GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 2, 1998, 177-200

ON THE BOUNDEDNESS OF CLASSICAL OPERATORS ON WEIGHTED LORENTZ SPACES

Y. RAKOTONDRATSIMBA

Abstract. Conditions on weights u(·), v(·) are given so that a classical operator T sends the weighted Lorentz space L^{rs}(vdx) into L^{pq}(udx). Here T is either a fractional maximal operator M_{α} or a fractional integral operator I_{α} or a Calderón–Zygmund operator. A characterization of this boundedness is obtained for M_{α} and I_{α} when the weights have some usual properties and max(r, s) ≤ min(p, q).

§ 0. Introduction

Let u(·), v(·), w_{1}(·), w_{2}(·) be weight functions on ℝ^{n}, n ∈ ℕ^{∗}, i.e., nonnegative locally integrable functions; and let T be a classical operator. The purpose of this paper is to determine when T is bounded from the weighted Lorentz space L_{v}^{rs}(w_{1}) into L_{u}^{pq}(w_{2}), i.e.,

∥w_{2}(·)(T f)(·)∥_{L_{u}^{pq}} ≤ C∥w_{1}(·) f(·)∥_{L_{v}^{rs}} for all functions f(·). (0.0)

Here C > 0 is a constant which depends only on n, p, q, r, s, and on the weight functions. Recall that

∥g(·)∥_{L_{u}^{pq}}^{q} = q ∫_{0}^{∞}( ∫_{\{y∈ℝ^{n}; |g(y)|>λ\}} u(y)dy)^{\frac{q}{p}} λ^{q−1}dλ,

for 1 ≤ p < ∞ and 1 ≤ q < ∞; and

∥g(·)∥_{L_{u}^{p∞}} = sup_{λ>0} λ( ∫_{\{y∈ℝ^{n}; |g(y)|>λ\}} u(y)dy)^{\frac{1}{p}}

177 for 1 ≤ p < ∞. It is always assumed that 1 < r, s, p, q < ∞. For convenience, the embedding defined by (0.0) will be denoted by T : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}).

1991 Mathematics Subject Classification. 42B25, 42B20.

Key words and phrases. Maximal operators, fractional integral, Calderón–Zygmund operators, weighted Lorentz spaces.

1072-947X/98/0300-0177$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

The classical operator under consideration is a fractional maximal operator or a fractional integral operator or a Calderón–Zygmund operator. The fractional maximal operator M_{α} of order α, 0 ≤ α < n, is defined as

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

Here Q is a cube with sides parallel to the coordinate planes. Thus M = M_{0} is the well-known Hardy–Littlewood maximal operator. The fractional integral operator I_{α}, 0 < α < n, is given by

(I_{α}f)(x) = ∫_{ℝ^{n}} |x − y|^{α−n}f(y)dy.

The Hilbert transform

(Hf)(x) = P.V. ∫_{ℝ^{1}} \frac{f(y)}{x − y} dy = lim_{ε→0} ∫_{|x−y|>ε} \frac{f(y)}{x − y} dy

is a particular case of the Calderón–Zygmund operator.

The boundedness M : L_{v}^{rs}(1) → L_{u}^{pq}(1) was considered and studied by many authors (see, for instance, [1], [2] and the references therein). However, as mentioned by Kokilashvili and Krbec [1], easy necessary and sufficient conditions on v(·), u(·) for which M_{α} : L_{v}^{rs}(1) → L_{u}^{pq}(1), 0 ≤ α < n, are not known. In this paper we find a sufficient condition for such a boundedness. For weight functions having some special properties (generally shared by usual weights), the condition found here is also a necessary one. One of the reasons which lead to considering M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) is the fact that weights cannot be combined as in the Lebesgue case where, for instance, ∥f(·)∥_{L_{u}^{pp}} = ∥u^{\frac{1}{p}} (·)f(·)∥_{L_{1}^{pp}}.

A weight function w(·) is constant on annuli if for a constant c > 0

sup_{R<|y|≤64R} w(y) ≤ c inf_{R<|z|≤64R} w(z) for all R > 0.

This latter condition can be denoted by w(·) ∈ A. If w(x) = |x|^{α} ln^{β}(e + |x|), with α ∈ ℝ and β ≥ 0, then w(·) ∈ A. A large class of weight functions w(·) for which w(·) ∈ A is given by those nondecreasing (resp. nonincreasing) radial w(·) which satisfy w(64t) ≤ Cw(t) (resp. w(t) ≤ Cw(64t)) for all t > 0. In the proof of Lemma 1 below, it is observed that M_{α} : L_{v}^{rs}(1) → L_{u}^{pq}(1) implies necessarily H : L_{v}^{rs}(1) → L_{u}^{pq}(w) with w(x) = |x|^{α−n} and (Hf)(x) = ∫_{|y|<|x|} f(y)dy. In view of this observation and also for convenience, it is always supposed that w_{1}(·) ∈ A and w_{2}(·) ∈ A.

As in Lemma 1 below, the boundedness M_{α} : L_{v}^{rs}(1) → L_{u}^{pq}(1) implies

∥(R + |cdot|)^{α−n}∥_{L_{u}^{pq}} ∥\frac{1}{v(·)}11_{|·|<R}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} ≤ C for all R > 0(0.1)

and

|x|^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}(u(x))^{\frac{1}{p}} ≤ c(v(x))^{\frac{1}{r}} for almost every x  (0.2)

for \frac{α}{n} + \frac{1}{p} − \frac{1}{r} = 0. Here 11_{E}(·) denotes the characteristic function of the measurable set E. Since (0.2) is a pointwise inequality, this condition can be easily checked for given weights u(·) and v(·). Contrary to the well-known standard conditions (see [2], [1]), (0.1) is expressed neither in terms of the operator M_{α} itself nor in terms of arbitrary cubes. This test condition needs only integrations on balls centered at the origin, which are well adapted for radial weight functions (the most useful weights in applications). Consequently, our idea is to derive M_{α} : L_{v}^{rs}(1) → L_{u}^{pq}(1) from conditions (0.1) and (0.2). However, when testing the problem in classical Lebesgue spaces, it is not reasonable to expect that the above embedding can be obtained only from these two conditions.

Roughly speaking, for max(r, s) ≤ min(p, q) we will prove that M_{α} : L_{v}^{rs}(1) → L_{u}^{pq}(1) whenever both (0.1) and a more stronger condition than (0.2) are satisfied (see Theorem 2). Eventually, for the Lebesgue case (i.e., r = s, p = q) the results we find are new. It is also of interest to note that the conditions used to get (0.0) are suitable for explicit computations.

The main results are presented in §1. The basic lemmas needed to prove them are given in §2. These latter are proved in §3. The final §5 is devoted to the proofs of the basic lemmas given in §2.

§ 1. The Results

Recall that our purpose is to study T : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}), i.e.,

∥w_{2}(·)(T f)(·)∥_{L_{u}^{pq}} ≤ C∥w_{1}(·)f(·)∥_{L_{v}^{rs}} for all functions f(·),

where T is a classical operator defined as above and w_{1}(·), w_{2}(·) ∈ A. Indeed, in considering this boundedness, restrictions on the range of r, s, p, q, and on the weight functions have to be done. To simplify the statement, consider the case of T = M_{α}.

Lemma 1. Let 0 ≤ α < n. Assume the embedding M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) is satisfied. Then

|Q|^{\frac{α}{n}} ∥w_{2}(·)11_{Q}(·)∥_{L_{u}^{pq}} ≤ C_{1}∥w_{1}(·)11_{Q}(·)∥_{L_{v}^{rs}} for all cubes Q. (1.1)

Consequently if w_{1}(·) = w_{2}(·) = 1, then \frac{1}{r} − \frac{1}{p} ≤ \frac{α}{n}. On the other hand,

∥\frac{1}{v(·)w_{1}(·)}11_{Q}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} < ∞ for all cubes Q.  (1.2)

The weight functions u(·), v(·) satisfy the Wheeden–Muckenhoupt condition

∥w_{2}(·) (R + | · |)^{α−n}∥_{L_{u}^{pq}} ∥\frac{1}{v(·)w_{1}(·)} 11_{\{|·|<R\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} ≤ C_{2} (1.3)

for all R > 0.

Let 1 < r < \frac{n}{α} and \frac{1}{r^{∗}} = \frac{1}{r} − \frac{α}{n}. If p = r^{∗} then

w_{2}(x) |x|^{n[ \frac{1}{p} − \frac{1}{r^{∗}}]} (u(x))^{\frac{1}{p}} ≤ c w_{1}(x) (v(x))^{\frac{1}{r}} for almost every x. (1.4)

This inequality is also satisfied for p ≠ r^{∗} if both u(·), v(·) ∈ A.

In view of this result, it will always be assumed that

v(·) and w_{1}(·) satisfy (1.2).

So by (1.1), the study of M_{α} : L_{v}^{rs}(1) → L_{u}^{pq}(1) for 0 ≤ α < n and 1 < r < \frac{n}{α} makes non-trivial sense only for the range p ≤ r^{∗}. For this reason and also for technical motivation it will be supposed that

1 < r < \frac{n}{α}, p ≤ r^{∗}, max(r, s) ≤ min(p, q).  (1.5)

In dealing with M_{α} when max(r, s) = p < q, it is useful to assume that

∑_{m=−∞}^{N−1} ∥ w_{2}(·)11_{{2^{m}<|·|≤2^{m+1}}}(·)∥_{L_{u}^{pq}}^{p} ≤ C ∥ w_{2}(·)11_{{|x|<2^{N} }}(·)∥_{L_{u}^{pq}}^{p} (1.6)

for all N ∈ ℤ. Such an inequality is always satisfied when q ≤ p (see Lemma 2 in §2). For the range p < q, (1.6) is true for some weight functions as in the case of w_{2}(·) = 1, or for power weights (see Proposition 8 below).

A stronger condition than (1.4) is

w_{2}(x) |x|^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]} ( sup_{4^{−1}|x|<|z|<4|x|} u(z))^{\frac{1}{p}} ≤

≤ c w_{1}(x) (v(x))^{\frac{1}{r}} for a.e. x,  (1.7)

or

w_{2}(x) |x|^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]} (u(x))^{\frac{1}{p}} ( sup_{4^{−1}|x|<|z|<4|x|} v(z))^{− \frac{1}{r}} ≤

≤ c w_{1}(x) for a.e. x,  (1.7′)

We are now in the position to state our first main result for the fractional maximal operator.

Theorem 2. (The fractional maximal operator M_{α} with 0 ≤ α < n) (A) Suppose M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}). Then the Wheeden–Muckenhoupt

condition (1.3) is satisfied.

(B) For the converse assume restrictions (1.5) and (1.6) hold. Then condition (1.3) implies M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) whenever the pointwise inequality (1.7) (or (1.7′)) is satisfied.

Remarks 3. (1) For the Hardy–Littlewood maximal operator M = M_{0}, this results deals with the embedding M : L_{v}^{rs}(w_{1}) → L_{u}^{rq}(w_{2}), since by restriction (1.5) we have α = 0, r^{∗} = r = p, and s ≤ r = p < q. For M_{α} with 0 < α < n, and in the Lebesgue case, i.e., p = q and r = s, restriction (1.5) means r ≤ p ≤ r^{∗}.

(2) Theorem 2 and Lemma 1 yield the following conclusion: With restrictions (1.5) and (1.6), both conditions (1.3) and (1.4) characterize the embedding M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{r^{∗}q}(w_{2}) whenever either u(·) or v(·) is constant on annuli. Similarly, if p ≠ r^{∗} and both u(·) and v(·) are constant on annuli, then M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) if and only if (1.3) and (1.4) are satisfied. Indeed, in the latter result condition (1.4) can be dropped by virtue of Proposition 9 and Remarks 11 below.

(3) The Wheeden–Muckenhoupt condition (1.3) is equivalent both to

R^{α−n}∥w_{2}(·) 11_{\{|·|<R\}}(·)∥_{L_{u}^{pq}} ∥\frac{1}{v(·)w_{1}(·)} 11_{\{|·|<R\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} ≤ A (1.8)

and to

∥w_{2}(·) | · |^{α−n} 11_{\{|·|>R\}}(·)∥_{L_{u}^{pq}} ∥\frac{1}{v(·)w_{1}(·)} 11_{\{|·|<R\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} ≤ H (1.9)

for all R > 0. The latter inequality is also useful to get the boundedness of some Hardy type operators in weighted Lorentz spaces (see Lemma 3).

It is of interest to identify some situations where the extra-condition (1.7) (or (1.7′)) can be obtained from the Wheeden–Muckenhoupt condition (1.3). Such a question will be discussed below.

But for the moment we state the main result for the fractional integral operator.

Theorem 4. (The fractional integral operator I_{α} with 0 < α < n)

(A) Suppose I_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}). Then the Hardy condition (1.9) is satisfied and so is its dual version

∥\frac{1}{v(·)w_{1}(·)} | · |^{α−n}11_{|·|>R}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} ∥w_{2}(·) 11_{\{|·|<R\}}(·)∥_{L_{u}^{pq}} ≤ H^{∗}. (1.9^{∗})

(B) For the converse, assume restriction (1.5) holds. Then both conditions (1.9) and (1.9^{∗}) imply I_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) whenever the pointwise inequality (1.7) (or (1.7′)) is satisfied.

As in Remark 3(2), by Theorem 4 and Lemma 1 we see that, with restriction (1.5), both conditions (1.9), (1.9^{∗}) and (1.4) characterize the embedding I_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{r^{∗}q}(w_{2}) whenever either u(·) or v(·) is constant on annuli.

Next, the weighted inequalities for Calderón–Zygmund operators T are considered. Each T is a linear operator which sends C_{c}^{∞}(ℝ^{n}) into L_{loc}^{1}(ℝ^{n},dx), is bounded on L^{2}(ℝ^{n}, dx), and has the representation

(T f)(x) = ∫_{ℝ^{n}} K(x, y)f(y)dy a.e. x ∉ supp f

for every f ∈ L_{c}^{∞}(ℝ^{n}). The kernel K(x, y) is a continuous function defined on {(x, y) ∈ ℝ^{n} × ℝ^{n}; x ≠ y} and satisfying the standard estimates

|K(x, y)| ≤ C|x − y|^{−n} for all x ≠ y,

|K(x, y) − K(x′, y)| + |K(y, x) − K(y, x′)| ≤ C(\frac{|x−x′|}{|x−y|})^{ϵ}|x − y|^{−n} whenever 2|x − x′| ≤ |x − y|. Here C > 0 and ϵ ∈]0, 1] are fixed constants. These operators were introduced by Coifman and Meyer in [3] and were known to be bounded on each space L^{p} for 1 < p < ∞.

Now we are in the position to state the sufficient conditions for these operators to be bounded on weighted Lorentz spaces.

Theorem 5. (The Calderon–Zygmund operator T)

Let s ≤ r ≤ q. Then conditions (1.9) and (1.9^{∗}) (with α = 0, p = r) imply T : L_{v}^{rs}(w_{1}) → L_{u}^{rq}(w_{2}) whenever the pointwise inequality (1.7) is satisfied.

For the Hilbert transform some of the above conditions become also necessary.

Proposition 6. (The Hilbert transform H)

Suppose H : L_{v}^{rs}(w_{1}) → L_{u}^{rq}(w_{2}). Then conditions (1.9), (1.9^{∗}), and (1.4) (with α = 0, r^{∗} = r = p) are satisfied.

Next we deal with a result which yields cases where the Muckenhoupt condition (1.8), with α ≥ 0, implies the Hardy inequality (1.9). For this purpose, some weight conditions are needed. Thus v(·) ∈ RD_{ν,r,s}(w_{1}), ν > 0, when for a constant c > 0

∥\frac{1}{v(·)w_{1}(·)}11_{\{|·|<λR\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} ≤cλ^{nν(1− \frac{1}{r})}∥\frac{1}{v(·)w_{1}(·)}11_{\{|x|<R\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}}

for all 0 < λ ≤ 1 and R > 0. Similarly, u(·) ∈ D_{ε,p,q}(w_{2}), ε ≥ 1, when

∥ w_{2}(·) 11_{\{|·|<λR\}}(·)∥_{L_{u}^{pq}} ≤ c λ^{nε \frac{1}{p}} ∥ w_{2}(·) 11_{\{|·|<R\}}(·)∥_{L_{u}^{pq}}

for all λ ≥ 1 and R > 0.

Proposition 7. The Muckenhoupt condition (1.8), with 0 ≤ α < n, implies the Hardy condition (1.9) whenever v(·) ∈ RD_{ν,r,s}(w_{1}) for some ν > 0. This implication is also true if u(·) ∈ D_{ε,p,q}(w_{2}) for 1 ≤ ε < (1− \frac{α}{n})p.

After some tedious computations, we obtain

Proposition 8. Let w(x) = |x|^{β−n}, w_{1}(x) = |x|^{β_{1}−n}, w_{2}(x) = |x|^{β_{2}−n}, u(x) = |x|^{γ−n}, v(x) = |x|^{δ−n} where β, β_{1}, β_{2}, γ, δ are nonnegative reals.

(A) If 0 < (β − n) + \frac{1}{p}γ, then

∥w(·)11_{|·|<R}(·)∥_{L_{u}^{pq}} ≈ R^{(β−n)+ \frac{1}{p} γ} ≈ (\frac{1}{R^{n}} ∫_{|y|<R} w(y)dy)(∫_{|y|<R} u(y)dy)^{\frac{1}{p}}

for all R > 0.

(B) The extra-assumption (1.6) is satisfied with w_{2}(·) = w(·).

(C) Let 0 ≤ α < n, 0 < (β_{2} − n) + \frac{1}{p}γ and (β_{1} − n) + \frac{1}{r}δ < n. Then the pointwise inequality (1.4) and the Muckenhoupt condition (1.8) are satisfied if and only if

α + (β_{2} − n) + \frac{1}{p}γ = (β_{1} − n) + \frac{1}{r}δ.  (1.10)

Moreover, since v(·) ∈ RD_{ν,r,s}(w_{1}) with ν = \frac{r}{r−1} \frac{1}{n} [(n − β_{1}) + (n − \frac{1}{r}δ)] > 0, then by Proposition 7 the Wheeden condition (1.3) is equivalent to (1.10).

We will end this section by studying some cases where the pointwise inequality (1.7) becomes a necessary condition for M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{rq}(w_{2}). For this purpose two weight conditions are introduced. Therefore we write that u(·) ∈ H whenever for some C > 0 and N ∈ ℕ^{∗}:

sup_{4^{−1}|x|<|y|<4|x|} u(y) ≤ C|x|^{−n} ∫_{2^{−N} |x|<|y|<2^{N} |x|} u(y)dy,  (1.11)

and v(·) ∈ H̃(r′, s′), r′ = \frac{r}{r−1}, s′ = \frac{s}{s−1}, whenever

|x|^{n}[\frac{1}{v(x)}]^{r′}v(x) ≤ C∥ \frac{1}{v(·)} 11_{2^{−N} |x|<|·|<2^{N} |x|}(·) ∥_{L_{u}^{r′s′}}^{r′} .

Without any difficulty we get

Proposition 9. The Muckenhoupt condition (1.8) implies the pointwise inequality (1.7) whenever u(·) ∈ H and v(·) ∈ H̃(r′, s′).

In this result the condition constant on annuli for w_{1}(·) and w_{2}(·) is taken in the sense that sup_{R<|y|≤2^{2N} R} w(y) ≤ c inf_{R<|z|≤2^{2N} R} w(z) with N ≥ 3. An immediate consequence of Proposition 9 can be stated as

Corollary 10. Let u(·) ∈ H and v(·) ∈ H̃(r′, s′); then

– Condition (1.7) (or (1.7′)) in Theorem 2 can be dropped;

– Condition (1.7) (or (1.7′)) in Theorems 4 and 5 can be replaced by the Muckenhoupt condition (1.8).

Remarks 11. (1) Property (1.11) holds for a large class of weight functions. For instance, w(·) satisfies (1.11) whenever w(·) ∈ A. Condition (1.11) is also true for any radial and monotone weight. But there also exists w(·) not necessarily monotone for which (1.11) is satisfied (take, for instance, w(x) = |x|^{δ−n}11_{|x|<1}(x) + |x|^{γ−n}11_{|x|>1}(x)).

(2) The condition v(·) ∈ H̃(r′, r′) holds if v^{1−r′}(·) satisfies condition (1.11). For general r and s, we have v(·) ∈ H̃(r′, s′) whenever there is C > 0 such that

|x|^{−n} ∫_{2^{−N} |x|<|y|<2^{N} |x|} v(y)dy ≤ cv(x).  (1.12)

Indeed, using the Hölder inequality and (1.12), with C(x, N) = {2^{−N}|x| < |y| < 2^{N}|x|}, we obtain

|x|^{\frac{n}{r′}} [\frac{1}{v(x)}]v^{\frac{1}{r′}} (x) ≈ (∫_{ℝ^{n}} \frac{1}{v(y)}11_{C(x,N)}(y)v(y)dy) × (|x|^{n}v(x))^{− \frac{1}{r}} ≤

≤ c_{1}∥ \frac{1}{v(·)}11_{C(x,N)}(·) ∥_{L_{v}^{r′s′}} (∫_{C(x,N)} v(z)dz)^{\frac{1}{r}} × (|x|^{n}v(x))^{− \frac{1}{r}} ≤

≤ c_{2}∥ \frac{1}{v(·)}11_{C(x,N)}(·) ∥_{L_{v}^{r′s′}} .

Any Muckenhoupt A_{1}-weight function v(·) satisfies condition (1.12). The same is true for v(·) ∈ A.

(3) Theorem 2, Proposition 9 and Remark 11(2) yield the following conclusions: With restrictions (1.5) and (1.6), the Wheeden–Muckenhoupt condition (1.3) characterizes the embedding M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) whenever both u(·) and v(·) are constants on annuli. Similarly, if u(·) and v(·) are constant on annuli, then I_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) if and only if both (1.8), (1.9) and (1.9^{∗}) are satisfied.

§ 2. Basic Lemmas

In this section we prove Lemma 1 and give some basic lemmas needed for the proofs of our results.

Proof of Lemma 1. Assume that M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}), i.e., ∥w_{2}(·)(M_{α}f)(·)∥_{L_{u}^{pq}} ≤ C∥w_{1}(·)f(·)∥_{L_{v}^{rs}} for all functions f(·). Let Q be a

cube and f(·) ≥ 0 with Q as its support. Since |Q|^{\frac{α}{n} −1}(∫_{Q} f(y)dy)11_{Q}(x) ≤ (M_{α}f)(x), therefore

|Q|^{\frac{α}{n} −1}( ∫_{Q} f(y)dy)∥w_{2}(·)11_{Q}(·) ∥_{L_{u}^{pq}} ≤ C∥w_{1}(·)(f11_{Q})(·)∥_{L_{v}^{rs}}. (2.1)

This is the key inequality for the inequalities of this lemma (except for (1.3)).

Taking f(·) = 11_{Q}(·) in (2.1), we obtain (1.1). In particular, for w_{1}(·) =

w_{2}(·) = 1 we have |Q|^{\frac{α}{n} + \frac{1}{p} − \frac{1}{r}} (|Q|^{−1} ∫_{Q} u(y)dy)^{\frac{1}{p}} ≤ C(|Q|^{−1} ∫_{Q} v(y)dy)^{\frac{1}{r}}.

The latter inequality implies \frac{s}{n} + \frac{1}{p} − \frac{1}{r} ≥ 0. Indeed, if this is not the case, then by the Lebesgue differentiation theorem and letting |Q| → 0 we necessarily have u(·) = 0 a.e..

To prove (1.2), suppose the contrary, i.e., ∥\frac{1}{v(·)w_{1}(·)}11_{Q}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} = ∞

for the cube Q. Then there is g(·) ≥ 0 for which ∥g(·)11_{Q}(·)∥_{L_{v}^{rs}} < ∞ and ∞ = ∫_{Q} g(y)(\frac{1}{v(y)w_{1}(y)})v(y)dy = ∫_{Q} g(y)w_{1}^{−1}(y)dy. Consequently inequality (2.1) cannot hold for the function f(·) = g(·)w_{1}^{−1}(·) unless u(·) = 0 a.e. (since the quantity on the right is finite).

The Wheeden–Muckenhoupt condition (1.3) can be derived from an inequality similar to (2.1) which is

( ∫_{|y|<R} f(y)dy)∥(R + | · |)^{α−n} w_{2}(·) ∥_{L_{u}^{pq}}≤ C∥w_{1}(·)f(·)∥_{L_{v}^{rs}}, (2.2)

for each f(·) ≥ 0 and whose support is the ball B = B(0, R) = {y; |y| < R} centered at the origin and with radius R. Inequality (2.2) can be obtained immediately from M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) and

(R + |x|)^{α−n} (∫_{|y|<R} f(y)dy) ≤ c (M_{α}f)(x).

Here c = c(α, n) > 0 depends only on α and n. This inequality is valid, since for |x| ≤ R we have B ⊂ B(x, 2R) and (R+|x|)^{α−n} ∫_{B} f(y)dy ≤ R^{α−n} ∫_{B(x,2R)} f(y)dy ≤ c(M_{α}f)(x), and for R < |x| we obtain B ⊂ B(x, 2|x|) and (R+|x|)^{α−n} ∫_{B} f(y)dy ≤ |x|^{α−n} ∫_{B(x,2|x|)} f(y)dy ≤ c(Mf)(x). Our purpose is to bound the quantity

T = ∥w_{2}(·) (R + | · |)^{α−n}∥_{L_{u}^{pq}} ∥\frac{1}{v(·)w_{1}(·)} 11_{\{|·|<R\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}}

by a constant which does not depend on R > 0. Since it can be assumed that 0 < ∥\frac{1}{v(·)w_{1}(·)} 11_{\{|·|<R\}}∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} < ∞, there is g(·) ≥ 0 such that

∥g(·)11_{|y|<R}(·)∥_{L_{v}^{rs}} ≤ 1 and

∥\frac{1}{v(·)w_{1}(·)}11_{\{|·|<R\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} =

= ∫_{|y|<R} \frac{1}{v(y)w_{1}(y)}g(y)v(y)dy = ∫_{|y|<R} w_{1}^{−1}(y)g(y)dy.

Finally, condition (1.3) appears by taking f(·) = w_{1}^{−1}(·)g(·) in (2.2). Indeed,

T =( ∫_{|y|<R} w_{1}^{−1}(y)g(y)dy)∥(R+|·|)^{α−n} w_{2}(·)∥_{L_{u}^{pq}}≤C∥g(·)11_{|y|<R}(·)∥_{L_{v}^{rs}}≤C.

To prove (1.4), an inequality similar to (2.1), with cubes replaced by balls, is used. First consider the case p = r^{∗}. Let x ≠ 0 and B = B(x, R) be the ball centered at x and with a small radius R, i.e., R < \frac{1}{2}|x|. Since w_{1}(·), w_{2}(·) ∈ A, for each y ∈ B: w_{1}(x) ≈ w_{1}(y) (in the sense that c^{−1}w_{1}(y) ≤ w_{1}(x) ≤ cw_{1}(y)) and w_{2}(x) ≈ w_{2}(y). Indeed, \frac{1}{2}|x| < |y| < 4\frac{1}{2}|x| and w_{1}(y) ≤ sup_{\frac{1}{2} |x|≤ |z|<64 \frac{1}{2} |x|} w_{1}(z) ≤ c inf_{\frac{1}{2} |x|≤ |z|<64 \frac{1}{2} |x|} w_{1}(z) ≤ c w_{1}(x). Analogously, w_{1}(x) ≤ cw_{1}(y). Taking f(·) = 11_{B}(·) in (2.1) (with balls instead of cubes) and using the above equivalences we obtain

w_{2}(x) |B|^{\frac{α}{n} + \frac{1}{p} − \frac{1}{r}} (|B|^{−1} ∫_{B} u(y)dy)^{\frac{1}{p}} ≤ Cw_{1}(x) (|B|^{−1} ∫_{B} v(y)dy)^{\frac{1}{r}} . (2.3)

Here \frac{α}{n} + \frac{1}{p} − \frac{1}{r} = 0 and |B| = R^{n}. Thus by (2.3) and the Lebesgue differentiation theorem (by letting R → 0) we have w_{2}(x) (u(x))^{\frac{1}{r^{∗}}} ≤ C w_{1}(x) (v(x))^{\frac{1}{p}}.

Next suppose p ≠ r^{∗} (\frac{α}{n} + \frac{1}{p} − \frac{1}{r} ≠ 0), and assume both u(·), v(·) ∈ A.

The purpose is to estimate I = I(x) = w_{2}(x) |x|^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}(u(x))^{\frac{1}{p}}. For the present case, the ball B = B(x, R) is taken with radius R = \frac{1}{9}|x|. Observe that \frac{8}{9}|x| < |y| < 64\frac{8}{9}|x| whenever y ∈ B. The conclusion appears as follows:

I = w_{2}(x) |x|^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}(u(x))^{\frac{1}{p}} ≤

≤ c_{1} w_{2}(x) R^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}( sup_{\frac{8}{9} |x|<|y|≤64 \frac{8}{9} |x|} u(y))^{\frac{1}{p}} ≤

≤ c_{1}G(u) w_{2}(x) R^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}(R^{−n} ∫_{B}[ inf_{\frac{8}{9} |x|<|z|≤64 \frac{8}{9} |x|} u(z)] dy)^{\frac{1}{p}} ≤

≤ c_{1}G(u) w_{2}(x) R^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}(R^{−n} ∫_{B} u(y)dy)^{\frac{1}{p}} ≤

≤ c_{2}G(u) w_{1}(x) (R^{−n} ∫_{B} v(y)dy)^{\frac{1}{r}} ≤ (by (2.3))

≤ c_{3}G(u) w_{1}(x) ( sup_{\frac{8}{9} |x|<|z|≤64 \frac{8}{9} |x|} v(z))^{\frac{1}{r}} ≤

≤ c_{3}G(u)G(v) w_{1}(x) ( inf_{\frac{8}{9} |x|<|y|≤64 \frac{8}{9} |x|} v(z))^{\frac{1}{r}} ≤

≤ c_{3}G(u)G(v) w_{1}(x) (v(x))^{\frac{1}{r}} .

In the proofs of our results we will have to perform some summations as stated in the following

Lemma 2. Suppose ∑_{k} 11_{E_{k}}(·) ≤ C 11_{∪E_{k}}(·) for a fixed constant C > 0, where E_{k}’s are measurable sets (so these sets are quasi-disjoint).

(A) Then

∑_{k}∥f(·)11_{E_{k}}(·)∥_{L_{w}^{rs}}^{λ} ≤ c_{1} ∥f(·)11_{∪E_{k}}(·)∥_{L_{w}^{rs}}^{λ} for all functions f(·)

whenever max(r, s) ≤ λ.

(B) For a constant c > 0, which depends only on C,

∥∑_{k} f(·)11_{E_{k}}(·)∥_{L_{u}^{pq}}^{γ} ≤ c ∑_{k}∥f(·)11_{E_{k}}(·)∥_{L_{u}^{pq}}^{γ}

whenever 0 < γ ≤ min(p, q).

Also, the proofs of our results will depend much on the boundedness of generalized Hardy type operators on weighted Lorentz spaces which are already introduced and studied by Edmunds, Gurka, and Pick [4]. The Hardy type operators under consideration are of the forms

(Hf)(x) = (H_{a,b}f)(x) = a(x) ∫_{|y|≤|x|} f(y) b(y)dy

and

(H^{∗}g)(x) = (H_{a,b}^{∗}g)(x) = b(x) ∫_{|x|≤|y|} g(y) a(y)dy,

where a(·) and b(·) are measurable nonnegative functions. It is supposed that

r = s = 1 or 1 < r < ∞ and ; 1 ≤ s ≤ ∞,
p = q = 1 or 1 < p < ∞ and 1 ≤ q ≤ ∞.

Lemma 3. Let r, s, p, q be as above and, moreover, max(r, s)≤min(p, q). Then

∥(Hf)(·)∥_{L_{u}^{pq}} ≤ C ∥f(·)∥_{L_{v}^{rs}} for all f(·) ≥ 0

if and only if

sup_{R>0} ∥ a(·) 11_{R<|·|}(·) ∥_{L_{u}^{pq}} ∥\frac{1}{v(·)} b(·) 11_{|·|<R}(·) ∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} < ∞.

Similarly,

∥(H^{∗}g)(·)∥_{L_{u}^{pq}} ≤ C ∥g(·)∥_{L_{v}^{rs}} for all g(·) ≥ 0

if and only if

sup_{R>0} ∥ b(·) 11_{|·|<R}(·) ∥_{L_{u}^{pq}} ∥\frac{1}{v(·)} a(·) 11_{R<|x|}(·) ∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} < ∞.

In order to get the weighted inequality for the maximal operators M_{α}, 0 ≤ α < n, the following cutting lemma is needed.

Lemma 4. Let 0 < λ ≤ min(p, q). Then for some constant C > 0 and for all f(·) ≥ 0:

∥ w_{2}(·) (M_{α}f)(·)∥_{L_{u}^{pq}}^{λ} ≤ C (S_{1}^{λ} + S_{2}^{λ} + S_{3}^{λ}),

where

S_{1}^{λ} = ∥ w_{2}(·)| · |^{α−n} (∫_{|y|≤|·|} f(y)dy) ∥_{L_{u}^{pq}}^{λ},
S_{2}^{λ} = ∑_{k∈ℤ}∥ w_{2}(·) (M_{α}f11_{G_{k}})(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{λ},

S_{3}^{λ} = ∑_{m∈ℤ} [2^{(α−n)m} (∫_{E_{m}} f(y)dy)]^{λ} ∥ w_{2}(·) 11_{|x|<2^{m}}(·)∥_{L_{u}^{pq}}^{λ}

and E_{k} = {2^{k} < |x| ≤ 2^{k+1}}, G_{k} = {2^{k−1} < |x| ≤ 2^{k+2}}.

In order to state in a condensed form a similar result for any fractional integral and any Calderón–Zygmund operators, define the linear operator T_{α}, 0 ≤ α < n, as sending C_{c}^{∞}(ℝ^{n}) into L_{loc}^{1}(ℝ^{n}, dx) and such that

(T_{α}f)(x) = ∫_{ℝ^{n}} K_{α}(x, y)f(y)dy a.e. x ∉ supp f

for every f(·) ∈ L_{c}^{∞}(ℝ^{n}), and with the kernel K_{s}(x, y) satisfying

|K_{α}(x, y)| ≤ C|x − y|^{(α−n)} for all x ≠ y.

It is also assumed that (T_{α}f)(·) is well defined almost everywhere for all bounded functions with compact supports. This is the case for 0 < α < n when T_{α} is the fractional integral operator I_{α}. For α = 0 this assumption will be realized if T_{0} : L^{p} → L^{p} for some p > 1 (which is the case for a Calderón–Zygmund operator).

Lemma 5. Let 0 ≤ α < n and 0 < λ ≤ min(p, q). Then for a constant C > 0 and for all functions f ∈ C_{c}^{∞}(ℝ^{n}):

∥ w_{2}(·) (T_{α}f)(·)∥_{L_{u}^{pq}}^{λ} ≤ C (S_{1}^{λ} + S_{2}^{λ} + S_{3}^{λ})

where

S_{1}^{λ} = ∥ w_{2}(·)| · |^{α−n} (∫_{|y|≤|·|} |f(y)|dy) ∥_{L_{u}^{pq}}^{λ},
S_{2}^{λ} = ∑_{k∈ℤ}∥ w_{2}(·) (T_{α}f11_{G_{k}})(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{λ},

S_{3}^{λ} = ∥ w_{2}(·) (∫_{|·|≤|y|} |f(y)||y|^{α−n}dy) ∥_{L_{u}^{pq}}^{λ}

and E_{k} and G_{k} are defined as in Lemma 4.

The proofs of these lemmas will be given in §4, and now we proceed to proving our main results.

§ 3. Proofs of the Main Results

Proof of Theorem 2. The real problem is to prove Part B. Since max(r, s) ≤ min(p, q) (see (1.5)), therefore one can find λ > 0 for which max(r, s) ≤ λ ≤ min(p, q). In view of cutting Lemma 4, we have to estimate each of S_{1}^{λ}, S_{2}^{λ}, and S_{3}^{λ} by

C ∥ w_{1}(·)f(·)∥_{L_{v}^{rs}}^{λ}

for a fixed constant C > 0 which, in general, depends on α, n, p, q, r, s, u(·), v(·), w_{1}(·) and w_{2}(·).

Estimate of S_{1}^{λ}. By taking g(·) = w_{1}(·)f(·), we obtain

∥ w_{2}(·)| · |^{α−n} (∫_{|y|≤|·|} \frac{1}{w_{1}(y)}g(y)dy) ∥_{L_{u}^{pq}} ≤ C ∥g(·)∥_{L_{v}^{rs}}.

Such an inequality can be considered as H : L_{v}^{rs} → L_{u}^{pq} where H = H_{a,b} is a Hardy type operator given by a(x) = w_{2}(x)|x|^{α−n} and b(y) = \frac{1}{w_{1}(y)}. In view of Lemma 3, this boundedness of H is equivalent to

sup_{R>0} ∥ a(·) 11_{R<|·|}(·) ∥_{L_{u}^{pq}} ∥\frac{b(·)}{v(·)} 11_{|·|<R}(·) ∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} < ∞

or to

∥ w_{2}(·)| · |^{α−n} 11_{R<|·|}(·) ∥_{L_{u}^{pq}} ∥\frac{1}{v(·)w_{1}(·)} 11_{|·|<R}(·) ∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} < C

for all R > 0. It is the Hardy condition (1.9) which is an immediate consequence of the Wheeden–Muckenhoupt condition (1.3).

Estimate of S_{3}^{λ}. Here the Muckenhoupt condition (1.8) (also an immediate consequence of (1.3)) is used. Now by the Hölder inequality and (1.8) we have

S_{3}^{λ} = ∑_{m∈ℤ} [2^{(α−n)m} (∫_{E_{m}} f(y)dy)]^{λ} ∥w_{2}(·)11_{|·|<2^{m}}(·)∥_{L_{u}^{pq}}^{λ} ≤

≤ C^{λ} ∑_{m∈ℤ}[2^{(α−n)m} ∥ w_{2}(·) 11_{|·|<2^{m}}(·)∥_{L_{u}^{pq}} ×

× ∥ \frac{1}{v(·)w_{1}(·)} 11_{|·|<2^{m}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} ]^{λ} ∥w_{1}(·)(f11_{E_{m}})(·)∥_{L_{u}^{rs}}^{λ} ≤

≤ (CA)^{λ} ∑_{m∈ℤ} ∥w_{1}(·)(f11_{E_{m}})(·)∥_{L_{v}^{rs}}^{λ} ≤

≤ (C′A)^{λ} ∥w_{1}(·)f(·)∥_{L_{v}^{rs}}^{λ} by Part A in Lemma 2.

Estimate of S_{2}^{λ}. To estimate S_{2}^{λ}it is sufficient to get

∥w_{2}(·)(M_{α}f11_{G_{k}})(·)11_{E_{k}}(·)∥_{L_{u}^{pq}} ≤ C∥w_{1}(·)(f11_{G_{k}})(·)∥_{L_{v}^{rs}}. (3.1)

Indeed, since ∑_{k} 11_{G_{k}}(·) ≤ 3 and max(r, s) ≤ λ, therefore by Part A in Lemma 2:

S_{2}^{λ} ≤ C^{λ} ∑_{k∈ℤ} ∥w_{1}(·)(f11_{G_{k}})(·)∥_{L_{u}^{rs}}^{λ} ≤ (cC)^{λ}∥w_{1}(·)f(·)∥_{L_{u}^{rs}}^{λ}.

To get (3.1) we will use the fact that M_{α} : L_{1}^{rs}(1) → L_{1}^{r^{∗}s}(1) (see [1], Theorem 5.2.2, p. 155), where 1 < r < ∞, 1 ≤ s ≤ ∞ and \frac{1}{r^{∗}} = \frac{1}{r} − \frac{α}{n}. The following three properties of Lorentz spaces (see [5]) are also used:

∥11_{E}(·)∥_{L_{w}^{ps}} = (∫_{E} w(y)dy)^{\frac{1}{p}} for all measurable sets E;
∥f∥_{L^{ps_{1}}} ≤ ∥f∥_{L^{ps_{2}}} for a fixed p, and s_{2} ≤ s_{1};

∥f_{1}f_{2}∥_{L^{ps}} ≤ c∥f_{1}∥_{L^{p_{1}s_{1}}} ∥f_{2}∥_{L^{p_{2}s_{2}}} with \frac{1}{p} = \frac{1}{p_{1}} + \frac{1}{p_{2}} and \frac{1}{s} = \frac{1}{s_{1}} + \frac{1}{s_{2}} .

ON THE BOUNDEDNESS OF CLASSICAL OPERATORS 191

For convenience, set

W_{1,k} = sup_{x∈G_{k}} w_{1}(x), W_{2,k} = sup_{y∈E_{k}} w_{2}(y), and U_{k} = sup_{z∈E_{k}} u(z).

Recall that w_{1}(·), w_{2}(·) ∈ A and the pointwise condition (1.7) is assumed. The chain of computations which leads to inequality (3.1) is as follows:

T_{k} = ∥w_{2}(·)(M_{α}f11_{G_{k}})(·)11_{E_{k}}(·)∥_{L_{u}^{pq}} ≤

≤ ∥w_{2}(·)(M_{α}f11_{G_{k}})(·)11_{E_{k}}(·)∥_{L_{u}^{ps}} ≤

(here s ≤ q since max(r, s) ≤ min(p, q))

≤ c_{0}W_{2,k} Uk^{\frac{1}{p}} ∥(M_{α}f11_{G_{k}})(·)11_{E_{k}}(·)∥_{L^{ps}} ≤

≤ c_{1}W_{2,k} Uk^{\frac{1}{p}} ∥(M_{α}f11_{G_{k}})(·)∥_{L^{r^{∗}s}} × ∥11_{E_{k}}(·)∥_{L^{r̃∞}} ≤

(where \frac{1}{r̃} = \frac{1}{p} − \frac{1}{r^{∗}} = \frac{1}{p} + \frac{α}{n} − \frac{1}{r})

≤ c_{2} 2^{nk[ \frac{1}{p} + \frac{α}{n} − \frac{1}{r}]}W_{2,k} Uk^{\frac{1}{p}} ∥(f11_{G_{k}})(·)∥_{L^{rs}} ≈

(since M_{α} : L_{1}^{rs}(1) → L_{1}^{r^{∗}s}(1))

≈ c_{2} 2^{nk[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}W_{2,k} Uk^{\frac{1}{p}} [∑_{j} 2^{js} (∫_{G_{k}∩\{f(·)>2^{j}\}} dx)^{\frac{s}{r}} ]^{\frac{1}{s}} =

= c_{2} [∑_{j} 2^{js} (∫_{G_{k}∩\{f(·)>2^{j}\}}[2^{nk[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]} W_{2,k} Uk^{\frac{1}{p}} ]^{r}dx)^{\frac{s}{r}} ]^{\frac{1}{s}} ≤

≤ c_{3} [∑_{j} 2^{js} (∫_{G_{k}∩\{f(·)>2^{j}\}}[|x|^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]} w_{2}(x) ×

× ( sup_{4^{−1}|x|<|z|<4|x|} u(z))^{\frac{1}{p}} ]^{r} dx)^{\frac{s}{r}} ]^{\frac{1}{s}} ≤

(here W_{2,k} ≤ sup_{(4^{−1}|x|)<|y|<16(4^{−1}|x|)} w_{2}(y) ≤ cw_{2}(x) since w_{2}(·) ∈ A)

≤ c_{4} [∑_{j} 2^{js} (∫_{G_{k}∩\{f(·)>2^{j}\}}(w_{1}(x))^{r} v(x)dx)^{\frac{s}{r}} ]^{\frac{1}{s}} ≤

(by hypothesis (1.7))

≤ c_{4} [∑_{j} W_{1,k}^{s}2^{js} (∫_{G_{k}∩\{f(·)>2^{j}\}} v(x)dx)^{\frac{s}{r}} ]^{\frac{1}{s}} ≤

≤ c_{5} [∑_{j} 2^{(j+N_{k})s} (∫_{G_{k}∩\{2^{N_{k}} f(·)>2^{(j+N_{k})}\}} v(x)dx)^{\frac{s}{r}} ]^{\frac{1}{s}} ≤

(since 2^{N_{k}} ≤ W_{1,k} < 2^{N_{k}+1} for some N_{k} ∈ ℤ)

≤ c_{5} [∑_{j} 2^{(j+N_{k})s} (∫_{G_{k}∩\{W_{1,k}f(·)>2^{(j+N_{k})}\}} v(x)dx)^{\frac{s}{r}} ]^{\frac{1}{s}} ≤

≤ c_{6} [∑_{j} 2^{ls} (∫_{G_{k}∩\{cw_{1}(·)f(·)>2^{l}\}} v(x)dx)^{\frac{s}{r}} ]^{\frac{1}{s}} ≤

(here W_{1,k} ≤ sup_{(8^{−1}|x|)<|z|<64(8^{−1}|x|)} w_{1}(z) ≤ cw_{1}(x) since w_{1}(·) ∈ A)

≤ c_{7} ∥w_{1}(·)(f11_{G_{k}})(·)∥_{L_{v}^{rs}}.

Now we will study how to obtain the same local estimate (3.1) if instead of (1.7) we use condition (1.7′). The main point is the existence of C > 0 for which

2^{nk[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]} W_{2,k} Uk^{\frac{1}{p}} ≤ Cw_{1}(x)(v(x))^{\frac{1}{r}} for all x ∈ G_{k}.

Indeed, by virtue of this inequality, a modification of the previous chain of computations leads to

T_{k} = ∥w_{2}(·)(M_{α}f11_{G_{k}})(·)11_{E_{k}}(·)∥_{L_{u}^{pq}} ≤

≤ c_{2}[∑_{j} 2^{js} (∫_{G_{k}∩\{f(·)>2^{j}\}}[2^{nk[ \frac{α}{n}+\frac{1}{p} − \frac{1}{r}]}W_{2,k}Uk^{\frac{1}{p}} ]^{r}dx)^{\frac{s}{r}} ]^{\frac{1}{s}}≤ (see above)

≤ c_{8}[∑_{j} 2^{js} (∫_{G_{k}∩\{f(·)>2^{j}\}}(w_{1}(x))^{r} v(x)dx)^{\frac{s}{r}} ]^{\frac{1}{s}}≤ (by this main point)

≤ c_{9}∥w_{1}(·)(f11_{G_{k}})(·)∥_{L_{v}^{rs}} (see again the details in the above estimate).

To prove the main point, it is essential to observe that

\frac{1}{sup_{4^{−1}|z|<|y|<4|z|}[\frac{1}{v(y)}]} ≤ v(x) for all x ∈ G_{k} and z ∈ E_{k}.

This inequality is true, since for x ∈ G_{k} and z ∈ E_{k} we have 4^{−1}|z| < 2^{k−1} < |x| < 2^{k+2} < 4|z| and 1 = v(x)\frac{1}{v(x)} ≤ v(x) sup_{4^{−1}|z|<|y|<4|z|}[\frac{1}{v(y)}].

So the main point appears since

2^{nk[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}W_{2,k} Uk^{\frac{1}{p}} ≤ C_{1} sup_{z∈E_{k}}\{|z|^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}W_{2,k} (u(z))^{\frac{1}{p}} \} ≤

≤ C_{2} sup_{z∈E_{k}}\{|z|^{n[ \frac{α}{n} + \frac{1}{p} − \frac{1}{r}]}w_{2}(z) (u(z))^{\frac{1}{p}} \} ≤ (since w_{2}(·) ∈ A)

≤ C_{3} sup_{z∈E_{k}} \{w_{1}(z) \frac{1}{sup_{4^{−1}|z|<|y|<4|z|}[\frac{1}{v(y)}]^{\frac{1}{r}}} \} ≤ (by condition (1.7′))

≤ C_{4} w_{1}(x) sup_{z∈E_{k}} \{\frac{1}{sup_{4^{−1}|z|<|y|<4|z|}[\frac{1}{v(y)}]^{\frac{1}{r}}} \} ≤ (since w_{1}(·) ∈ A)

≤ C_{4} w_{1}(x) (v(x))^{\frac{1}{r}} (by the above observation).

Proof of Theorems 4 and 5. Part A of Theorem 4 is proved. Next a condensed proof for Part B of Theorem 4 and Theorem 5 is given.

Suppose I_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}). Since (M_{α}f)(·) ≤ c(s, n)(I_{α}f)(·), we have M_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) and the necessity of condition (1.3) (given by Lemma 1) implies the Hardy condition (1.9). On the other hand, observe that I_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) is equivalent to I_{α} : L_{u}^{p′q′}(\frac{1}{uw_{2}} ) → L_{v}^{r′s′}(\frac{1}{vw_{1}} ) where p′ = \frac{p}{p−1}, q′ = \frac{q}{q−1}, . . .. So the dual condition (1.9^{∗}) appears from the embedding M_{α} : L_{u}^{p′q′}(\frac{1}{uw_{2}} ) → L_{v}^{r′s′}(\frac{1}{vw_{1}} ) as above.

To get Part B of Theorem 4 and Theorem 5, it is sufficient to derive the embedding T_{α} : L_{v}^{rs}(w_{1}) → L_{u}^{pq}(w_{2}) with 0 ≤ α < n by using conditions (1.9), (1.9^{∗}) and hypothesis (1.7). As in the proof of Theorem 2 and in view of Lemma 5, the problem is reduced to estimating each of S_{1}^{λ}, S_{2}^{λ},

and S_{3}^{λ} by C ∥w_{1}(·)f(·)∥_{L_{v}^{rs}}^{λ}. Here C > 0 is a constant which depends

eventually on α, n, p, q, r, s, u(·), v(·), w_{1}(·), w_{2}(·), and λ is chosen such that max(r, s) ≤ λ ≤ min(p, q).

The estimation of S_{1}^{λ} can be carried out as in the proof of Theorem 2 by using the Hardy condition (1.9) and Lemma 3.

Also, S_{2}^{λ} can be estimated as in the proof of Theorem 2. Indeed, in the present case for α > 0 we have I_{α} : L_{1}^{rs}(1) → L_{1}^{r^{∗}s}(1) (see [1], Theorem 6.3.3, p. 191), where 1 < r < ∞, 1 ≤ s ≤ ∞ and \frac{1}{r^{∗}} = \frac{1}{r} − \frac{α}{n}; and, on the other hand, for α = 0 we have T_{α} : L_{1}^{rs}(1) → L_{1}^{rs}(1) which can be obtained by interpolation.

The estimate for S_{3}^{λ} is equivalent to H^{∗} : L_{v}^{rs} → L_{u}^{pq}, with H^{∗} = H_{a,b}^{∗} and b(x) = w_{2}(x), a(y) = \frac{1}{w_{1}(y)}|y|^{α−n}. By Lemma 3, this boundedness of H^{∗} is equivalent to the dual Hardy condition (1.9^{∗}).

Proof of Proposition 6. Suppose H : L_{v}^{rs}(w_{1}) → L_{u}^{rq}(w_{2}). Our purpose is just to get conditions (1.4) and (1.9) (with α = 0, r^{∗} = r = p), since (1.9^{∗}) can be obtained by using a duality argument. The main key to the proof is

( ∫_{I} f(y)dy) ∥w_{2}(·)(| · −x_{I}| + |I|)^{−1}∥_{L_{u}^{rq}} ≤ C ∥w_{1}(·)(f11_{I})(·)∥_{L_{v}^{rs}} (3.2)

for each interval I centered at x_{I}, and for each f(·) ≥ 0 whose support is I. Indeed, (3.2) first implies ∥w_{2}(·)11_{I}(·)∥_{L_{u}^{rq}} ≤ C′ ∥w_{1}(·)11_{I}(·)∥_{L_{v}^{rs}}. So

using w_{1}(·), w_{2}(·) ∈ A, we obtain

w_{2}(x) (|I|^{−1} ∫_{I} u(y)dy)^{\frac{1}{r}} ≤ cw_{1}(x)(|I|^{−1} ∫_{I} v(y)dy)^{\frac{1}{r}} ,

for each interval I centered at x ≠ 0 and with a length |I| sufficiently small. Then by the Lebesgue differentiation theorem: w_{2}(x)(u(x))^{\frac{1}{r}} ≤ cw_{2}(x)(v(x))^{\frac{1}{r}}, which is actually condition (1.4).

On the other hand, applying (3.2) for intervalls I =]−R, R[ (i.e., x_{I} = 0) we obtain condition (1.9), since

∥\frac{1}{v(·)w_{1}(·)}11_{|·|<R}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} ∥w_{2}(·)(| · | + R)^{−1} ∥_{L_{u}^{rq}} ≤

≤ c (∫_{|x|<R} \frac{1}{v(y)w_{1}(y)} g(y)v(y)dy)∥w_{2}(·)(| · | + R)^{−1} ∥_{L_{u}^{rq}} ≤

(where ∥g(·)∥_{L_{v}^{rs}} ≤ 1)

≤ cC ∥w_{1}(·)\frac{1}{w_{1}(·)}g(·)∥_{L_{v}^{rs}} = cC ∥g(·)∥_{L_{v}^{rs}} ≤ cC.

In view of Part B of Lemma 2, inequality (3.2) will be obtained immediately for some N > 0:

( ∫_{I} f(y)dy) ∥w_{2}(·)s_{I}(·)11_{]a+N,∞[}(·)∥_{L_{u}^{rq}} ≤ C ∥ w_{1}(·)(f11_{I})(·)∥_{L_{v}^{rs}} (3.3)

and

( ∫_{I} f(y)dy) ∥w_{2}(·)s_{I}(·)11_{]−∞,a+N[}(·)∥_{L_{u}^{rq}} ≤ C ∥w_{1}(·)(f11_{I})(·)∥_{L_{v}^{rs}}, (3.4)

where s_{I}(x) = (|x − x_{I}| + |I|)^{−1} and I is any intervall centered at x_{I} and having the form I = [a, a + R], R > 0. Inequalities (3.3) and (3.4) will be immediate consequences of the pointwise estimates

(Hf11_{[a,a+N]})(·) 11_{]a+N,∞[}(·) ≥ c s_{I}(·) (∫_{I} f(y)dy) 11_{]a+N,∞[}(·) (3.5)

and

(Hf11_{[a+N,a+R]})(·) 11_{]−∞,a+N[}(·) ≥ c s_{I}(·) (∫_{I} f(y)dy) 11_{]−∞,a+N[}(·) (3.6)

for a some N > 0 and N < R. So the problem of obtaining inequality (3.2) is just reduced to obtaining (3.5) and (3.6).

Let f(·) ≥ 0 be supported by I = [a, a + R], and let Q = ∫_{I} f(y)dy. One can take 0 < N < R such that \frac{1}{2}Q = ∫_{[a,a+N]} f(y)dy. Clearly, 0 < x − y = |x − y| ≤ |x − x_{I}| + |I| = s_{I}^{−1}(x) for each x ∈]a + N, ∞[ and y ∈ [a, a + N]. Inequality (3.5) is satisfied, since

(Hf11_{[a,a+N]})(x) 11_{]a+N,∞[}(x) ≥ s_{I}(x) (∫_{[a,a+N]} f(y)dy) 11_{]a+N,∞[}(x) =

= s_{I}(x) \frac{1}{2}Q 11_{]a+N,∞[}(x) = s_{I}(x) \frac{1}{2}(∫_{I} f(y)dy) 11_{]a+N,∞[}(x).

If we note that

∫_{[a+N,a+R]} f(y)dy = ∫_{[a,a+R]} f(y)dy − ∫_{[a,a+N]} f(y)dy = \frac{1}{2}Q,

inequality (3.6) can be proved in the same way Proof of Proposition 7. To get the Hardy condition (1.9) from the Muckenhoupt condition (1.8), first consider the case v(·) ∈ RD_{ν,r,s}(w_{1}), ν > 0. Applying Lemma 2 with 0 < θ ≤ min(p, q), we have

∥w_{2}(·)| · |^{α−n}11_{\{|·|>R\}}(·)∥_{L_{u}^{pq}}^{θ} ∥\frac{1}{v(·)w_{1}(·)}11_{\{|·|<R\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}}^{θ} ≤

≤ c_{1} ∑_{k≥0}∥w_{2}(·) | · |^{α−n} 11_{{2^{k}R<|·|≤2^{k+1}R}}(·)∥_{L_{u}^{pq}}^{θ} ×

× ∥\frac{1}{v(·)w_{1}(·)}11_{\{|·|<2^{−(k+1)}2^{(k+1)}R\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}}^{θ} ≤

≤ c_{2} ∑_{k≥0} 2^{−nνθk(1− \frac{1}{r})}(2^{k+1}R)^{α−n}(∥w_{2}(·) 11_{\{|·|≤2^{k+1}R\}}(·)∥_{L_{u}^{pq}} ×

× ∥\frac{1}{v(·)w_{1}(·)} 11_{\{|·|<2^{(k+1)}R\}}(·)∥_{L_{v}^{\frac{r}{r−1} \frac{s}{s−1}}} )^{θ} ≤ c_{3}A^{θ}.

For u(·) ∈ D_{ε,p,q}(w_{2}), 1 ≤ ε < (1 − \frac{α}{n})p and with the same choice of θ, then

∥w_{2}(·)| · |^{α−n}11_{{|·|>R}}(·)∥_{L_{u}^{pq}}^{θ} ≤c_{4}∑_{k≥0}∥w_{2}(·)| · |^{α−n}11_{{2^{k}R<|·|≤2^{k+1}R}}(·)∥_{L_{u}^{pq}}^{θ} ≤

≤c52R^{(α−n)θ} ∑_{k≥0} 2^{k(α−n)θ} ∥w_{2}(·) 11_{{|·|≤2^{k+1}R}}(·)∥_{L_{u}^{pq}}^{θ} ≤

≤c_{6}R^{(α−n)θ} ∑_{k≥0} 2^{−knθ[(1− \frac{α}{n})− \frac{1}{p} ε]} ∥w_{2}(·) 11_{\{|·|<R\}}(·)∥_{L_{u}^{pq}}^{θ} ≤

≤c_{7} (R^{(α−n)} ∥w_{2}(·) 11_{{|·|<R}}(·)∥_{L_{u}^{pq}})^{θ}.

Clearly, by the latter estimate, condition (1.8) implies (1.9).

§ 4. Proofs of the Basic Lemmas and Propositions 8 and 9

In this section we prove Lemmas 2, 3, and 4 (which were used for the proofs of our main results) and also Propositions 8 and 10. Proof of Lemma 2. A proof of Part A with C = 1 was given in [6]. The present case can be easily obtained by using a duality argument.

To prove Part B, assume that 0 < γ ≤ min(p, q). For p = q or q < p, the key is based on the fact that ∥ · ∥_{L_{u}^{\frac{p}{γ} \frac{q}{γ}}} is equivalent to a norm. Hence

∥f(·) ∑_{k} 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{γ} = ∥(f(·) ∑_{k} 11_{E_{k}}(·))^{γ} ∥_{L_{u}^{\frac{p}{γ} \frac{q}{γ}}} ≤

≤c_{1}∥f^{γ}(·)∑_{k}11_{E_{k}}(·)∥_{L_{u}^{\frac{p}{γ} \frac{q}{γ}}} ≤c_{2}∑_{k}∥f^{γ}(·)11_{E_{k}}(·)∥_{L_{u}^{\frac{p}{γ} \frac{q}{γ}}} =c_{3}∑_{k}∥f(·)11_{E_{k}}(·)∥_{L_{u}^{pq}}^{γ}.

Now consider the case p < q, so γ ≤ p < q or \frac{γ}{p} ≤ 1 and 1 < \frac{q}{γ}. Thus for a nonnegative sequence of reals (b_{j})_{j} ∈ l^{θ}, with ∑_{j} b_{j}^{θ} ≤ 1 and θ = \frac{q}{γ−q} we obtain

∥f(·) ∑_{k} 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{γ} ≤ c_{4}[∑_{j} 2^{jq} (∑_{k} ∫_{E_{k}∩\{f(·)>2^{j}\}} u(y)dy)^{\frac{γ}{p} × \frac{q}{γ}} ]^{\frac{γ}{q}} ≤

≤c_{4}[∑_{j} [2^{jγ} ∑_{k}(∫_{E_{k}∩\{f(·)>2^{j}\}} u(y)dy)^{\frac{γ}{p}} ]^{\frac{q}{γ}} ]^{\frac{γ}{q}} ≤ (since \frac{γ}{p} ≤ 1)

≤c_{4} ∑_{k} ∑_{j} 2^{jγ} b_{j} (∫_{E_{k}∩\{f(·)>2^{j}\}} u(y)dy)^{\frac{γ}{p}} ≤

≤c_{4} ∑_{k} [ ∑_{j} 2^{jq} (∫_{E_{k}∩\{f(·)>2^{j}\}} u(y)dy)^{\frac{q}{p}} ]^{\frac{γ}{q}} ≈ c_{4} ∑_{k} ∥f(·)11_{E_{k}}(·)∥_{L_{u}^{pq}}^{γ}.

The first part of this result was proved by Edmunds, Gurka, and Pick [4]. The second part can be derived by the first one with a duality argument.

Proof of Lemma 4. Since M_{α} is subadditive, for a fixed constant c > 0

∥ w_{2}(·) (M_{α}f)(·)∥_{L_{u}^{pq}}^{λ} ≤ c(P_{1}^{λ} + P_{2}^{λ} + P_{3}^{λ}) for all f(·) ≥ 0

with

P_{1}^{λ} = ∥ ∑_{k∈ℤ} w_{2}(·) (M_{α}f11_{{|y|≤2^{k−1}}})(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{λ},

P_{2}^{λ} = ∥ ∑_{k∈ℤ} w_{2}(·) (M_{α}f11_{G_{k}})(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{λ},

P_{3}^{λ} = ∥ ∑_{k∈ℤ} w_{2}(·) (M_{α}f11_{{2^{k+2}<|y|}})(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{λ}.

We have only to estimate P_{1}^{λ} and P_{3}^{λ}. Estimate of P_{1}^{λ}. The key is to prove

(M_{α}f_{{|·|≤2^{k−1}}})(x) ≤ c |x|^{(α−n)}[∫_{{|y|≤|x|}} f(y)dy] for all x ∈ E_{k}. (4.1)

Here c > 0 is a constant which depends only on α and n. Indeed, using (4.1) we have

P_{1}^{λ} ≤ c_{1}∥ ∑_{k∈ℤ} w_{2}(·)| · |^{α−n} (∫_{|y|≤|·|} f(y)dy) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{λ} =

= c_{1}∥ w_{2}(·)| · |^{α−n} (∫_{|y|≤|·|} f(y)dy) ∥_{L_{u}^{pq}}^{λ} = c_{1}S_{1}^{λ}.

To get inequality (4.1) observe that |y| ≤ |x|, for x ∈ E_{k} and |y| ≤ 2^{k−1}. Thus the support of g(·) = f(·)11_{{|y|≤2^{k−1}}}(·) is contained in the set {y; |y| ≤ |x|} = {|y| ≤ |x|}. On the other hand, the term ∫_{B(x,r)} g(y)dy does not vanish whenever r ≥ 2^{k+2}. Consequently for such a real r

r^{α−n} ∫_{B(x,r)} g(y)dy ≤ c 2^{k(α−n)} ∫_{B(x,r)} g(y)dy ≤

≤ c′ |x|^{k(α−n)} ∫_{B(x,r)∩supp g} g(y)dy ≤ c′ |x|^{k(α−n)} ∫_{{|y|≤|x|}} f(y)dy;

so (4.1) is proved.

Estimate of P_{3}^{λ}. To estimate P_{3}^{λ} we claim that

(M_{α}f_{{2^{k+2}≤|·|}})(x) ≤ c sup_{l≥2}{d_{k+l}^{(α−n)}(∫_{E_{k+l}} f(y)dy)} for each x ∈ E_{k}, (4.2)

where d_{k+l} = 2^{k+l} and c = c(α, n) > 0. The proof of this claim is given below.

First, the cases λ = p = q, λ ≤ q < p, and λ < p ≤ q are treated. Observe that

[∑_{k∈ℤ} (M_{α}f11_{2^{k+2}<|·|})(·) 11_{E_{k}}(·)]^{λ} ≤ ∑_{k} (M_{α}f11_{2^{k+2}<|·|})^{λ}(·) 11_{E_{k}}(·) ≤

≤ c ∑_{k∈ℤ} [sup_{l≥2}{d_{k+l}^{(s−n)}(∫_{E_{k+l}} f(y)dy)}]^{λ} 11_{E_{k}}(·) ≤

≤ c ∑_{l=2}^{∞} ∑_{k∈ℤ} [d_{k+l}^{(s−n)}(∫_{E_{k+l}} f(y)dy)]^{λ} 11_{E_{k}}(·) =

= c ∑_{m∈ℤ} [d_{m}^{(s−n)}(∫_{E_{m}} f(y)dy)]^{λ} ∑_{k=−∞}^{m−2} 11_{E_{k}}(·) =

= c ∑_{m∈ℤ} [d_{m}^{(s−n)}(∫_{E_{m}} f(y)dy)]^{λ} 11_{{|·|<2^{m}}}(·).

Since with the above hypotheses ∥ · ∥_{L_{u}^{\frac{p}{λ} \frac{q}{λ}}} is equivalent to a norm, we have

P_{3}^{λ} ≤ c^{λ}∥ w_{2}(·)^{λ} ∑_{m∈ℤ} [d_{m}^{(s−n)}(∫_{E_{m}} f(y)dy)]^{λ} 11_{\{|·|<2^{m}\}}(·)∥_{L_{u}^{\frac{p}{λ} \frac{q}{λ}}} ≤

≤ C ∑_{m∈ℤ} [d_{m}^{(s−n)}(∫_{E_{m}} f(y)dy)]^{λ} ∥ w_{2}(·) 11_{{|·|<2^{m}}}(·)∥_{L_{u}^{pq}}^{λ}.

Next the case λ = p < q is considered. The key point is hypothesis (1.6). Thus

P_{3}^{p} ≤ c ∥ ∑_{k∈ℤ}[sup_{l≥2} d_{k+l}^{(s−n)}(∫_{E_{k+l}} f(y)dy)] w_{2}(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{p} ≤

≤ C ∑_{k∈ℤ} [sup_{l≥2} d_{k+l}^{(s−n)}(∫_{E_{k+l}} f(y)dy)]^{p} ∥ w_{2}(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{p} ≤

(by the second part of Lemma 2 (λ = p ≤ min(p, q)))

≤ C ∑_{l=2}^{∞} ∑_{k∈ℤ} [d_{k+l}^{(s−n)}(∫_{E_{k+l}} f(y)dy) ]^{p}∥ w_{2}(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{p} ≤

≤ C ∑_{m} [d_{m}^{(s−n)}(∫_{E_{m}} f(y)dy) ]^{p} ∑_{k=−∞}^{m−2}∥ w_{2}(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{p} ≤

≤ C′ ∑_{m} [d_{m}^{(s−n)}(∫_{E_{m}} f(y)dy) ]^{p}∥ w_{2}(·) 11_{|·|<2^{m}}(·)∥_{L_{u}^{pq}}^{p} (by (1.6)).

Now claim (4.2) can be proved. It is assumed that

S = sup_{l≥2} 2^{(α−n)(k+l)}[∫_{{2^{k+l}<|y|≤2^{k+l+1}}} f(y)dy] < ∞.

Let x ∈ E_{k}. The claim is reduced to finding a constant c > 0 for which

r^{α−n} ∫_{B(x,r)} f(y)11_{{2^{k+2}<|y|}}(y)dy ≤ c S  (4.3)

whenever ∫_{B(x,r)} f(y)11_{{2^{k+2}<|y|}}(y)dy is a non-vanishing term.

Consider r > 0 with ∫_{B(x,r)} f(y)11_{{2^{k+2}<|y|}}(y)dy ≠ 0. There is an integer m ≥ 2 for which B(x, r) ∩ {2^{k+m} < |y| ≤ 2^{k+m+1}} ≠ ∅ and B(x, r) ∩ {2^{k+m+1} < |y| ≤ 2^{k+m+2}} = ∅. Since 2^{k+m} − 2^{k+1} < r ≤ 2^{k+m+1} and m ≥ 2, we obtain \frac{1}{2}2^{k+m} ≤ r < 22^{k+m}. With these preliminaries

∫B(x,r)f(y)11_{{2^{k+2}<|y|}}(y)dy = ∑_{l=2}^{m} ∫_{B(x,r)∩{2^{k+l}<|y|≤2^{k+l+1}}} f(y)dy ≤

≤ ∑_{l=2}^{m} ∫_{{2^{k+l}<|y|≤2^{k+l+1}}} f(y)dy ≤ S ∑_{l=2}^{m} 2^{(k+l)(n−α)} ≤

≤S \frac{2^{2(n−α)}}{2^{(n−α)} − 1} × 2^{[k+(m−1)](n−α)} = c(α, n)S r^{(n−α)}.

The latter inequality immediately implies (4.3). Proof of Lemma 5. First it is clear that for a fixed constant c > 0:

∥w_{2}(·)(T_{α}f)(·)∥_{L_{u}^{pq}}^{λ} ≤ c(P_{1}^{λ} + P_{2}^{λ} + P_{3}^{λ}) for all f(·) ∈ C_{c}^{∞}

with

P_{1}^{λ} = ∥ ∑_{k∈ℤ} w_{2}(·) (T_{α}f11_{{|y|≤2^{k−1}}})(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{λ},

P_{2}^{λ} = ∥ ∑_{k∈ℤ} w_{2}(·) (T_{α}f11_{G_{k}})(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{λ},

P_{3}^{λ} = ∥ ∑_{k∈ℤ} w_{2}(·) (T_{α}f11_{{2^{k+2}<|y|}})(·) 11_{E_{k}}(·)∥_{L_{u}^{pq}}^{λ},

where E_{k} and G_{k} are defined as in the proof of Lemma 4. By the assumption on T_{α}, the expressions (T_{α}f11_{{|·|≤2^{k−1}}})(·), (T_{α}f11_{G_{k}})(·), (T_{α}f11_{{2^{k+2}<|·|}})(·) are well defined.

As in the proof of Lemma 4, the estimate for P_{1}^{λ} will be obtained at once:

|(T_{s}f_{{|·|≤2^{k−1}}})(x)| ≤ c(α, n) |x|^{(α−n)}[∫_{{|y|≤|x|}} |f(y)| dy] for each x ∈ E_{k}.

To get this inequality observe that |y| ≤ \frac{1}{2}|x| < |x| and \frac{1}{2}|x| ≤ |x − y| for x ∈ E_{k} and |y| ≤ 2^{k−1}. So using the standard estimate for the kernel K, we have |K(x, y)| ≤ c|x − y|^{(α−n)} ≤ c′|x|^{(s−n)}. Since x does not belong to the support of the function (f_{{|y|≤2^{k−1}}})(·), we obtain |(T_{s}f11_{{|·|≤2^{k−1}}})(x)| ≤

∫_{{|y|≤2^{k−1}}} |K(x, y)||f(y)|dy ≤ C |x|^{(α−n)}[∫_{{|y|<|x|}} |f(y)|dy].

Similarly, the estimate for P_{3} is a consequence of

|(T_{s}f_{{2^{k+2}<|·|}})(x)| ≤ c(s, n)[∫_{{|x|≤|y|}} |f(y)||y|^{(s−n)} dy] for all x ∈ E_{k}.

Indeed, |x| < 2|x| ≤ |y| and \frac{1}{2}|y| ≤ |x − y|, for x ∈ E_{k} and 2^{k+2} < |y|. So |K(x, y)| ≤ C|x − y|^{(s−n)} ≤ C|y|^{(s−n)}. On the other hand, since x does not belong to the support of the function (f11_{{2^{k+2}<|y|}})(·), we have

|(T_{s}f11_{{2^{k+2}<|·|}})(x)| ≤ C [∫_{{|x|<|y|}} |f(y)||y|^{(s−n)}dy].

Acknowledgement

I would like to thank the referee for his helpful comments and suggestions.

References

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2. M. Carro and J. Soria, The Hardy–Littlewood maximal function and weighted Lorentz spaces. Preprint, 1993.

3. R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, 2nd ed. Astérisque 57(1979),

4. D. Edmunds, P. Gurka, and L. Pick, Compactness of Hardy-type integral operators in weighted Banach function spaces. Studia Math. 109 (1994), No. 1, 73–90.

5. A. Hunt, On L(p, q) spaces. L’enseign. Math. 12(1966), 249–294.

6. E. Sawyer, Weighted Lebesgue and Lorentz norm inequality for the Hardy operators. Trans. Amer. Math. Soc. 281(1984), 329–337.

(Received 3.11.1995; revised 25.12.1996)

Author’s address:

Institut Polytechnique St. Louis

Ecole de Physique et de Mathématiques Industrielles

13, boulevard de l’Hautil, 95 092 Cergy-Pontoise Cedex

France