A two-weight inequality for the Bessel potential operator

A two-weight inequality for the Bessel potential operator

Y. Rakotondratsimba

Abstract. Necessary conditions and sufficient conditions are derived in order that Bessel potential operator Js,λ is bounded from the weighted Lebesgue spaces L^pv = L^p(Rⁿ, v(x)dx) intoL^quwhen 1< p≤q <∞.

Keywords: weighted inequalities, Bessel potential operators, Riesz potential operators Classification: Primary 42B25

§1. Introduction

The Bessel potential operatorJ_s,λ is defined via the Fourier transform by (J[_s,λf)(ξ) =

4π²|ξ|²+λ^1s−^s₂

fb(ξ) whereλ >0, 0< s < n,n∈N^∗ andbg(ξ) =R

y∈Rⁿe^−2iπy·ξg(y)dy. Our purpose is to characterize the weight functionsu(·) andv(·) for which there isC >0 such that

(1.1) Z

x∈Rⁿ

(J_s,λf)^q(x)u(x)dx¹

q

≤C Z

x∈Rⁿ

f^p(x)v(x)dx¹_p

for all f(·)≥0, and with 1 < p ≤ q < ∞. A weight means a nonnegative locally integrable function. This inequality implies J_s,λ is bounded from the weighted Lebesgue spaceL^p_v =L^p(Rⁿ, vdx) intoL^q_u. For the convenience (1.1) will also be denoted byJ_s,λ:L^p_v →L^q_u.

Inequality (1.1) plays a fundamental role in Analysis since it is closely connected with spectral properties of Schr¨odinger operators [Ch-Wh], [Ke-Sa] and it leads to applications in partial differential equations ([Ad-Pi], [Ma-Ve]), theory of Sobolev spaces ([Ma]), complex analysis, etc. For instance estimate like

Z

y∈Rⁿg^p(y)u(y)dy

≤c Z

y∈Rⁿ

(−∆ +λ^1s)^s2gp

(y)v(y)dy for all smooth functions g(·),

which also appears in partial differential equation related to the operators (−∆ + λ^1s)^s2, can be derived fromJ_s,λ:L^p_u→L^p_u, since ((−∆+λ^1s)^s2J_s,λ)g= (J_s,λ(−∆+

λ^1s)^s2g) =g.

Compared to the Riesz potential operatorsI_s, 0< s < n, defined by (Isf)(x) =

Z

y∈Rⁿ

|x−y|^s−nf(y)dy,

few works (see for instance [Ad1], [Sc]) are devoted to the study ofJ_s,λ:L^pv → L^q_u; and people had to be content oneself on J_s,λ ≤cI_s so that a condition for Is :L^p_v →L^q_u is also right forJ_s,λ :L^p_v →L^q_u. The strongest up to date results are those of Kerman-Sawyer [Ke-Sa], and Maz’ya-Verbitsky [Ma-Ve]. Indeed a characterization of weightsu(·) for which J_s,λ:L^p₁→L^q_u (i.e. v(·) = 1) is given in [Ke-Sa], and investigations of weightsw(·) which ensureJs,1 :L^p₁→L^q

(Js,1w)^p′

are presented in [Ma-Ve]. Although a necessary and sufficient condition for Is : L^pv →L^quis known ([Sa-Wh]), the analog condition characterizingJ_s,λ:L^pv →L^qu

is not clear in the literature. Consequently our intention is to fill this gap.

Although a result due to Sawyer and Wheeden [Sa-Wh] related toT :L^pv →L^qu, whereTis a potential operator given by a positive kernelK(x, y), could be applied directly to get J_s,λ : L^pv → L^qu, the fast decrease at infinity of the kernel K_s,λ of J_s,λ (see §3) leads to conditions more refined than the standard ones used forT. Therefore the boundednessJ_s,λ:L^p_v →L^q_u deserves its own study which is performed in this paper.

The main results are presented in the next section. And§3 is devoted to basic lemmas used for the results whose proofs are given in§4.

§2. The main results

In this paper we always assume:

0< s < n, λ >0, 1< p≤q <∞, p^′= p

p−1, q^′ = q q−1, u(·), v(·) are weight functions with σ(·) =v^−p−11 (·)∈L¹_loc(Rⁿ, dx).

Our first main result is

Theorem 1. The boundedness J_s,λ : L^pv → L^qu holds if and only if there are C, c >0such that

(2.1) Z

Q

(Isg)^q(x)u(x)dx¹_q

≤C Z

(3Q)

g^p(x)v(x)dx¹_p

for each g(·)≥0whose support is3Q

and (2.2) Z

y /∈(3Q)

|x_Q−y|^(s−n)p′exp{−cλ^21s|x_Q−y|}σ(y)dy_p′¹

× Z

y∈Q

u(y)dy¹_q

≤C for all cubesQcentered atx_Q and with |Q|ⁿ¹ =λ^−2s1 . Remind that a cubeQ(centered atx_Q= (x_i)∈Rⁿ) is a product ofnintervals of the form [x_i−l, x_i+l] wherel >0. And forR >0,RQis the cube given by the product of [x_i−Rl, x_i+Rl]. The Lebesgue measureR

y∈Q dyofQis denoted by|Q|.

Next we give some remarks whose proofs are given in§4.

Remarks.

(1) A necessary condition forJ_s,λ :L^p_v →L^q_u, which is consequently assumed is 0≤ _n^s +¹_q−¹_p. So for 1< p < ⁿ_s this boundedness has a sense forp≤q≤p^∗ with _p¹∗ =¹_p −_n^s.

(2) Theorem 1 remains true if in conditions (2.1) and (2.2) the cubes Q are chosen such that|Q|ⁿ¹ ≈λ^−21s. This equivalence meansc₁λ^−21s ≤ |Q|¹ⁿ ≤c₂λ^−21s for some fixed constantsc−1, c₂>0.

(3) Condition (2.1) in Theorem 1 can be replaced by

(2.3) Z

Q₂

(Ish)^q(x)u(x)dx¹_q

≤C Z

Q₁

h^p(x)v(x)dx¹_p

for each functionh(·)≥0 whose support is Q₁; and whereQ₁ andQ₂ are cubes with|Q₁|ⁿ¹ =|Q₂|ⁿ¹ = [resp. ≈]λ^−2s1 andQ₁∩Q₂6=∅.

(4) Also the condition (2.2) can be replaced by (2.4) exp(−cm)m^(s−n)(λ^−2s1 )^(s−n) Z

y∈Q₂

σ(y)dy_p′¹ Z

x∈Q₁

u(x)dx¹_q

≤C for all integers m ≥ 4 and cubes Q₁, Q₂ with |Q₁|¹ⁿ = |Q₂|ⁿ¹ = λ^−21s and dist(Q₁, Q₂) = inf{|x−y|;y ∈Q₁, x∈Q₂} ≈(mλ^−2s1 )>0. So here we are in the caseQ₁∩Q₂=∅.

(5) The weight functionw(·) satisfies the doubling condition ifR

(2Q)w(y)dy≤ CR

Qw(y)dy for someC >0 and all cubes Q. If one ofu(·) andσ(·) =v^−p−11 (·) is a doubling weight then an easy condition which ensures (2.4) is

(2.5) (λ^−2s1 )^(s−n) Z

y∈Qσ(y)dy_p′¹ Z

x∈Qu(x)dx¹

q ≤C

for all cubesQwith |Q|ⁿ¹ =λ^−2s1 . Now we can state the following

Theorem 2. Assume that one ofu(·)andσ(·) =v^−p−11 (·)is a doubling weight function. ThenJ_s,λ:L^p_v→L^q_u if and only if the condition(2.3)is satisfied.

With this theorem and the well known results on the (global) boundedness Is:L^pv→L^qu, we obtain the following more useful statement.

Proposition 3. Let u(·) and σ(·) as in Theorem 2. Then J_s,λ : L^p_v → L^q_u whenever, for somet >1 with1< t <

1 q−¹_p+1

s n+¹_q−_p¹, (2.6) |Q|^ns+1q−1p 1

|Q|

Z

Q

u^t(y)dy_tq¹ 1

|Q|

Z

Q

σ^t(y)dy ¹

tp′

≤A

for all cubesQwith |Q|ⁿ¹ ≤λ^−2s1 . Moreover for u(·) andσ(·)satisfying A_∞ condition, thenJ_s,λ :L^p_v →L^q_u if and only if

(2.7) |Q|^ns+1q−1p 1

|Q|

Z

Q

u(y)dy¹_q 1

|Q|

Z

Q

σ(y)dy_p′¹

≤A

for all cubesQwith |Q|ⁿ¹ ≤λ^−21s. Conditions (2.6) and (2.7) have nontrivial senses since_n^s+¹_q−¹_p ≥0 is assumed (see Remark 1). Recall thatw(·) satisfies the A_∞ condition if, for some r > 1:

|Q|^ns−1(_|Q|¹ R

Qw(y)dy)^1r(_|Q|¹ R

Qw^1−r′(y)dy)^r′1 ≤cfor all cubesQ.

As an example, for each weight function w(·) then J_s,1 : L^p₁ → L^p

(Js,1w)^p′

whenever for a t >1: |Q|^ns(_|Q|¹ R

Q(J_s,1w)^tp′(y)dy)^tp1 ≤ C for all cubesQ with

|Q|ⁿ¹ ≤1. Such a result was proved by a different method in [Ma-Ve]. We will present below another application of Proposition 3.

Although this result yields sufficient condition for J_s,λ : L^p_v → L^q_u, we are able to state a necessary and sufficient condition for this embedding. However, compared with (2.6), the corresponding characterizing condition is not easy to check in general.

Proposition 4. Letu(·), σ(·)as in the hypotheses of Theorem 2. Then J_s,λ : L^p_v →L^q_u if and only if for someC >0:

(2.8) Z

(3Q)

(I_sσI_Q)^q(y)u(y)dy¹_q

≤C Z

Q

σ(y)dy¹_p and

(2.8^∗) Z

(3Q)

(I_suI_Q)^p′(y)σ(y)dy_p′¹

≤C Z

Q

u(y)dy_q′¹

for all dyadic cubesQwith|Q|¹ⁿ ≤λ^−21s.

A dyadic cubeQis a product of nintervals of the form [2^k(x_i−l),2^k(x_i+l)]

wherel >0, andI_E(·) is the characteristic function of the measurable setE.

Letw(·) be a weight function, and 0< s < _prⁿ with 1< r <∞. By a result due to Adams [Ad2], there isC >0 such that

(2.9) Z

x∈Rⁿ

(Isf)^p(x)w(x)dx≤C Z

x∈Rⁿ

f^p(x) Msprw^r1_r (x)dx

for all f(·)≥0.

Here M_β, 0 ≤ β < n, is the usual fractional maximal operator defined as (M_βg)(x) = sup{|Q|^βn−1R

Q|g(y)|dy;Q∋ x}. Since J_s,λ is pointwise majorized byI_sthen inequality (2.9) remains true withI_sreplaced byJ_s,λ, and it becomes natural to ask whether (2.9) holds withJ_s,λand the weight in the second member defined by a smaller operator thanM_β. Therefore we will be interested to get an inequality like

(2.10) Z

x∈Rⁿ

(J_s,λf)^p(x)w(x)dx≤C Z

x∈Rⁿ

f^p(x) M_spr,λw^r1

r(x)dx for all f(·)≥0, where (M_β,λg)(x) = sup{|Q|^βn−1R

Q|g(y)|dy;Q∋xand|Q|¹ⁿ ≤λ^−2s1}.

Unfortunately (2.10) is false in general. Indeed take n = 1, λ = 1, w(·) = I[0,1](·) and f(·) = I

[3,4](·). Clearly (M_β,1w^r)(x) = 0 for all |x| ≥ 3 and R

x∈Rⁿf^p(x)(M_β,1w^r)^1r(x)dx = 0. On the other hand (J_s,λf)(·)≈(I_sf)(·)≈1, on [0,1] andR

x∈Rⁿ(J_s,λf)^p(x)w(x)dx≈1.

Consequently to get (2.10), some restriction on the weight function w(·) is needed. Really, by Proposition 3, we have

Corollary 5. Let r > 1, 0 < s < _prⁿ and λ > 0. Suppose that one of w(·) andσ(·)is a doubling weight function, whereσ(·) = (M_spr,λw^r)^1r(1−p′)(·). Then there is C >0 for which(2.10)is true. This constantC depends on n, p, sand the constant on the doubling condition.

§3. Preliminaries lemmas

As we have alluded in§1, by arguments in [Ar-Sm] the kernelK_s,λ(·) of J_s,λ satisfies

(3.1) K_s,λ(R)≈R^s−n if R≤λ^−21s, else K_s,λ(R)≈R^12(s−n+1)exp(−Rλ^21s).

These equivalences lead to a better knowledge of the behaviour ofJ_s,λ.

Lemma 1. Let0≤s < n,λ >0. Then

(3.2) C₁(T_s,c2sλf)(·)≤(J_s,λf)(·)≤C₂(T_s,c−2sλf)(·).

HereC₁,C₂,cdepend only onnands. And the operatorT_s,µ(µ >0)is defined as(T_s,µf)(x) =R

y∈Rⁿ|x−y|^(s−n)exp{−µ^2s1 |x−y|}f(y)dy.

Obviously (J_s,λf)(·)≤C(I_sf)(·).

Lemma 2. Let L >0. One can find a family(Q_l)_l∈I of cubes with|Q_l|ⁿ¹ =L and disjoint interiors such that

(3.3) Rⁿ=[

l∈I

Q_l,

and there is an integerN >1 (depending only onn)for which the following holds:

(3Q_l) = [

l^′∈Il

Q_l′ where l^′∈ I_l if Q_l∩Q_l′ 6=∅, and card{l;l^′ ∈ I_l} ≤N; (3.4)

(3Q_l)^c={y;y /∈(3Q_l)}= [∞ m=4

[(m+ 1)Q_l\(m−1)Q_l] = [∞ m=4

[

j∈Jm,l

Q_j, (3.5)

where j ∈ J_m,l iff dist(Q_j, Q_l) ≈ (mL), and card{j;j ∈ J_m,l} ≤ N ×mⁿ or card{l;j∈ J_m,l} ≤N×mⁿ;

|x−y| ≈ |x_Ql−y| ≈(mL) for all x∈Q_l, y∈Q_j and j∈ J_m,l; (3.6)

X

l∈I

I_(3Q

l)(·)≤N; (3.7)

X

l∈I

I[(m+1)Ql\(m−1)Ql](·)≤N mⁿ for each integer m≥4.

(3.8)

Proof of Lemma 1: Using the property of the exponential like

lim_R→∞R^αexp{−βR} = 0, and estimates (3.1) for K_s,λ then we can find C₁, C₂, c > 0 depending only on s and n such that C₁R^s−nexp{−cλ^21sR} ≤ K_s,λ(R) ≤ C₂R^s−nexp{−c⁻¹λ^2s1 R} for all R > 0. With the definition of the

operatorT_s,µ, these inequalities imply (3.2).

Proof of Lemma 2: This geometrical lemma will be a consequence of the homogeneity property of the euclidean space Rⁿ. Thus the points (3.3) to (3.6) are standard and can be easily seen.

Inequality (3.7) is a consequence of (3.4) since X

l∈I

I(3Ql)(·) =X

l∈I

X

l^′∈Il

I_Q

l′(·) =X

l^′∈I

I_Q

l′(·) X

l;l^′∈Il

1≤NX

l^′∈I

I_Q

l′(·)≤N.

Inequality (3.8) comes from the cardinality property (3.5) since X

l∈I

I[(m+1)Ql\(m−1)Ql](·) =X

l∈I

X

j∈Jm,l

I_Q

j(·) =X

j∈I

I_Q

j(·) X

l;j∈Jm,l

1

≤N mⁿX

j∈I

I_Q

j(·)≤N mⁿ.

§4. Proofs of results

Proof of Theorem 1: We begin by the sufficient part. By Lemma 1, the proof of J_s,λ : L^p_v →L^q_u is reduced to that ofT_s,c−2sλ :L^p_v → L^q_u. Without a loss of generality it can be assumed that c = 1. Take a family of cubes (Q_l)_l∈I with common sizeL=λ^−2s1 (=|Q|ⁿ¹) as in Lemma 2. So forf(·)≥0 we have

Z

x∈Rⁿ

(T_s,λf)^q(x)u(x)dx=X

l∈I

Z

Ql

(T_s,λf)^q(x)u(x)dx≤C{S₁+S₂} where

S₁=X

l∈I

Z

Ql

(T_s,λfI

(3Ql))^q(x)u(x)dx, S₂=X

l∈I

Z

Ql

(T_s,λfI_(3Q

l)^c)^q(x)u(x)dx,

and C > 0 is a constant which depends on n and q. The estimates for S₁ are done as follows

S₁≤X

l∈I

Z

Ql

(IsfI_(3Q

l))^q(x)u(x)dx by the definition ofT_s,λ

≤CX

l∈I

Z

(3Ql)

f(x)^pv(x)dx^q_p

by the condition (2.1)

≤C Z

x∈Rⁿ

h X

l∈I

I_(3Q

l)(x)i

f(x)^pv(x)dx_p^q

since q p≥1

≤CN^qp Z

x∈Rⁿ

f(x)^pv(x)dx^q_p

by (3.7).

Let H(c, Q)

= Z

y∈(3Q)^c

|x_Q−y|^(s−n)p′exp{−cL⁻¹|x_Q−y|}σ(y)dy_p′¹ Z

x∈Qu(x)dx¹_q . Consequently

S₂=X

l∈I

Z

x∈Ql

h Z

y∈(3Ql)^c

|x−y|^s−nexp{−L⁻¹|x−y|}f(y)dyiq

u(x)dx by the definition ofT_s,λ

≤CX

l∈I

h Z

y∈(3Ql)^c

|x_Ql−y|^s−nexp{−cL⁻¹|x_Ql−y|}f(y)dyiq

×

× Z

x∈Ql

u(x)dx

by property (3.6)

≤CX

l∈I

[H(c, Q_l)]^q Z

y∈(3Ql)^c

exp{−cL⁻¹|x_Ql−y|}f(y)^pv(y)dy^q_p by the H¨older inequality

≤CH^qX

l∈I

Z

y∈(3Ql)^c

exp{−cL⁻¹|x_Ql−y|}f(y)^pv(y)dy^q

p

by the condition (2.2)

≤CH^q X

l∈I

Z

y∈(3Ql)^c

exp{−cL⁻¹|x_Ql−y|}f(y)^pv(y)dy_p^q

since q p ≥1

=CH^q X

l∈I

X∞ m=4

Z

y∈[((m+1)Ql)\((m−1)Ql)]

exp{−cL⁻¹|x_Ql−y|}f(y)^pv(y)dx^q_p

≤CH^qX^∞

m=4

exp{−c^′m}

Z

y∈Rⁿ

h X

l∈I

I[((m+1)Ql)\((m−1)Ql)]yi

f(y)^pv(y)dx^q_p

≤N CH^qhX^∞

m=4

exp{−c^′m}i Z

y∈Rⁿ

f(y)^pv(y)dx^q

p by (3.8)

≤N C^′H^q Z

y∈Rⁿ

f(y)^pv(y)dx^q_p

by the fast decreasing of the exponential function.

Conversely supposeJ_s,λ :L^p_v →L^q_u. ThenT_s,c2sλ :L^p_v →L^q_u (by Lemma 1).

LetQbe a cube with|Q|ⁿ¹ =λ^−2s1, andh(·)≥0 a function whose support is 3Q.

The last boundedness implies

(4.1) Z

Q

(T_s,c2sλh)^q(x)u(x)dx¹

q ≤C Z

(3Q)

h(x)^pv(x)dx¹

p

for a constantC >0 which does not depend onh(·) andQ. Then for allx∈Q (T_s,c2sλh)(x) =

Z

y∈(3Q)

|x−y|^s−nexp{−(c^2sλ)^2s1 |x−y|}h(y)dy

≥exp{−c^′} Z

y∈(3Q)

|x−y|^s−nh(y)dy since |x−y| ≤c^′λ^−2s1

= exp{−c^′}(I_sh)(x) since the support ofh(·) is (3Q).

This last inequality with (4.1) yields the point (2.1) in Theorem 1. To get (2.2) observe that, by duality,T_s,c2sλ:L^pv→L^qu is equivalent to

Z

y∈Rⁿ

h Z

x∈Rⁿ

|x−y|^s−nexp{−(c^2sλ)^21s|x−y|}f(x)u(x)dxip^′

(x)σ(y)dy_p′¹

≤C Z

x∈Rⁿ

f(x)^q′u(x)dx_q′¹ .

Now take a cubeQwith|Q|ⁿ¹ =λ^−2s1 , andf(·)≥0 equal to 1 in its supportQ.

Then Z

y∈(3Q)^c

h Z

x∈Q

|x−y|^s−nexp{−(c^2sλ)^21s|x−y|}u(x)dxip^′

(x)σ(y)dy_p′¹

≤C Z

x∈Q

u(x)dx_q′¹ . Since|x−y| ≈ |x_Q−y|, for allx∈Q,y∈(3Q)^c, andR

Qu(x)dx <∞then Z

y∈(3Q)^c

|x_Q−y|^(s−n)p′exp{−c^′p^′(c^2sλ)^2s1 |x_Q−y|}σ(y)dy¹

p′

× Z

x∈Qu(x)dx¹

q ≤C

which is the condition (2.2)

Proof of Remark 1: Suppose J_s,λ : L^p_v → L^q_u. Then T_s,c2sλ : L^p_v → L^q_u and Is : L^p(Q, vdx) → L^q(Q, udx) for all cubes Q with |Q|ⁿ¹ ≤ λ^−2s1. So

|Q|^ns+1q−1p(_|Q|¹ R

Qσ(y)dy)^p′1(_|Q|¹ R

Qu(y)dy)^1q ≤C for a constantC >0 not de- pending onQ. By the Lebesgue differentiation theorem, this last inequality yields

0≤_n^s +¹_q−¹_p unlessu(·) = 0 orσ(·) = 0.

Proof of Remark 3: If J_s,λ : L^p_v → L^q_u then T_s,c2sλ : L^p_v → L^q_u and the condition (2.3) is satisfied. Indeed ifQ1andQ2 are cubes with|Q₁|¹ⁿ =|Q₂|ⁿ¹ = λ^−2s1 and Q₁∩Q₂ 6=∅, then |x−y| ≤c^′λ^−2s1 for x∈Q₁ and y ∈Q₂, and the operatorT_s,c2sλ can be replaced byIs. To see that (2.3) implies (2.1), letQbe a cube with|Q|ⁿ¹ =λ^−2s1 andh(·)≥0 supported in (3Q). By (3.4), (3Q) =S

lQ_l with|Q_l|ⁿ¹ =|Q|ⁿ¹,Q=Q_l6=∅, and so by (2.3) the condition (2.1) appears since

Z

Q

(I_sh)^q(x)u(x)dx≤CX

l

Z

Q

(I_shI_Q

l)^q(x)u(x)dx

≤CX

l

Z

Ql

h(x)^pv(x)dx^q

p

≤C Z

x∈Rⁿ

h X

l

I_Q

l(x)i

h(x)^pv(x)dx^q_p

=C Z

(3Q)

h(x)^pv(x)dx^q_p .

Proof of Remark 4: Suppose (2.2) is true. To get (2.4) letQ₁, Q₂ be cubes with |Q₁|¹ⁿ =|Q₂|ⁿ¹ =λ^−21s and dist(Q₁, Q₂)≈(mλ^−21s) wherem ≥4. Since Q₂⊂(3Q₁)^c then, takingQ=Q₁ in (2.2) and using|x_Q1 −y| ≈dist(Q₁, Q₂)≈ (mλ^−2s1) for ally∈Q2, we obtain (2.4). Conversely suppose this last condition is satisfied for some constantc₀>0. For a cubeQwith|Q|ⁿ¹ =λ^−21s andc=c₀c⁻¹₁ , withc₁ a fixed constant depending only on n, then

Z

(3Q)^c

|x_Q−y|^(s−n)p′exp{−(2c)λ^−2s1 |x_Q−y|}σ(y)dy Z

Q

u(x)dx^p_q^′

= X∞ m=4

X

l∈J(m,Q)

Z

Ql

|x_Q−y|^(s−n)p′exp{−(2c)λ^−21s|x_Q−y|}σ(y)dy

× Z

Q

u(x)dx^p_q^′

≤C X∞ m=4

X

l∈J(m,Q)

(mλ^−2s1)^s−nexp{−(2cc₁)m} Z

Ql

σ(y)dy Z

Q

u(x)dx^p′

q

here |x_Q−y| ≈dist(Q, Q₁)≈(mλ^−21s)

≤C X∞ m=4

m^s−nexp{−c₀m} X

l∈J(m,Q)

exp{−c₀m}(mλ^−2s1)^s−n Z

Ql

σ(y)dy

× Z

Q

u(x)dx^p′

q

≤C^′C X∞ m=4

m^s−nexp{−c₀m} X

l∈J(m,Q)

1 by the condition (2.4)

≤N C^′C X∞ m=4

m^sexp{−c₀m}=N C^′CC^′′ by (3.5).

Proof of Remark 5: Since the arguments are the same, we can suppose that σ(·) satisfies the doubling condition. This hypothesis implies R

(tQ)σ(y)dy ≤ C₁t^nρR

Qσ(y)dy for all t >1. The constantρ, C₁ >0, depend on the doubling condition. Suppose (2.5) is satisfied. To get (2.4) let Q₁, Q₂ with |Q₁|ⁿ¹ =

|Q₂|ⁿ¹ =λ^−2s1 and dist(Q₁, Q₂)≈(mλ^−2s1 ), m≥4. SinceQ₂ ⊂(c₁mQ₁) for a fixed constant c₁ (depending only onn), thenR

Q2σ(y)dy ≤R

(c1mQ1)σ(y)dy ≤ C₁m^nρR

Q1σ(y)dy. With this last inequality the conclusion appears, since for all c >0

exp(−cm)m^(s−n)(λ^−2s1 )^(s−n) Z

y∈Q₂σ(y)dy_p′¹ Z

x∈Q₁u(x)dx¹

q

≤C2exp(−cm)m^[s−n(1−

ρ p′)]

(λ^−2s1 )^(s−n) Z

Q₁σ(y)dy_p′¹ Z

Q₁u(x)dx¹_q

≤C₀C₃C₂

where C₀ is from the condition (2.5) and C₃ a constant which exists by the property of the exponential function (lim_R→∞R^βexp{−γR} = 0, γ > 0) and

does not depend onm.

Proof of Theorem 2: By Theorem 1, Remarks 3 and 4 thenJ_s,λ:L^p_v →L^q_uiff both (2.3) and (2.4) hold. So we have just to prove that (2.3) implies (2.4). Taking Q1 =Q2 =Q (with |Q|ⁿ¹ = λ^2s1 ) in (2.3) then Is : L^p(Q, vdx) → L^q(Q, udx), with a constant independent of Q. So, as in the proof of Remark 1, (2.5) is satisfied. By Remark 5, this last condition implies (2.4).

Proof of Proposition 3: By Theorem 2 and Remark 2, to getJ_s,λ:L^p_v →L^q_u it is sufficient to get (2.3), which can be written as

(4.2) Z

x∈Rⁿ

(Isf)^q(x)u(x)e dx¹_q

≤C Z

x∈Rⁿ

f(x)^pev(x)dx¹_p

for all f(·)≥0.

Here eu(·) = u(·)I_Q

2(·), ev(·) = v(·)I_Q

1(·) and Q₁, Q₂ are cubes with |Q₁|ⁿ¹ =

|Q₂|ⁿ¹ = ¹₃λ^−21s andQ₁∩Q₂6=∅. We emphasize that C >0 is a constant which does not depend onQ₁andQ₂. Sawyer and Wheeden [Sa-Wh] proved that (4.2) holds if for somet >1 andS >0

(4.3) |Q|^ns+1q−1p 1

|Q|

Z

Qeu(y)^tdy¹

tq 1

|Q|

Z

Qeσ(y)^tdy_tp′¹

≤S

for any cubeQof arbitrary size and where eσ(·) =σ(·)I_Q

1(·).

Precisely they foundC =cS where c >0 depends only on s, n, p, q. Of course the constantS >0 in (4.3) must depend onu(·) ande eσ(·). Thus to get (4.2), by using this Sawyer-Wheeden’s result, we have to prove that in our context really S in (4.3) depends only onu(·) andv(·) but not on the cubes Q₁ andQ₂.

Call A(u,e eσ, Q) the left member of (4.3), and where Q is an arbitrary cube.

First consider the case|(3Q₁)|ⁿ¹ ≤ |Q|ⁿ¹. Note thatR

Que^t(y)dy≤R

Q₂u^t(y)dy≤ R

(3Q₁)u^t(y)dy and R

Qeσ^t(y)dy ≤ R

(3Q₁)σ^t(y)dy. Using these estimates and 1< t <

1 q−¹_p+1

s

n+¹_q−_p¹ then

A(eu,eσ, Q)≤ A(u, σ,(3Q1))≤A.

This last inequality is true since |3Q₁|ⁿ¹ = 3|Q₁|ⁿ¹ ≤ λ^−2s1, and A > 0 which depends on u(·), v(·) comes from (2.6). Next suppose |Q|ⁿ¹ ≤ |3Q₁|ⁿ¹. Since R

Que^t(y)dy≤R

Qu^t(y)dyandR

Qeσ^t(y)dy≤R

Qσ^t(y)dy then, again by (2.6), A(eu,eσ, Q)≤ A(u, σ, Q)≤A here |Q|ⁿ¹ ≤ |3Q₁|ⁿ¹ ≤λ^−21s.

ThereforeA(eu,eσ, Q)≤Afor any cube of arbitrary size, and withA >0 indepen- dent ofQ₁,Q₂. Then (4.2) is satisfied and soJ_s,λ:L^p_v →L^q_u.

If moreover bothu(·) andσ(·) satisfy the MuckenhouptA_∞condition then, as above, bothu(·) ande eσ(·) satisfyA_∞ with constants depending onu(·) andσ(·) but not onQ₁ andQ₂. It is known from [Sa-Wh] that condition (4.3), witht= 1, is a sufficient condition which ensures the embedding (4.2). Condition (4.3) with t= 1 and a constant S >0 not depending on Q₁ and Q₂ can be obtained from

(2.7).

Proof of Proposition 4: Choose the family of dyadic cubes (Q_l)_l∈I, in Lemma 2, with common size equal to 2^k, where k is an integer such that 2^k ≤λ^−2s1 <

2^k+1. Again we have to get (4.2) (where Q₁ and Q₂ are dyadic cubes with

|Q₁|ⁿ¹ =|Q₂|ⁿ¹ = 2^k andQ₁∩Q₂6=∅). By the Sawyer’s theorem [Sa-Wh], then (4.2) holds iff for someS >0

(4.4) Z

y∈Rⁿ

(IsσeI_Q)^q(y)eu(y)dy¹_q

≤S Z

Qeσ(y)dy¹_p

and

(4.4^∗) Z

y∈Rⁿ

(IsueI_Q)^p′(y)eσ(y)dy_p′¹

≤S Z

Qeu(y)dy_q′¹

for each dyadic cubeQwith an arbitrary size. Therefore it remains to prove that the condition (2.8) (respectively (2.8^∗)) implies (4.4) (respectively (4.4^∗)) and the corresponding contactS >0 depends only onu(·), σ(·) but not onQ₁ and Q₂. Conditions (4.4) and (4.4^∗) can be written as

(4.5) Z

Q∩Q₂

(I_sσI_Q∩Q

1)^q(y)u(y)dy¹_q

≤S Z

Q∩Q₁

σ(y)dy¹_p and

(4.5^∗) Z

Q∩Q2

(IsuI_Q∩Q

2)^p′(y)σ(y)dy_p′¹

≤S Z

Q∩Q2

u(y)dy_q′¹

The crucial fact we use is the well-known property of dyadic cubes which asserts that: for a given closed dyadic cubesQ,Q₀ the only cases which can occur are:

(a)Q∩Q₀=∅; (b)Q∩Q₀=∂Q∩∂Q₀; (c)Q⊂Q₁; (d)Q₁⊂Q.

First take a dyadic cubeQwith|Q|ⁿ¹ ≤2^k. We can assumeR

Q∩Q1σ(y)dy6= 0 (respectively R

Q∩Q2u(y)dy 6= 0) else (4.5) (respectively (4.5^∗)) is trivially satis- fied. Suppose Q⊂Q₁ (respectively Q⊂Q₂). For Q₁ 6=Q₂ then (4.5) (respec- tively (4.5^∗)) is trivially satisfied since necessarilyR

Q∩Q₂u(y)dy= 0 (respectively R

Q∩Q₁σ(y)dy= 0). But forQ1=Q2thenQ∩Q₁=Q(respectivelyQ∩Q2=Q) and (4.5) (respectively (4.5^∗)) is reduced to

(4.6) Z

Q

(I_sσI_Q)^q(y)u(y)dy¹_q

≤S Z

Q

σ(y)dy¹_p (respectively

(4.6^∗) Z

Q

(IsuI_Q)^p′(y)σ(y)dy¹

p′

≤S Z

Q

u(y)dy¹

q′

).

Since |Q|ⁿ¹ ≤2^k≤λ^−21s, then by (2.8) (respectively (2.8^∗)) the condition (4.6) (respectively (4.6^∗)) is satisfied with S =C, a constant which depends only on u(·) andσ(·).

Next consider 2^k<and assumeR

Q∩Q1σ(y)dy6= 0 (respectivelyR

Q∩Q2u(y)dy 6= 0) else there is nothing to prove. IfQ₁ ⊂Q(respectively Q₂ ⊂Q) then (4.5) (respectively (4.5^∗)) is reduced to

(4.7) Z

Q∩Q2

(IsσI_Q

1)^q(y)u(y)dy¹_q

≤S Z

Q1

σ(y)dy¹_p

(respectively (4.7^∗) Z

Q∩Q₁

(I_suI_Q

2)^p′(y)σ(y)dy_p′¹

≤S Z

Q₂

u(y)dy_q′¹ ).

If moreoverR

Q∩Q₂u(y)dy6= 0 (respectively R

Q∩Q₁σ(y)dy6= 0) then necessarily Q₂⊂Q(respectively Q₁⊂Q) and (4.7) (respectively (4.7^∗)) is the same as

(4.8) Z

Q₂

(I_sσI_Q

1)^q(y)u(y)dy¹_q

≤S Z

Q₁

σ(y)dy¹_p

(respectively (4.8^∗) Z

Q₁

(IsuI_Q

2)^p′(y)σ(y)dy_p′¹

≤C Z

Q₂

u(y)dy_q′¹ ).

Since |Q₁|¹ⁿ =|Q₂|¹ⁿ = 2^k ≤λ^−21s and Q₂ ⊂(3Q₁) (respectively Q₁ ⊂(3Q₂)) then the condition (4.8) (respectively (4.8^∗)) is satisfied with S=C by (2.8) (re- spectively (2.4^∗)).

Proof of Corollary 5: Letv(·) = (M_spr,λw^r)^1r(·). It remains to proveJ_s,λ: L^p_v →L^p_w. Since one ofw(·) and σ(·) =v^−p−11 (·) is a doubling weight function, then by Proposition 3, (2.6) is a sufficient condition in order to get the above em- bedding. So we have to estimate|Q|^ns(_|Q|¹ R

Qw^r(y)dy)^rp1 (_|Q|¹ R

Qσ^r(y)dy)^rp′1 = F_r(Q) by a constant which does nor depend on Q with |Q|¹ⁿ ≤ λ^−2s1. By the definition ofM_β,λthenσ^r(x)≤(|Q|^sprn _|Q|¹ R

Qw^r(y)dy)^1−p′ for each cubeQwith

|Q|ⁿ¹ ≤λ^−2s1 and for allx∈Q. ConsequentlyF_r(Q)≤ |Q|^sn(_|Q|¹ R

Qw^r(y)dy)^rp1 (|Q|^sprn _|Q|¹ R

Qw^r(y)dy)^−rp1 = 1.

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Current address:

Institut Polytechnique St.-Louis (EPMI), 13, boulevard de l’Hautil, 95 092 Cergy-Pontoise Cedex, France

E-mail: [email protected] [email protected]

(Received May 13, 1996)