Real Interpolation method, Lorentz spaces and refined Sobolev inequalities

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Real Interpolation method, Lorentz spaces and refined Sobolev inequalities

Diego Chamorro, Pierre-Gilles Lemarié-Rieusset

To cite this version:

Diego Chamorro, Pierre-Gilles Lemarié-Rieusset. Real Interpolation method, Lorentz spaces and

refined Sobolev inequalities. 2012. �hal-00751688�

Real Interpolation method, Lorentz spaces and refined Sobolev inequalities

Diego CHAMORRO

and Pierre-Gilles LEMARI´ E–RIEUSSET

November 14, 2012

Abstract

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.

Key words: Lorentz spaces, refined Sobolev inequalities, real interpolation spaces, Besov spaces.

1 Introduction

This paper is a generalization of recent results by H. Bahouri and A. Cohen [2] on Lorentz spaces and refined Sobolev inequalities. Let us recall the setting: if q >1, the classical Sobolev inequalities kfkL^p ≤ CkfkW˙^s,q

with 1/p = 1/q−s/n, had been refined by P. G´erard, Y. Meyer and F. Oru [7] using a Besov space in the right-hand side of the inequality:

kfkL^p≤Ckfk^q/p_W˙s,qkfk^1−q/p_˙

B∞^s−n/q,∞

. (1)

Similarly the inequalitykfkL^p≤CkfkB˙^s,qq forq≥1 may be refined by kfkL^p≤Ckfk^q/p_B˙s,q

q kfk^1−q/p_˙

B∞^s−n/q,∞

. (2)

In the previous formula, and for all the following theorems and inequalities, since α, β > 0, we will say that f ∈B˙_q^α,p₀ ⁰∩B˙_q^−β,p₁ ¹ iff can be writen using the Littlewood-Paley decompositionf =X

j∈Z

∆jf and if the semi- normsk · kB˙^α,pq0 ⁰ andk · kB˙q^−β,p1 ¹ are bounded: this way we will choose a natural representation in such spaces.

In[2], H. Bahouri and A. Cohen show that it is possible to improve the estimate kfkL^p,q≤CkfkB˙^s,qq

whereL^p,q is a Lorentz space, into the following inequality kfkL^p,q≤Ckfk^q/p_B˙s,q

q kfk^1−q/p_˙

Bq^s−n/q,∞ (3)

and they prove that this estimate is sharp since it is not possible to replace the norm ˙B^s−n/q,∞q above by a weaker Besov norm ˙Br^s−n/q,∞ with r > q. They also ask the following question: it is possible to improve the inequality

kfkL^p,r≤CkfkB˙^s,qq

into

kfkL^p,r≤Ckfk^q/p_B˙s,q

r kfk^1−q/p_˙

Br^s−n/q,∞ whenr6=q? (4)

The proof they gave for (3) involves difficult estimations of Lorentz norms and could not easily be extended to the caser6=q.

∗†Laboratoire d’Analyse et de Probabilit´es, Universit´e d’Evry Val d’Essonne, 23 Boulevard de France, 91037 Evry Cedex - France

In this article we are going to provide an elementary proof of (3), show that the inequality (4) is valid and we will also study a wider family of related inequalities. The main idea is, in the spirit of [11], to make a systematic use of the characterization of Lorentz spaces as interpolation spaces and to never use the traditional definition of Lorentz spaces as Orlicz spaces.

Our first theorem deals with the two first inequalities given above in a quite general form. As we shall see in the proof given in section 3, this is just a variant of Hedberg’s inequality [8].

Theorem 1 Let α, β >0 andq0, q1∈[1 +∞]. Let θ= _α+β^α ∈(0,1)and let ¹_p = ^1−θ_q0 +_q^θ₁. Then there exists a constant C0 such that, for every f ∈F˙_∞^α,q0 ∩F˙_∞^−β,q1(Rⁿ), we have

|f(x)| ≤C0

sup

j∈Z

2^jα|∆jf(x)|

1−θ sup

j∈Z

2^−jβ|∆jf(x)|

θ

(5) In particular, we get :

kfkL^p≤C0kfk^1−θ_F˙^α,q0

∞

kfk^θ_F˙−β,q1

∞

(6) Let us show briefly how to obtain the two first inequalities from this theorem. We observe that, for 0< s < n/q and for 1< q < +∞, we have ˙W^s,q= ˙F₂^s,q⊂F˙_∞^s,q⊂F˙^s−

n q,∞

∞ . Thus, if we setα=s > 0,−β =s−n/q < 0, q0=qandq1= +∞, we deduce from (6) the inequality (1). In the same way, for 0< s < n/qand 1≤q≤+∞, we have ˙B_q^s,q= ˙F_q^s,q⊂F˙∞^s,q⊂F˙^s−

n q,∞

∞ and we get inequality (2) from inequality (6).

Note also that we obtain by similar arguments the following useful inequality:

kfkL^p≤C0kfk^1−θ_˙

B^α,qq0⁰

kfk^θ_˙

B^−β,qq1 ¹

(7) with ¹_p = ^1−θ_q0 +_q^θ₁ andθ=_α+β^α ∈]0,1[.

Our next result studies inequalities (3) and (4). The proof of this theorem reduces to a few lines once we have in mind the characterization of Lorentz spaces as interpolation spaces.

Theorem 2 Letα, β >0,q0, q1∈[1 +∞]withq06=q1. Letθ= _α+β^α ∈]0,1[, ¹_p =^1−θ_q0 +_q^θ₁ and letr∈[1,+∞].

If f ∈B˙_r^α,q0∩B˙_r^−β,q1(Rⁿ)then f ∈L^p,r(Rⁿ)and we have kfkL^p,r ≤C0kfk^1−θ_B˙α,q0

r kfk^θ_˙

Br^−β,q1. (8)

In the expression above we have the same indexr in Lorentz and Besov spaces, so the the next step is to deal with general Besov spaces ˙B_r^α,q₀ ⁰ and ˙B^−β,q_r1 ¹ and to try to control a Lorentz norm in L^p,r. For this we define θ=_α+β^α ∈]0,1[ and we investigate the validity of the inequality

kfkL^p,r ≤C0kfk^1−θ_B˙α,q0 r0

kfk^θ_B˙−β,q1 r1

. (9)

A scaling argument¹ gives us that necessarily we have ¹_p = ^1−θ_q0 +_q^θ₁. For the index r, we will have a similar condition as it is explained in the next theorem.

It is important to observe that inequality (9) can be studied from the point of view of interpolation theory by the following equivalent problem: B˙_r^α,q₀ ⁰,B˙_r^−β,q₁ ¹

θ,1⊂L^p,r.

However the interpolation between these spaces with these parameters is a delicate issue as it is explained in [10] and, to the best of our knowledge, it was not treated before.

The next result studies the validity of inequality (9) in some particular cases.

Theorem 3 Let α, β, r^∗>0 andθ= _α+β^α with _p¹= ^1−θ_q0 +_q^θ₁ and _r¹∗ =^1−θ_r0 +_r^θ₁.

1The normsk k^X involved in (9) are homogeneous: λ >07→ kf(λx)k^X is an homogeneous function ofλ.

1) For f ∈B˙^α,q_r0 ⁰∩B˙^−β,q_r1 ¹(Rⁿ) and ifr > r^∗, we have f ∈L^p,r(Rⁿ)and kfkL^p,r ≤Ckfk^1−θ_B˙α,q0

r0

kfk^θ_˙

Br^−β,q1 ¹

. 2) Moreover, this inequality is valid forr=r^∗ in the following cases :

a) r=r0=r1,

b) r0=q0 andr1=q1, c) 1< p≤2andr^∗ =p.

3) Finally, the condition r≥r^∗ is sharp.

As we shall see in section 3, theorems 2 & 3 are obtained by direct interpolation. However, it is possible to go one step further with the following results:

Theorem 4 Let α, β > 0, q0, q1 ∈ [1 +∞], q0 < q1. Let θ = _α+β^α ∈]0,1[ and let ¹_p = ^1−θ_q

0 + _q^θ

1. Let q0≤r0≤r1≤q1 and let ¹_r =^1−θ_r0 +_r^θ₁. Then we have

kfkL^p,r ≤C0kfk^1−θ_B˙α,q0 r0

kfk^θ_˙

Br^−β,q1 ¹

.

As we shall see, theorem 4 can be obtained by the use of inequalities (7) and (8) following the ideas given in theorem 2.

Note now that in all these inequalities we have _p¹ =^1−θ_q0 +_q^θ₁ for 1≤q0< q1≤+∞and a similar condition forr, r0andr1: namely ¹_r = ^1−θ_r0 +_r^θ₁. We also assumed the following relationship between these parameters:

q0≤r0≤r1≤q1. A more general result is given with the next theorem:

Theorem 5 Let α, β > 0, q0, q1 ∈[1 +∞], q0 6=q1. Letθ = _α+β^α ∈]0,1[and let ¹_p = ^1−θ_q0 +_q^θ₁. If p≤2, we define

I=

(x, y)∈[0,1]² / x(1−θ) +θy= 1 p

= [(x0, y0),(x1, y1)]

withx0< y0 andx1< y1, then ifr0 andr1satisfyx0≤_r¹₀ ≤ _r¹₁ ≤y0ory1≤_r¹₁ ≤ _r¹₀ ≤x1, and if ¹_r = ^1−θ_r0 +_r^θ₁ then we have

kfkL^p,r ≤C0kfk^1−θ_˙

Br^α,q0 ⁰

kfk^θ_˙

Br^−β,q1 ¹

.

Finally, all these inequalities are sharp in the sense that it is not possible to remove the condition ¹_r = ^1−θ_r0 +_r^θ₁. The plan of the paper is the following: in section 2 we recall some facts about Lorentz and Besov spaces and we will pay a special attention to the interpolation definition of Lorentz spaces. In section 3 we will give the proofs and finally, in section 4 we will treat the part 3) of theorem 3 where we will study the sharpness of these inequalities.

2 Functional spaces and real interpolation method

For Besov and Triebel-Lizorkin spaces we will use the characterization based on a Littlewood-Paley decompo- sition. We start with a nonnegative function ϕ ∈ D(Rⁿ) such that ϕ(ξ) = 1 over |ξ| ≤ 1/2 and ϕ(ξ) = 0 if

|ξ| ≥1. Letψ be defined asψ(ξ) =ϕ(ξ/2)−ϕ(ξ). We define the operatorsSj and ∆j in the Fourier level by the formulas Sdjf(ξ) =ϕ(ξ)fb(ξ) and ∆djf(ξ) =ψ(ξ)fb(ξ). The distribution ∆jf is called thej-th dyadic block of the Littlewood-Paley decomposition off. If lim

j→−∞Sjf = 0 inS^′(Rⁿ), then the equalityf =X

j∈Z

∆jf is called the homogeneous Littlewood-Paley decomposition off.

Definition 2.1 For 1≤p, q≤+∞ ands∈R, we define the homogeneous Besov spaces as the set of distribu- tionsB˙^s,p_q (Rⁿ) =

f ∈ S^′(Rⁿ) :kfkB˙q^s,p<+∞ , where

kfkB˙^s,pq =



X

j∈Z

2^jsqk∆jfk^q_Lp





1/q

with the usual modifications when q= +∞. For 1< p <+∞, 1≤q≤+∞ ands∈R, we define in the same way the homogeneous Triebel-Lizorkin spaces byF˙_q^s,p(Rⁿ) =

f ∈ S^′(Rⁿ) :kfkF˙q^s,p<+∞ with

kfkF˙q^s,p=



X

j∈Z

2^jsq|∆jf|^q





1/q_Lp

with the usual modifications when q= +∞.

Note that the quantities k · kB˙^s,pq and k · kF˙q^s,p are only semi-norms since forj ∈ Z we have ∆jP = 0 for all polynomialsP.

We turn now to Lorentz spaces which are a generalization of the Lebesgue spaces. For (X, µ) a measurable space, they are usually defined in terms of the distribution and rearrangement functionsdf(t) andf^∗(s) given by the formulas

df(t) =µ({x:|f(x)| ≥t}) and f^∗(s) = inf{t:df(t)≤s},

where µ(A) denotes the measure of a set A. Then for 1 ≤ p < +∞ and 1 ≤ r ≤ +∞, the Lorentz spaces L^p,r(X, µ) are traditionally defined in the following way

L^p,r(X, µ) =

f :X −→R:kfkL^p,r<+∞

where

kfkL^p,r =

Z +∞

0

s^p1f^∗(s)rds s

^1/r

, (10)

with the usual modifications whenr= +∞, which corresponds to the weak-L^pspaces. With this characterization is not complicated to see that we haveL^p,p=L^p and that forr0< r1 we have the embeddingL^p,r0 ⊂L^p,r1.

However, the previous formula is not very useful since it depends on the rearrangement functionf^∗ and we will use a more helpful characterization which is given in the lines below.

We recall now some classical results from interpolation theory concerning the real interpolation method.

See [3] for a detailed treatment. IfA0 and A1 are two Banach spaces which are continuously embedded into a common topological vector spaceV, if 0< θ <1 and 1≤r≤+∞, then the real interpolation space [A0, A1]θ,r

may be defined in the following way: f ∈[A0, A1]θ,r if and only iff ∈V andf can be written inV asf =X

j∈Z

fj, withfj∈A0∩A1and (2^−jθkfjkA0)j∈Z∈ℓ^r, (2^j(1−θ)kfjkA1)j∈Z∈ℓ^r. This space is normed with

kfk[A0,A1]θ,r = inf

f=P fj

X

j∈Z

2^−jθrkfjk^r_A0^1/r

+ X

j∈Z

2^j(1−θ)rkfjk^r_A1^1/r

(11)

Forf =X

j∈Z

fj andρ >0, withρ6= 1, we have the following inequality that will be very helpful in the sequel:

kfk[A0,A1]θ,r≤Cρ,θ,r

X

j∈Z

ρ^−jθrkfjk^r_A0(1−θ)/r X

j∈Z

ρ^j(1−θ)rkfjk^r_A1θ/r

. (12)

An important property of the real interpolation method is the reiteration theorem:

Proposition 2.1

1) If θ06=θ1, we have

[A0, A1]θ0,r0,[A0, A1]θ1,r1

θ,r= [A0, A1](1−θ)θ0+θθ1,r. (13) 2) If θ0=θ1, (13) is still valid if ¹_r= ^1−θ_r0 +_r^θ₁.

We saw with the expression (10) how to define Lorentz spaces L^p,r(X, µ) for 1< p <+∞, 1≤r≤+∞as an Orlicz space. However, it will be simpler to use their characterization as real interpolates of Lebesgue spaces [3]:

Proposition 2.2 (Lorentz spaces as interpolation spaces)

1) For 1< p <+∞,1≤r≤+∞

L^p,r= [L¹, L^∞]θ,r withθ= 1−1

p. (14)

2) For p06=p1, we have

[L^p0, L^p1]θ,r= [L^p0,r0, L^p1,r1]θ,r =L^p,r with 1

p =1−θ p0 + θ

p1. (15)

3) In the casep0=p1=pwe have

[L^p,r0, L^p,r1]θ,r =L^p,r if 1

r =1−θ r0 + θ

r1. (16)

Of course, (15) and (16) are consequences of (14) through the reiteration theorem. In this paper, we shall use decompositions (11) and estimates (12) when we deal with functions in Lorentz spaces, and we will mainly con- sider the cases (X, µ) = (Rⁿ, λ) whereλis the Lebesgue measure, or (X, µ) = (Z, µ) withµthe counting measure.

In the case of Lorentz spaces, we can use decomposition (11) with an useful extra property (see [11]) : Lemma 2.1 Let 1< p <+∞, 1≤r≤+∞. Then there exists a constant C0 such that every f ∈L^p,r(X, µ) can be decomposed asf =X

j∈Z

fj where

• (2^{−j(p−1)/p}kfjkL¹)

ℓ^r+(2^j/pkfjkL^∞)

ℓ^r ≤C0kfkL^p,r

• the fj have disjoint supports : ifj6=k,fjfk = 0.

Inequality (12) is very useful to provide an upper bound for the Lorentz norm off and it will be systematically used here. In order to get a lower bound, we shall use the following duality result :

Lemma 2.2 Let1< p <+∞,1≤r≤+∞. Then there exists a constantC0such that for everyf ∈L^p,r(X, µ) and every g∈L^{p−1p ,r−1r} (X, µ), we have f g∈L¹(X, µ)and

Z

f g dµ

≤C0kfkL^p,rkgk

L

p p−1, r

r−1.

Remark 2.1 As we shall see in the proofs given in the section below, the characterization of Lorentz and Besov spaces based on real interpolation is a useful tool since the problem we are dealing with can be studied in terms of weighted sequences. See [3], [5] or [10] for more details concerning the interpolation of Besov spaces.

3 Refined inequalities: the proofs

Proof of theorem 1. We just writeAα(x) = sup

j∈Z

2^jα|∆j(x)|andAβ(x) = sup

j∈Z

2^−jβ|∆j(x)|to obtain

|f(x)| ≤X

j∈Z

|∆jf(x)| ≤X

j∈Z

min

2^−jαAα(x),2^jβAβ(x)

.

We definej0(x) as the largest index such that 2^jβAβ(x)≤2^−jαAα(x) and we write

|f(x)| ≤ X

j≤j0(x)

2^jβAβ(x) + X

j>j0(x)

2^−jαAα(x)≤CAα(x)^α+ββ Aβ(x)^α+βα , (17) thus, inequality (5) is proved. In order to obtain (6), it is enough to apply H¨older inequality in the expression above since we haveθ= _α+β^α and ¹_p =^1−θ_q

0 +_q^θ

1.

Remark 3.1 Inequality (17) is a little more precise than Hedberg’s inequality [1, 8] : iff =Iαg(x)whereIαis a Riesz potential² with 0< α < nand if g∈B˙_∞^−β,∞, then, if Mg is the Hardy–Littlewood maximal function of g, we have Aα(x)≤CMg(x)and Aβ(x)≤CkgkB˙∞^−β,∞. Thus, we find easily the refined Sobolev inequality (1).

See more details in [14, 4].

2defined in the Fourier level byIdαg(ξ) =|ξ|^−αbg(ξ).

Proof of theorem 2. We start pickingp0andp1such that 1≤q0< p0< p < p1< q1≤+∞with ²_p =_p¹₀+_p¹₁. We have then _p¹_i = ^1−a_q0ⁱ +^a_q1ⁱ with 0< ai <1 andi= 0,1. We write

k∆jfkLpi ≤ k∆jfk^1−a_L^q0ⁱk∆jfk^a_L^iq1 = 2^jαk∆jfkL^q01−ai

2^−jβk∆jfkL^q1ai

2^j

−α(1−ai)+βai

.

Recalling that ¹_p = ^1−θ_q0 + _q^θ₁ and θ = _α+β^α we have −α(1−a0) +βa0 = α(1−a1)−βa1. Thus, noting ρ= 2^{−2[α(1−a0)−βa0]}>0 and using the H¨older inequality we obtain

X

j∈Z

ρ^−jr/2k∆jfk^r_L^p0 ≤ kfk^r(1−a_B˙^α,q0⁰⁾ r kfk^ra_˙⁰

B^−β,qr ¹ and X

j∈Z

ρ^jr/2k∆jfk^r_L^p1 ≤ kfk^r(1−a_B˙^α,q0¹⁾ r kfk^ra_˙ ¹

Br^−β,q1.

From this, and applying proposition 2.2, we deduce that iff ∈B˙^α,q_r ⁰∩B˙^−β,q_r ¹(Rⁿ) thenf ∈[L^p0, L^p1]¹

2,r =L^p,r. Furthermore, using inequality (12) we finally have:

kfkL^p,r≤Cρ,rkfk^1−θ_˙

B^α,qr ⁰kfk^θ_˙

B^−β,qr ¹

Proof of theorem 3.

1) Case r > r^∗: With no loss of generality, we may assume thatq0< q1 and we fixε >0 such that 1

q1

< 1 p−ε(1

q0

− 1 q1

) = 1 p1

< 1 p+ε(1

q0

− 1 q1

) = 1 p0

< 1 q0

.

The proof follows essentially the same ideas used in the previous theorem. Indeed, we have, forγj = 2^jαk∆jfkL^q0

andηj = 2^−jβk∆jfkL^q1, and forǫ0= 1 and ǫ1=−1,

k∆jfkLpi ≤ k∆jfk^1−θ+ǫ_Lq0 ^iεk∆jfk^θ−ǫ_Lq1^iε=γ_j^1−θ+ǫiεη_j^θ−ǫiε2^{−jǫiε(α+β)}.

Asr06=r1, we can only say that (γ_j^1−θ+ǫiεη_j^θ−ǫiε)j∈Z∈ℓ^ρi where _ρ¹_i = ^1−θ+ǫ_r0 ^iε+^θ−ǫ_r1^iε. We may use inequality (12), but we get only that f ∈ [L^p0, L^p1]1/2,ρ = L^p,ρ with ρ = max(ρ0, ρ1), and satisfies inequality (9) with r=ρ. However, we may chooseεas small as we want, and thusρas close tor^∗ as we want; thusf satisfies (9) for everyr > r^∗.

2) Case r =r^∗:

a) ifr=r0=r1 : this case was treated in theorem 2.

b) ifr0=q0 andr1=q1 : This is a direct consequence of (6) since we havekfkB˙_qi^α,qi =kfkF˙_qi^α,qi ⊂ kfkF˙∞^α,qi

andkfk_B˙⁻β,qi

qi =kfk_F˙⁻β,qi

qi ⊂ kfk_F˙⁻β,qi

∞ , we obtain (7).

c) Case 1< p≤2 andr^∗=p: We just write

k∆jfkL^p≤ k∆jfk^1−θ_L^q0k∆jfk^θ_L^q1 = (2^αk∆jfkL^q0)^1−θ(2^−jβk∆jfkL^q1)^θ and get by H¨older inequality:

kfkB˙p^0,p≤Ckfk^1−θ_B˙α,q0 r0

kfk^θ_B˙−β,q1 r1

.

We then use the embedding ˙B_p^0,p⊂L^p=L^p,p, which is valid forp≤2.

Proof of theorem 4. We see how direct interpolation has given us theorem 3, but we only obtained partial results for theorems 4 and 5. Indeed, in theorem 4, we want a positive result for q0 ≤r0 ≤r1 ≤q1 but thus far we have proven the result only for (r0, r1) = (q0, q1) and for q0 ≤r0 =r1 ≤q1. To complete the proof of theorem 4, we must reiterate interpolations to those new estimates.

This will be done through the following lemma :

Lemma 3.1 Let α, β >0,q0, q1∈[1 +∞],q0< q1. Letθ= _α+β^α ∈]0,1[and let ¹_p =^1−θ_q

0 +_q^θ

1. 1) If q0≤r0≤q1 and let ¹_r =^1−θ_r

0 +_q^θ

1, then we have kfkL^p,r ≤C0kfk^1−θ_B˙α,q0

r0

kfk^θ_˙

Bq^−β,q1 ¹

. (18)

2) If q0≤r1≤q1 and let ¹_r =^1−θ_q0 +_r^θ₁, then:

kfkL^p,r ≤C0kfk^1−θ_B˙α,q0 q0

kfk^θ_B˙−β,q1 r1

. (19)

Proof of the lemma 3.1. We only prove the first inequality, as the proof for the second one is similar. Since f ∈B˙^α,q_r0 ⁰, notingλj= 2^jαk∆jfkL^q0 we have (λj)j∈Z∈ℓ^r0. Thus, using lemma 2.1 for the interpolation

ℓ^r0= [ℓ^q0, ℓ^q1]η,r, (20)

with _r¹₀ = ^1−η_q0 +_q^η₁, we see that we have a partitionZ=X

k∈Z

Zk such that

2^−kη X

j∈Zk

λ^q_j⁰_q¹₀

ℓ^r0

+

2^k(1−η) X

j∈Zk

λ^q_j¹_q¹₁

ℓ^r0

≤Ckλjkℓ^r0 (21)

Moreover sincef ∈B˙_q^−β,q₁ ¹ we have X

j∈Zk

2^−jβq1k∆jfk^q_L^1q1

1/q1

k∈Z

∈ℓ^q1.

Let us noteαk = X

j∈Zk

2^−jβq1k∆jfk^q_L¹q1

1/q1

,βk= 2^−kη X

j∈Zk

λ^q_j⁰_q¹₀

,γk = 2^k(1−η) X

j∈Zk

λ^q_j¹)^q11 andfk= X

j∈Zk

∆jf. We apply now inequality (7) and theorem 2 to obtain

kfkkL^p≤CAkfkk^1−θ_˙

Bq^α,q0 ⁰

kfkk^θ_˙

B^−β,qq1 ¹

≤Cα^θ_kβ_k^1−θ2^kη(1−θ) (22)

and

kfkkL^p,q1 ≤CBkfkk^1−θ_˙

B^α,qq1 ⁰

kfkk^θ_˙

Bq^−β,q1 ¹

≤Cα^θ_kγ_k^1−θ2−k(1−η)(1−θ). (23) Since we have f =X

k∈Z

fk, with these two inequalities at hand, and using (12), we find thatf ∈[L^p, L^p,q1]η,r

with ¹_r= ^1−η_p +_q^η₁. But, since_r¹₀ =^1−η_q0 +_q^η₁ and ¹_p =^1−θ_q0 +_q^θ₁, we obtain [L^p, L^p,q1]η,r=L^rwith¹_r = ^1−θ_r0 +_q^θ₁.

Remark 3.2 Note that we use twice interpolation arguments: first in estimate (21) and then with inequalities (22) and (23) in order to obtain f ∈[L^p, L^p,q1]η,r.

Once this lemma is proved, it is enough to reapply similar arguments to obtain theorem 4. Indeed, since we haveq0< r0 < r1 < q1, we start usingℓ^r0 = [ℓ^q0, ℓ^r1]η,r0 instead of (20) and we obtain a partitionZ=X

k∈Z

Zk

such that 2^−kη X

j∈Zk

λ^q_j⁰_q¹_0ℓr0

+

2^k(1−η) X

j∈Zk

λ^r_j¹_r¹_1ℓr0

≤Ckλjkℓ^r0

with _r¹₀ = ^1−η_q0 +_r^η₁ and where the sequence (λj)j∈Nwithλj= 2^jαk∆jfkL^q0 belongs toℓ^r0 sincef ∈B˙_r^α,q₀ ⁰. Sincef ∈B˙_r^−β,q₁ ¹we have P

j∈Zk2^−jβq1k∆jfk^q_L^1q1

1/q1

k∈Z∈ℓ^q1, and we note againαk = X

j∈Zk

2^−jβq1k∆jfk^q_L^1q1

1/q1

, βk = 2^−kη X

j∈Zk

λ^q_j⁰_q¹₀

,γk = 2^k(1−η) X

j∈Zk

λ^q_j¹)^q11 and fk = X

j∈Zk

∆jf. Next, we only need to apply (19) and (8) instead of (22) and (23) to obtain

kfkkL^p,s≤CAkfkk^1−θ_B˙^α,q0 q0

kfkk^θ_B˙−β,q1 r1

≤Cα^θ_kβ^1−θ_k 2^kη(1−θ) where ¹_s = ^1−θ_q0 +_r^θ₁, and

kfkkL^p,r1 ≤CBkfkk^1−θ_B˙α,q0 r1

kfkk^θ_B˙−β,q1 r1

≤Cα^θ_kγ_k^1−θ2−k(1−η)(1−θ).

Finally, we have via inequality (12) that f ∈ [L^p,s, L^p,r1]η,r with ¹_r = ^1−η_s +_r^η₁. To conclude, we use the fact that ¹_s= ^1−θ_q0 +_r^θ₁ and _r¹₀ = ^1−η_q0 +_r^η₁ in order to obtain thatf ∈L^p,r with ¹_r =^1−θ_r0 +_r^θ₁ . Proof of theorem 5. Similarly, in theorem 5, we want a positive result on the triangles 1/y0≤r1≤r0≤1/x0

and 1/x1≤r0 ≤r1≤1/y1, and we have already obtained that the theorem is true for 1/y0≤r1=r0 ≤1/x0

and 1/x1≤r0=r1≤1/y1 as well as for (1/x0,1/y0) and (1/x1,1/y1). To complete the proof of theorem 5, we must reiterate interpolations to those new estimates. This will be achieved with the following lemma.

Lemma 3.2 Let Σ be the set of points (x, y) ∈ [0,1]×[0,1] such that, for z^∗ = (1−θ)x+θy, we have the inequality

kfk_L^p,1/z∗ ≤Cx,ykfk^1−θ_B˙α0,q0 1/x

kfk^θ_B˙−β,q1 1/y

.

and let A, B, C∈[0,1]×[0,1]such that[A, B] is horizontal(yA=yB)and[A, C]is vertical (xA=xC). Then :

• if A∈ΣandB∈Σthen[A, B]⊂Σ,

• if A∈ΣandC∈Σthen [A, C]⊂Σ,

• if A∈Σand[B, C]⊂Σ, then the triangleABC is contained inΣ.

Proof of the lemma 3.2. The proof of this inequality follows closely the ideas of lemma 3.1. We give the details here for the sake of completness.

We begin with the case of M ∈[A, B]: setxA, z0, ρsuch that z0= (1−θ)xA+θρ andxB, z1, ρ such that z1= (1−θ)xB+θρ. Then, if we definexM such thatxM = (1−η)xA+ηxB, we must show that we have the inequality

kfkL^p,1/z≤C0kfk^1−θ_B˙^α,q0 1/xM

kfk^θ_B˙−β,q1 1/ρ

withz= (1−θ)xM +θρ.

• Sincef ∈B˙_1/x^α,q0_M, we have that (λj)j∈Z∈ℓ^1/xM, withλj = 2^jαk∆jfkL^q0. Moreover, by hypothesis we have xM = (1−η)xA+ηxB, so we can writeℓ^1/xM = [ℓ^1/xA, ℓ^1/xB]η,1/xM. Thus, with lemma 2.1 we obtain a partitionZ=X

k∈Z

Zk such that

2^−kη X

j∈Zk

λ^1/x_j ^AxA

_ℓ1/xM

+

2^k(1−η) X

j∈Zk

λ^1/x_j ^BxB

_ℓ1/XM

≤Ckλjk_ℓ¹/xM.

We will noteβk= 2^−kη X

j∈Zk

λ^1/x_j ^AxA

,γk = 2^k(1−η) X

j∈Zk

λ^1/x_j ^BxB

andfk= X

j∈Zk

∆jf.

• Sincef ∈B˙^−β,q_1/ρ ¹, we can writeαk = X

j∈Zk

2^−jβ1/ρk∆jfk^1/ρ_L^q1

ρ

k∈Z

∈ℓ^1/ρ.

Now, sincez0= (1−θ)xA+θρandz1= (1−θ)xB+θρ we can apply theorem 4 to the functionsfk to obtain:

kfkk_L^p,1/z0 ≤ CAkfkk^1−θ_B˙α,q0 1/xA

kfkk^θ_B˙−β,q1 1/ρ

≤Cα^θ_kβ_k^1−θ2^kη(1−θ)

kfkk_L^p,1/z1 ≤ CBkfkk^1−θ_B˙^α,q0 1/xB

kfkk^θ_B˙−β,q1 1/ρ

≤Cα^θ_kγ_k^1−θ2−k(1−η)(1−θ)

From these two estimates we deduce, sincef =X

k∈Z

fk, thatf ∈[L^p,1/z0, L^p,1/z1]η,1/z withz= (1−η)z0+ηz1. But by hypothesis we havez0= (1−θ)xA+θρandz1= (1−θ)xB+θρ, so we find thatz= (1−θ)xM+θρ.

We have proven that we may interpolate along horizontal lines and the vertical case is totally similar. To finish the proof of the lemma suppose that A∈Σ and [B, C]⊂Σ and take a point P ∈Σ. Write P ∈[M, N] where [M, N] is horizontal, M ∈[A, C] and N ∈[B, C]. Then we find that M ∈ Σ by vertical interpolation betweenAandC, and thatP ∈Σ by horizontal interpolation between M andN.

Thus, Lemma 3.2 is proven, and this finishes the proof of theorem 5.

4 Sharpness and optimality of the inequalities.

In this section, we adapt Bahouri and Cohen’s example [2] to the general case. Their idea is to use the analysis of the regularity of chirps by Jaffard and Meyer [9]. They express their example in terms of wavelets, as it is easy to estimate Besov norms in wavelet bases [12, 13]. They pick just one wavelet per scale and adjust its support so that they are able to compute the Lorentz norm of the sum. As we shall see, their example can be extended to the general case but we will not use wavelets, but atoms for Besov spaces [6, 16], as we shall use only upper estimates for Besov norms. We consider a functionωsuch that it is supported in the ballB(0,1), is C^N(Rⁿ) and

Z

Rn

x^γω(x)dx= 0 for allγ∈Nⁿ with|γ|< N, for someN such that N >max(|α|,|β|). Then we have, for all 1≤q≤+∞, 1≤r≤+∞and|s|< N,

X

j∈Z

X

k∈Zⁿ

λj,kω(2^jx−k) _B˙s,q

r

≤CX

j∈Z

2^jsr2^−jnr/q X

k∈Zⁿ

|λj,k|^qr/q1/r

. (24)

For our example, we shall fix someX∈R, someY ∈Rand someγ∈Rand we define fL=

jL

X

j=1

2^jX X

k∈Kj

ω(2^jx−k) and gL=

jL

X

j=j1

2^jY X

k∈Kj

ω(2^jx−k)

where theKj are finite sets such that

• the supports of the functions (ω(2^jx−k))j1≤j≤jL,k∈Kj are disjoint each one from each other

• Kj is a set of cardinalj with 2^δj≤Aj≤ 2^δ(j+1)(which is possible if δj1≥0 andδjL≥0).

For everyr∈[1,+∞], we then have the following estimates (from (24)):

kfLkB˙^α,qr ⁰ ≤ Cr

X^jL

j=j1

2^{jr(X+α−n/q0+δ/q0)}1/r

kfLkB˙r^−β,q1 ≤ Cr

X^jL

j=j1

2^{jr(X−β−n/q1+δ/q1)1/r}

kgLk_B˙^−α,q0/(q0−1)

r ≤ Cr

X^jL

j=j1

2^{jr(Y−α−n(1−1/q0)+δ(1−1/q0))1/r}

kgLk_B˙β,q1/(q1−1)

r ≤ Cr

X^jL

j=j1

2^{jr(Y+β−n(1−1/q1)+δ(1−1/q1))1/r} We thus fixX, Y, γ such that 





X+α−n/q0+δ/q0= 0 X−β−n/q1+δ/q1= 0

Y −α−n(1−1/q0) +δ(1−1/q0) = 0

(25) Sinceα−n/q0+δ/q0=−β−n/q1+δ/q1, we have as well Y +β−n(1−1/q1) +δ(1−1/q1) = 0. This gives that, for allr∈[1,+∞] we have

kfLkB˙^α,qr ⁰ ≤ CrL^1/r kfLkB˙r^−β,q1 ≤ CrL^1/r kgLk_B˙^−α,q0/(q0−1)

r ≤ CrL^1/r

kgLk_B˙^β,q1/(q1−1)

r ≤ CrL^1/r

whereCr does not depend onL. Theorem 2 shows that

kfLkL^p,r≤CrL^1/r and kgLk_L^p/(p−1),r ≤CrL^1/r. Moreover, since the supports of the functionsω(2^jx−k) are disjoint, we have

Z

fLgLdx=

jL

X

j=j1

2^j(X+Y) X

k∈Kj

2^−jnkωk²_L2 ≥

jL

X

j=j1

2^{j(X+Y−n+δ)}kωk²_L2

From (25), we haveX+Y −n+δ= 0, so that, using lemma 2.2 we obtain:

kωk²_L² L≤CkfLkL^p,rkgLk

L

p p−1, r

r−1 ≤C^′kfLkL^p,rL^1−1r If we asssume that the interpolation inequality (8) is valid, we get that

L^1/r≤CkfLkL^p,r ≤C^′kfLk^1−θ_B˙^α,q0 r0

kfLk^θ_B˙−β,q1 r1

≤C^′′L^{(1−θ)/r0+θ/r1}

LettingL−→+∞, we find that 1/r≤(1−θ)/r0+θ/r1: we have thus proven the optimality of these inequalities.

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