GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 6, 1996, 583-600

WEIGHTED L_{Φ} INTEGRAL INEQUALITIES FOR MAXIMAL OPERATORS

Y. RAKOTONDRATSIMBA

Abstract. Sufficient (almost necessary) conditions are given on the weight functions u(·), v(·) for

Φ_{2}^{−1}[ ∫_{ℝ^{n}} Φ_{2}(C_{2}(M_{s}f)(x))u(x)dx] ≤ Φ_{1}^{−1}[C_{1} ∫_{ℝ^{n}} Φ_{1}(|f(x)|)v(x)dx]

to hold when Φ_{1}, Φ_{2} are φ-functions with subadditive Φ_{1}Φ_{2}^{−1}, and M_{s} (0 ≤ s < n), is the usual fractional maximal operator.

§ 1. Introduction

Let n ∈ ℕ^{∗}, 0 ≤ s < n. The fractional maximal operator M_{s} of order s is defined as

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

Throughout this paper, Q will denote a cube with the sides parallel to the coordinate planes.

As in [1], a real function Φ(·) defined on [0, ∞[ is called a φ-function if it is a nondecreasing continuous function which satisfies Φ(0) = 0 and lim_{s→∞} Φ(s) = ∞. The φ-function Φ(·) is subadditive if Φ(t_{1} + t_{2}) ≤ Φ(t_{1}) + Φ(t_{2}) for all t_{1}, t_{2} ∈ [0, ∞[. Let u(·), v(·) be weight functions (i.e., nonnegative locally integrable functions). In this paper we study the integral inequality

Φ_{2}^{−1}[ ∫_{ℝ^{n}} Φ_{2}(C_{2}(M_{s}f)(x))u(x)dx] ≤ Φ_{1}^{−1}[C_{1} ∫_{ℝ^{n}} Φ_{1}(|f(x)|)v(x)dx] (1)

583 for all functions f(·). Here Φ_{1}(·), Φ_{2}(·) are φ-functions with subadditive (Φ_{1}Φ_{2}^{−1})(·), and C_{1}, C_{2} are nonnegative constants which do not depend on each function f(·). For convenience we also denote this inequality by M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}}.

1991 Mathematics Subject Classification. 42B25, 46E30.

Key words and phrases. Fractional maximal operators, Weighted inequalities, Orlicz spaces.

1072-947X/96/1100-0583$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

As mentioned in a recent monograph of Kokilashvili and Krbec [2], this is an open problem to characterize a pair of weight functions u(·), v(·) for which M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}} holds. Such an integral inequality can be useful in studying the boundary value problems for quasilinear partial differential equations (see [3]).

In the Lebesgue case, i.e., Φ_{1}(t) = t^{p}, Φ_{2}(t) = t^{q} with 1 < p ≤ q < ∞, inequality (1) can be rewritten as

( ∫_{ℝ^{n}} (M_{s}f)^{q}(x)u(x)dx)^{\frac{1}{q}} ≤ C( ∫_{ℝ^{n}} |f(x)|^{p}v(x)dx)^{\frac{1}{p}} .  (2)

Sawyer [4] characterized the weight functions u(·), v(·) for which (2) held. One of the crucial keys he used to solve this problem is to note the equivalence of (2) to

( ∫_{ℝ^{n}} (M_{s}σg)^{q}(x)u(x)dx)^{\frac{1}{q}} ≤ C( ∫_{ℝ^{n}} |g(x)|^{p}σ(x)dx)^{\frac{1}{p}} ,  (3)

where σ(·) = v^{− \frac{1}{p−1}} (·). Indeed, ∫_{ℝ^{n}} |(fσ^{−1})(x)|^{p}σ(x)dx = ∫_{ℝ^{n}} |f(x)|^{p}v(x)dx. Thus in the Orlicz setting, inequality (3) leads naturally to the following first generalization:

Φ_{2}^{−1}[ ∫_{ℝ^{n}} Φ_{2}(C_{2}(M_{s}σg)(x))u(x)dx] ≤ Φ_{1}^{−1}[C_{1} ∫_{ℝ^{n}} Φ_{1}(|g(x)|)σ(x)dx] (4)

where σ(·) is some weight function. Such integral inequalities were studied by L. Qinsheng [5], and, independently, by the author [6]. Roughly speaking, inequality (4) is not technically too far from the Lebesgue setting, and so this problem can be handled by Sawyer’s ideas [4]. The second generalization of (2) is the two-weight modular inequality (1) or M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}} introduced above. This inequality is more difficult to study than (4). Indeed, from the latter, we cannot easily derive (1), since contrary to the Lebesgue case, there is no obvious connection between ∫_{ℝ^{n}} Φ_{1}|(f(x)σ^{−1})(x)|)σ(x)dx and ∫_{ℝ^{n}} Φ_{1}(|f(x)|)v(x)dx. The problem M : L_{w}^{Φ} → L_{w}^{Φ}, where M = M_{0} is the classical Hardy–Littlewood maximal operator, was studied by Kerman and Torchinsky [7]. They considered an N-function Φ(·) with Φ(·), Φ^{∗}(·) ∈ Δ_{2}. Recall that an N-function is a convex φ-function satisfying lim_{t→0} \frac{Φ(t)}{t} = lim_{t→∞} \frac{t}{Φ(t)} = 0, and the associated conjugate function Φ^{∗}(·) is defined by Φ^{∗}(t) = sup_{s>0}{st − Φ(s)}. The condition Φ(·) ∈ Δ_{2} means Φ(2t) ≤ CΦ(t) for all t > 0.

As in the Lebesgue setting, the problem M : L_{v}^{Φ} → L_{u}^{Φ} for u ≠ v is completely different. Using the Lebesgue arguments [4], Chen [8] and Sun [9] studied this problem independently. They obtained the following result:

Suppose Φ(·) is an N-function with Φ(·), Φ^{∗}(·) ∈ Δ_{2} and there is a weight function σ(·) satisfying the extra-assumption:

sup_{y∈ℝ^{n}} ∫_{ℝ^{n}} Φ([τ_{y}^{−1}Ñ_{τ_{y}σ}(τ_{y}f)](x))v(x)dx ≤ C ∫_{ℝ^{n}} Φ(|f(x)|)v(x)dx (∗)

for all functions f(·); then M : L_{v}^{Φ} → L_{u}^{Φ} holds if and only if

∫_{ℝ^{n}} Φ((Mεσ1I_{Q})(x))u(x)dx ≤ c ∫_{ℝ^{n}} Φ((εσ1I_{Q})(x))v(x)dx  (5)

for all cubes Q and ε > 0.

Here (τ_{y}f)(x) = f(x − y), Ñ_{σ}f(x) = σ(x)N_{σ}(fσ^{−1})(x), and N_{σ} is the maximal function defined by

(N_{σ}g)(x) = sup {|Q|_{σ}^{−1} ∫_{Q} |g(y)|σ(y)dy; Q is a cube with Q ∋ x}.

Unfortunately, this result has two main drawbacks. Firstly, for a general N-function Φ(·), no exact condition is known to be imposed on a weight function v(·) for which there is another weight function σ(·) satisfying the extra-condition (∗). Secondly, as in the Lebesgue setting, the Sawyer’s condition (5) is not easy to check since it is expressed in terms of the maximal function M itself. Thus it was a challenging problem for specialists in weighted inequalities to obtain a sufficient (almost necessary) condition on weight functions u(·), v(·) which ensures M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}}. An attempt in this direction was made by the author in [10]. For completeness, we recall here the result he obtained.

Let 1 ≤ r < ∞, Ψ_{1}(t) = Φ_{1}(t^{\frac{1}{r}}) be an N-function and (Φ_{1}Φ_{2}^{−1})(·) be subadditive. Moreover, let us assume Φ_{2}(·), Ψ_{1}^{∗}(·) ∈ Δ_{2}. Then M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}} holds if and only if

Φ_{2}^{−1}[(A_{2}ε|Q|^{\frac{s}{n}} ∫_{Q} u(y)dy)] ≤ Φ_{1}^{−1}[(A_{1}Φ_{1}(ε) ∫_{Q} v(y)dy)] (∗∗)

with constants A_{1} and A_{2} for all cubes Q and all ε > 0, whenever the weight function v(·) belongs to the Muckenhoupt class A_{p} for 1 < p ≤ r.

With the definition given below, w ∈ A_{p} if and only if (w, w) ∈ A_{1}(0, p, p). Although the test condition (∗∗) is more computable than the above Sawyer one (5), the restriction on the weight function v(·) is an inconvenience.

Therefore our main purpose in this paper is to derive M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}} by a similar test condition without restrictions on the weight functions u(·), v(·). This condition is denoted by (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}) (r ≥ 1) and means

\frac{1}{|Q|} ∫_{Q} (Φ1^{\frac{1}{r}})^{∗} [|Q|^{\frac{s}{n}} \frac{A_{2}}{Φ_{2}^{−1}[\frac{Φ_{2}Φ_{1}^{−1}(A_{1}ε^{−1}|Q|)}{ε^{−1}|Q|} _{\frac{1}{\frac{1}{|Q|} ∫_{Q} εu(y)dy}} ]} \frac{1}{(εv(x))^{\frac{1}{r}}} ] ×

×(εv(x))^{\frac{1}{r}} dx ≤ 1  (6)

for all cubes Q and all ε > 0. The presence of ε > 0 can be explained by the lack of homogeneities of the φ-functions Φ_{1}(·) and Φ_{2}(·). Assuming that Φ1^{\frac{1}{r}} (·) is an N-function for some r > 1, we will prove that (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}) is the sufficient condition in order that M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}} (see Proposition 3). This sufficient condition we introduce is almost necessary in the sense that (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2},) is a necessary condition for this embedding (see Proposition 1). Our result includes one result due to Perez [11]. We were really inspired by this author’s technique, which we will develop in the Orlicz setting. In order to include the modular inequality (4), we also deal with the general embedding M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2}}(w_{2}), i.e.,

Φ_{2}^{−1}[ ∫_{ℝ^{n}} Φ_{2}(C_{2}w_{2}(x)(M_{s}f)(x))u(x)dx] ≤

≤ 1Φ_{1}^{−1}[C_{1} ∫_{ℝ^{n}} Φ_{1}(w_{1}(x)|f(x)|)v(x)dx].  (7)

Let us consider again the inequality M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}}. The corresponding weak version M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2},∞} is defined by

Φ_{2}^{−1}[Φ_{2}(C_{2}λ)  ∫_{{(M_{s}f)(x)>λ}} u(x)dx] ≤ Φ_{1}^{−1}[C_{1} ∫_{ℝ^{n}} Φ_{1}(|f(x)|)v(x)dx] (8)

for all functions f(·) and all λ > 0. Using the ideas from [12], we will prove that if (Φ_{1}Φ_{2}^{−1})(·) is subadditive and Φ_{1}(·) is an N-function, then M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2},∞} holds if and only if (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2},). This weak inequality was already solved and considered by many authors [2], [13] for the case s = 0 and Φ_{1}(·) = Φ_{2}(·). We emphasize that in this paper we never use the Δ_{2} condition for the φ-functions Φ_{1}(·) and Φ_{2}(·).

We leave open the problem I_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}}, where I_{s} (0 < s < n) is the fractional integral defined by I_{s}f(x) = ∫_{ℝ^{n}} |x − y|^{s−n}f(y)dy. A characterization of the weight functions u(·), v(·) for which the corresponding weak inequality was proved was done by the author in [14].

We state our main results in §2, and we will give the basic lemma in the next section. Weak inequalities are proved in §4, and §5 is devoted to proving strong inequalities. The last section deals with our test condition (6).

§ 2. The Results

Weak Inequalities

We first characterize the weight functions u(·), v(·) for which the weak inequality (8) holds.

Proposition 1. Let 0 ≤ s < n, Φ_{2}(·) be a φ-function and Φ_{1}(·) be an N-function with subadditive (Φ_{1}Φ_{2}^{−1})(·).

Assume that the weak inequality M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2},∞} is satisfied for some constants C_{1}, C_{2} > 0. Then (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}) with the constants A_{1} = C_{1} and A_{2} = C_{2}.

Conversely, let (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}) for some constants A_{1}, A_{2} > 0. Then M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2},∞} is satisfied with the constants C_{1} = NA_{1}, C_{2} = CA_{2}.

Here N = N(n), C = C(s, n) are nonnegative constants which depend respectively on n and s, n.

In fact, this weak inequality can be considered as the particular case of M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2},∞}(w_{2}), i.e.,

Φ_{2}^{−1}[ ∫_{{(M_{s}f)(x)>λ}} Φ_{2}(C_{2}λw_{2}(x))u(x)dx] ≤

≤ Φ_{1}^{−1}[C_{1} ∫_{ℝ^{n}} Φ_{1}(w_{1}(x)|f(x)|)v(x)dx]  (9)

for all functions f(·) and all λ > 0.

Here v(·), u(·), w_{1}(·), w_{2}(·) are weight functions. Recall (see [4]) that for each N-function Φ(·) and for each weight function w(·), the quantity ∥·∥_{Φ,w} defined by

∥f(·)∥_{Φ,w} = sup { ∫_{ℝ^{n}} |f(y)g(y)|w(y)dy; ∫_{ℝ^{n}} Φ^{∗}(|g(y)|)w(y)dy ≤ 1}

yields a norm in the Orlicz space L_{w}^{Φ} (which is the set of all measurable functions f(·) satisfying ∫_{ℝ^{n}} Φ(λ|f(y)|)w(y)dy < ∞ for some λ > 0). And for each cube Q we also define ∥f(·)∥_{Φ,Q,w} as ∥f(·)∥_{Φ,|Q|^{−1}1I_{Q}w(·)}.

Our characterization of the weight functions u(·), v(·), w_{1}(·), w_{2}(·) for the weak inequality (9) can be stated as

Theorem 2. Let 0 ≤ s < n, Φ_{2}(·) be a φ-function and Φ_{1}(·) be an N-function with subadditive (Φ_{1}Φ_{2}^{−1})(·).

Suppose that the weak inequality M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2},∞}(w_{2}) is satisfied for some constants C_{1}, C_{2} > 0. Then (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}), i.e.,

Φ_{2}^{−1}[ ∫_{Q} Φ_{2}(A_{2}w_{2}(x)|Q|^{\frac{s}{n}} ∥\frac{1}{w_{1}(·)εv(·)}∥_{Φ_{1}^{∗},Q,εv})u(x)dx] ≤ Φ_{1}^{−1}[A_{1}ε^{−1}|Q|]

for all cubes Q and all ε > 0. Here A_{1} = C_{1}, and A_{2} = C_{2}.

Conversely, if (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}), for some constants A_{1}, A_{2} > 0, then M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2},∞}(w_{2}) holds with the constants C_{1} = NA_{1}, C_{2} = CA_{2}.

For w_{2}(·) = 1, the condition (u, v) ∈ A(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) can be reduced to the easy form

\frac{1}{|Q|} ∫_{Q} Φ_{1}^{∗} [|Q|^{\frac{s}{n}} \frac{A′_{2}}{Φ_{2}^{−1}[\frac{Φ_{2}Φ_{1}^{−1}(A′_{1}ε^{−1}|Q|)}{ε^{−1}|Q|} _{\frac{1}{\frac{1}{|Q|} ∫_{Q} εu(y)dy}} ]} \frac{1}{w_{1}(x)εv(x)}] ×

×εv(x)dx ≤ 1.

Indeed, for each N-function Ψ(·) and each weight function w(·) we have ∥f∥_{[Ψ],w} ≤ 1 equivalent to ∫_{ℝ^{n}} Ψ(|f(x)|)w(x)dx ≤ 1. Observe that (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}, 1, 1) is the same as the condition (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}) introduced above.

Strong Inequalities

Proposition 3. Let 0 ≤ s < n, Φ_{1}(·), Φ_{2}(·) be φ-functions with subadditive (Φ_{1}Φ_{2}^{−1})(·).

Suppose Φ_{1}(·) is an N-function and the strong inequality M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}} is satisfied for some constants C_{1}, C_{2} > 0. Then (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}) with the constants A_{1} = C_{1} and A_{2} = C_{2}.

Conversely, we assume that Φ1^{\frac{1}{r}} (·) is an N-function for some r > 1 and (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}) for some constants A_{1}, A_{2} > 0. Then M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}} is satisfied with the constants C_{1} = N(n)A_{1}, C_{2} = c(s, n)A_{2}.

In the Lebesgue case Φ_{1}(t) = t^{p} (p > 1) we take r as r = \frac{p}{(p′s)′} = \frac{ps−p+1}{s}

(s > 1). Here r > 1, and Φ1^{\frac{1}{r}} (t) = t^{(p′s)′}. Thus our proposition includes a Perez’ result [5]. In fact, the strong inequality considered here is the particular case of M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2}}(w_{2}) or inequality (7). Thus the main result of this paper is

Theorem 4. Let 0 ≤ s < n, Φ_{1}(·), Φ_{2}(·) be φ-functions with subadditive (Φ_{1}Φ_{2}^{−1})(·).

Let Φ_{1} be an N-function and suppose that the strong inequality M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2}}(w_{2}) is satisfied for some constants C_{1}, C_{2} > 0. Then

(u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}),

with the constants A_{1} = C_{1} and A_{2} = C_{2}.

Conversely, we assume that Φ1^{\frac{1}{r}} (·) is an N-function for some r > 1 and (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) for some constants A_{1}, A_{2} > 0, i.e.,

Φ_{2}^{−1}[ ∫_{Q} Φ_{2}(A_{2}w_{2}(x)|Q|^{\frac{s}{n}} ∥\frac{1}{w_{1}(·)(εv(·))^{\frac{1}{r}}} ∥_{(Φ \frac{1}{r} ; 1 )^{∗},Q,(εv(·))^{\frac{1}{r}}} )u(x)dx] ≤

≤ Φ_{1}^{−1}[A_{1}ε^{−1}|Q|]

for all cubes Q and all ε > 0. Then M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2}}(w_{2}) is satisfied with the constants C_{1} = N(n)A_{1}, C_{2} = c(s, n)A_{2}.

For w_{2}(·) = 1, the condition (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) can be reduced to the easy form

\frac{1}{|Q|} ∫_{Q} (Φ1^{\frac{1}{r}})^{∗} [|Q|^{\frac{s}{n}} \frac{A_{2}}{Φ_{2}^{−1}[\frac{Φ_{2}Φ_{1}^{−1}(A_{1}ε^{−1}|Q|)}{ε^{−1}|Q|} _{\frac{1}{\frac{1}{|Q|} ∫_{Q} εu(y)dy}} ]} \frac{1}{w_{1}(x)(εv(x))^{\frac{1}{r}}} ] ×

×(εv(x))^{\frac{1}{r}} dx ≤ 1.

Due to the Hölder inequality, it is clear that for each r > 1

∥\frac{1}{w_{1}(·)εv(·)}∥_{Φ_{1}^{∗},Q,εv(·)} ≤ ∥\frac{1}{w_{1}(·)[εv(·)]^{\frac{1}{r}}} ∥_{(Φ \frac{1}{r} ; 1 )^{∗},Q,[εv(·)]^{\frac{1}{r}}};

thus (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) becomes a necessary and sufficient condition for the strong inequality M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2}}(w_{2}) if for some r > 1

∥\frac{1}{w_{1}(·)εv(·)}∥_{Φ_{1}^{∗},Q,εv(·)} ≥ C∥\frac{1}{w_{1}(·)[εv(·)]^{\frac{1}{r}}} ∥_{(Φ \frac{1}{r} ; 1 )^{∗},Q,[εv(·)]^{\frac{1}{r}}} .

The latter inequality was studied by Perez [5] in the Lebesgue case.

As we have seen in the first part of Proposition 3, (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}) is a necessary condition in order that M_{s} : L_{v}^{Φ_{1}} → L_{u}^{Φ_{2}}. The argument we will use to get Theorem 4 also involves

Proposition 5. Let 0 ≤ s < n, Φ_{1}(·) be an N-function, Φ_{2}(·) a φfunction with subadditive (Φ_{1}Φ_{2}^{−1})(·). Suppose (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}). Then

Φ_{2}^{−1}[ ∫_{ℝ^{n}} Φ_{2}(C_{2}(M_{s}f)(x))u(x)dx] ≤ Φ_{1}^{−1}[C_{1} ∫_{ℝ^{n}} Φ_{1}(|f(x)|)v(x)dx]

for all functions f(·) satisfying the reverse Hölder inequality

(\frac{1}{|Q|_{ω}} ∫_{Q} [Φ_{1}(|f(x)|)v(x)\frac{1}{ω(x)}]ω(x)dx)^{\frac{1}{r}} ≤

≤ c\frac{1}{|Q|_{ω}} ∫_{Q} [Φ_{1}(|f(x)|)v(x)\frac{1}{ω(x)}]^{\frac{1}{r}} ω(x)dx,  (10)

where r > 1 and ω(·) is an A_{∞} weight function.

Of course, C_{1}, C_{2} will depend on c, n and on the constants of the condition A_{1}(s, Φ_{1}, Φ_{2}).

Now we will discuss the means for checking the test condition (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}, w_{1}, 1). For simplicity we will consider only the case where Φ_{1}(·) = Φ_{2}(·) = Φ(·), since the general case can be treated likewise with minor modifications. Let us denote by A[Q, ε, A_{1}, A_{2}] the quantity

\frac{1}{|Q|} ∫_{Q} (Φ^{\frac{1}{r}})^{∗} [|Q|^{\frac{s}{n}} \frac{A_{2}}{Φ^{−1}[A_{1} _{\frac{1}{\frac{1}{|Q|} ∫_{Q} uε}}]} \frac{1}{w_{1}(x)(εv(x))^{\frac{1}{r}}} ] (εv(x))^{\frac{1}{r}} dx.

With this notation, (u, v) ∈ A_{r}(s, Φ, Φ, w_{1}, 1) if and only if there are A_{1}, A_{2} > 0 for which sup_{ε>0} sup_{Q} A[Q, ε, A_{1}, A_{2}] ≤ 1. It is clear that

c_{0} sup_{ε>0} sup_{R>0} A[B(0, R), ε, c_{1}A_{1}, c_{2}A_{2}] ≤ sup_{ε>0} sup_{Q} A[Q, ε, A_{1}, A_{2}]

where B(0, R) is the ball centered at the origin and with radius R > 0, and c_{0}, c_{1}, c_{2} are nonnegative constants which depend on s and n.

In applications it is easier to compute A[B(0, R), ε, c_{1}A_{1}, c_{2}A_{2}] than the expression A[Q, ε, A_{1}, A_{2}], especialy when u(·), v(·) are radial weight functions. Thus it is interesting to know when the reverse of the last inequality is satisfied. In order to answer this question, we say that a weight function w(·) satisfies the growth condition (C), when there are constants c, C > 0 such that

sup_{R<|x|≤2R} w(x) ≤ \frac{C}{R^{n}} ∫_{|y|≤cR} w(y)dy

for all R > 0. Many of the usual weight functions w(·) satisfy this growth condition, since both nonincreasing and nondecreasing radial functions are admissible. This is also the case when the weight w(·) is essentially constant on annuli, i.e., w(y) ≤ cw(x) for \frac{1}{2}|y| ≤ |x| ≤ 2|y|.

Now we can state

Proposition 6. Let 1 ≤ r < ∞, and (Φ^{\frac{1}{r}})(·) be an N-function. Assume that u(·) is a weight satisfying (C). Also suppose that the weights (Φ^{\frac{1}{r}})^{∗}(λ_{\frac{1}{w_{1}(·)[v(·)]^{\frac{1}{r}}}} )[v(·)]^{\frac{1}{r}} satisfy the growth condition (C) uniformly in

λ > 0. Then there are nonnegative constants c_{0}, c_{1}, c_{2} such that

sup_{ε>0} sup_{Q} A[Q, ε, A_{1}, A_{2}] ≤ c_{0} sup_{ε>0} sup_{R>0} A[B(0, R), ε, c_{1}A_{1}, c_{2}A_{2}].

Indeed, c_{0}, c_{1}, c_{2} depend on the constants in the growth conditions involving u(·), v(·), w_{1}(·) but not on these individual weights.

We only have to prove Theorems 2, 4, and Propositions 5, 6. We first begin by the basic result underlying the proof of these results.

§ 3. A Basic Lemma

Lemma 7. Let r ≥ 1, Φ1^{\frac{1}{r}} (·) be an N-function.

Suppose (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) for some constants A_{1}, A_{2} > 0. Then

Φ_{2}^{−1}[ ∫_{Q} Φ_{2}(C_{2}w_{2}(x)|Q|^{\frac{s}{n}} [\frac{1}{|Q|} ∫_{Q} |f(y)|dy])u(x)dx] ≤

≤ Φ_{1}^{−1}[C_{1}|Q|(\frac{1}{|Q|} ∫_{Q} [Φ_{1}(w_{1}(x)|f(x)|)v(x)]^{\frac{1}{r}} dx)^{r}]  (11)

for all cubes Q and all locally integrable functions f(·) with supports contained in Q. Here C_{1} = A_{1}, C_{2} = A_{2}.

Conversely, inequality (11) implies (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) with the constants A_{1} = C_{1}, A_{2} = C_{2}.

Let Φ(·) be an N-function and w(·) a weight function. The Hölder inequality asserts that

∫_{ℝ^{n}} |f(y)g(y)|w(y)dy ≤ ∥f∥_{[Φ],w}∥g∥_{Φ^{∗},w},

where ∥f∥_{[Φ],w} = inf{s > 0; ∫_{ℝ^{n}} Φ(λ^{−1}|f(y)|)w(y)dy ≤ 1}.

Proof of Lemma 7. Suppose the condition (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) is satisfied for some constants A_{1}, A_{2} > 0.

Let Q be a cube and f(·) a locally integrable function whose support is contained in Q. Without loss of generality we can assume

0 < ∫_{Q} [Φ_{1}(w_{1}(x)|f(x)|)v(x)]^{\frac{1}{r}} dx < ∞.

Note that w_{1}(·)v(·) is not identically zero on Q. Then by the Hölder inequality we have

\frac{1}{|Q|} ∫_{Q} |f(x)|dx = \frac{1}{|Q|} ∫_{Q} (w_{1}(x)|f(x)|)(\frac{1}{w_{1}(x)(εv(x))^{\frac{1}{r}}} )(εv(x))^{\frac{1}{r}} dx ≤

≤ ∥w_{1}(·)f(·)∥_{[Φ \frac{1}{r} ; 1 ],Q,(εv) \frac{1}{r}} ∥\frac{1}{w_{1}(·)(εv(·))^{\frac{1}{r}}} ∥_{(Φ \frac{1}{r} ; 1 )^{∗},Q,(εv)^{\frac{1}{r}}}

for all ε > 0. Choosing ε > 0 such that

(\frac{1}{|Q|} ∫_{Q} [Φ_{1}(w_{1}(x)|f(x)|)εv(x)]^{\frac{1}{r}} dx)^{r} = 1

or ε^{−1} = (\frac{1}{|Q|} ∫_{Q}[Φ_{1}(w_{1}(x)|f(x)|)v(x)]^{\frac{1}{r}} dx)^{r}, we have

∥w_{1}(·)f(·)∥_{[Φ \frac{1}{r} ; 1 ],Q,(εv)^{\frac{1}{r}}} ≤ 1.

Next, using the condition (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) and the above two inequalities, we get

M = Φ_{2}^{−1}[ ∫_{Q} Φ_{2}(A_{2}w_{2}(x)|Q|^{\frac{s}{n}} [\frac{1}{|Q|} ∫_{Q} |f(y)|dy])u(x)dx] ≤

≤ Φ_{2}^{−1}[ ∫_{Q} Φ_{2}(A_{2}w_{2}(x)|Q|^{\frac{s}{n}} ∥\frac{1}{w_{1}(·)(εv(·))^{\frac{1}{r}}} ∥_{(Φ \frac{1}{r} ; 1 )^{∗},Q,(εv)^{\frac{1}{r}}} )u(x)dx] ≤

≤ Φ_{1}^{−1}[A_{1}|Q|ε^{−1}] = Φ_{1}^{−1}[A_{1}|Q|(\frac{1}{|Q|} ∫_{Q} [Φ_{1}(w_{1}(x)|f(x)|)v(x)]^{\frac{1}{r}} dx)^{r}].

Therefore inequality (11) is satisfied with the constants C_{1} = A_{1}, C_{2} = A_{2}.

Conversely, suppose that inequality (11) is satisfied for some constants C_{1}, C_{2} > 0. Let Q be a cube and ε > 0. By the definition of the Orlicz’s norm, there is a nonnegative function g(·) such that

\frac{1}{|Q|} ∫_{Q} [Φ_{1}(w_{1}(x)g(x))εv(x)]^{\frac{1}{r}} dx ≤ 1

and therefore

∥\frac{1}{w_{1}(·)(εv(·))^{\frac{1}{r}}} ∥_{(Φ \frac{1}{r} ; 1 )^{∗},Q,(εv)^{\frac{1}{r}}} = \frac{1}{|Q|} ∫_{Q} g(y)dy.

Now using inequality (11) with such a function g(·) we obtain

M = Φ_{2}^{−1}[ ∫_{Q} Φ_{2}(C_{2}w_{2}(x)|Q|^{\frac{s}{n}} ∥\frac{1}{w_{1}(·)(εv(·))^{\frac{1}{r}}} ∥_{(Φ \frac{1}{r} ; 1 )^{∗},Q,(εv)^{\frac{1}{r}}} )u(x)dx] ≤

≤ Φ_{2}^{−1}[ ∫_{Q} Φ_{2}(C_{2}|Q|^{\frac{s}{n}} [\frac{1}{|Q|} ∫_{Q} g(y)dy]w_{2})u(x)dx] ≤

≤ Φ_{1}^{−1}[C_{1}|Q|(\frac{1}{|Q|} ∫_{Q} [Φ_{1}(w_{1}(x)g(x))v(x)]^{\frac{1}{r}} dx)^{r}] ≤ Φ_{1}^{−1}[C_{1}|Q|ε^{−1}].

Thus (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) with the constants A_{1} = C_{1}, A_{2} = C_{2}.

§ 4. Proof of Theorem 2

Assume that Φ_{1}(·) is an N-function, and the weak inequality M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2},∞}(w_{2}) is satisfied with the constants C_{1}, C_{2} > 0. Let Q be a cube, and f(·) be a locally integrable function whose support is in Q. Since (M_{s}f)(x) ≥ |Q|^{\frac{s}{n}} (\frac{1}{|Q|} ∫_{Q} |f(y)|dy)1I_{Q}(x), by taking the real λ = |Q|^{\frac{s}{n}} (\frac{1}{|Q|} ∫_{Q} |f(y)|dy) in the above weak inequality we get inequality (11) with the constants C_{1} and C_{2}. Therefore by the first part of Lemma 7, (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) with the constants A_{1} = C_{1}, A_{2} = C_{2}.

To prove the converse, we follow the same lines we used in [12]. Thus, suppose (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) with some constants A_{1}, A_{2} > 0. Let N be a nonnegative integer and M_{s}^{N} the truncated maximal operator defined by

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

Q is a cube with Q ∋ x and |Q| ≤ N}.

Let λ > 0, f(·) be a locally integrable function and Ω_{λ,N} = {(M_{s}^{N}f)(·) > λ}. Since ∫_{{(M_{s}f)(x)>λ}} u(x)dx = lim_{N→∞} ∫_{Ω_{λ,N}} u(x)dx, it is sufficient to prove

∫_{Ω_{λ,N}} Φ_{2}(C_{2}λw_{2}(x))u(x)dx ≤ Φ_{2}Φ_{1}^{−1}[C_{1} ∫_{ℝ^{n}} Φ_{1}(w_{1}(x)|f(x)|)v(x)dx],

where C_{1}, C_{2}, do not depend on the integer N.

For every x ∈ Ω_{λ,N}, there is a cube Q(x) centered at x such that

|Q(x)|^{\frac{s}{n}} (\frac{1}{|Q(x)|} ∫_{Q(x)} |f(y)|dy) > cλ,

where c = c(s, n) > 0 depends only on s and n. Since sup{|Q(x)|; x ∈ Ω_{λ,N}} < ∞, by the classical Besicovitch covering theorem we can choose from the set {Q(x); x ∈ Ω_{λ,N}} a sequence of cubes (Q_{k})_{k} satisfying

Ω_{λ,N} ⊂ ∪_{k}Q_{k}; ∑_{k} 1I_{Q_{k}} ≤ K(n)1I_{∪_{k}Q_{k}}; |Q_{k}|^{\frac{s}{n}} (\frac{1}{|Q_{k}|} ∫_{Q_{k}} |f(y)|dy) > cλ.

By the second part of Lemma 7, the condition (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) implies inequality (11). Using the latter and the fact that (Φ_{1}Φ_{2}^{−1})(·) is subadditive, we obtain

∫_{Ω_{λ,N}} Φ_{2}(cA_{2}λw_{2}(x))u(x)dx ≤ ∑_{k} ∫_{Q_{k}} Φ_{2}(cA_{2}λw_{2}(x))u(x)dx ≤

≤ ∑_{k} ∫_{Q_{k}} Φ_{2}(A_{2}w_{2}(x)|Q_{k}|^{\frac{s}{n}} [\frac{1}{|Q_{k}|} ∫_{|Q_{k}|} |f(y)|dy])u(x)dx ≤

≤ ∑_{k} Φ_{2}Φ_{1}^{−1}[A_{1} ∫_{Q_{k}} Φ_{1}(w_{1}(x)|f(x)|)v(x)dx] ≤

≤ Φ_{2}Φ_{1}^{−1}[A_{1} ∑_{k} ∫_{Q_{k}} Φ_{1}(w_{1}(x)|f(x)|)v(x)dx] ≤

≤ Φ_{2}Φ_{1}^{−1}[C(n)A_{1} ∫_{R^{n}} Φ_{1}(w_{1}(x)|f(x)|)v(x)dx],

and therefore M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2},∞}(w_{2}) holds with the constants C_{1} = C(n)A_{1}, C_{2} = c(s, n)A_{2}.

§ 5. Proofs of Theorem 4 and Proposition 5

Proof of the necessity part of Theorem 4. Assume that Φ_{1} is an N-function and suppose the strong inequality M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2}}(w_{2}) is satisfied for some constants C_{1}, C_{2} > 0. This implies the weak inequality M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2},∞}(w_{2}) with the same constants C_{1}, C_{2} > 0. Thus, as in the proof of Theorem 2, (u, v) ∈ A(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) with the constants A_{1} = C_{1}, A_{2} = C_{2}.

To prove the converse, to get M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2}}(w_{2}), we first do some preliminaries so as to discretize the operator M_{s}.

Preliminaries for the proof of sufficiency part of Theorem 4. Recall that the dyadic version M_{s}^{d} of the maximal operator M_{s} is defined as

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

We need

Lemma 8. Let f(·) be a function with a compact support and λ > 0. If {(M_{s}f)(·) > λ} is a nonempty set then we can find a family of nonoverlapping maximal dyadic cubes (Q_{j})_{j} for which

\{(M_{s}f)(·) > λ\} ⊂ ∪_{j}(3Q_{j}), 2^{s−2n}λ < |Q_{j}|^{\frac{s}{n}} (\frac{1}{|Q_{j}|} ∫_{Q_{j}} |f(y)|dy) ≤ 2^{−n}λ,

{(M_{s}^{d}f)(·) > 2^{s−2n}λ} = ∪_{j}Q_{j}.

This lemma is the standard one and was proved in [5]. As a consequence we obtain

Lemma 9. Let f(·) be a function with a compact support. For each integer k let

Ω_{k} = {(M_{s}f)(·) > a^{k}}, Γ_{k} = {(M_{s}^{d}f)(·) > 2^{−2n}a^{k}}, where a = 2^{−3n}.

There is a family of maximal nonoverlapping dyadic cubes (Q_{kj})_{j} for which

\begin{matrix}Ω_{k} ⊂ ∪_{j}(3Q_{kj}), Γ_{k} = ∪_{j}Q_{kj}, \\ a^{k} < |Q_{kj}|^{\frac{s}{n}} (\frac{1}{|Q_{kj}|} ∫_{Q_{kj}} |f(y)|dy) ≤ 2^{n−s}a^{k}. \end{matrix}  (12)

Let E_{k,j} = Q_{kj}\(Q_{kj} ∩ Γ_{k+1}). Then (E_{k,j})_{k,j} is a disjoint family of sets for which

|Q_{k,j}| < c|E_{k,j}| with c = c(s, n) = \frac{2^{2(n+s)}}{2^{2(n+s)} − 1}.  (13)

The first part of this result is an immediate consequence of Lemma 8, for λ = 2^{2n−s}a^{k}, while the second one follows from the estimate

|Q_{k,j} ∩ Γ_{k+1}| < 2^{−2(n+s)}|Q_{k,j}|.

The latter one can be obtained from (1) as follows,

|Q_{k,j} ∩ Γ_{k+1}| = ∑_{i} |Q_{k,j} ∩ Q_{k+1,i}| ≤ ∑_{i;Q_{k+1,i}⊂Q_{k,j}} |Q_{k+1,i}| ≤

≤ \frac{1}{a^{k+1}} ∑_{i;Q_{k+1,i}⊂Q_{k,j}} |Q_{k+1,i}|^{\frac{s}{n}} ∫_{Q_{k+1,i}} |f(y)|dy ≤

≤ \frac{2^{−s}}{a^{k+1}} ∑_{i;Q_{k+1,i}⊂Q_{k,j}} |Q_{k,j}|^{\frac{s}{n}} ∫_{Q_{k+1,i}} |f(y)|dy ≤

≤ \frac{2^{−s}}{a^{k+1}} |Q_{k,j}|^{\frac{s}{n}} ∫_{Q_{k,j}} |f(y)|dy ≤ \frac{2^{n−2s}}{a} |Q_{k,j}| = 2^{−2(n+s)}|Q_{k,j}|.

Now we can proceed to

Proof of the sufficiency part of Theorem 4. We assume that Φ1^{\frac{1}{r}} (·) is an N-function for some r > 1 and (u, v) ∈ A_{r}(s, Φ_{1}, Φ_{2}, w_{1}, w_{2}) for some constants A_{1}, A_{2} > 0. We have to prove M_{s} : L_{v}^{Φ_{1}}(w_{1}) → L_{u}^{Φ_{2}}(w_{2}), i.e.,

∫_{ℝ^{n}} Φ_{2}(C_{2}w_{2}(x)(M_{s}f)(x))u(x)dx ≤ Φ_{2}Φ_{1}^{−1}[C_{1} ∫_{ℝ^{n}} Φ_{1}(w_{1}(x)|f(x)|)]v(x)dx

for all locally integrable functions f(·).

We assume that f(·) is bounded and it has a compact support. We do not lose generality, since the estimates we obtain do not depend on the bound of f(·), and the monotone convergence theorem yields the conclusion. With the notations of Lemma 9, we have

S = ∫_{ℝ^{n}} Φ_{2}(3^{−n}2^{−3n}A_{2}w_{2}(x)(M_{s}f)(x))u(x)dx =

= ∑_{k} ∫_{Ω_{k}−Ω_{k+1}} Φ_{2}(3^{−n}2^{−3n}A_{2}w_{2}(x)(M_{s}f)(x))u(x)dx ≤

≤ ∑_{k,j} ∫_{3Q_{k,j}} Φ_{2}(3^{−n}2^{−3n}A_{2}a^{k+1}w_{2}(x))u(x)dx ≤

≤ ∑_{k,j} ∫_{3Q_{k,j}} Φ_{2}(3^{−n}A_{2}w_{2}(x)|Q_{k,j}|^{\frac{s}{n}} [\frac{1}{|Q_{k,j}|} ∫_{Q_{k,j}} |f(y)|dy])u(x)dx ≤

INTEGRAL INEQUALITIES FOR MAXIMAL OPERATORS 597

(see estimate (12) in Lemma 9)

≤ ∑_{k,j} ∫_{3Q_{k,j}} Φ_{2}(A_{2}w_{2}(x)|3Q_{k,j}|^{\frac{s}{n}} [\frac{1}{|3Q_{k,j}|} ∫_{3Q_{k,j}} |f(y)|dy])u(x)dx ≤

≤ ∑_{k,j} Φ_{2}Φ_{1}^{−1}[A_{1}|3Q_{k,j}|(\frac{1}{|3Q_{k,j}|} ∫_{3Q_{k,j}} [Φ_{1}(w_{1}(x)|f(x)|)v(x)]^{\frac{1}{r}} dx)^{r}] ≤

(by the first part of Lemma 7)

≤ Φ_{2}Φ_{1}^{−1}[A_{1} ∑_{k,j} |3Q_{k,j}|(\frac{1}{|3Q_{k,j}|} ∫_{3Q_{k,j}} [Φ_{1}(w_{1}(x)|f(x)|)v(x)]^{\frac{1}{r}} dx)^{r}] ≤

(recall that Φ_{1}^{−1}Φ_{2} is subadditive)

= Φ_{2}Φ_{1}^{−1}[3^{n}A_{1} ∑_{k,j} |Q_{k,j}|(\frac{1}{|3Q_{k,j}|} ∫_{3Q_{k,j}} [Φ_{1}(w_{1}(x)|f(x)|)v(x)]^{\frac{1}{r}} dx)^{r}] ≤

≤ Φ_{2}Φ_{1}^{−1}[3^{n}c(s, n)A_{1} ∑_{k,j} |E_{k,j}| ×

× (\frac{1}{|3Q_{k,j}|} ∫_{3Q_{k,j}} [Φ_{1}(w_{1}(x)|f(x)|)v(x)]^{\frac{1}{r}} dx)^{r}] ≤

(see estimate (13) in Lemma 9)

≤ Φ_{2}Φ_{1}^{−1}[3^{n}c(s, n)A_{1} ∑_{k,j} ∫_{E_{k,j}} (M[Φ_{1}(w_{1}|f|)v]^{\frac{1}{r}} )^{r}(x)dx] ≤

(M = M_{0} is the Hardy–Littlewood maximal operator)

≤ Φ_{2}Φ_{1}^{−1}[3^{n}c(s, n)A_{1} ∫_{ℝ^{n}} (M[Φ_{1}(w_{1}|f|)v]^{\frac{1}{r}} )^{r}(x)dx] ≤

(E_{k,j}’s are disjoint sets)

≤ Φ_{2}Φ_{1}^{−1}[3^{n}c(s, n)A_{1} ∫_{ℝ^{n}} Φ_{1}(w_{1}(x)|f(x)|)v(x)dx]

(by the well known maximal theorem (r > 1)). Thus the inequality is fulfilled with the constants C_{1} = 3^{n} \frac{2(n+s)}{2^{2(n+s)}−1}A_{1}, C_{2} = 2^{−3n}3^{−n}A_{2}.

Proof of proposition 5. Since this result can be obtained by using a few changes in the above estimates, we will outline the essential arguments. With the above notation, the condition ω(·) ∈ A_{∞} implies the existence of a constant c = c(ω) > 0 such that

|E_{kj}|_{ω} ≤ c|Q_{kj}|_{ω}.

Now, as above, by Lemma 7 (recall that (u, v) ∈ A_{1}(s, Φ_{1}, Φ_{2})) we obtain

S = ∫_{ℝ^{n}} Φ_{2}(c_{2}A_{2}(M_{s}f)(x))u(x)dx ≤

≤ Φ_{2}Φ_{1}^{−1}[c_{1}A_{1}∑_{k,j}|E_{kj}|_{ω}(\frac{1}{|3Q_{kj}|_{ω}} ∫_{3Q_{k,j}} [Φ_{1}(|f(x)|)v(x)\frac{1}{ω(x)}]ω(x)dx)] ≤

≤ Φ_{2}Φ_{1}^{−1}[c_{1}A_{1}∑_{k,j}|E_{kj}|_{ω}(\frac{1}{|3Q_{kj}|_{ω}} ∫_{3Q_{k,j}} [Φ_{1}(|f(x)|)v(x)\frac{1}{ω(x)}]^{\frac{1}{r}} ω(x)dx)^{r}] ≤

≤ Φ_{2}Φ_{1}^{−1}[cc_{1}A_{1} ∑_{k,j} ∫_{E_{kj}} (M_{ω}[Φ_{1}(|f|)v \frac{1}{ω} ]^{\frac{1}{r}} )^{r}(x)ω(x)dx] ≤

≤ Φ_{2}Φ_{1}^{−1}[cc_{1}A_{1} ∫_{R^{n}} ([M_{ω}[Φ_{1}(|f|)v \frac{1}{ω} ]^{\frac{1}{r}} )^{r}(x)ω(x)dx] ≤

≤ Φ_{2}Φ_{1}^{−1}[cc′_{1}A_{1} ∫_{R^{n}} Φ_{1}(|f(x)|)v(x)dx].

§ 6. Proof of Proposition 6

Let ε > 0 and Q be a cube centered at x_{0} and having a side with length R > 0.

First we suppose |x_{0}| ≤ 2R. Then for a constant c_{1} ≥ 1 which depends only on the dimension n we obtain Q ⊂ B(0, c_{1}R), which results in

A[Q, ε, A_{1}, A_{2}] ≤ (c_{1}^{n})\frac{1}{(c_{1}R)^{n}} ∫_{B(0,c_{1}R)} (Φ^{\frac{1}{r}})^{∗} ×

× [(c_{1}R)^{s} \frac{(c_{1})^{−s}A_{2}}{Φ^{−1}[(c_{1})^{−n}A_{1} _{\frac{1}{\frac{1}{(c_{1}R)^{n}} ∫_{B(0,c_{1}R)}εu(y)dy}}]} \frac{1}{w_{1}(x)(εv(x))^{\frac{1}{r}}} ](εv(x))^{\frac{1}{r}} dx.

Next we consider the case 2R < |x_{0}|. Then |x| ≈ |x_{0}| whenever x ∈ Q, and Q ⊂ B(0, c_{3}|x_{0}|) with a constant c_{3} ≥ 1 which depends only on n. Thus, the growth condition on u(·) yields

\frac{1}{|Q|} ∫_{Q} u(y)dy ≤ \frac{c_{4}(n, u)}{(c_{3}|x_{0}|)^{n}} ∫_{B(0,c_{3}|x_{0}|)} u(y)dy.

Since the family of weights (Φ^{\frac{1}{r}})^{∗}(λ_{\frac{1}{w_{1}(·)[v(·)]^{\frac{1}{r}}}} )[v(·)]^{\frac{1}{r}} satisfies uniformly

(in λ) the growth condition (C), we obtain

A[Q, ε, A_{1}, A_{2}] ≤ \frac{1}{|Q|} ∫_{Q} (Φ^{\frac{1}{r}})^{∗} ×

×[(c_{3}|x_{0}|)^{s}\frac{c′_{2}A_{2}}{Φ^{−1}[c′_{1}A_{1} _{\frac{1}{\frac{1}{(c_{3}|x_{0}|)^{n}} ∫_{B(0,c_{3}|x_{0}|)} εu(y)dy}}]} \frac{1}{w_{1}(x)(εv(x))^{\frac{1}{r}}} ](εv(x))^{\frac{1}{r}} dx≤

≤ sup_{x∈Q}(Φ^{\frac{1}{r}})^{∗} ×

×[(c_{3}|x_{0}|)^{s}\frac{c′_{2}A_{2}}{Φ^{−1}[c′_{1}A_{1} _{\frac{1}{\frac{1}{(c_{3}|x_{0}|)^{n}} ∫_{B(0,c_{3}|x_{0}|)} εu(y)dy}}]} \frac{1}{w_{1}(x)(εv(x))^{\frac{1}{r}}} ](εv(x))^{\frac{1}{r}}≤

≤ c_{5} \frac{1}{(c_{3}|x_{0}|)^{n}} ∫_{B(0,c_{3}|x_{0}|)} (Φ^{\frac{1}{r}})^{∗} ×

×[(c_{3}|x_{0}|)^{s}\frac{c′_{2}A_{2}}{Φ^{−1}[c′_{1}A_{1} _{\frac{1}{\frac{1}{(c_{3}|x_{0}|)^{n}} ∫_{B(0,c_{3}|x_{0}|)} εu(y)dy}}]} \frac{1}{w_{1}(x)(εv(x))^{\frac{1}{r}}} ](εv(x))^{\frac{1}{r}} dx.

Finally, by these estimates we get

sup_{ε>0} sup_{Q} A[Q, ε, A_{1}, A_{2}] ≤ c_{0} sup_{ε>0} sup_{R>0} A[B(0, R), ε, c′′_{1}A_{1}, c′′_{2}A_{2}].

References

1. J. Musielak, Orlicz spaces and modular spaces. Lecture Notes in Math. 1034, Springer, Berlin etc., 1983.

2. V. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and Orlicz spaces. World Scientific, Singapore, 1991.

3. M. Schechter, Potential estimates in Orlicz spaces. Pacific J. Math. 133(2)(1988), 381–389.

4. E. Sawyer, A characterization of a two weight norm inequality for maximal operators. Studia Math. 75(1982), 1–11.

5. L. Qinsheng, Two weight mixed Φ-inequalities for the Hardy operator and the Hardy–Littlewood maximal operator. J. London Math. Soc. 46(2)(1992), 301–318.

6. Y. Rakotondratsimba, Inégalités à poids pour des opérateurs maximaux et des opérateurs de type potentiel. Thèse de Doctorat. Univ. Orléans, France, 1991.

7. R. Kerman and A. Torchinsky, Integral inequality with weights for the Hardy maximal function. Studia Math. 71(1981-82), 272–284.

8. J. C. Chen, Weights and L_{Φ}-boundedness of the Poisson integral operator. Israel J. Math. 81(1993), 193–202.

9. Q. Sun, Weighted norm inequalities on spaces of homogeneous type. Studia Math. 101(3)(1992), 241–251.

10.Y. Rakotondratsimba, Weighted strong inequalities for maximal functions in Orlicz spaces. Preprint Univ. Orléans, France, 1992.

11. C. Perez, Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43(1994), 31–45.

12. Y. Rakotondratsimba, Weighted weak inequalities for some maximal functions in Orlicz spaces. Preprint Univ. Orléans, France, 1991.

13. L. Pick, Two-weight weak type maximal inequalities in Orlicz classes. Studia Math. 100(3)(1991), 207–218.

14. Y. Rakotondratsimba, Weighted weak inequalities for fractional integrals in Orlicz spaces. Preprint Univ. Orléans, France, 1991.

(Received 20.12.1994)

Author’s address:

Uneversité d’Orléans Departement de Mathematiques U.F.R. Faculté des Sciences B.P. 6759 45067 Orléans cedex 2 France

Current address:

Centre Polytecnique St-Louis Ecole de Physique et de Mathematiques Industrielles 13, boulevard de l’Hautil 95 092 Cergy-Pontoise cedex France