition 2.4, it suces to check that E is K-monotone. Similarly to the case of function spaces, the hypothesis of the theorem imply that `<sup>p</sup> = `^p∩`^q ⊂ E ⊂

`<sup>p</sup>+`^q=`<sup>q</sup> and that for anyu∈E andv ∈`^q:

u^q. v^q ⇒v∈E,kvk_E ≤Ckuk_E and u^p v^p⇒v∈E,kvk_E ≤Ckuk_E.

Letu∈E and v∈`<sup>q</sup>,u6= 0. Assume that for allt >0, K<sub>t</sub>(u, `^p, `<sup>q</sup>)≤K<sub>t</sub>(v, `^p, `<sup>q</sup>). The formula for the K-functional used previously still holds in this context. Indeed, for any sequencew∈`^q andt >0:

K_t(w, p, q)≈ Z t^α

0

(w^∗)^{p 1/p}

+t Z ∞

t^α

(w^∗)^q 1/q

.

So there exists a constantc independent of uand v so that for all t >0: Z t^α

0

(cu^∗)^p 1/p

+t Z ∞

t^α

(cu^∗)^q 1/q

≥ Z t^α

0

(v^∗)^p 1/p

+t Z ∞

t^α

(v^∗)^q 1/q

,

By Lemma 4.2, there exists nonegative functions h and l such that, h+l = v^∗, h is non-increasing,(cu)^p h^p and (cu)^q. l^q. Let h=E(h^p)^1/p and l=E(l^q)^1/q.

We will start by showing that (cu)^ph^p and(cu)^q. l^q. The functions:

t7→R_t

0(cu^∗)^p and t7→R_t

0h^p

are ane on intervals of the form [i, i+ 1], i∈N. So it suces to check the inequality at integers. Let n∈N,

Z n 0

(cu^∗)^p ≥ Z n

0

h^p = Z n

0

h^p. The other inequality is proven in the same manner.

Recall that h and l can be seen as sequences. Note that a_p := max(1,2^p−1) is the optimal constant such that for all a, b∈R⁺,(a+b)^p ≤ap(a^p+b^p). We have:

v^∗=E((v^∗)^p)^1/p≤apE(h^p+l^p)^1/p≤apa_1/p

E(h^p)^1/p+E(l^p)^1/p

≤apa_1/p(h+l), where we used the fact thatE(l^p)^1/p≤E(l^q)^1/q sinceq ≥p. By hypothesis on E,h∈E and l∈E with

h

E .kuk_E and l

E .kuk_E.

So v∈E andkvk_E .kuk_E with an implied constant depending onE,p and q.

Remark 4.8. As was pointed out to me by M. Cwikel, the couple(`<sup>p</sup>, `^q) is not known in general to be a Calderón couple. This is why we cannot show the reverse implication in Theorem 4.7. However, for 1≤ p ≤q ≤ ∞, the couple (`<sup>p</sup>, `^q) is a Calderón couple ([27]) and we can recover the following:

Corollary 4.9. Let1≤p≤q <∞andE be a quasi-Banach symmetric sequence space.

ThenEis left-p-monotone and right-q-monotone if and only ifEis an interpolation space between `<sub>p</sub> and `_q.

This is what was obtained in [7] by Cwikel and Nilsson.

5. A link between p-monotonicity and p-convexity

In this subsection we prove some lemmas linkingp-convexity (resp. q-concavity) and left-p-monotonicity (resp. right-q-monotonicity). First, we show thatp-convexity implies some weak form of the left-p-monotonicity. Then, we show that under the assumption thatEhas the Fatou property, this weak form of left-p-monotonicity is, in fact, equivalent to left-p-monotonicity by a simple approximation argument.

Recall thatE is said to bep-convex with constantC if for alln∈Nand(xi)i≤n∈Eⁿ,

n

X

i=1

|x_i|^p

!1/p E

≤C

n

X

i=1

kx_ik^p_E

!1/p

.

Similarly,E is said to beq-concave with constantC if for all n∈N and(xi)i≤n∈Eⁿ,

n

X

i=1

kx_ik^q_E

!1/q

≤C

n

X

i=1

|x_i|^q

!1/q E

.

We will denote by F the space of dyadic step functions:

F :={f ∈L∞(0,∞) :∃n∈N,∃N ∈N,∃a∈(R⁺)^N, f =

N

X

i=1

ai1_[(i−1)2⁻ⁿ_,i2⁻ⁿ₎}.

Note that this space will also be useful in section 6.

Lemma 5.1. Letf, g∈F be non-increasing andp∈(0,∞).

(1) If f^p g^p then there exists h∈F such thath≥g,f^p h^p andkfk_p=khk_p. (2) If f^p. g^p then there exists h∈F such thath≥g,f^p. h^p and kfk_p =khk_p. Proof. We believe that it is not dicult to convince oneself that this lemma is true. We provide a possible construction for h in both cases and leave the details to the reader.

Note the by consideringf^p and g^p, we can suppose thatp= 1.

(1). Fixa∈(0, α],a <∞ and dyadic, such thatf and g are supported in (0, a). For any s≥0, dene,

h_s∈F :t7→max(g(t), s)1_[0,α).

Clearly,hsis non-increasing andhs≥g. Choose the uniques0 such thatkh_s0k₁ =kfk₁. It can be checked that f h_s0. Hence, h_s0 satises the conditions of the lemma.

(2). Since f, g∈F, we can choose a dyadic a≤α such thatf and g are constant on (0, a). For anys≥0, dene,

hs=g+s1_[0,a)∈F,

and proceed as in (1).

Remark 5.2. Let p∈(0,∞). Ifkfk_p=kgk_p <∞ then|f|^p |g|^p ⇔ |g|^p.|f|^p. 5.1. Convexity implies left-monotonicity. Our rst lemma is a direct consequence of the geometric form of the Schur-Horn theorem (Theorem 1.5).

Lemma 5.3. LetEbe a quasi-Banach symmetric function space,p-convex with constant C. Let f, g∈F. Then,

f^p g^p ⇒ kgk_E ≤Ckfk_E.

Proof. Let f, g∈F suppose that f^p g^p. Since k.k_E is increasing, we can also assume that kfk_p =kgk_p by Lemma 5.1. Since f and g belong toF, there exist N, n∈N and vectorsa= (ai)i≤N and b= (bi)i≤N in(R⁺)^N such that:

f =

N

X

i=1

a_i1_[(i−1)2⁻ⁿ_,i2⁻ⁿ₎ andg=

N

X

i=1

b_i1_[(i−1)2⁻ⁿ_,i2⁻ⁿ₎.

The hypothesis on f and g means that a^p b^p and kak_p = kbk_p. This means by the Schur-Horn theorem thatb^p is in the convex hull of the permutationsa^pσ ofa^p where for σ∈S_N, we writea^pσ := (a^p_σ(i))1≤i≤N. Let

f_σ =

N

X

i=1

a_σ(i)1_[(i−1)2⁻ⁿ_,i2⁻ⁿ₎.

This means that there exist non-negative coecients(λ_σ)σ∈S_N adding up to1such that g^p = X

σ∈S_N

λσf_σ^p = X

σ∈S_N

λ^1/pfσ

p

.