Refined Sobolev inequalities in Lorentz spaces

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Refined Sobolev inequalities in Lorentz spaces

Hajer Bahouri, Cohen Albert

To cite this version:

Hajer Bahouri, Cohen Albert. Refined Sobolev inequalities in Lorentz spaces. Journal of Fourier Analysis and Applications, Springer Verlag, 2011, 17 (4), pp.662-673. �hal-00795960�

Refined Sobolev inequalities in Lorentz spaces

Hajer Bahouri and Albert Cohen March 1, 2013

Abstract

We establish refined Sobolev inequalities between the Lorentz spaces and homogeneous Besov spaces. The sharpness of these inequalities is illustrated on several examples, in particular based on non-uniformly oscillating functions known as chirps. These results are also used to derive refined Hardy inequalities.

1 Introduction

The Sobolev inequality states that for any 0< s < ^d₂ there exists a constant C >0 such that for any functionf defined onR^d,

kfkL^p≤CkfkH˙^s,

with ¹_p := ¹₂−^s_d (thereforepis strictly between 2 and∞). A refined version of this inequality was proved in [10] :

kfkL^p≤Ckfk¹⁻_H˙s^2sdkfk^2sd

B˙^s−

d

∞,∞2

,

where ˙B^s−∞,∞^d2 is the homogeneous Besov space. Similar results also hold when the smoothness of the function is measured inL^q for 1≤q <∞: for any 0< s < ^d_q,

kfkL^p≤Ckfk¹⁻

qs d

B˙_q,q^s kfk

qs d

B˙^s−

d q

∞,∞

, (1.1)

with¹_p := ¹_q−^s_d, or the same kind of inequalities with ˙W^s,qin place of ˙B_q,q^s . We also refer to [3, 4, 13, 14, 16]

for similar results, involving in particular the spaceBV.

These inequalities are generalizations of the usual Sobolev embedding in Lebesgue spaces, their addi- tional feature being that they are invariant under oscillations and fractal transforms see [1]. A significant application is the description of defect of compactness in Sobolev embedding (see [9] in theL²framework and [11] inL^pframework).

On the other hand, it is also known that Sobolev and Besov spaces also embed in Lorentz spaces L^p,q(R^d) : with ¹_p = ¹_q −^s_d, one has

kfkL^p,q ≤CkfkB˙_q,q^s . (1.2)

This may be derived from the standard inequalities inL^p spaces by a real interpolation argument. This justifies looking for refined inequalities of the type (1.1) withL^p,q(R^d) in place of L^p(R^d).

In this paper, such inequalities will be established. The “price to pay” will be a slight modification in one index of the Besov space of negative smoothness, namely

kfkL^p,q≤Ckfk¹⁻

qs d

B˙_q,q^s kfk

qs d

B˙^s−

d q

∞,q

, (1.3)

for the same range of indices (s, p, q) as in (1.1). It will be shown that this change is unavoidable and that such inequalities are sharp.

The structure of the paper is the following. First, in §2, we briefly recall some facts on Lorentz and Besov spaces, we establish estimates for Besov spaces which are used in the sequel. In§3 we give the

proof of the refined Sobolev inequalities (1.3), which follows from the same line of reasoning as in [10] or [13]. The sharpness of the inequality is discussed in§4. As with the improved Sobolev inequality (1.1), these inequalities remain sharp for oscillatory functions, but more interestingly the necessity of the above mentioned change of index can be proved by considering non-uniformly oscillating functions known as chirps. Such functions have been deeply studied by Meyer and Jaffard, see [12, 16] and arise naturally in signal processing. Finally, as a by-product of (1.3) we obtain in §5 a refined version of the Hardy inequalities in theL^q framework, namely

Z |f(x)|^q

|x|^{sq 1}_q

≤Ckfk¹⁻

qs d

B˙_q,q^s kfk

qs d

B˙^s−

d q

∞,q

. (1.4)

A similar inequality was derived in [1] in the caseq = 2. However, our method of proof is significantly simpler and applies to arbitrary values ofq. Let us stress that Hardy inequalities are of important use in analysis: among other applications, we can mention blow-up methods or the study of pseudo-differentional operators with singular coefficients.

Acknowledgement: the authors are grateful to the anonymous reviewers for their constructive sug- gestions in improving the paper.

2 Lorentz spaces and Besov spaces

For any measurable functionf onR^d, we define its distribution and rearrangement functions µf(t) :=|{x; |f(x)| ≥t} and f^∗(s) := inf{t; µf(t)≤s}.

For 1 ≤ p < ∞ and 1 ≤ q ≤ ∞, the Lorentz space L^p,q(R^d) is defined as the set of all measurable functionsf such that the functions^1pf^∗(s) belongs toL^q(R+,^ds_s). TheL^p,q quasi-norm is defined by

kfkL^p,q :=Z ^+∞

0

(s^1pf^∗(s))^qds s

¹_q

, (2.5)

with the standard modification when q = ∞ which corresponds to the weak L^p space wL^p = L^p,∞. It is well known that L^p,p = L^p and that Lorentz spaces can be derived from L^p spaces by the real interpolation method. In particular, when 1< p <∞ we haveL^p,q = [L¹, L^∞]θ,q with ¹_p = 1−θ. We refer to [5, 2] for these classical results.

In this paper, we shall use the expression of theL^p,q quasi-norm in terms of the distribution function µf, namely

kfkL^p,q = p

Z +∞

0

(tµf(t)^1p)^qdt t

¹_q

. (2.6)

To derive this expression, one first writes

f^∗(s)^q =q Z f^∗(s)

0

t^q−1dt, so that

kfk^q_L^p,q =R+∞

0 s^qp−1f^∗(s)^qds

=qR+∞

0 s^qp−1[Rf^∗(s)

0 t^q−1dt]ds

=qR+∞

0 t^q−1[Rµf∗(t)

0 s^qp−1ds]dt

=pR+∞

0 t^q−1µf^∗(t)^qpdt.

Observing thatµf =µf^∗ we obtain (2.6).

There exists several equivalent definitions of Besov spaces either based on Littlewood-Paley decompo- sitions, moduli of smoothness, approximation procedures, wavelet decompositions. We refer to [18] and [7] for a general treatment, and only recall here the definition based on Littlewood-Paley theory.

Letϕbe a function ofdvariables such that its Fourier transform ˆϕis non-negative,C^∞, supported in the ball{|ω| ≤M} for someM >1 and such that ˆϕ(ω) = 1 if|ω| ≤1. Forj ∈Zwe define the operator Sj acting on tempered distributions ofdvariables by

Sjf = 2^djϕ(2^j·)∗f and ∆j =Sj+1−Sj or equivalently

∆jf = 2^djψ(2^j·)∗f,

with ˆψ(ω) = ˆϕ(^ω₂)−ϕ(ω). Forˆ s < ^d_p andq≥1, we define the homogeneous Besov space ˙B_p,q^s (R^d) as the space of tempered distributionsf such thatPJ

j=−J∆jf converges towardsf inS^′ asJ →+∞and such that

kfkB˙_p,q^s :=k(2^sjk∆jfkL^p)j∈Zkℓ^q

is finite. It is known and not difficult to check that this definition is independent of the initial choice of the functionϕwith the above mentioned property, the resulting norms being equivalent. The condition that s <^d_p is necessary to ensure the fact that we have defined a proper Banach space (otherwise convergence should be interpreted in S^′/Πn where Πn is the space of polynomials of total degree n =⌊s−^d_p⌋, for more details see [8] and the discussion therein p. 153–155).

We shall consider Besov spaces of regularity indexswhich is either positive or negative. Using discrete Hardy inequalities, it is also possible to characterize these spaces using only the operatorSj : fors >0, we have

kfkB˙_p,q^s ∼ k(2^sjk(I−Sj)fkL^p)j∈Zkℓ^q, and fors <0,

kfkB˙^s_p,q∼ k(2^sjkSjfkL^p)j∈Zkℓ^q.

Equivalent norms for Besov spaces can also be obtained with the discrete frequencies 2^j replaced by a continuous one: introducing for anyλ >0 the operators

Sλf =λ^dϕ(λ·)∗f and ∆λf =S2λf−Sλf =λ^dψ(λ·)∗f, we then find thatkfkB˙_p,q^s is equivalent to the norm of the function

F(λ) :=λ^sk∆λfkL^p (2.7)

inL^q(R+,^dt_t). By similar arguments as when working with discrete frequency, we can equivalently choose

F(λ) :=λ^sk(I−Sλ)fkL^p (2.8)

fors >0 or

F(λ) :=λ^skSλfkL^p, (2.9)

fors <0. For the purpose of our analysis, we shall also need estimates on the derivativeF^′(λ) =^dF_dλ, as expressed by the following result.

Lemma 2.1 There exists a constantK which depends only ons, d, p, qand the choice ofϕsuch that for allf ∈B˙_p,q^s , the functionG(λ) :=λF^′(λ)belongs toL^q(R+,^dt_t)with

kGk_Lq(R+,^dt_t)≤KkfkB˙_p,q^s , (2.10) whereF is defined by(2.7), or by(2.8)whens >0, or by (2.9)whens <0.

Proof: we only consider the case whereF is defined by (2.7), the two others being treated in the same way. We first estimate the derivative of the functiong(λ) :=k∆λfkL^p. Clearly, we have

|g^′(λ)| ≤ k d

dλ(λ^dψ(λ·))∗fkL^p=kλ^d−1ψ(λ·)˜ ∗f+dλ^d−1ψ(λ·)∗fkL^p,

with ˜ψ(x) =x· ∇ψ(x). Since we have

F^′(λ) =sλ^s−1g(λ) +λ^sg^′(λ), it follows that

|G(λ)| ≤(s+d)F(λ) +λ^skλ^dψ(λ·)˜ ∗fkL^p:= (s+d)F(λ) +Fe(λ) The function ˜ψsatisfiesψ(ω) =b˜ ϕ(b˜ ^ω₂)−ϕ(ω) withb˜

b˜

ϕ(ω) := ˆϕ(ω) +ω· ∇ϕ(ω).ˆ

Since ˆ˜ϕisC^∞, compactly supported in the ball{|ω| ≤M}, and such that ˆϕ(ω) = 1 if|ω| ≤1, we obtain from the definition of Besov spaces that

kGk_L^q_(R+,^dt

t)≤(s+d)kFk_L^q_(R+,^dt

t)+kFek_L^q_(R+,^dt

t)≤KkfkB˙^s_p,q,

for someK which depends on (s, d, p, q, ϕ). ⋄

3 Proof of the refined inequality

We first observe that we may prove (1.3) in the particulary case wheref is aC^∞ compactly supported function, and derive by density the general validity of the inequality.

We decomposef into low and high frequencies according to f =Sλf + (I−Sλ)f

whereSλis the convolution operator defined in the previous section. From the characterization of Besov spaces which was recalled in§2.2, we have

Z ∞ 0

[F(λ)]^qdλ

λ ≤C0kfk^q

B˙^s−

d q

∞,q

whereC0 is an absolute constant and

F(λ) :=λ^s−dqkSλfkL^∞.

We also have Z +∞

0

λ^qsk(I−Sλ)fk^q_Lq

dλ

λ ≤C1kfk^q_B˙s q,q, whereC1 is also an absolute constant.

We now consider the functionλ7→t(λ) defined by

t(λ) = 2λ^dpF(λ) = 2λ^s+dp−dqkSλfkL^∞ = 2kSλfkL^∞.

This function may not be monotone but it tends towards 0 asλ→0 and towardsM := 2kfkL^∞≥ kfkL^∞

asλ→+∞. We may thus use it as a change of variable in the expression (2.6) of theL^p,q quasi-norm, which yields

kfk^q_Lp,q=p Z +∞

0

t^qµf(t)^qpdt=p Z 2M

0

t^qµf(t)^qpdt=p Z +∞

0

t(λ)^q−1t^′(λ)

µf(t(λ))^q_p

dλ. (3.11) From the triangle inequality we have

x; |f(x)|> t ⊂

x; |Sλf(x)|> t 2 ∪

x; |(I−Sλ)f(x)|> t 2 Since we have|Sλf(x)| ≤ ^t(λ)₂ for allx, it follows that

{x; |f(x)|> t(λ)} ⊂ {x; |(I−Sλ)f|>t(λ) 2 },

i.e. µf(t(λ))≤µ(I−Sλ)f(^t(λ)₂ ). Combining this observation together with (3.11), we obtain kfk^q_Lp,q ≤ pR+∞

0 t(λ)^q−1t^′(λ)[µ(I−Sλ)f(^t(λ)₂ )]^qpdλ

= 2^qdR+∞

0 [λ^dpF(λ)]^q−1(λ^dp−1F(λ))[µ(I−Sλ)f(^t(λ)₂ )]^qpdλ + 2^qpR+∞

0 [λ^dpF(λ)]^q−1(λ^dpF^′(λ))[µ(I−Sλ)f(^t(λ)₂ )]^qpdλ

= 2^qdR+∞

0 λ^dqp−1F(λ)^q[µ(I−Sλ)f(^t(λ)₂ )]^qpdλ + 2^qpR+∞

0 λ^dqpF(λ)^q−1F^′(λ)[µ(I−Sλ)f(^t(λ)₂ )]^qpdλ := T0+T1

In order to estimate further the two above terms, we remark that µ(I−Sλ)f(t(λ)

2 )≤(t(λ)

2 )^−qk(I−Sλ)fk^q_Lq = (λ^dpF(λ))^−qk(I−Sλ)fk^q_Lq. ForT0, this gives

T0 ≤2^qdR+∞

0 λ^dqp−1F(λ)^q[(λ^dpF(λ))^−qk(I−Sλ)fk^q_Lq]^qpdλ

= 2^qdR+∞

0 F(λ)^q(1−qp)[λ^d−dqpk(I−Sλ)fk^q_Lq]^pqdλ_λ

= 2^qdR+∞

0 F(λ)^q(1−qp)[λ^qsk(I−Sλ)fk^q_Lq]^qpdλ_λ. Applying H¨older inequality, we thus obtain

T0 ≤2^qdR+∞

0 F(λ)^{q dλ}_λ1−^q_pR+∞

0 λ^qsk(I−Sλ)fk^q_Lqdλ λ

^q_p

≤2^qdR+∞

0 F(λ)^{q dλ}_λ^qs_dR+∞

0 λ^qsk(I−Sλ)fk^q_Lqdλ λ

1−^qs_d

≤C kfk

qs d

B˙^s−

d

∞,qq

kfk¹⁻

qs d

B˙_q,q^s

q

.

whereC is an absolute constant. Proceeding in an exactly similar manner forT1, we obtain T1≤2^qpZ ^+∞

0

F(λ)^q−1λF^′(λ)dλ λ

^qs_dZ ^+∞

0

λ^qsk(I−Sλ)fk^q_Lq

dλ λ

1−^qs_d

.

Using Lemma 2.1 together with a H¨older inequality, we see that the integral in the first factor can again be controlled bykfk^q

B˙^s−

d q

∞,q

, and therefore we also obtain

T1≤C kfk

qs d

B˙^s−

d

∞,qq

kfk¹⁻

qs d

B˙^s_q,q

q

,

whereCis an absolute constant. Combining our estimates for T0 andT1 we have proved that kfk^q_Lp,q≤C

kfk

qs d

B˙^s−

d q

∞,q

kfk¹⁻

qs d

B˙^s_q,q

q

,

which is the desired result.

Remark 3.1 For our range of indices (s, p, q), we also have a continuous embedding of B˙^s_q,r intoL^p,r for any1≤r≤ ∞(this can easily be proved from the standard Sobolev embedding by a real interpolation argument). This suggests a correponding improved inequality of the form

kfkL^p,r ≤Ckfk¹⁻_˙ ^qsd

B_q,r^s kfk^qsd

B˙^s−

dq

∞,r

,

but the proof of this does not seem obvious to us, neither by interpolation arguments, nor by an adaptation of our current proof (that corresponds to the caser=q).

4 Sharpness of the inequality

We shall discuss the sharpness of (1.3) through two examples. The first one shows that the refined estimates are sharp with respect to oscillating functions: consider for all positiveω, the function

ϕω(x) =e^iωx1ϕ(x),

whereϕ is a fixed function in S(R^d). For such a function, kϕωkL^p,q is independent of ω since theL^p,q quasi-norm only depends on the modulus. On the other hand for|ω|>1,

kϕωkB˙_q,q^s ∼ω^s.

Therefore, the Sobolev inequality (1.2) is not sharp when ω is large. It is also easily checked that for

|ω|>1

kϕωk

B˙^s−

d q

∞,q

∼ω^s−dq,

(see [1] for the proof of these equivalences) and therefore kϕωk¹⁻

qs d

B˙_q,q^s kϕωk

qs d

B˙^s−d/q∞,q

∼ω^{s(1−qsd)+(s−dq)qsd} = 1, which shows that (1.3) is sharp for such oscillatory functions.

The defect of this first example is that it does not make much distinction between (1.3) and the improved Sobolev inequality (1.1) inL^p spaces : indeed, we also have thatkϕωkL^p is independent ofω and thatkϕωk

B˙^s−

d q

∞,∞

∼ω^s−dq. In particular, this example does not reveal the necessity of using ˙B^s−

d

∞,qq in place of ˙B^s−

d

∞,∞q when we want to control theL^p,q quasi-norm. A finer example is needed order to show that this cannot be avoided. This example should contain oscillations in a wide range of frequencies in order to make the distinction between ˙B^s−

d

∞,qq and ˙B^s−

d

∞,∞q , whereas it should also contain a wide range of magnitudes in order to distinguish between L^p and L^p,q. Our example has roughly the form of a hyperbolic “chirp”

f(x) =x^−αsin(1 x),

with α > 0. For the sake of simplicity, we have placed ourselves in the one-dimensional case d = 1 although this example can be generalized to arbitrary dimensions. In order to simplify our analysis, we shall define a variant off in the form of a wavelet series. We recall that the Haar system is defined by the functions

ψj,k(x) = 2^j2ψ(2^jx−k), j, k∈Z, whereψ:=χ_[0,¹

2[−χ_[¹

2,1]. Besov spaces ˙B_p,q^s can be characterized as follows by wavelets (see e.g. [15]

or [6]): iff =P

j,kdj,kψj,k, then

kfkB˙^s_p,q ∼ k

2^(s+12−p1)jk(dj,k)k∈Zkℓ^p

j∈Zkℓ^q.

In the case of the Haar system, this characterisation holds if ¹_p−1< s < ¹_p (due to the lack of smoothness of the functionsψj,k) which is sufficient for our purpose.

Letq≥1 and 0< s < ¹_q, and let ¹_p := ¹_q −s. ForJ >0, we consider the function fJ(x) =

XJ

j=1

2^(p1−12)jψj,1(x) = XJ

j=1

2^jpψ(2^jx−1). (4.12)

Note that the wavelets in the expansion offJ have disjoint supports. We first notice that we have kfJkB˙^s_q,q ∼X^J

j=1

2^(qs+q2−1)j2^(pq−q2)j1_q

=J^1q.

For the Besov norms of negative indices, we have kfJk

B˙^s−

1 q

∞,∞

∼sup^J

j=1

2^(s−1q+12)j2^(1p−12)j = 1, and

kfJk

B˙^s−

1 q

∞,q

∼X^J

j=1

[2^(s−1q+12)j2^(1p−12)j]^q_q¹

=J^1q.

Since the wavelets in the expansion offJ have disjoint supports, theL^pnorm is of the order kfJkL^p=X^J

j=1

k2^jpψ(2^j· −1)k^p_Lp

¹_p

∼J^1/p.

Remarking that the rearrangement function offJ is simply given by f_J^∗(s) =|fJ(s+ 2^−J)|, we can also easily evaluate itsL^p,q quasi-norm using (2.5) which gives

kfJkL^p,q =PJ

j=12^jqp R2^−j

0 (s+ 2^−j−2^−J)^qp−1ds¹_q

∼PJ

j=12^jqp2^−j2^{−j(qp−1)1}_q

=J^1q. From all these estimates, we obtain that

kfJk^1−qs_B˙s q,q kfJk^qs

B˙^s−

1 q

∞,∞

∼J^1q−s=J^1p ∼ kfJkL^p, and

kfJk^1−qs_B˙s

q,q

kfJk^qs

B˙^s−

1

∞,qq

∼J^1q−sJ^s=J^1q ∼ kfJkL^p,q. Again (1.3) and (1.1) are sharp, but this example also shows that the use of ˙B^s−

d

∞,qq in place of ˙B^s−

d

∞,∞q is unavoidable for the validity of (1.3).

5 Refined Hardy inequalities

Let as in the previous sections 1< p < ∞and s ∈]0,^d_q[, with ¹_p = ¹_q −^s_d. For such indices, we claim that the refined Hardy inequality (1.4) holds. It is actually a direct consequence of (1.3) combined with a generalized version of H¨older’s inequality known as O’Neil inequalities : if 1≤p1, p2, q1, q2≤ ∞, then for anyf ∈L^p1,q1(R^d) andg∈L^p2,q2(R^d),

kf gkL^p,q(R^d)≤CkfkL^p1,q1(R^d)kgkL^p2,q2(R^d), (5.13) where _p¹ = _p¹

1 +_p¹

2 and ¹_q = _q¹

1 +_q¹

2 and C =C(p1, p2, q1, q2) (a first proof of these inequalities using rearrangements can be found in [17], while a more modern proof would proceed by bilinear interpolation between the usual inequalities for Lebesgue spaces).

We now takeg(x) = _|x|¹s and apply (5.13), in the specific form kf gkL^q,q(R^d)≤CkfkL^p,q(R^d)kgkL^r,∞(R^d)

where ¹_q = ¹_p+¹_r, i.e. r= ^d_s. Since obviouslyg∈L^r,∞(R^d), we have Z |f(x)|^q

|x|^{sq 1}_q

≤Ckfk_L^p,q_(R^d₎.

Combining this with (1.3), we obtain (1.4). Let us mention that this inequality is obtained in [1] for q= 2 by a different method which can not be generalized to the generalL^q framework.

Concerning the sharpness of (1.4), we can make similar observations as for (1.3). In particular, this inequality is sharp for oscillatory functions of the typeϕω(x) =e^iωx1ϕ(x). It is also easily checked that the sequencefJ defined by (4.12) satisfies

Z |fJ(x)|^q

|x|^{sq 1}_q

∼J^1q ∼ kfJkL^p,q(R^d),

making (1.4) sharp for such functions, and justifying the use of ˙B^s−

d

∞,qq in place of ˙B^s−

d

∞,∞q .

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[17] R. O’Neil,Convolution operators on L(p, q)spaces, Duke Math.J. (30) 129-142, 1963.

[18] H. Triebel,Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, 1978.

Hajer Bahouri

Centre de Math´ematiques - Facult´e de Sciences et Technologie Universit´e Paris XII - Val de Marne

61, avenue du G´en´eral de Gaulle 94 010 Creteil Cedex, France bahouri@univ-paris12.fr

Albert Cohen

UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

cohen@ann.jussieu.fr