GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 6, 1998, 583-600

CHARACTERIZATION OF A TWO-WEIGHTED VECTOR-VALUED INEQUALITY FOR FRACTIONAL MAXIMAL OPERATORS

Y. RAKOTONDRATSIMBA

Abstract. We give a characterization of the weights u(·) and v(·) for which the fractional maximal operator M_{s} is bounded from the weighted Lebesgue spaces L^{p}(l^{r}, vdx) into L^{q}(l^{r}, udx) whenever 0 ≤ s < n, 1 < p, r < ∞, and 1 ≤ q < ∞.

1. Introduction

The fractional maximal operator M_{s} of order s, 0 ≤ s < n, is defined by

(M_{s}f)(x) = sup \{|Q|^{\frac{s}{n} −1} ∫_{Q} |f(y)| dy; Q cube with Q ∋ x\}.

The cubes considered always have sides parallel to the coordinate axes. Here M = M_{0} is the well-known Hardy–Littlewood maximal operator.

Our main purpose is to characterize the weights u(·) and v(·) for which there is C > 0 such that

∥( ∑_{k}(M_{s}f_{k})^{r}(·))^{\frac{1}{r}} ∥_{L_{u}^{q}} ≤ C∥( ∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{v}^{p}}  (1.1)

for all sequences of nonnegative functions (f_{k}(·))_{k}, with 1 < p, r < ∞ and 1 ≤ q < ∞. The quantity ∥f(·)∥_{L_{w}^{t}} is defined as (∫_{ℝ^{n}} |f(x)|^{t}w(x) dx)^{\frac{1}{t}}. For convenience, (1.1) will often be denoted by M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}).

Such vector-valued inequalities are very important in many areas of analysis. For instance, they can be used to get weighted inequalities for other operators [1], and they are also involved in the study of some functional spaces [2].

Fefferman–Stein’s well-known inequality states that M : L_{1}^{p}(l^{r})→ L_{1}^{p}(l^{r}) for 1 < p, r < ∞. The one-weighted inequality M : L_{w}^{p}(l^{r}) → L_{w}^{p}(l^{r}) was

583 solved by Andersen and John [3], and also independently by Kokilashvili [4]. In [5], a characterization of the two-weighted inequality M : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}) is obtained by applying the extrapolation theory of J. Garcia-Cuerva and Rubio de Francia and by adapting Sawyer’s method for the two-weighted scalar inequality. For s > 0 it seems that only the boundedness M_{s} : L_{1}^{p}(l^{r}) → L_{u}^{q}(l^{r}), with p < q, was characterized by F. Ruiz and J. L. Torrea [6].

1991 Mathematics Subject Classification. 42B25.

Key words and phrases. Vector-valued inequalities, maximal operators, tent spaces.

1072-947X/98/1100-0583$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

Our necessary and sufficient condition for M : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}) is completely different from the one given in [5], since the approach is not based on the extrapolation theory. We will use the atomic decomposition of tent spaces introduced by Coifman, Meyer, and Stein [7]. Such a technique has been introduced by the author in [8] to deal with

∥(M_{s}f)(·)∥_{L_{u}^{q}} ≤ C∥f(·)∥_{L_{v}^{p}} for all f(·) ≥ 0  (1.2)

which is a scalar version and particular case of inequality (1.1). For 1 < p ≤ q < ∞, Sawyer [9] proved that (1.2) holds if and only if for some S > 0

∥(M_{s}v^{− \frac{1}{p−1}} 11_{Q})(·)11_{Q}(·)∥_{L_{u}^{q}} ≤ S∥(v^{− \frac{1}{p−1}} 11_{Q})(·)∥_{L_{v}^{p}} for all cubes Q. (1.3)

Here 11_{Q}(·) denotes the characteristic function of the cube Q. In other words, (1.2) is true if this inequality is satisfied only for all particular functions of the form f(·) = v^{− \frac{1}{p−1}} (·)11_{Q}(·). So it can be expected that M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) if and only if (1.1) is true for all sequences f_{k}(·) = v^{− \frac{1}{p−1}} (·)11_{Q_{k}}(·). In this paper (see Theorem 1) we are able to derive such a boundedness if (1.1) is true for all f_{k}(·) = λ_{k}v^{− \frac{1}{p−1}} (·)11_{Q_{k}}(·) with λ_{k} > 0.

Our results are presented in Section 2. The next Section 3 is devoted to their proofs except Theorem 1 and Proposition 8 whose proofs are given in Section 4. The proofs of the basic lemmas are given in Section 5.

2. Main Results

In this paper it is always assumed that 0 ≤ s < n, 1 < p, r < ∞, 1 ≤ q < ∞ with 1/p − 1/q ≤ s/n and u(·), v(·), σ(·) = v^{− \frac{1}{p−1}} (·) are weight functions.

Our main result, which can be seen as a generalization of Sawyer’s theorem quoted above, is

Theorem 1. Suppose that for some constant S > 0

∥( ∑_{k} [λ_{k}(M_{s}v^{− \frac{1}{p−1}} 11_{Q_{k}})(·)11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{L_{u}^{q}} ≤

≤ S∥( ∑_{k} [λ_{k}(v^{− \frac{1}{p−1}} 11_{Q_{k}})(·)]^{r})^{\frac{1}{r}} ∥_{L_{v}^{p}}  (2.1)

for all λ_{k} and all cubes Q_{k}; then M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}). Conversely, this boundedness implies condition (2.1).

For the boundedness M : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}), Theorem 1 is only interesting when r < p. Indeed, by the interpolation theory for linearizable operators [1], it is known that the vector-valued inequality M : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}) becomes equivalent to the scalar inequality (1.2) (with p = q) when p ≤ r (see [5] for details).

Next we apply Theorem 1 to treat some particular cases for which p = q. These cases correspond to significant and critical (for s > 0) situations in applications.

Corollary 2. Assume that w(·) is a weight function with w^{1−p}(·) ∈ L_{loc}^{1}(ℝ^{n}, dx). Then M_{s} : L_{w^{1−p}}^{p}(l^{r}) → L_{w}^{p}(l^{r}) if and only if for some constant C > 0

∫_{Q} w(y) dy ≤ C|Q|^{1− \frac{s}{n}} for all cubes Q.  (2.2)

Corollary 3. Suppose that for some constant C > 0

u(·)(M_{s}v^{− \frac{1}{p−1}} )^{p}(·) ≤ Cv^{− \frac{1}{p−1}} (·);

then M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}). In particular, the boundedness

M_{s} : L_{w}^{p}(l^{r}) → Lp_{\frac{w^{1−p′}}{(Msw^{1−p′})^{p}}} (l^{r})

holds for any weight w(·) with w^{1−p′}(·) ∈ L_{loc}^{1}(ℝ^{n}, dx).

Proposition 4. Let r < p and 1 < t < ∞ with 0 ≤ s < min\{\frac{n}{r}, \frac{n}{pt}\}. If for some constant C > 0 (M_{spt}u^{t})^{\frac{1}{t}} (·) ≤ Cv(·), then M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}). In particular, M_{s} : L_{(M_{spt}w^{t})^{\frac{1}{t}}}^{p} (l^{r}) → L_{w}^{p}(l^{r}) for any weight w(·)

with w^{t}(·) ∈ L_{loc}^{1}(ℝ^{n}, dx).

The latter boundedness is a substitute for Fefferman–Stein’s well-known inequality

∫_{ℝ^{n}}(M_{s}f)^{r}(x)w(x) dx ≤ C ∫_{ℝ^{n}} f^{r}(x)(M_{sr}w)(x) dx for all f(·) ≥ 0.

As proved in [5], it is possible to obtain a better estimate in the sense that (M_{spt}w)^{\frac{1}{t}} (·) can be replaced by a smaller operator. But we do not go in this direction.

In general, a condition like (2.1) is difficult to check for given weights u(·) and v(·). So our purpose is now to examine this condition more thoroughly for the case r < p = q. A general condition which implies (2.1) can be stated.

Proposition 5. Let r < p and (\frac{p}{r})′ = \frac{p}{p−r}. Then condition (2.1) is satisfied whenever, for a fixed constant C > 0 and for each g(·) ∈ L^{( \frac{p}{r})′}(ℝ^{n}, dx), one can find G(·) ∈ L^{( \frac{p}{r})′}(ℝ^{n}, dx) such that

∥G(·)∥_{L^{( \frac{p}{r})′} (ℝ^{n},dx)} ≤ ∥g(·)∥_{L^{( \frac{p}{r})′} (ℝ^{n},dx)}

and

∫_{Q}(M_{s}σ11_{Q})^{r}(x)u^{\frac{r}{p}} (x)g(x) dx≤C∫_{Q}σ^{\frac{r}{p}} (x)G(x) dx for all cubes Q. (2.3)

As mentioned above, C. Pérez [5] has recently proved that M : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}) is equivalent to (2.3). For s > 0 it is also expected that (2.3) is equivalent to our condition (2.1), but this fact is not needed for the sequel. Proposition 5 was inspired by the results of [5].

Obviously, (2.1) implies Sawyer’s condition (1.3). As for (2.3), it is strictly contained in (1.3), since, by duality,

( ∫_{Q}(M_{s}σ11_{Q})^{p}(x)u(x) dx)^{\frac{r}{p}} = ∫_{Q}(M_{s}σ11_{Q})^{r}(x)u^{\frac{r}{p}} (x)g(x) dx

and

∫_{Q} σ^{\frac{r}{p}} (x)G(x) dx ≤ ( ∫_{Q} σ(x) dx)^{\frac{r}{p}}

for some g(·) ∈ L^{( \frac{p}{r})′}(ℝ^{n}, dx) with the unit norm.

However, to get the boundedness M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}) it would be desirable to have a sufficient condition whose form is closer to (1.3) rather than to (2.3) and (2.1).

Proposition 6. Let r < p. Condition (2.3) or (2.1) is satisfied whenever for some t > 1 and C > 0

(\frac{1}{|Q|} ∫_{Q}(M_{s}σ11_{Q})^{tp}(x)u^{t}(x) dx)^{\frac{1}{tp}} ≤ C(\frac{1}{|Q|} ∫_{Q} σ^{\frac{r}{p}} (x) dx)^{\frac{1}{r}} (2.4)

for all cubes Q.

Clearly, by the Hölder inequality, (2.4) implies Sawyer’s condition (1.3). Conditions like (2.4) or (1.3) are not useful for computations. Indeed, the presence of the maximal operator M_{s} in the condition makes it difficult or sometimes impossible to determine by computation whether M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}) for given weights u(·) and v(·). Therefore it would be more preferable to have a sufficient condition for this boundeness without using M_{s}. We shall give an example.

Proposition 7. Let r < p. Condition (2.3) or (2.1) is satisfied if for some t_{2} > t_{1} > 1, and C > 0

(\frac{1}{|Q|}∫_{Q}(|x|^{s}σ(x))^{pδ}dx)^{\frac{1}{pδ}} (\frac{1}{|Q|}∫_{Q}u^{t_{2}}(x) dx)^{\frac{1}{pt_{2}}} ≤C(\frac{1}{|Q|}∫_{Q}σ^{\frac{r}{p}} (x)dx)^{\frac{1}{r}} (2.5)

for all cubes Q; here \frac{1}{δ} = \frac{1}{t_{1}} − \frac{1}{t_{2}} .

By Proposition 7 the Muckenhoupt condition

(\frac{1}{|Q|} ∫_{Q} u(x) dx)^{\frac{1}{p}} (\frac{1}{|Q|} ∫_{Q} σ(x) dx)^{\frac{1}{p′}} ≤ C for all cubes Q

characterizes the boundedness M : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}) whenever u(·), σ^{\frac{r}{p}} (·) ∈ RH_{∞}. Also, for σ^{\frac{r}{p}} (·) ∈ RH_{∞} we have M : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}) if for some C > 0 and t > 1

(\frac{1}{|Q|} ∫_{Q} u^{t}(x) dx)^{\frac{1}{tp}} (\frac{1}{|Q|} ∫_{Q} σ(x) dx)^{\frac{1}{p′}} ≤ C for all cubes Q.

Recall that w(·) ∈ RH_{∞} if sup_{x∈Q} w(x) ≤ C \frac{1}{|Q|} ∫_{Q} w(y) dy for a fixed constant C > 0 and all cubes Q. Other sufficient conditions for M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) which do not involve the maximal operator M_{s} can be found in [10] (see also [5] for p = q and s = 0). No more results in this direction are given here, since our main purpose is to prove the characterization in Theorem 1.

We end with a necessary and sufficient condition for M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) with special weights v(·). For this, recall that w(·) ∈ RD_{ρ}, ρ > 0, if for some c > 0

∫_{Q′} w(y) dy ≤ c(\frac{|Q′|}{|Q|})^{ρ} ∫_{Q} w(y) dy for all cubes Q′, Q with Q′ ⊂ Q.

Any doubling weight w(·) (and, in particular, any A_{∞} weight) satisfies the reverse doubling condition RD_{ρ}.

Proposition 8. Suppose that for some constant A > 0

∥( ∑_{k} [λ_{k}|Q_{k}|^{\frac{s}{n} −1}( ∫_{Q_{k}} v^{− \frac{1}{p−1}} (y) dy)11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{L_{u}^{q}} ≤

≤ A∥( ∑_{k} [λ_{k}(v^{− \frac{1}{p−1}} 11_{Q_{k}})(·)]^{r})^{\frac{1}{r}} ∥_{L_{v}^{p}} for all cubes Q_{k} and λ_{k} > 0. (2.6)

Then M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) whenever v^{− \frac{1}{p−1}} (·) ∈ RD_{ρ} with 1 − \frac{s}{n} ≤ ρ. Conversely, the boundedness M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) implies condition (2.6).

Therefore for r < p and σ(·) = v^{− \frac{1}{p−1}} (·) ∈ RD_{ρ} with 1 − \frac{s}{n} ≤ ρ we have M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}) whenever for some t > 1 and A > 0

|Q|^{\frac{s}{n}} (\frac{1}{|Q|} ∫_{Q} σ(y) dy)(\frac{1}{|Q|} ∫_{Q} u^{t}(y) dy)^{\frac{1}{pt}} ≤

≤ A(\frac{1}{|Q|} ∫_{Q} σ^{\frac{r}{p}} (y) dy)^{\frac{1}{r}} for all cubes Q.  (2.7)

Theorem 1 can be seen as an extension of Sawyer’s result [9] for scalar functions to vector functions. Note that, whenever s > 0 and 1 < p < q < ∞, for the scalar inequality (1.2) a better criterion than (1.3) has been established by Wheeden for the Euclidean space [11] and by Gogatishvili and Kokilashvili in a more general case, namely, for homogeneous type spaces [12]. For this range of s, p, and q it is interesting to find a criterion for M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) which will replace condition (2.1).

3. Proofs of Corollaries 2, 3, and Propositions 6, 7

To get M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}), by Theorem 1 with p = q, it is sufficient to estimate

S = ∥( ∑_{k} [λ_{k}(M_{s}σ11_{Q_{k}})(·)11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{L_{u}^{p}}  (3.1)

by C∥(∑_{k}[λ_{k}(σ(·)11_{Q_{k}})(·)]^{r})^{\frac{1}{r}} ∥_{L_{v}^{p}}, for all sequences of nonnegative (λ_{k})_{k} and all sequences of cubes (Q_{k})_{k}. Here C > 0 is a constant which depends only on s, n, p, q, u(·), and v(·).

Proof of Corollary 2. Suppose that M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{p}(l^{r}). Take the sequence of functions (f_{k}(·))_{k} defined as f_{0}(·) = σ(·)11_{Q}(·) and f_{k}(·) = 0 for all k ≠ 0. Since v(·) = w^{1−p}(·) and u(·) = σ(·) = w(·), condition (2.2) follows immediately.

To prove the converse statement, with the above values of u(·) and v(·), condition (2.2) implies that (M_{s}σ)(·) ∈ L^{∞}(ℝ^{n}, dx) and thus

S ≤C∥(∑_{k}[λ_{k}11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{L_{u}^{p}}=C∥(∑_{k}[λ_{k}11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{L_{σ}^{p}}(since u(·)=σ(·))

= C∥( ∑_{k} [λ_{k}σ(·)11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{L_{v}^{p}} (since σ(·) = σ^{p}(·)v(·)).

Proof of Corollary 3. To get the first part of the corollary, we write (3.1) as

S ≤ ∥( ∑_{k} [λ_{k}11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{L_{u(·)(Msσ)^{p}(·)}^{p}} ≤

≤ C∥( ∑_{k} [λ_{k}11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{L_{σ}^{p}} = C∥( ∑_{k} [λ_{k}σ(·)11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{L_{v}^{p}}.

The proof of the second part is trivial, since in that case v(·) = w(·) and u(·) = σ(·)(M_{s}σ)^{−p}(·).

Proof of Proposition 4. The second part of the proposition is an immediate consequence of the first one if we take u(·) = w(·) and v(·) = (M_{spt}u^{t})^{\frac{1}{t}} (·).

The first part is proved by assuming r < p, μ = \frac{p}{r}, μ′ = \frac{p}{p−r} and again considering (3.1). By duality, there is a nonnegative function g(·) ∈ L^{μ′}(ℝ^{n}, dx) with the unit norm such that

S^{r} = ∫_{ℝ^{n}} [ ∑_{k} λ_{k}^{r}(M_{s}σ11_{Q_{k}})^{r}(x)11_{Q_{k}}(x)]u^{\frac{r}{p}} (x)g(x) dx =

= ∑_{k} λ_{k}^{r} ∫_{Q_{k}}(M_{s}σ11_{Q_{k}})^{r}(x)u^{\frac{r}{p}} (x)g(x) dx.  (3.2)

The key to the proof is the existence of a constant C > 0 for which

∫_{Q}(M_{s}σ11_{Q})^{r}(x)u^{\frac{r}{p}} (x)g(x) dx≤C∫_{Q}σ^{r}(x)(M_{spt}u^{t})^{\frac{r}{tp}} (x)(Mg^{ν′})^{\frac{1}{ν′}} (x) dx(3.3)

for all cubes Q. Here M = M_{0} is the Hardy–Littlewood maximal operator, ν = t\frac{p}{r} (> 1), and ν′ = \frac{ν}{ν−1}. By inequality (3.3) the rest of the estimate of S^{r} is

S^{r} ≤ C ∑_{k} λ_{k}^{r} ∫_{Q_{k}} σ^{r}(x)(M_{spt}u^{t})^{\frac{r}{tp}} (x)(Mg^{ν′})^{\frac{1}{ν′}} (x) dx = (by (3.3))

= C ∫_{ℝ^{n}} [ ∑_{k} λ_{k}^{r}11_{Q_{k}}(x)]σ^{r}(x)(M_{spt}u^{t})^{\frac{r}{tp}} (x)(Mg^{ν′})^{\frac{1}{ν′}} (x) dx =

= C ∫_{ℝ^{n}} [ ∑_{k} λ_{k}^{r}11_{Q_{k}}(x)] [σ^{p}(x)(M_{spt}u^{t})^{\frac{1}{t}} (x)]^{\frac{r}{p}} (Mg^{ν′})^{\frac{1}{ν′}} (x) dx ≤

≤ CC1^{\frac{r}{p}} ∫_{ℝ^{n}} [ ∑_{k} λ_{k}^{r}11_{Q_{k}}(x)]σ^{\frac{r}{p}} (x)(Mg^{ν′})^{\frac{1}{ν′}} (x) dx ≤

(since (M_{spt}u^{t})^{\frac{1}{t}} (·) ≤ C_{1}v(·))

≤ CC1^{\frac{r}{p}} ( ∫_{ℝ^{n}} [ ∑_{k} λ_{k}^{r}11_{Q_{k}}(x)]^{\frac{p}{r}} σ(x) dx)^{\frac{r}{p}} ( ∫_{ℝ^{n}}(Mg^{ν′})^{\frac{μ′}{ν′}} (x) dx)^{\frac{1}{μ′}} ≤

590 Y. RAKOTONDRATSIMBA

≤ CC1^{\frac{r}{p}} C_{2}( ∫_{ℝ^{n}} [ ∑_{k} λ_{k}^{r}11_{Q_{k}}(x)]^{\frac{p}{r}} σ(x) dx)^{\frac{r}{p}} ( ∫_{ℝ^{n}} g^{μ′}(x) dx)^{\frac{1}{μ′}} ≤

(by the maximal inequality theorem and since 1 < μ′/ν′ or μ < ν)

≤ CC1^{\frac{r}{p}} C_{2}∥( ∑_{k} [λ_{k}(σ(·)11_{Q_{k}})(·)]^{r})^{\frac{1}{r}} ∥_{L_{v}^{p}}^{r} (recall that ∥g∥_{L^{μ′} (ℝ^{n},dx)}=1).

To get (3.3) it is sufficient to use

(M_{sr}u^{\frac{r}{p}} g)(·) ≤ (M_{spt}u^{t})^{\frac{r}{tp}} (·)(Mg^{ν′})^{\frac{1}{ν′}} (·).  (3.4)

Indeed, by Fefferman–Stein’s classical inequality and (3.4), we have

∫_{Q}(M_{s}σ11_{Q})^{r}(x)u^{\frac{r}{p}} (x)g(x) dx ≤ C ∫_{Q} σ^{r}(x)(M_{sr}u^{\frac{r}{p}} g)(x) dx ≤

≤ C ∫_{Q} σ^{r}(x)(M_{spt}u^{t})^{\frac{r}{tp}} (x)(Mg^{ν′})^{\frac{1}{ν′}} (x) dx.

Inequality (3.4) is an immediate consequence of the Hölder inequality, since

|Q|^{\frac{sr}{n}} \frac{1}{|Q|} ∫_{Q}ϕ(y)ψ(y) dy ≤(|Q|^{\frac{srν}{n}} \frac{1}{|Q|} ∫_{Q}ϕ^{ν}(y) dy)^{\frac{1}{ν}} (\frac{1}{|Q|} ∫_{Q}ψ^{ν′}(y) dy)^{\frac{1}{ν′}}

for ϕ(·) = u^{\frac{r}{p}} (·), ψ(·) = g(·), ν = t\frac{p}{r}, and ν′ = \frac{ν}{ν−1}. Proof of Proposition 5. Suppose condition (2.3) is satisfied. To get (2.1), it is sufficient to estimate S^{r} for r < p. Similarly to (3.2), consider a nonnegative function g(·) ∈ L^{μ′}(ℝ^{n}, dx) with the unit norm. Then

S^{r} = ∑_{k} λ_{k}^{r} ∫_{Q_{k}}(M_{s}σ11_{Q_{k}})^{r}(x)u^{\frac{r}{p}} (x)g(x) dx ≤ (by (3.2))

≤ C ∑_{k} λ_{k}^{r} ∫_{Q_{k}} σ^{\frac{r}{p}} (x)G(x) dx = (by condition (2.3))

= C ∫_{ℝ^{n}} [ ∑_{k} λ_{k}^{r}11_{Q_{k}}(x)]σ^{\frac{r}{p}} (x)G(x) dx ≤

≤ C( ∫_{ℝ^{n}} [ ∑_{k} λ_{k}^{r}11_{Q_{k}}(x)]^{\frac{p}{r}} σ(x) dx)^{\frac{r}{p}} ( ∫_{ℝ^{n}} G^{μ′}(x) dx)^{\frac{1}{μ′}} ≤

≤ C( ∫_{ℝ^{n}}[ ∑_{k} λ_{k}^{r}11_{Q_{k}}(x)]^{\frac{p}{r}} σ(x) dx)^{\frac{r}{p}} ( ∫_{ℝ^{n}}g^{μ′}(x) dx)^{\frac{1}{μ′}} ≤ (by (2.3))

≤ C∥( ∑_{k} [λ_{k}(σ(·)11_{Q_{k}})(·)]^{r})^{\frac{1}{r}} ∥_{L_{v}^{p}}^{r} (recall that ∥g∥_{L^{μ′} (ℝ^{n},dx)} =1).

Proof of Proposition 6. Assume condition (2.4) is satisfied for some constant C > 0. To get (2.3), let g(·) ∈ L^{μ′}(ℝ^{n}, dx). The above notation (μ = \frac{p}{r}, ν = t\frac{p}{r}, …, etc.) is again used. Let C_{2} > 0 be a constant asso-

ciated with the boundedness of M : L^{\frac{μ′}{ν′}} (ℝ^{n}, dx) → L^{\frac{μ′}{ν′}} (ℝ^{n}, dx), and set

G(·) = C_{2}^{− \frac{1}{ν′}} (Mg^{ν′})^{\frac{1}{ν′}} (·). Clearly, ∥G(·)∥_{L^{μ′} (ℝ^{n},dx)} ≤ ∥g(·)∥_{L^{μ′} (ℝ^{n},dx)} and we obtain (2.3), since

\frac{1}{|Q|} ∫_{Q}(M_{s}σ11_{Q})^{r}(x)u^{\frac{r}{p}} (x)g(x) dx ≤

≤ (\frac{1}{|Q|} ∫_{Q}(M_{s}σ11_{Q})^{tp}(x)u^{t}(x) dx)^{\frac{r}{tp}} (\frac{1}{|Q|} ∫_{Q} g^{ν′}(y) dy)^{\frac{1}{ν′}} ≤

≤ C^{r}(\frac{1}{|Q|} ∫_{Q} σ^{\frac{r}{p}} (x)[\frac{1}{|Q|} ∫_{Q} g^{ν′}(y) dy]^{\frac{1}{ν′}} dx) ≤ (by (2.4))

≤ C^{r}(\frac{1}{|Q|} ∫_{Q} σ^{\frac{r}{p}} (x)(Mg^{ν′})^{\frac{1}{ν′}} (x) dx = C2^{\frac{1}{ν′}} C^{r} \frac{1}{|Q|} ∫_{Q} σ^{\frac{r}{p}} (x)G(x) dx.

Proof of Proposition 7. Assume condition (2.5) is satisfied for some constant C > 0. To get (2.3), let g(·) ∈ L^{μ′}(ℝ^{n}, dx) with μ = \frac{p}{r}, and take ν = t_{1} \frac{p}{r} with t_{1} > 1. Choose G(·) as in the proof of Proposition 6. Here we use the fact that M_{s} : L^{α}(ℝ^{n}, |x|^{sα}dx) → L^{α}(ℝ^{n}, dx) for all α > 1. Let t_{2}, δ with t_{1} < t_{2} and \frac{1}{δ} = \frac{1}{t_{1}} − \frac{1}{t_{2}} . Then 1 = \frac{r}{pδ} + \frac{r}{pt_{2}} + \frac{1}{ν′}, and by the Hölder inequality we get the conclusion, since

\frac{1}{|Q|} ∫_{Q}(M_{s}σ11_{Q})^{r}(x)u^{\frac{r}{p}} (x)g(x) dx ≤

≤ (\frac{1}{|Q|}∫_{Q}(M_{s}σ11_{Q})^{pδ}(x) dx)^{\frac{r}{pδ}}(\frac{1}{|Q|}∫_{Q}u^{t_{2}}(x) dx)^{\frac{r}{pt_{2}}}(\frac{1}{|Q|}∫_{Q}g^{ν′}(y) dy)^{\frac{1}{ν′}}≤

≤ c_{1}(\frac{1}{|Q|}∫_{Q}(|x|^{s}σ(x))^{pδ} dx)^{\frac{r}{pδ}}(\frac{1}{|Q|}∫_{Q}u^{t_{2}}(x) dx)^{\frac{r}{pt_{2}}}(\frac{1}{|Q|}∫_{Q}g^{ν′}(y) dy)^{\frac{1}{ν′}}≤

≤ c_{1}C^{\frac{r}{p}} (\frac{1}{|Q|} ∫_{Q} σ^{\frac{r}{p}} (x)[\frac{1}{|Q|} ∫_{Q} g^{ν′}(y) dy]^{\frac{1}{ν′}} dx) ≤ by (2.5)

≤ c_{1}C^{\frac{r}{p}} (\frac{1}{|Q|}∫_{Q}σ^{\frac{r}{p}} (x)(Mg^{ν′})^{\frac{1}{ν′}} (x) dx)=C2^{\frac{1}{ν′}} C^{r} \frac{1}{|Q|}∫_{Q}σ^{\frac{r}{p}} (x)G(x) dx.

4. Proof of Theorem 1 and Proposition 8

Since M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) immediately implies condition (2.1), our purpose in this section is to prove the converse. We shall state some basic lemmas and use them to derive the above boundedness.

As in the scalar case [9], the main problem is reduced to obtaining the dyadic version of (1.1) denoted by M_{s}^{dya} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) and having the form

∥( ∑_{k}(M_{s}^{dya}f_{k})^{r}(·))^{\frac{1}{r}} ∥_{L_{u}^{q}} ≤ C∥( ∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{v}^{p}}  (4.1)

for all sequences of nonnegative functions (f_{k}(·))_{k}. Here C = cS with c > 0 depending only on s, n, p, and q;

(M_{s}^{dya}f)(x) = sup \{|Q|^{\frac{s}{n} −1} ∫_{Q} |f(y)| dy; Q dyadic cube with Q ∋ x\}.

To get M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) we need three basic lemmas.

Lemma 4.1. Suppose that for some constant S > 0

∥( ∑_{k} [λ_{k}(M_{s}^{dya}σ̃11_{Q_{k}})(·)11_{Q_{k}}(·)]^{r})^{\frac{1}{r}} ∥_{Lq ; _{u}} ≤ S∥( ∑_{k} [λ_{k}(σ̃11_{Q_{k}})(·)]^{r})^{\frac{1}{r}} ∥_{Lp ; _{v}}

for all sequences of nonnegative (λ_{k})_{k} and all sequences of dyadic cubes (Q_{k})_{k}, where σ̃(·) = v^{− \frac{1}{p−1}} (·). Then M_{s}^{dya} : Lp ; _{v}(l^{r}) → Lq ; _{u}(l^{r}) or, more precisely,

∥( ∑_{k}(M_{s}^{dya}f_{k})^{r}(·))^{\frac{1}{r}} ∥_{Lq ; _{u}} ≤ cS∥( ∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{Lp ; _{v}}

where c > 0 is a constant which depends only on s, n, p, q, and r but not on the weights ṽ(·) and ũ(·).

This leads to

Lemma 4.2. Condition (2.1) implies that

∥( ∑_{k}(^{z}M_{s}f_{k})^{r}(·))^{\frac{1}{r}} ∥_{L_{u}^{q}} ≤ cS∥( ∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{v}^{p}}  (4.2)

for all z ∈ ℝ^{n} and for all sequences of nonnegative functions (f_{k}(·))_{k}, where

(^{z}M_{s}f)(x) =

= sup \{|Q|^{\frac{s}{n} −1} ∫_{Q} |f(y)| dy; Q ∋ x and Q − z is a closed dyadic cubes\}.

The constant c > 0 depends only on s, n, p, q, and r but not on the weights v(·) and u(·). Here S is the constant appearing in condition (2.1).

This connection of this result with M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) is given by Lemma 4.3. For all r > 1 there is c > 0 such that

[ ∑_{k}(M_{s}^{2^{N}} f_{k})^{r}(·)]^{\frac{1}{r}} ≤ \frac{c}{2^{n(N+3)}} ∫_{[−2^{N+2},2^{N+2}]^{n}} [ ∑_{k}(^{z}M_{s}f_{k})^{r}(·)]^{\frac{1}{r}} dz (4.3)

for all N ∈ ℤ and all sequences of nonnegative functions (f_{k}(·))_{k}. Here the truncated maximal operator M_{s}^{R}, R > 0, is defined by

(M_{s}^{R}f)(x) = sup \{|Q|^{\frac{s}{n} −1} ∫_{Q} |f(y)| dy; Q ∋ x with |Q|^{\frac{1}{n}} ≤ R\}.

Now let us see how these lemmas imply M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}). It is sufficient to get M_{s}^{R} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) uniformly for all R > 0, since the conclusion appears after using the monotone convergence theorem. Thus let R > 0 such that 2^{N−1} < R ≤ 2^{N} for some integer N. Then

∥( ∑_{k}(M_{s}^{R}f_{k})^{r}(·))^{\frac{1}{r}} ∥_{L_{u}^{q}} ≤ ∥( ∑_{k}(M_{s}^{2^{N}} f_{k})^{r}(·))^{\frac{1}{r}} ∥_{L_{u}^{q}} ≤

≤ \frac{c}{2^{n(N+3)}} ∥ ∫_{[−2^{N+2},2^{N+2}]^{n}} [ ∑_{k}(^{z}M_{s}f_{k})^{r}(·)]^{\frac{1}{r}} dz∥_{L_{u}^{q}} ≤ (by (4.3))

≤ \frac{c}{2^{n(N+3)}} ∫_{[−2^{N+2},2^{N+2}]^{n}} ∥( ∑_{k}(^{z}M_{s}f_{k})^{r}(·))^{\frac{1}{r}} ∥_{L_{u}^{q}} dz ≤

(by the Minkowski inequality)

≤ c′S∥( ∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{u}^{q}} (by inequality (4.2)).

The latter inequality is obtained after using (4.2) for which the constant cS does not depend on z. So the proof of M_{s} : L_{v}^{p}(l^{r}) → L_{u}^{q}(l^{r}) will be completed after proving these basic lemmas, which will be done in the next section.

Proof of Proposition 8. The first part of this result follows from Theorem 1, since under the assumption on σ(·)

(M_{s}σ11_{Q})(·)11_{Q}(·) ≤ c[|Q|^{\frac{s}{n} −1}( ∫_{Q} σ(y) dy)]11_{Q}(·)

for all cubes Q and for a fixed constant c > 0. The proof of this inequality can be found in [8] (Proposition 2, p.93).

The second part is a consequence of Proposition 6 and the above inequality.

5. Proofs of Basic Lemmas 4.1, 4.2, 4.3

Proof of Lemma 4.3. Inequality (4.3) is based on

(M_{s}^{2^{N}} g)(·) ≤ \frac{c}{2^{n(N+3)}} ∫_{[−2^{N+2},2^{N+2}]^{n}} (^{z}M_{s}g)(·) dz for all g(·) ≥ 0, (5.1)

where c > 0 does not depend on N and g(·). This inequality was proved in [9] (Lemma 2, p. 8). Consider a sequence of nonnegative scalars (α_{k})_{k} with (∑_{k} α_{k}^{r′})^{\frac{1}{r′}} = 1 and such that (∑_{k}(M_{s}^{2^{N}} f_{k})^{r}(x))^{\frac{1}{r}} = ∑_{k}(M_{s}^{2^{N}} f_{k})(x)α_{k}. Then inequality (4.3) appears after using (5.1), since

( ∑_{k}(M_{s}^{2^{N}} f_{k})^{r}(x))^{\frac{1}{r}} ≤ \frac{c}{2^{n(N+3)}} ∫_{[−2^{N+2},2^{N+2}]^{n}} [ ∑_{k}(^{z}M_{s}f_{k})(x)α_{k}] dz ≤

≤ ( ∑_{k} α_{k}^{r′})^{\frac{1}{r′}} \frac{c}{2^{n(N+3)}} ∫_{[−2^{N+2},2^{N+2}]^{n}} [ ∑_{k}(^{z}M_{s}f_{k})^{r}(x)]^{\frac{1}{r}} dz =

= \frac{c}{2^{n(N+3)}} ∫_{[−2^{N+2},2^{N+2}]^{n}} [ ∑_{k}(^{z}M_{s}f_{k})^{r}(x)]^{\frac{1}{r}} dz.

Proof of Lemma 4.2. The key to deriving inequality (4.2) is that for some constant c > 0

∥( ∑_{k}(M_{s}^{dya}g_{k})^{r}(x))^{\frac{1}{r}} ∥_{L^{q}(u(x−z)dx)} ≤ cS∥( ∑_{k} g_{k}^{r}(x))^{\frac{1}{r}} ∥_{L^{p}(v(x−z)dx)} (5.2)

for all z ∈ ℝ^{n} and all sequences of nonnegative functions (g_{k}(·))_{k}. Indeed, with f_{k,z}(y) = f_{k}(y − z) we have that

∥( ∑_{k}(^{z}M_{s}f_{k})^{r}(x))^{\frac{1}{r}} ∥_{L^{q}(u(x)dx)}=∥( ∑_{k}(M_{s}^{dya}f_{k,z})^{r}(x + z))^{\frac{1}{r}} ∥_{L^{q}(u(x)dx)}=

= ∥( ∑_{k}(M_{s}^{dya}f_{k,z})^{r}(x))^{\frac{1}{r}} ∥_{L^{q}(u(x−z)dx)} ≤

≤ cS∥( ∑_{k} f_{k,z}^{r}(x))^{\frac{1}{r}} ∥_{L^{p}(v(x−z)dx)} = (by (5.2))

= cS∥( ∑_{k} f_{k}^{r}(x − z))^{\frac{1}{r}} ∥_{L^{p}(v(x−z)dx)} = cS∥( ∑_{k} f_{k}^{r}(x))^{\frac{1}{r}} ∥_{L^{p}(v(x)dx)}.

Now to get (5.2), by Lemma 4.1 with ũ(·) = u(· − z) and ṽ(·) = v(· − z), it is sufficient to check that

∥( ∑_{k} [λ_{k}(M_{s}^{dya}σ_{z}11_{Q_{k}})(x)11_{Q_{k}}(x)]^{r})^{\frac{1}{r}} ∥_{L^{q}(u(x−z)dx)} ≤

≤ S∥( ∑_{k} [λ_{k}(σ_{z}11_{Q_{k}})(x)]^{r})^{\frac{1}{r}} ∥_{L^{p}(v(x−z)dx)}  (5.3)

and all dyadic cubes (Q_{k})_{k}; here σ_{z}(y) = σ(y − z) = v^{− \frac{1}{p−1}} (y − z). To obtain this test condition, observe that

∥( ∑_{k} [λ_{k}(σ_{z}11_{Q_{k}})(x)]^{r})^{\frac{1}{r}} ∥_{L^{p}(v(x−z)dx)} =

=∥( ∑_{k} [λ_{k}(σ11_{Q_{k}−z})(x)]^{r})^{\frac{1}{r}} ∥_{L^{p}(v(x)dx)}

and

∥( ∑_{k} [λ_{k}(M_{s}^{dya}σ_{z}11_{Q_{k}})(x)11_{Q_{k}}(x)]^{r})^{\frac{1}{r}} ∥_{L^{q}(u(x−z)dx)} =

= ∥( ∑_{k}[λ_{k}(M_{s}^{dya}σ_{z}11_{Q_{k}})(x + z)11_{Q_{k}−z}(x)]^{r})^{\frac{1}{r}} ∥_{L^{q}(u(x)dx)} ≤

≤ ∥( ∑_{k} [λ_{k}(M_{s}σ11_{Q_{k}−z})(x)11_{Q_{k}−z}(x)]^{r})^{\frac{1}{r}} ∥_{L^{q}(u(x)dx)}.

With these observations, condition (2.1) (for all cubes) clearly implies the test condition (5.3) (for all dyadic cubes).

Proof of Lemma 4.1. It is sufficient to find a constant c > 0 such that

∥( ∑_{k}(M_{s}^{dya,R}f_{k})^{r}(·))^{\frac{1}{r}} 11_{]0,R[^{n}}(·)∥_{L_{u}^{q}} ≤ cS∥( ∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{v}^{p}} (5.4)

for all R > 0 and all nonnegative functions (f_{k}(·))_{k}. Here (M_{s}^{dya,R}f)(x) = sup\{|Q|^{\frac{s}{n} −1} ∫_{Q} |f(y)| dy; Q is a dyadic cube with x ∈ Q ⊂]0, R[^{n}\}. Inequality (5.4) is based on

Lemma 5.1. There is a constant c > 0 such that for all (f_{k}(·))_{k} ∈ L_{v}^{p}(l^{r}) one can find nonegative scalars (λ_{jk})_{k,j}, and dyadic cubes (Q_{jk})_{k,j} satisfying

(M_{s}^{dya,R}f_{k})^{r}(·)11_{]0,R[^{n}}(·) ≤ ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} (M_{s}^{dya}σ11_{Q_{jk}})^{r}(·)11_{Q_{jk}}(·) (5.5)

for all r > 0 and

∥( ∑_{k} ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} 11_{Q_{jk}}(·))^{\frac{1}{r}} ∥_{L_{σ}^{p}} ≤ c∥( ∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{v}^{p}}. (5.6)

It is for the proof of such results that the atomic decomposition of tent spaces is useful. Assuming Lemma 5.1 to be true, inequality (5.4) can be derived as follows:

∥( ∑_{k}(M_{s}^{dya,R}f_{k})^{r}(·)11_{]0,R[^{n}}(·))^{\frac{1}{r}} ∥_{L_{u}^{q}}^{r} =∥ ∑_{k}(M_{s}^{dya,R}f_{k})^{r}(·)11_{]0,R[^{n}}(·)∥_{L \frac{q}{r} ; u} ≤

≤ ∥ ∑_{k} ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} (M_{s}^{dya}σ11_{Q_{jk}})^{r}(·)11_{Q_{jk}}(·)∥_{L \frac{q}{r} ; u} ≤ (by (5.5))

≤ S^{r}∥( ∑_{k} ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} σ(·)11_{Q_{jk}}(·))^{\frac{1}{r}} ∥_{L_{v}^{p}}^{r} = (by condition (2.1))

= S^{r}∥( ∑_{k} ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} 11_{Q_{jk}}(·))^{\frac{1}{r}} ∥_{L_{σ}^{p}}^{r} ≤ (since σ^{p}(·)v(·) = σ(·))

≤ cS^{r}∥( ∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{σ}^{p}}^{r} (by (5.6)).

Here c > 0 does not depend on R.

We are going to introduce the notions of tent spaces needed for our purpose as in [8]. Our definitions will be slightly different from those introduced in Coifman, Meyer, and Stein in [7], since we use the dyadic versions of the spaces they defined.

Let X =]0, ∞[^{n}−{(2^{−l}k_{j})_{j}} ((k_{j})_{j} ∈ ℕ^{n}), and X̃ = X × 2^{ℤ}. For each couple (x, w) ∈ X̃ there is a unique (open) dyadic cube Q = Q_{yw} containing y and having the side length w = 2^{−l}. Let

(y, w) ∈ Γ̃(x) if and only if x ∈ Q_{yw}.  (5.7)

Also define

Ω̂ = ( ⋃ {Γ̃(x); x ∈ Ω^{c}})^{c}  (5.8)

for each measurable set Ω ⊂]0, ∞[^{n}. Thus

(y, w) ∈ Ω̂ if and only if Q_{yw} ⊂ Ω.  (5.9)

Finally, define the functional A_{∞} acting on each measurable function f̃ on X̃ by

(A_{∞}f̃)(x) = sup {|f̃(y, w)|; (y, w) ∈ Γ̃(x)}.  (5.10)

The following atomic decomposition Lemma was proved in [8] (Lemma 2, p. 96).

Lemma 5.2. Let 0 < p < ∞. There is C > 0 such that for all functions g̃(y, w = 2^{−l}) with a support contained in { \atop Q[0, R]} and ∥(A_{∞}g̃)(·)∥_{L_{v}^{p}} < ∞, one can find scalars λ_{j} > 0, dyadic cubes Q_{j}, and functions ã_{j}(y, w) which satisfy the following conditions:

supp ã_{j} are disjoint and |ã_{j}(y, w)| ≤ |Q_{j}|_{σ}^{− \frac{1}{p}} ˜11_{Q_{j}}(y, w); (5.11)
g̃(y, w) = ∑_{j} λ_{j}ã_{j}(y, w) a.e.;  (5.12)
∑_{j} λ_{j}^{r}|Q_{j}|_{σ}^{− \frac{r}{p}} 11_{Q_{j}}(·) ≤ C^{r}(A_{∞}g̃)^{r}(·) for all r > 0.  (5.13)

Here Q[0, R] denotes the cube ]0, R[^{n}, R > 0. Contrary to λ_{j}, Q_{j}, and ã_{j}(y, w)’s, the constant C > 0 does not depend on R > 0 and the function g̃(y, w).

A fundamental observation is that

(M_{s}^{dya,R}f)(x) =

= sup \{|Q_{yw}|^{\frac{s}{n} −1} ∫_{Q_{yw}} |f(y)| dy; Q_{yw} ∋ x and Q_{yw} ⊂ (]0, R[)^{n}\} =

= sup{ω̃(y, w)g̃(y, w); Q_{yw} ∋ x and Q_{yw} ⊂ (]0, R[)^{n}},

where ω̃(y, w) = |Q_{yw}|^{\frac{s}{n} −1}|Q_{yw}|_{σ} = |Q_{yw}|^{\frac{s}{n} −1} ∫_{Q_{yw}} σ(z) dz and

g̃(y, w) = \begin{cases}|Q_{yw}|_{σ}^{−1} ∫_{Q_{yw}} g(z)σ(z) dz if (y, w) ∈ { \atop Q[0, R]} \\ 0 else \end{cases}

and g(·) = σ^{−1}(·)f(·). Moreover,

(A_{∞}g̃)(·) ≤ (N_{σ}g)(·)  (5.14)

with (N_{σ}g)(x) = sup{|Q|_{σ}^{−1} ∫_{Q} |g(y)|σ(y) dy; Q dyadic with Q ∋ x}. Consequently, for 1 < p < ∞,

∥(A_{∞}g̃)(·)∥_{L_{σ}^{p}} ≤ ∥(N_{σ}g)(·)∥_{L_{σ}^{p}} ≤ c(n, p)∥σ^{−1}(·)f(·)∥_{L_{σ}^{p}} =
= c(n, p)∥f(·)∥_{L_{v}^{p}}.  (5.15)

The latter inequality can be obtained by interpolation, that is, by proving with the aid of the classical arguments that N_{σ} : L^{1}(ℝ^{n}, σ(x)dx) → L^{1,∞}(ℝ^{n}, σ(x)dx) and N_{σ} : L^{∞}(ℝ^{n}, σ(x)dx) → L^{∞}(ℝ^{n}, σ(x)dx).

Now we are ready to give Proof of Lemma 5.1. This result is essentially based on Lemma 5.2. Indeed, if (f_{k}(·))_{k} is a sequence of nonnegative functions with ∥(∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{v}^{p}} < ∞, then, in particular, f_{k}(·) ∈ L_{v}^{p} for each k. So by (5.15) the associated function g̃_{k}(y, w) defined as above satisfies ∥(A_{∞}g̃_{k})(·)∥_{L_{v}^{p}} < ∞. Hence by Lemma 5.2 one can find scalars λ_{jk} > 0, dyadic cubes Q_{jk}, and functions ã_{jk}(y, w = 2^{−l}) satisfying the following conditions:

supp ã_{jk} are disjoint and |ã_{jk}(y, w)| ≤ |Q_{jk}|_{σ}^{− \frac{1}{p}} ˜11_{Q_{jk}}(y, w); (5.16)
g̃_{k}(y, w) = ∑_{j} λ_{jk}ã_{jk}(y, w) a.e.;  (5.17)
∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} 11_{Q_{jk}}(·) ≤ C^{r}(A_{∞}g̃_{k})^{r}(·) for all r > 0.  (5.18)

Let us check inequality (5.5). Since

(M_{s}^{dya,R}f_{k})^{r}(x)=sup{ω^{r}(y, w)g̃_{k}^{r}(y, w); Q_{yw} ∋x and Q_{yw} ⊂(]0, R[)^{n}},

for all (y, w) with x ∈ Q_{yw} ⊂ Q[0, R] we have

ω^{r}(y, w)g_{k}^{r}(y, w) ≤ ω^{r}(y, w) ∑_{j} λ_{jk}^{r}|a_{jk}^{r}(y, w)| ≤ (supp ã_{j} are disjoint)

≤ ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} [|Q_{yw}|^{\frac{s}{n} −1}|Q_{yw}|_{σ} ˜11_{Q_{j}}(y, w)]^{r} ≤

(by the definition of ω̃(y, w) and (5.16))

≤ ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} [|Q_{yw}|^{\frac{s}{n} −1}|Q_{yw} ∩ Q_{jk}|_{σ} ˜11_{Q_{jk}}(y, w)]^{r} =

(since by (5.9) : Q_{yw} ⊂ Q_{jk})

= ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} [(|Q_{yw}|^{\frac{s}{n} −1} ∫_{Q_{yw}} σ(z)11_{Q_{jk}}(z) dz)˜11_{Q_{jk}}(y, w)]^{r} ≤

≤ ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} (M_{s}^{dya}σ11_{Q_{jk}})^{r}(x)11_{Q_{jk}}(x) recall that x ∈ Q_{yw} ⊂ Q_{jk}.

In checking inequality (5.6) the main key is

∥( ∑_{k}(N_{σ}g_{k})^{r}(·))^{\frac{1}{r}} ∥_{L_{σ}^{p}} ≤ c∥( ∑_{k} g_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{σ}^{p}},  (5.19)

where c > 0 does not depend on the sequence (g_{k}(·))_{k}. Indeed, using (5.18), (5.14), and the above inequality, we obtain

∥( ∑_{k} ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} 11_{Q_{jk}}(·))^{\frac{1}{r}} ∥_{L_{σ}^{p}} = ∥ ∑_{k} ∑_{j} λ_{jk}^{r}|Q_{jk}|_{σ}^{− \frac{r}{p}} 11_{Q_{jk}}(·)∥_{L \frac{p}{r} ; σ} ≤

≤ C^{r}∥ ∑_{k}(A_{∞}g̃_{k})^{r}(·)∥_{L \frac{p}{r} ; σ} = C^{r}∥( ∑_{k}(A_{∞}g̃_{k})^{r}(·))^{\frac{1}{r}} ∥_{L_{σ}^{p}}^{r} ≤

≤ C^{r}∥( ∑_{k}(N_{σ}g_{k})^{r}(·))^{\frac{1}{r}} ∥_{L_{σ}^{p}}^{r} ≤(cC)^{r}∥( ∑_{k} g_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{σ}^{p}}^{r} =

= (cC)^{r}∥( ∑_{k} f_{k}^{r}(·))^{\frac{1}{r}} ∥_{L_{v}^{p}}^{r},

since g_{k}(·) = σ^{−1}(·)f_{k}(·) and σ^{1−p}(·) = v(·).

The boundedness (5.19) can be obtained by using the same classical arguments [1] as for M_{0} : L_{1}^{p}(l^{r}) → L_{1}^{p}(l^{r}). Indeed the main points are (5.15) and Fefferman–Stein’s inequality

∫_{ℝ^{n}}(N_{σ}f)^{r}(x)g(x)σ(x) dx≤c∫_{ℝ^{n}}f^{r}(x)(N_{σ}g)(x)σ(x) dx, 1 < r < ∞, (5.20)

where c > 0 is a constant which does not depend on the nonnegative functions f(·), g(·). Similarly to (5.15), inequality (5.20) can be obtained by interpolation, since N_{σ} : L^{∞}[ℝ^{n}, (N_{σ}g)(x)σ(x)dx] → L^{∞}[ℝ^{n}, g(x)σ(x)dx] and N_{σ} : L^{1}[ℝ^{n}, (N_{σ}g)(x)σ(x)dx] → L^{1∞}[ℝ^{n}, g(x)σ(x)dx]. In this last boundedness no doubling condition σ(·) is needed, since we work with dyadic cubes. By (5.15) it is clear that N_{σ} : L_{σ}^{p}(l^{p}) → L_{σ}^{p}(l^{p}). And since N_{σ} : L_{σ}^{p}(l^{∞}) → L_{σ}^{p}(l^{∞}), by interpolation [1] we get inequality (5.19) for p ≤ r < ∞. The corresponding result for 1 < r < p can be obtained by duality using inequality (5.20) (see also [1] for details with the nonweighted case M_{0} : L_{1}^{p}(l^{r}) → L_{1}^{p}(l^{r})).

Acknowledgement

I would like to thank C. Pérez for the helpful discussion we had on vectorvalued inequalities and for communicating his paper [5]. I also thank to the referee for his comments and suggestions about this paper.

References

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(Received 01.02.1996; revised 30.10.1997)

Author’s address: Institut Polytechnique St-Louis, EPMI 13 Boulevard de I’Hautil 95 092 Cergy-Pontoise France