Equivalent expressions for norms in classical Lorentz spaces

Abstract We characterize the weights w such that ∫0 ∞ ƒ*(s)p w(s) ds≃ ∫0 ∞ (ƒ**(s) – ƒ*(s))p w(s) ds. Our result generalizes a result due to Bennett–De Vore–Sharpley, where the usual Lorentz L p,q norm is replaced by an equivalent expression involving the functional ƒ ** – ƒ *. Sufficient conditions for the boundedness of maximal Calderón–Zygmund singular integral operators between classical Lorentz spaces are also given.


Introduction
Let (Ω, Σ(Ω), µ) be a nonfinite totally σ−finite resonant measure space, and let w be a strictly nonnegative Lebesgue measurable function on R + = (0, ∞) (briefly a weight).For 1 ≤ p < ∞ the classical Lorentz space Λ p µ (w) (see [10] and [6]) is defined by those measurable functions in Ω such that where f * µ (t) = inf s : λ µ f (s) ≤ t is the decreasing rearrangement of f , and λ µ f (y) = µ {x ∈ Ω : |f (x)| > y} is the distribution function of f with respect to the measure µ (we refer the reader to [4] for further information about distribution functions and decreasing rearrangements).
Similarly, the weak Lorentz space Λ p,∞ µ (w) (see [6]) is defined by the condition where W (t) = t 0 w(s)ds.Obviously, the above spaces are invariant under rearrangement and generalize the Lorentz spaces L p,q µ since if w(t) = t q/p−1 , (1 ≤ q, p < ∞) then Λ q µ (w) = L p,q µ and Λ q,∞ µ (w) coincides with L p,∞ µ ; in particular the Lebesgue space L p µ is the space Λ p µ (w) when w = 1.
Let us denote by f * * µ the maximal function of f * µ defined by It is proved in [3] (see also [4], Proposition 7.12) that in the case p > 1 the usual Lorentz L p,q µ norm can be replaced by an equivalent expression in terms of the functional where as usual, by A B we mean that c −1 A ≤ B ≤ cA, for some constant c > 0 independent of appropriate quantities.
The main purpose of this paper is to extend (1) in the context of the classical Lorentz spaces and describe the weights w for which The work is organized as follows: in Section 2 we provide a brief review of the parts of the theory of B p and B * ∞ weights that we shall use in this paper and prove some properties of the weights w that belong to B p ∩ B * ∞ .In Section 3 we characterize the weights w for which (2) holds, and as application, we obtain sufficient conditions for the boundedness of maximal Calderón-Zygmund singular integral operators between Lorentz spaces Λ p µ (w), if µ is an absolutely continuous measure on R n defined by µ(A) where u belongs to the class of weights A p 0 , for some p 0 ≥ 1 (see [8] as a general reference of this class of weights).

Preliminaries
If h is a Lebesgue measurable function defined on R + the Hardy operator P and its adjoint Q are defined by Results by M. Ariño and B. Muckenhoupt (see [1]) and C. J. Neugebauer (see [11]) which extend Hardy's inequalities, ensure that: The boundedness of P on Λ p,∞ µ (w) was also considered by J. Soria (see [14] Theorem 3.1).Soria's result ensures that: Lemma 2.1 Let 1 ≤ p < ∞ and w be a weight on R + .Then, the following are equivalent, For any a > 1 we have that where the last inequality follows from (4).Since we have that Now if we take a = e 2c we obtain a constant C (depending only on p) such that Finally, since P w(r) ≤ pP Q p w (r) it follows that We observe that condition ii) is hence by Fubini's theorem which by [7] and by Sagher's Lemma (see [12]), this happens if and only if On the other hand, as we have seen before, condition 3 The main result Theorem 3.1 Let 1 ≤ p < ∞ and w be a weight in R + .Then, the following are equivalent, , where the equivalence constants do not depend on µ.
Thus by Hardy's Lemma (see [4] Proposition 3.6, pag.56) and Fubini [15], Theorem 3.11.pag 192) we have that Collecting terms, we get The reverse inequality follows by the triangular inequality and condition B p .
ii) ⇒ i).This is a direct consequence of Lemma 2.1 since if we apply condition ii) to the characteristic function χ A with µ(A) = r, we obtain µ (w) and then w / ∈ B * ∞ ) and hence if f ∈ Λ p,∞ µ (w) we get lim t→∞ f * * µ (t) = 0. Now using the elementary identity (see [3]) and letting s → ∞ we find that if f ∈ Λ p,∞ µ (w) Hence On the other hand, since w ∈ B p the boundedness of the Hardy operator in Λ p,∞ µ (w) implies that iii) ⇒ i).Since 1/W 1/p is decreasing and lim x→∞ 1/W 1/p (x) = 0; as a consequence of Ryff's Theorem (see [4] Corollary 7.6.pag.83) there is a µ-measurable function f on Ω such that f * µ = 1/W 1/p , then by hypothesis and by (3) w ∈ B p .Given a > 0 and s > 1, define and let g(t) = Qh(t).Since g is decreasing and lim x→∞ g(t) = 0, again by Corollary 7.6.pag.83 of [4], we can find we get that in particular which implies that .
Summarizing we have proved that which by Lemma 2.1 implies that w ∈ B p ∩ B * ∞ .Observe that in the above theorem we have proved the norm equivalence between f (in the classical Lorentz space Λ p µ (w)) and f * * µ − f * µ in the weighted L p (w) space.In fact we have the following Proposition 3.1 The following statements are equivalent, where the rearrangement (f * * µ − f * µ ) * is taken with respect the Lebesgue measure in R + .
Proof.i) ⇒ ii).Since w ∈ B p ∩ B * ∞ , it follows from Theorem 3.1 that lim t→∞ f * * µ (t) = 0, for every f ∈ Λ p µ (w).Hence, where S := P • Q is the Calderón operator.Since if h is a nonnegative function on R + then S h (t) is decreasing, for each t > 0, by taking rearrangement (with respect the Lebesgue measure in R + ) we get (see [4] Proposition 5.2.pag.142) Applying condition ii) we get Given a > 0 and s > 1, define and let g(t) = Qh(t).Since g is decreasing and lim x→∞ g(t) = 0, using again Ryff's Theorem (see [4]  Remark 3.1 If 1 < p 0 < ∞ using the same proof that above and Theorem 3.3.9 of [5], one can easily check that Theorem 3.2 holds for every w in the biggest class B p/p 0 ,∞ ∩ B * ∞ where w ∈ B q,∞ ⇔ W (r)/r q ≤ cW (s)/s q 0 < s < r < ∞, (q > 0).
Corollary 7.6, pag.83) we can find f such that f * µ = g.Then f * * µ − f * µ = P Qh − Qh = P h + Qh − Qh = P h, thus, by condition ii), since h is decreasing and since w ∈ B p , we get that ∞ 0 Qh(x) p w(x) dx ) p w(x) dx = W (a) + a ≤ c W (a) + a p ∞ a w(x) x p dx ≤ cW (a)