Necessary condition for measures which are <span class="mathjax-formula">$(L^{q},L^{p})$</span> multipliers

BLAISE PASCAL

Bérenger Akon Kpata, Ibrahim Fofana and Konin Koua Necessary condition for measures which are (L^q, L^p)

multipliers

Volume 16, n^o2 (2009), p. 339-353.

<http://ambp.cedram.org/item?id=AMBP_2009__16_2_339_0>

© Annales mathématiques Blaise Pascal, 2009, tous droits réservés.

L’accès aux articles de la revue « Annales mathématiques Blaise Pas- cal » (http://ambp.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://ambp.cedram.org/legal/). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Publication éditée par le laboratoire de mathématiques de l’université Blaise-Pascal, UMR 6620 du CNRS

Clermont-Ferrand — France

cedram

Article mis en ligne dans le cadre du

Centre de diffusion des revues académiques de mathématiques

Necessary condition for measures which are (L

, L

) multipliers

Bérenger Akon Kpata Ibrahim Fofana

Konin Koua

Abstract

LetG be a locally compact group andρ the left Haar measure onG. Given a non-negative Radon measureµ, we establish a necessary condition on the pairs (q, p)for whichµis a multiplier fromL^q(G, ρ)toL^p(G, ρ). Applied toRⁿ, our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7].

WhenGis the circle group, we obtain a generalization of a condition stated by Oberlin [15] and improve on it in some cases.

Résumé

Soit Gun groupe localement compact et ρ la mesure de Haar à gauche sur G. Etant donné une mesure de Radon positive µ, nous établissons une condition nécessaire sur les couples(q, p)pour lesquelsµest un multiplicateur deL^q(G, ρ) dansL^p(G, ρ). Appliqué àRⁿ, notre résultat est plus fort que la condition néces- saire établie par Oberlin dans [14] et est très lié à une classe de mesures définie par Fofana dans [7].

LorsqueGest le tore, nous obtenons une généralisation d’une condition énoncée par Oberlin [15] et l’améliorons dans certains cas.

1. Introduction

We suppose that G is a locally compact group and ρ is the left Haar measure on G.

For 1≤q <∞, a Radon measure µon Gis said to be L^q-improving if there exists a real number p > q such that

µ∗f ∈L^p(G, ρ) and kµ∗fk

Lp(G, ρ) ≤ckfk

Lq(G, ρ)

Keywords: Cantor-Lebesgue measure, L^q-improving measure, non-negative Radon measure.

Math. classification:43A05, 43A15.

for all f ∈L^q(G, ρ), wherecis a real number not depending onf. Of course absolutely continuous measures with Radon-Nikodym deriva- tives with respect to ρinL^r(G, ρ) with ¹_q+¹_r−1>0are L^q-improving.

But L^q-improving singular measures also exist.

Bonami [2] showed that all tame Riesz products on the Walsh group areL^q-improving, and that was extended to all compact abelian groups by Ritter [16]. Moreover it is well known that on the circle groupT=R/Z, the Cantor-Lebesgue measure µ²_δ associated with the Cantor set of constant ratio of dissectionδ >2 isL^q-improving for1< q <∞.(See Section 4 for a precise definition of this measure.) This result was proved by Oberlin [12] forδ = 3. Ritter [17], Beckner, Janson and Jerison [1] proved the same for δ rational and Christ [3] for δ irrational.

In fact, Christ has extended the result to Cantor-Lebesgue measures with variable but bounded ratios 2< δt≤c of dissection.

In this note, we are interested in the following problem: given a non- negative Radon measure µonG, determine the indices1≤q < p <∞for which there exists a non-negative constant c(µ, q, p) such that

kµ∗fk

Lp(G, ρ) ≤c(µ, q, p)kfk

Lq(G, ρ), f ∈L^q(G, ρ). (1.1) In [15] Oberlin stated the following

Proposition 1.1. If the Cantor-Lebesgue measure µ²₃ associated to the middle third Cantor set satisfies (1.1), then

1 q +

1−log 2

log 3 1−1 p

≤1. (1.2)

Graham, Hare and Ritter obtained in [9] the following

Proposition 1.2. Letµbe a measure on the circle groupTand1≤q <2.

If there exists a non-negative constant c(µ, q) such that kµ∗fk_L2(T)≤c(µ, q)kfk_Lq(T), f ∈L^q(T),

then there exists a positive real number K such that for any interval I whose endpoints are x and x+h, we have

|µ(I)| ≤K|h|^1q−12. (1.3)

Inequality (1.3) means that µ satisfies a Lipschitz condition of order

1 q −¹₂.

Replacing T by Rⁿ, Oberlin proved a similar necessary condition (see the proof of Proposition 2 in [14]).

Proposition 1.3. If a non-negative Radon measure onRⁿ satisfies (1.1), then there exists a positive real number K such that

µ(R)≤K|R|^1q−1p (1.4)

for all rectangles R in Rⁿ.

In the present paper, we establish the following necessary condition:

Proposition 1.4. Suppose that µis a non-negative Radon measure on G satisfying (1.1). Then for any subsets V and{x_i / i∈I} of Gsuch that i) V is relatively compact,

ii) I is countable and (xiV)∩(xjV) =∅ for i6=j, we have

ρ(V)^{1p X}

i∈I

µ(xiV)^p

!¹_p

≤c(µ, q, p)ρV⁻¹V

1

q . (1.5)

We show that all the necessary conditions stated in Proposition 1.1, Proposition 1.2 and Proposition 1.3 follow from Proposition 1.4.

Moreover any non-negative Radon measure µ on T or Rⁿ satisfying the conclusion of Proposition 1.4 belongs to the space M^{p, α}, _α¹ = 1−

1

q +¹_p (see Notation 3.4 and Section 4 for the definition of M^{p, α}). In [7], Fofana used these spaces of measures and their subspaces (L^q, l^p)^α to express a necessary condition for Fourier multipliers. He also obtained a generalization of Hausdorff-Young inequality. For other results related to these spaces see [6], [8] and [11].

Inequality (1.2) means exactly that µ²₃ belongs to M^{p, α} where _α¹ = 1−

1

q +¹_p (see the comment after the proof of Proposition 4.2).

Applied to the Cantor-Lebesgue measure associated to the Cantor set of constant ratio of dissection δ >3, Proposition 1.4 yields the following Proposition 1.5. Let δ >3 and 1< q < p < ∞. Assume that

µ²_δ∗f

L^p(T)≤cµ²_δ, p, qkfk_Lq(T), f ∈L^q(T).

Then

p≤

log^δ₂

log^δ₃q (1.6)

and

1 q +

1−log 2

logδ 1− 1 p

≤1. (1.7)

Notice that (1.6) is stronger than (1.7) if q > ^{log 3}_{log 2}.

The remainder of this paper is organized as follows: in Section 2 we prove Proposition 1.4 and apply it to G = Rⁿ in Section 3. In Section 4 we examine the case G=T.

2. Proof of Proposition 1.4

Proof. LetV be a relatively compact subset ofG. Thenf =χ_V⁻¹_V belongs toL^q(G, ρ). We have, for alli∈I and allx∈xiV,

µ∗f(x) = Z

G

fy⁻¹xdµ(y)≥ Z

xiV

fy⁻¹xdµ(y),

y∈xiV =⇒y⁻¹x∈V⁻¹V and fy⁻¹x= 1 and therefore µ∗f(x)≥µ(x_iV). It follows that

Z

G

(µ∗f(x))^pdρ(x)≥^X

i∈I

Z

xiV

(µ∗f(x))^pdρ(x)≥^X

i∈I

µ(xiV)^pρ(xiV). Therefore

ρ(V)^{1p X}

i∈I

µ(xiV)^p

!¹

p

≤ kµ∗fk

Lp(G, ρ)

≤ c(µ, q, p)kfk

Lq(G, ρ)

= c(µ, q, p)ρV⁻¹V

1 q .

This completes the proof.

3. Case G=Rⁿ

Notation 3.1. Let R be a rectangle in Rⁿ with sides aivi, i = 1, ... , n, where (v_i)_1≤i≤n is a direct orthonormal basis in Rⁿ and a_i > 0, i = 1, ... , n.

For any r >0 and k= (k1, ... , k_n)∈Zⁿ, set R^r_k=

( _n X

i=1

(kirai+xi)vi / 0≤xi< rai,i= 1, ... , n )

.

In other words, R^r_k is a rectangle which i-th edge is parallel to the vector v_i and of length ra_i. Notice that forr >0, the family{R^r_k / k∈Zⁿ} is a partition of Rⁿ.

Proposition 3.2. Let 1≤q ≤p <∞. If a non-negative Radon measure µ on Rⁿ satisfies (1.1), then for all rectangles R in Rⁿ

sup

r>0

(rⁿ|R|)^α1−1



 X

k∈Zⁿ

µ(R^r_k)^p





1 p

≤c(µ, q, p)2^nq where _α¹ = 1−¹_q +¹_p.

Proof. Letr >0. Notice that for everyk∈Zⁿwe have thatR^r_k=R0+u_k, where R0 =

_n P

i=1

x_iv_i / 0≤x_i < ra_i,i= 1, ... , n

and u_k = ^Pn

i=1

k_ira_iv_i. It follows from Proposition 1.4 that

|R₀−R0|^−1q |R₀|^1p



 X

k∈Zⁿ

µ(R^r_k)^p





1 p

≤c(µ, q, p).

Since |R0|=rⁿ|R|, we have 2^−nq (rⁿ|R|)^−1q (rⁿ|R|)^1p



 X

k∈Zⁿ

µ(R^r_k)^p





1 p

≤c(µ, q, p).

Hence

(rⁿ|R|)^α1−1



 X

k∈Zⁿ

µ(R^r_k)^p





1 p

≤c(µ, q, p)2^nq.

The assertion follows.

Remark 3.3. Proposition 1.3 is a direct consequence of Proposition 3.2.

In fact, suppose that µ satisfies (1.1) and let R be any rectangle. As R¹_k / k∈Zⁿ is a partition of Rⁿ,R ⊂ ∪

k∈MR¹_k, where M is a subset of Zⁿ which number of elements does not exceed 2ⁿ. Soµ(R) ≤ ^X

k∈M

µ(R_k¹) and by Hölder inequality we have

|R|^1p−1q µ(R) ≤ 2

n(p−1)

p |R|^1p−1q



 X

k∈M

µR¹_k^p





1 p

≤ 2

n(p−1)

p |R|^1p−1q



 X

k∈Zⁿ

µR¹_k^p





1 p

≤ 2

n(p−1)

p sup

r>0

(rⁿ|R|)^α1−1



 X

k∈Zⁿ

µ(R^r_k)^p





1 p

where _α¹ = 1−¹_q +¹_p. Thus, by Proposition 3.2 we obtain

|R|^1p−1q µ(R)≤2^{n 1−}

1 p+¹

q

c(µ, q, p).

Notation 3.4. For anyk∈Zⁿ,x∈Rⁿ andr >0, set I_k^r=

n

i=1Π [k_ir, (k_i+ 1)r) and J_x^r =

n i=1Π

x_i− r

2, x_i+ r 2

.

LetM⁰denote the space of Radon measures (not necessarily non-negative) on Rⁿ. For µ∈M⁰,|µ|stands for its total variation. Let1≤α, p≤ ∞.

For µ∈M⁰ and r >0, we set

rkµk_p=









 P

k∈Zⁿ

|µ|(I_k^r)^p

!¹_p

if1≤p <∞, sup

x∈Rⁿ

|µ|(J_x^r) ifp=∞ and kµk_{p, α}= sup

r>0

rⁿ(¹_α⁻¹)

rkµk_p.

We define M^{p, α}(Rⁿ) =ⁿµ∈M⁰ / kµk_{p, α}<∞^o.

Another consequence of Proposition 3.2 is the following

Corollary 3.5. Assume that 1≤q≤p <∞ and µ satisfies (1.1). Then µ belongs to M^{p, α}(Rⁿ) where _α¹ = 1− ¹_q+¹_p.

Proof. It follows by choosing a_i = 1 for i ∈ {1, . . . , n} and (v_i)_1≤i≤n = (e_i)_1≤i≤n the usual basis ofRⁿ in the definition of R^r_k in Proposition 3.2.

4. Case G=T

In this section we suppose that m ≥ 2 is an integer. Let us describe the construction of the Cantor set with variable ratios of dissection and its associated Cantor-Lebesgue measure. We take the interval [0, 1)as a model for T. Letδ_t> mfor t= 1, 2, ... . Delete from [0, 1),(m−1)left closed intervals of equal length _m−1¹ 1−^m_δ

1

so that themremaining left closed intervals denoted by E_l¹, 1 ≤l ≤ m, are equally spaced and have the same length _δ¹

1. From each intervalE_l¹,1≤l≤m, delete(m−1)left closed intervals of equal length _(m−1)δ¹

1

1−_δ^m

1δ2

so that themremaining left closed subintervals E_l², 1≤l ≤m², are equally spaced and have the same length _δ¹

1δ2. At this stage, the remaining subset of[0, 1)isC_(δ^m

1, δ2)=

m²

∪

l=1E_l². By iteration, we obtain a sequence of subsetsC_(δ^m

1, δ2, ... , δj)= ^m

j

∪

l=1E_l^j, where each E_l^j is a left closed interval of length r_j =

j

Π

t=1δ_t⁻¹. C_(δ^m

t) =

∞∩

j=1C_(δ^m

1, δ2, ... , δj)is the(m, (δ_t))-Cantor set and theδ_t’s are called its ratios of dissection. Associated toC_(δ^m

t)in a natural way is a probability measure µ^m_(δ

t) satisfying µ^m_(δ

t)

E_l^j= _m¹j forj= 1, 2, ...and for l= 1, 2, ... , m^j. This measure is the Cantor-Lebesgue measure associated to the(m, (δt))- Cantor set. Whenδ_t=δ,t= 1, 2, ..., we writeµ^m_(δ

t)=µ^m_δ . It follows that µ²₃ is the usual Cantor-Lebesgue measure associated to the middle third Cantor set. For a detailed exposition on Cantor sets see Zygmund [19].

Notice that ifµis a non-negative Radon measure onT, then in a natural way, we may identifyµwith a non-negative Radon measureνonRhaving support in the interval [0, 1). In addition, we have the following result established by Ritter in [17].

Proposition 4.1. Let 1 ≤q ≤ p < ∞, and suppose there is a constant K >0 such that

kµ∗fk_Lp(T)≤Kkfk_Lq(T), f ∈L^q(T). Then there is a constant K0>0 such that

kν∗fk_Lp(R)≤K0kfk_Lq(R), f ∈L^q(R). Defining, for 1≤α, p≤ ∞,

M^{p, α}(T) ={µ∈M^{p, α}(R) /supp(µ)⊂[0, 1)}

where supp(µ) denotes the support of µ, it is easy to see that Corollary 3.5 holds in this setting.

The following result gives a characterization of measuresµ^m_(δ

t)which belong toM^{p, α}(T).

Proposition 4.2. Letδt> m,t= 1, 2, .... Assume that1< α≤p <∞.

Then µ^m_(δ

t) belongs toM^{p, α}(T) if and only if there exists a constantc >0 such that

j

t=1Πδt≤cm

α(p−1)j

p(α−1), j= 1, 2, ... .

In particular, the Cantor-Lebesgue measure µ^m_δ of constant ratio of dis- section δ belongs to M^{p, α}(T) if and only if

1− 1

α −logm logδ

1−1

p

≤0. (4.1)

Proof. a) For allr ≥1

r^α1−1 _rµ^m_(δt)

p=r^α1−1 ≤1.

b) Let j be a positive integer and rj =

j

t=1Πδ⁻¹_t . Recall that for l = 1, 2, ... , m^j,E_l^j=r_j and µ^m_(δ

t)

E_l^j= _m¹j. For each fixed l, put K_l = nk∈N / E_l^j ∩I_k^rj 6=∅^o. Then K_l has at most 2 elements. In the same way, for each fixed k inNsetL_k=ⁿl∈1, 2, ... , m^j / E_l^j ∩I_k^rj 6=∅^o.

Then the number of elements of L_k is at most 2. We have m^jm^−jp =

m^j

X

l=1

µ^m_(δt)E_l^jp

=

m^j

X

l=1



 X

k∈K_l

µ^m_(δt)E_l^j∩I_k^rj





p

≤ 2^p−1X

l∈L_k

X

k∈K_l

µ^m_(δt)E_l^j ∩I_k^rjp

= 2^p−1X

k∈N

X

l∈L_k

µ^m_(δt)E_l^j∩I_k^rjp

≤ 2^pX

k∈N

µ^m_(δ

t)

I_k^rjp.

Then

r

1 j(_α¹⁻¹)

j m^1p−1 j

=

j t=1Πδt

!1−¹

α

m^{−j 1−1p}

≤2r

1 α−1 j rj

µ^m_(δt)

p. c) Let r ∈(0, 1).There exists an integer j ≥1 such that r_j ≤r < rj−1

where r0 = 1and r_n =

n

t=1Πδ_t⁻¹ forn≥1. Furthermore, each I_k^r intersects at most m intervals E^j_l. So µ^m_(δ

t)(I_k^r) ≤ m^−jm. The number of I_k^r which intersect the intervals E_l^j is at most 2m^j.It follows that

X

k∈N

µ^m_(δ

t)(I_k^r)^p≤2m^j(1−p)m^p. Hence

r^α1−1 r

µ^m_(δt)

p ≤ 2^1pr^α1−1m^{j 1p−1}

m

≤ 2^1pr

1 α−1

j m^{j 1p−1}

m

= 2^1pm





j t=1Πδt

!¹_j(¹⁻_α¹) m^1p−1





j

.

Finally, µ^m_(δ

t)∈M^{p, α}(T) ⇐⇒ sup

j





j t=1Πδt

!¹

j(¹⁻_α¹) m^1p−1





j

<∞ µ^m_(δ

t)∈M^{p, α}(T) ⇐⇒ Π^j

t=1δt≤cm

α(p−1)j

p(α−1), j = 1, 2, ...

where cis a positive constant not depending onj.

d) Now, let δ_t=δ for allt≥1. From c) we know that:

µ^m_δ ∈M^{p, α}(T)⇐⇒δ^j ≤cm

α(p−1)j

p(α−1), j= 1, 2, ...

where cis a positive constant not depending onj. That means:

µ^m_δ ∈M^{p, α}(T) ⇐⇒ logδ ≤ ^α(p−1)_p(α−1)logm µ^m_δ ∈M^{p, α}(T) ⇐⇒ 1−_α¹ −^log_log^m_δ(1−¹_p)≤0.

Notice that for 1− _α¹ = ¹_q − ¹_p, (4.1) reduces to (1.2) when m= 2 and δ = 3.

Proposition 4.3. Letµ^m_(δ

t) be the Cantor-Lebesgue measure with variable ratios δ_t> m of dissection. Let1< q < p <∞. Assume that

µ^m_(δ

t)∗f

L^p(T)≤cµ^m_(δ

t), p, qkfk_Lq(T), f ∈L^q(T). Then there exists a constant c >0 such that

j

t=1Πδt≤cm

q(p−1)j

p−q , j = 1, 2, ... . In particular, if δt=δ for all t≥1, then

1 q +

1−logm

logδ 1−1 p

≤1.

Proof. Let1−_α¹ = ¹_q −¹_p. Then the desired result follows from Corollary

3.5 and Proposition 4.2.

Proposition 1.1 is obtained from Proposition 4.3 by taking m= 2 and δ_t= 3 for allt≥1.

Proof of Proposition 1.5. We are in the case m = 2 and δt = δ > 3 for all t ≥1. Let j be a positive integer. Observe that for any non-negative integer k, any l ∈ ⁿ1, 2, ... , 2^j+ko, E_l^j+k = x^j+k_l +^h0, δ^−j−k and µ²_δE_l^j+k= 2^−j−k. SetA0 =0, δ^−jandB0 =A0−A0 = −δ^−j, δ^−j. From Proposition 1.4 we obtain

|B₀|^−1q |A₀|^p1





2^j

X

l=1

µ²_δE_l^jp





1 p

≤cµ²_δ, p, q.

Observe that for fixed l in 1, 2, ... , 2^j , E_l^j contains two intervals E_l^j+1

1 and E_l^j+1

2 satisfying

µ²_δE^j_l=µ²_δE_l^j+1

1 ∪E_l^j+1

2

and

E_l^j+1

1 ∪E_l^j+1

2 =x^j_l +^h0, δ^−j−1∪^hδ^−j−δ^−j−1, δ^−j.

Setting A1 = 0, δ^−j−1∪δ^−j−δ^−j−1, δ^−j and applying Proposition 1.4 we obtain

|A₁−A1|^−1q |A₁|^1p





2^j

X

l=1

µ²_δE_l^jp





1 p

≤cµ²_δ, p, q. But each preceding intervalE^j+1_l

i ,i∈ {1, 2}, contains two intervalsE_l^j+2

i,1

and E_l^j+2

i,2 such that µ²_δE_l^j+1

i

=µ²_δE_l^j+2

i,1 ∪E_l^j+2

i,2

= 1 2^j+1. Moreover ∪²

i=1

E_l^j+2

i,1 ∪E_l^j+2

i,2

=x^j_l +A2 where A2 = ^h0, δ^−j−2∪^hδ^−j−1− δ^−j−2, δ^−j−1∪

∪^hδ^−j− δ^−j−1, δ^−j− δ^−j−1+δ^−j−2∪^hδ^−j−δ^−j−2, δ^−j. This remark enables us to apply again Proposition 1.4. Thus we obtain

|A2−A2|^−1q |A2|^1p





2^j

X

l=1

µ²_δE_l^jp





1 p

≤cµ²_δ, p, q.

The iteration of the process leads us to two sequences of sets (A_k)_k≥0 and

Af_k

k≥0 defined by:

A_k+1 = 1 δA_k∪

δ^−j−1 δA^f_k

, A]_k+1 = 1 δA^f_k∪

δ^−j−1 δA_k

with A0 =0, δ^−j,Af0= 0, δ^−jand satisfying

|B_k|^−1q |A_k|^p1





2^j

X

l=1

µ²_δE_l^jp





1 p

≤cµ²_δ, p, q, (4.2) where Bk=Ak−Ak for all k≥0.

Notice that A0 −A0 = A^f0 −A^f0 and |A0| = A^f0

. Furthermore, for any k≥0, clearlyA_k+1−A_k+1 =A]_k+1−A]_k+1and since ¹_δA_k∩δ^−j−¹_δAf_k=

∅= ¹_δAf_k∩δ^−j− ¹_δA_kwe have |A_k+1|=A]_k+1. Thus

A_k−A_k=Af_k−Af_k and |A_k|=Af_k, k≥0. (4.3) Observe that: |A₀| = δ^−j, |A₁| = 2δ^−j−1 and |A₂| = 2²δ^−j−2. Suppose that for some integerk≥0,|A_k|= 2^kδ^−j−k. By the preceding remarks we get |A_k+1|= ¹_δ|A_k|+¹_δA^f_k= ²_δ|A_k|= 2^k+1δ^−j−(k+1). We conclude that

|A_k|= 2^kδ^−j−k, k≥0.

Notice thatA0+Af0 = (0, 2δ^−j) =δ^−j−(−δ^−j, δ^−j) =δ^−j−(A0−A0) = δ^−j−B0. Furthermore, for any k≥0, on the one hand

B_k+1 = 1

δA_k∪

δ^−j −1 δA^f_k

− 1

δA_k∪

δ^−j−1 δA^f_k

= 1

δ(Ak−Ak)∪ 1

δ

Ak+Afk

−δ^−j

∪

∪

δ^−j− 1 δ

Af_k+A_k

∪1 δ

Af_k−Af_k

= 1

δ(A_k−A_k)∪ 1

δ

A_k+Af_k−δ^−j

∪

δ^−j−1 δ

Af_k+A_k

(because of(4.3))

and on the other hand A_k+1+A]_k+1 = 1

δ

A_k+Af_k∪

δ^−j+ 1

δ(A_k−A_k)

∪

∪

δ^−j +1 δ

Af_k−Af_k

∪

2δ^−j −1 δ

Af_k+A_k

= 1 δ

Ak+Afk

∪

δ^−j+ 1

δ(Ak−Ak)

∪

∪

2δ^−j−1 δ

Af_k+A_k

(because of(4.3))

and so A_k+1+A]_k+1=δ^−j −B_k+1. Thus

|A_k−Ak|=Ak+Afk

, k≥0. (4.4)

Notice that for allk≥0, the sets¹_δ(Ak−Ak),¹_δAk+Afk

−δ^−jandδ^−j−

1 δ

Af_k+A_k form a partition of B_k+1. Thus, by(4.4)we have |B_k+1| =

3

δ|A_k−A_k| = ³_δ|B_k|, k ≥ 0. As |B₀| = 2δ^−j, we conclude that for all k≥0,|B_k|=³_δ^k2δ^−j.

Finally, using inequality (4.2) we get:

2 3

δ k

δ^−j

!−¹_q

2^kδ^−j−k

1

p2^{j 1p−1}

≤cµ²_δ, p, q, k≥0, j≥1

2^−1q 3^−1qδ^1q−p12^1pkδ^1q−1p2^1p−1j ≤cµ²_δ, p, q, k≥0, j≥1

3^−1qδ^1q−1p2^1p ≤1 and δ^1q−1p2^1p−1 ≤1

p≤

log₂^δ

log₃^δq and 1 q +

1−log 2

logδ 1−1 p

≤1.

References

[1] William Beckner, Svante Janson, and David Jerison,Convolution in- equalities on the circle, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math.

Ser., Wadsworth, Belmont, CA, 1983, pp. 32–43. MR MR730056 (85j:42016)

[2] A. Bonami,Étude des coefficients de Fourier des fonctions deL^p(G), Ann. Inst. Fourier (Grenoble) 20(1970), 335–402.

[3] M. Christ,A convolution inequality concerning Cantor-Lebesgue mea- sures, Revista Mat. Iberoamericanavol. 1, n.^◦ 4(1985), 79–83.

[4] K. J Falconer, The geometry of fractal sets, Cambridge University Press, London/New York, 1985.

[5] ,Fractal geometry, Wiley, New York, 1990.

[6] I. Fofana,Continuité de l’intégrale fractionnaire et espaces (L^q, l^p)^α, C. R. A. S. Paris t. 308, série I(1989), 525–527.

[7] , Transformation de Fourier dans (L^q, l^p)^α et M^{p, α}, Afrika matematika série 3, vol. 5(1995), 53–76.

[8] ,Espaces (L^q, l^p)^α et continuité de l’opérateur maximal frac- tionnaire de Hardy-Littlewood, Afrika matematika série 3, vol. 12 (2001), 23–37.

[9] C. C. Graham, K. Hare, and D. Ritter, The size of L^p-improving measures, J. Funct. Anal. 84(1989), 472–495.

[10] R. Larsen, An introduction to the theory of multipliers, Springer- Verlag, Berlin, Heidelberg, New York, 1971.

[11] K-S. Lau, Fractal Measures and Mean ρ-Variations, J. Funct. Anal.

108 (1992), 427–457.

[12] D. M. Oberlin, A convolution property of the Cantor-Lebesgue mea- sure, Colloq. Math.47 (1982), 113–117.

[13] , Convolution with measure on hypersurfaces, Math. Proc.

Camb. Phil. Soc. 129(2000), 517–526.

[14] , Affine dimension : measuring the vestiges of curvature, Michigan Math. J.51(2003), 13–26.