Refined Hardy inequalities

Refined Hardy inequalities

HAJERBAHOURI, JEAN-YVESCHEMIN ANDISABELLEGALLAGHER

Abstract. The aim of this article is to present “refined” Hardy-type inequalities.

Those inequalities are generalisations of the usual Hardy inequalities, their ad- ditional feature being that they are invariant under oscillations: when applied to highly oscillatory functions, both sides of the refined inequality are of the same or- der of magnitude. The proof relies on paradifferential calculus and Besov spaces.

It is also adapted to the case of the Heisenberg group.

Mathematics Subject Classification (2000):43A80 (primary); 42B99 (second- ary).

1.

Introduction

The aim of this article is to prove a “refined” version of the Hardy inequalities [11, 12]. Those inequalities have some importance in Analysis (among other applica- tions we can mention blow-up methods or the study of pseudodifferential operators with singular coefficients). Many works have been devoted to those inequalities, and our goal is first to provide an elementary proof of the standard Hardy inequality, and then to prove a refined inequality in the spirit of the refined Sobolev inequal- ity proved in [10]. The setting will be both the classicalR^N space, as well as the Heisenberg groupH^d(for an application of the Hardy inequality on the Heisenberg group we refer for instance to [1]).

1.1. Elementary Hardy inequality

The simple case ofR^N withN ≥3 with one derivative gives the following inequal-

ity:

R^N

u²(x)

|x|² d x ≤C∇u²_L2. (1.1) In order to prove this inequality, it is enough to observe that we have

1

|x|² = −1 2R

1

|x|²

with R=x· ∇.

Received February 22, 2006; accepted in revised form July 21, 2006.

An integration by parts joint with the fact that the divergence of R is equal to N gives the result.

Let us now present the case of the Heisenberg group. The Heisenberg groupH^d is the spaceR^2d+1endowed with the following product group law:

w·w=(x+x,y+y,s+s+(y|x)−(y|x))

where w = (x,y,s)andw = (x,y,s). Let us notice that H^d is a non com- mutative group and that the inverse ofwisw⁻¹ = (−x,−y,−s). The Lebesgue measure on H^d seen as R^2d+1 is invariant by translation with respect to this law.

We define the convolution of two functions by f g(w)=

H^d

f(ww⁻¹)g(w)dw.

Let us emphasize that this convolution product is, asH^d itself, not commutative.

We say that a vector field X is left invariant if X(f(a·)) = (X f)(a·). The Lie algebra of left invariant vector fields is spanned by the vector fields

X_j=∂x_j+y_j∂s, Y_j=∂y_j−x_j∂s with j ∈ {1, . . . ,d} and S=∂s=1

2[Y_j,X_j]. In all that follows, we shall denote by

Z

Hdef

= ^2d

j=1

Z²_j and forα∈ {1, . . . ,2d}^k,

Z

Z

Definition 1.1. Letkbe a non negative integer, we denote by H˙^k(H^d)the homo- geneous Sobolev space of orderkwhich is the space of functionsusuch that

u²_H˙k(H^d)

def=

α∈{1,...,2d}^k

Z

Let us also introduce the distance to the origin ρ(w)^def=

(|x|²+ |y|²)²+s^{2 1}₄

with w=(x,y,s)

and the dilation δ_λ(w) ^def= (λx, λy, λ²s). Let us point out that the function ρ is homogenenous of degree 1 in the sense that

ρ◦δ_λ=λρ

and the vector fieldsZ_j change the homogeneity as Z_j(f ◦δ_λ)=λ(Z_jf)◦δ_λ. Moreover, we have

Z_jρ^σ≤C_σρ^σ−1. (1.4)

Let us also introduce the homogeneous dimension N = 2d +2 noticing that the Jacobian of the dilationδ_λisλ^N. The Hardy inequality with one derivative in this context is

H^d

u²(w)

ρ²(w)dw≤C∇Hu²_L2 where ∇Hu^def= (Z₁u, . . . ,Z_2du).

The proof (as written for instance in [1]) of this inequality relies mainly on the fact that

1 ρ² = −1

2R 1

ρ²

with R^def= d

j=1

(x_jX_j +y_jY_j)+2s∂s. An integration by parts and the fact that divR= N essentially gives the result.

1.2. More general Hardy inequalities

Now we want to state Hardy inequalities with any number of derivatives less thanN/2.

Theorem 1.2. Let s ∈]0,N/2[. There exists a constant C such that

R^N

|u|²(x)

|x|^2s d x ≤Cu²_H˙s(R^N) and

H^d

|u|²(w)

ρ^2s(w)dw≤Cu²_H˙s(H^d)

where the spacesH˙^sare defined by complex interpolation.

Classically, the way of proving this consists in proving that the operators 1

|x|^s(−)^−s2 or 1

ρ^s(−H)^−s2

are bounded onL²(R^N)orL²(H^d). The purpose of this paper is first to give a more direct proof of these inequalities, which will be the same forR^N orH^d. Moreover, in the case ofR^N, let us apply the above Hardy inequality withs = 1 to the fam- ily(f_ε)_ε>0of functions defined by

f_ε(x)=e^ixε1θ(x)

whereθis a given function in the Schwartz class

S

The second purpose of this paper is to improve Hardy inequalities into inequalities which in particular will be invariant under the multiplication by oscillating functions likee^i(x|ω)ε .

This requires the introduction of Besov spaces of negative index and thus Lit- tlewood Paley theory. In the case ofR^N, this is quite classical. In the case of the Heisenberg group, it was constructed by H. Bahouri, P. G´erard and C.-J. Xu in [2]

(see also [3]). We can summarize this theory in the following properties, which hold regardless of the space which can be R^N or H^d; one of the features of this paper is to write unified statements and proofs, which hold independently of the space. It is therefore natural to introduce unified notation. In the same way as on the Heisenberg group we have defined a family

Z

forα∈ {1, . . . ,N}^k,

Z

∀w∈R^N, w⁻¹=−w, ρ(w)^def= _N

j=1

|wj|²

1 2

, and ∀a∈R, δaw=aw.

Using that notation, the elements of Littlewood-Paley theory we will need are the following.

Both in the case ofR^N andH^d, there exists a family(S_j)j∈Zof operators such that for any p belonging to[1,∞[,

∀u∈L^p, lim

j→−∞S_juL^p =0 and lim

j→∞S_ju−uL^p =0. (1.5) Moreover, for any multi-indexα, there exists a constant C such that, for any(p,q)∈ [1,∞]²satisfying p≤q, we have

Z

1 p−_q¹

+|α|j

S_juL^p. (1.6) Moreover, ifj

def= S_j+1−S_j, two integers N₀and N₁exist such that

|j − j| ≥N₀=⇒

jj =0 and j

S_j−N₀ujv

=0

, (1.7)

|k−k| ≤N₀ and j ≥k+N₁

=⇒j(kukv)=0. (1.8) For any positive integer k, there exists a constant C such that, for any p∈[1,∞],

juL^p ≤C2^{−2j k}(−)^kjuL^p. (1.9)

The operatorsj are of the form

ju=uh_j with h_j(w)=2^{j N}h(δ2^jw) and h∈

S

We remark that as j is a function of the Laplacian (respectively sublaplacian) onR^N (respectivelyH^d), it commutes with the latter operator.

Definition 1.3. Lets ∈ R be given, as well as p andr, two real numbers in the interval [1,∞]. Then we define the space B˙^s_p,r of tempered distributions u such that

lim

j→−∞S_ju=0 and u_B˙s p,r

def= (2^{j s}juL^p)

^r(Z) <∞.

Let us notice that Inequality (1.6) implies immediately that, whenq≥ pandr≥r, we have

u

˙ B^s−N

1 p−1

q q,r

≤Cu_B˙s

p,r. (1.11)

The result we will prove is the following. It is stated and proved indifferently inR^N andH^d.

Theorem 1.4. Let s be a real number in the interval]0,N/2[and let p and q be two real numbers in[1,∞]such that

2≤q< 2N

N −2s < p≤ ∞.

There is a constant C such that, for any function u ∈ ˙B^s−N(

1 2−¹_q)

q,2 , the following inequality holds:

|u(w)|² ρ^2s(w) dw

1 2

≤Cu^α

B˙^s−N 1

2−1 p p,2

u^1−α

˙ B^s−N

1 2−1

q q,2

with α= pq p−q

1 q − 1

2+ s N

. Let us remark that, when p= ∞andq =2, the above theorem implies that

|u(w)|² ρ^2s(w) dw

1 2

≤Cu^2sN

B˙^s−

N2

∞,2

u¹_H˙⁻_s^2sN. (1.12) This inequality should be compared to the following similar result derived by P.

G´erard, Y. Meyer and F. Oru in [10], in the case of the Sobolev inequalities onR^N (see [3] for the Heisenberg case), namely

uL^r ≤Cu^2sN

˙ B^s−

N2

∞,∞

u¹_H˙⁻_s^2sN with 1 r = 1

2− s

N· (1.13)

The following result indicates the invariance of (1.12) and (1.13) under oscillations.

Proposition 1.5. Letθbe a function in

S

∀ε≤ε0, f_εB˙^σ_p,1≤Cε^−σ. (1.14) This proposition implies immediately the following corollary.

Corollary 1.6. There exists a family (f_ε)_ε>0 of smooth functions such that, for any s in]0,N/2[and anyβ >2s/N , we have

ε→lim0

f_ε

L^N−2s2N

f_ε^β

˙ B^s−

N

∞,∞2

f_ε¹_H˙^−β_s = +∞ and lim

ε→0

1 f_ε^β

˙ B^s−

N

∞,∞2

f_ε¹_H˙^−β_s f_ε²

ρ^2sdw= +∞.

1.3. Structure of the paper and idea of the proof

The idea of the proof of Theorems 1.2 and 1.4 is to see them from a non linear point of view. More precisely, we write

u²(w)

ρ^2s(w)dw= ρ^−2s,u².

Then it is enough to prove thatρ^−2sandu²belongs to a pair of spaces in duality.

In the second section, we shall prove thatρ^−2s belongs to the space B˙₁^N_,∞^−2s. Then using a product law, we shall conclude the proof of Theorem 1.2.

In the third section, we shall use paradifferential calculus to prove Theorem 1.4.

In the fourth section, we shall prove Proposition 1.5. We shall also investigate if it is possible to extend Corollary 1.6 for a family of non negative functions.

2.

The behavior of negative powers of

Proposition 2.1. Let s be a real number in the interval]0,N/2[. Then the func- tionρ^−2sbelongs to the Besov space B˙₁^N_,∞^−2s.

Proof of Proposition 2.1. Let us introduce a smooth compactly supported func- tionχwhich is identically equal to 1 near the unit ball and let us write

ρ^−2s=ρ0+ρ1 with ρ0

def= χρ^−2s and ρ1

def= (1−χ)ρ^−2s.

It is obvious thatρ^−2s∈L¹+L^qwithq>N/2swhich implies that lim

j→−∞S_jρ^−2s= 0 inL¹+L^q. Then, the homogeneity of the functionρgives

jρ^−2s = 2^{j N}ρ^−2sh(δ2^j·)

= 2^j(N+2s)ρ^−2s(δ₂^j·) h(δ₂^j·)

= 2^{2j s}(0ρ^−2s)(δ2^j·).

Therefore jρ^−2sL¹ = 2^j(2s−N)0ρ^−2sL¹ which reduces the problem to proving that the function 0ρ^−2s is in L¹. As ρ0 is in L¹, 0ρ0 is also in L¹ thanks to the continuity of the operator0 on Lebesgue spaces. In order to esti- mateρ1inL¹, we shall use Inequality (1.9) to write that

0ρ1L¹≤C_k(−)^k0ρ1L¹ ≤C_k(−)^kρ1L¹.

By the Leibniz formula,(−)^kρ1−(1−χ)(−)^kρis a smooth compactly sup- ported function. Then, we achieve the proof by using (1.4) and choosingk such that 2k> N−2s.

As an application, we shall prove Theorem 1.2. Whenubelongs toH˙^s, then u²∈ ˙B^2s−

N 2

2,1 and u²

B˙^2s−

N 2,1 2

≤Cu²_H˙s.

That result is classical in R^N and was proved inH^d by two of the authors in [3].

Now writing that

ρ^−2s,u² =

|j−j|≤N0

jρ^−2s, ju²,

we infer, thanks to Proposition 2.1 and embeddings (1.11), that ρ^−2s,u² ≤ u²_H^s

|j−j|≤N₀

2^−j

N 2−2s

d_j2^−j

2s−^N₂

with (d_j)j∈Z∈¹(Z).

This proves Theorem 1.2.

Remark 2.2. Let us point out that, in Theorem 1.2, the functionρ^−2s can be any function in B˙

N 2−2s 2,∞ .

3.

Paradifferential calculus and refined inequalities

In order to prove Theorem 1.4, let us recall the paraproduct algorithm introduced by J.-M. Bony in [4] in the case ofR^N and by two of the authors in the case ofH^d

in [3]. In both cases, this allows to write that

u² =2T_uu+R(u,u), with T_uu^def=

j

S_j−N0uju and R(u,u)^def=

|j−j|≤N₀

juju.

Using (1.7) and (1.6), we get jT_uuL^∞ =

|j−j|≤N₀

j(S_j−N₀uju) L^∞

≤

|j−j|≤N0

S_j−N₀uL^∞juL^∞.

Now let us write that 2^−j

N 2−s

S_juL^∞≤

k≤j−1

2^(j−k)

s−^N₂

2^k

s−^N₂

kuL^∞.

Young’s inequality on series implies that S_juL^∞≤Cc_j2^j

N 2−s

u

˙ B^s−

N

∞,22

with

j

c²_j =1.

This gives

j(T_uu)L^∞ ≤ Cu

˙ B^s−

N

∞,22 j+N0 j=j−N₀

2^j(N−2s)c_j2^−j

N 2−s

juL^∞

≤ Cu²

B˙^s−

N2

∞,2

2^j(N−2s)

j+N₀ j=j−N₀

d_j with

j

d_j =1. Thanks to (1.11), and Proposition 2.1, we have,

ρ^−2s,T_uu ≤Cu^2α

B˙^s−N 1

2−1 q q,2

u^2−2α

˙ B^s−N

1 2−1

p p,2

(3.1)

for any 0≤α≤1 and p,q≥1.

The estimate ofρ^−2s,R(u,u)relies on the following elementary interpola- tion lemma.

Proposition 3.1. Let s be a real number in the interval]0,N/2[and let p and q be two real numbers in[1,∞]such that

2≤q< 2N

N −2s < p≤ ∞.

There is a constant C such that for any functions f and g which belongs to L^p∩L^q, we have

ρ^−2s,f g ≤Cf^α_L^pg^α_L^pf¹_L^−αq g¹_L^−αq with α= pq p−q

1 q −1

2 + s N

·

Proof of Proposition 3.1.Let us write that, for any positiveR, ρ^−2s, f g = I₁(R)+I₂(R) with I₁(R)^def=

(ρ≤R)

f g ρ^2sdw and I₂(R)^def=

(ρ≥R)

f g ρ^2sdw.

The condition onpandqimplies thatρ^−2sis locallyL^p−2p and isL^q−2q outside any compact neighbourhood of 0. By H¨older’s inequality, we infer that

I₁(R)≤ 1_(ρ≤R)ρ^−2s

L

p−2p fL^pgL^p and I₂(R)≤ 1_(ρ≥R)ρ^−2s

L

q−2q fL^qgL^q.

As the functionρis homogeneous of order 1, we get, by the change of variablew= δ_R−1w,

1_(ρ≤R)ρ^−2s

L

p−2p = R^N−2s−2Np 1_(ρ≤1)ρ^−2s

L

p−2p and 1_(ρ≥R)ρ^−2s

L

q−2q = R^N−2s−2Nq 1_(ρ≥1)ρ^−2s

L q−2q .

Thus we have, for any positive R, ρ^−2s, f g ≤C R^N−2s

R^−2Np fL^pgL^p+R^−2Nq fL^qgL^q

.

Choosing the optimal

R=

fL^qgL^q

fL^pgL^p

_2N(p−q)^pq

concludes the proof of the proposition.

Let us go back to the proof of Theorem 1.4. By definition of R(u,u), we have ρ^−2s,R(u,u) =

||≤N₀

j∈Z

ρ^−2s, juj−u.

Proposition 3.1 implies that ρ^−2s,R(u,u) ≤

||≤N₀

j∈Z

2^{2j(s−N(12−1p))}juL^pj−uL^p

_α

×

2^{2j(s−N(12−1q))}juL^qj−uL^q

1−α

.

By definition of the Besov norms, this implies that two series(c_j)j∈Zand(c_j)j∈Z

exist in the unit sphere of²(Z)such that ρ^−2s,R(u,u) ≤Cu^2α

B^s−N 1

2−1 p p,2

u^2(1−α)

B^s−N 1

2−1 q q,2

||≤N₀

j∈Z

(c_jc_j−)^α(c_jc_j−)^1−α.

From H¨older inequalities, it follows that ρ^−2s,R(u,u) ≤Cu^2α

B^s−N 1

2−1 p p,2

u^2(1−α)

B^s−N 1

2−1 q q,2

.

Together with (3.1), this gives Theorem 1.4.

4.

Oscillations and fractal transforms in refined inequalities

The purpose of this section is to provide examples which show that the refined estimates are sharp. The first one deals with oscillating functions.

4.1. Oscillations

Here we want to prove Proposition 1.5, namely, by definition of Besov spaces, that for any functionθ in

S

j

2^jσj f_εL^p≤Cε^−σ with f_ε(w)^def= e^iwε1θ(w). (4.1) We shall treat differently the high frequencies (indices j such that 2^jε is greater than 1) and low frequencies (indices j such that 2^jεis less than 1).

Let us first estimate the low frequencies. Denoting the vectorZ₁ =∂x1 in the case ofR^N andZ₁=∂x₁−y₁∂sin the case ofH^d, we have

iεZ₁e^iwε1 = −e^iwε1.

By integration by parts, we get j f_ε(w) = (−iε)^N2^{j N}

(−Z₁)^N(eⁱ

w 1−w1

ε )θ(ww⁻¹)h(δ2^j(w))dw

= (−iε)^N2^{j N}

eⁱ

w 1−w1

ε (Z₁)^N(θ(ww⁻¹)h(δ2^j(w))dw

= (−iε)^N2^{j N} N

=0

C_N

eⁱ

w 1−w1

ε Z₁^N−(θ(ww⁻¹))Z₁(h(δ₂^j(w))dw

where the vector fieldZ₁acts on the variablew. As

Z₁(h(δ₂^j(w))=2^j(Z₁h)(δ₂^j(w)) and −Z₁(θ(ww⁻¹))=(Z₁θ)(ww⁻¹), we infer that

|jf_ε(w)|

= ε^N2^{j N}

N

=0

C_N2^j(−1)^N−

e^iw

1−w1

ε (Z₁^N−θ)(ww⁻¹)(Z₁h)(δ2^j(w))dw

≤ ε^N2^{j N} N

=0

C_N2^j

|Z₁^N−θ||(Z₁h)(δ₂^j·)|

(w).

Young inequalities imply that |Z₁^N−θ||(Z₁h)(δ₂^j·)|

L^p

≤min

2^−jNpZ₁^N−θ_L¹Z₁hL^p, 2^{−j N}Z₁^N−θL^pZ₁h_L¹ . Therefore, asσ >−N

1− ¹_p

,

2^j≤¹_ε

2^jσjf_εL^p ≤ Cε^N





2^j≤1

2^j

σ+N 1−¹_p

+

1≤2^j≤¹_ε

2^j(σ+N)





≤ Cε^−σ.

In order to estimate high frequencies, let us use (1.9). We get, for any non negative integer M,

j f_εL^p ≤ C2^j(N−2M)((−)^Mf_ε) h(δ₂^j·)L^p

≤ C2^{−2j M}(−)^Mf_εL^p.

The Leibniz formula implies that, for anyε∈]0,ε0],(−)^Mf_εL^p≤Cε^−2MθL^p. Thus we infer, thanks to (1.6), that

2^j≥¹_ε

2^jσj f_εL^p≤Cε^−2M

2^j≥¹_ε

2^j(σ−2M).

ChoosingM such thatσ −2M is negative gives

2^j≥¹_ε

2^jσj f_εL^p ≤Cε^−σ. (4.2)

This ends the proof of Proposition 1.5.

4.2. Fractal transform and Besov norms

In this subsection we will show that oscillations are not the sole responsible for the smallness of a Besov norm. Below we present another situation, of a sequence of non negative functions for which theL^pnorms and the Besov norms are balanced as the family(f_ε)of Proposition 1.5. Again, we shall present statements and proofs common to the case ofR^NandH^d. In order to do so, let us define the distancedas

∀(w, w)∈R^N×R^N, d(w, w)^def= max

1≤j≤N|wj −w_j| and, for any(w, w)∈H^d×H^d,

d(w, w)^def=max

max

1≤j≤d|x_j −x_j|, max

1≤j≤N|y_j −y_j|,|s−s+(y|x)−(y|x)|¹²

, where in the case ofH^d we have notedw= (x,y,s)andw =(x,y,s). Let us denote by Q the ball ford centered at zero and of radius 1/2. Now let us define the following quantities. Let D andL such that D = L = N in the case ofR^N andD= N−1 andL =N+1 in the case ofH^d. For Jin{−1,1}^L, we define the pointwJ ofQand the cubeQ_J by

wJ def= δ³

8J and Q_J ^def= wJ ·δ¹

4Q=

w /d(w, wJ)≤ 1 8

.

Omitted elementary computations show that Q_J ⊂ Q and

J = J=⇒d(Q_J,Q_J)≥

√3

4 . (4.3)

Now let us define the transformT which duplicates (after dilation and translation) functions defined on Q.

Definition 4.1. Let us denote byT the following transform T





D

D

f → T f ^def= 2^D

J∈{−1,1}^L

f_J with f_J(w)^def= f(δ4(w⁻_J¹w)).

For a subsetAofQ, we denote byT Athe set defined by T A^def=

J∈{−1,1}^L

wJδ¹

4A.

Let us notice thatT A⊂ Qand that Supp(T f)=T(Supp f). Let us also observe that, using (4.3), we have

T f^p_Lp = 2^Dp

J∈{−1,1}^L

f_JL^pp

= 2^Dp





J∈{−1,1}^L

2^−2N



f_L^pp

= 2^Dp+L−2Nf_L^pp

= 2^D(p−1)f^p_Lp. Thus, we have

T fL^p =2^D

1−¹_p

fL^p. (4.4)

The wayT acts on Besov spaces is described by the following proposition.

Proposition 4.2. For any(p,r)∈[1,+∞]²and anyσ in −N

1− ¹_p

, + ∞ , there exists a constant C such that

T f_B˙σ p,r ≤2^D

1−¹_p

+2σf_B˙σ

p,r+CfL¹.

Proof of Proposition 4.2.For the sake of simplicity, we only prove this proposition in the case whenr =1. By definition of the Besov norm, we have

T f_B˙^σ

p,1 =T₁f +T₂f with T₁f ^def=

j≤0

2^jσjT fL^p

and T₂f ^def=

j≥1

2^jσjT fL^p. On the one hand, using Bernstein’s inequality (1.6), the fact thatN

1− ¹_p

+σ >0 and (4.4) with p=1, we get

T₁f ≤C

j≤0

2^{jσ+j N}

1−¹_p

jT fL¹ ≤CT fL¹ ≤CfL¹. (4.5)