PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 3, March 1996

WEAK-TYPE WEIGHTS AND NORMABLE LORENTZ SPACES

MAR´IA J. CARRO, ALEJANDRO GARC´IA DEL AMO, AND JAVIER SORIA

(Communicated by J. Marshall Ash)

Abstract. We show that the Lorentz space Λ1(w) is a Banach space if and only if the Hardy-Littlewood maximal operator M satisfies a certain weak-type estimate. We also consider the case of general measures. Finally, we study some properties of several indices associated to these spaces.

1. Introduction

We are going to study weighted Lorentz spaces of functions defined in Rn as follows (for standard notation we refer to [BS] and [GR]): If u is a weight in Rn, + w is a weight in R , fu∗ denotes the decreasing rearrangement of f with respect to the measure u(x) dx and 0 norm, if and only if, w is a decreasing function. The problem of findingk·k conditions on w so that Λp(w) is a Banach space (that is, there exists a norm equivalent to Λp(w)) was solved, for p>1, by E. Sawyer ([Sa]). This condition is that the Hardy-Littlewoodk·k maximal operator is bounded on Λp(w). The weights for which this holds were first characterized by M.A. Ari˜no and B. Muckenhoupt ([AM]), and it is known as the Bp condition: there exists C>0 such that, for all r>0,

r ∞ w(x) (1) rp dx C w(x) dx. xp ≤ Zr Z0 It is clear that (1) is not the right condition for p =1,sincewithw 1, we have that Λ1(w)=L1 is a Banach space, but w does not satisfy (1). It is≡ well known that the weighted strong-type and weak-type boundedness of the Hardy-Littlewood maximal operator coincide for p>1. This motivates to consider the same kind of p weak-type estimates for the spaces Λu(w) . To this end we recall the following definition:

Received by the editors September 14, 1994. 1991 Mathematics Subject Classification. Primary 42B25, 46E30. The first and third authors were partially supported by DGICYT PB94–0879. The second author was partially supported by DGICYT PB94–0243.

c 1996 American Mathematical Society

849

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Definition 1.1 ([CS1]). Let u and w beweightsasaboveand0

u λf (y) 1/p p, p, Λu ∞(w)= f; f Λu ∞(w) =supy w(t)dt < , k k y>0 ∞  Z0   u where λf is the distribution function of f with respect to the measure u(x) dx.

x 1/p Observe that f p, =sup f ∗(x)( w(t) dt) . Also, for the weight k kΛu ∞(w) x>0 u 0 w(t)=t(p/q) 1,wehavethatΛp, (w)=Lq, . Our main result is the fact that − u ∞ u ∞R in order to fully characterize when Λp(w) is a Banach space, for the whole range 1 p< , we should replace the boundedness of M in Λp(w) (as in [Sa]) by the a ≤ ∞ p p, priori weaker condition on the maximal operator M :Λ (w) Λ ∞(w). We give the details in section 2. In section 3 we study the case of a general−→ measure and in section 4 we define an index associated to Λp(w) and show several characterizations of this index in terms of the Bp condition.

2. Weak-type weights

Let us recall the following characterization of the weak-type boundedness of M for p>1 (see Theorem 3.9 in [CS3]):

p p, Theorem 2.1. If p>1,thenM:Λ (w) Λ ∞(w), if and only if, there exists C>0such that for all r>0 −→

r x p0 1/p0 r 1/p 1 − (2) w(t) dt w(x) dx w(t) dt Cr. x ≤  Z0  Zo    Z0  It is easy to show that (2) is equivalent to (1) (see [Sa]) and hence, it is also equivalent to the strong-type boundedness M :Λp(w) Λ p ( w ). The case p =1 of (2) motivates the following definition: −→

Definition 2.2. We say that w B1, if there exists C>0 such that for all 0 0, rt− χ t r 1, C χ(0,r) 1 . { ≥ } L ∞(w) L (w) 1 1 , ≤ k (v) S : Ldec(w) L ∞ ( w ) . −→

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Proof. Let us first show the equivalence of (ii), (iii) and (v). By definition, M : 1 1 , Λ (w) Λ ∞ ( w ) is bounded, if and only if −→

λMf(y) (3) sup y w(t)dt C f Λ1(w) = C f ∗ L1(w). y>0 ≤ k k k k Z0

∗ Now since Mf f∗∗ and λMf = λ(Mf)∗ (y) (see [BS]), we obtain that (3) is equivalent to ≈

λSf (y) (4) sup y w(t)dt C f L1(w), y>0 ≤ k k Z0 for all f L1 (w). Therefore (4) is the same as saying that ∈ dec 1 1 , S : L (w) Λ ∞ ( w ) . dec −→

Now it is easy to observe that if f is a decreasing function, then Sf 1, = k kΛ ∞(w) 1, Sf L ∞(w) (which proves (v)). But Theorem 3.3-(b) in [CS2] shows that this is k k 1 equivalent to w B1, (take w0 = w1 = w, p0 = p1 =1andk(x, t)=x− χ(0,x)(t) ∈ ∞ in that theorem). Assume now (i) holds and let us show (ii). If Λ1(w)isaBanach 1 space, then there exists an equivalent norm on Λ (w), Λ1(w) and hence, k·k≈k·k1 there exists C>0 such that for all N N and g1, ,gN Λ (w), g1 + + ∈ ··· ∈k k ··· gN Λ1(w) C( g1 Λ1(w)+ + gN Λ1(w)). Suppose first r =2 s,withk=1,2, . k ≤ k k ··· k k k ··· Set f = χ(0,2ks) and fj = χ(js,(j+1)s), j =0,1, ,2 1. Let F and Fj be functions n ··· − defined in R such that F ∗ = f, Fj∗ = fj∗ = χ(0,s) and F = j Fj .Now, P 2k 1 k − k W(2 s)= F 1 C Fj 1 = C2 W (s). k kΛ (w) ≤ k kΛ (w) j=0 X k 1 k Finally, for a general r>s,letk=1,2, be such that 2 − s r<2 s.Then ··· ≤ 1 r 1 C2k 2C s w(t)dt W (2ks) W (s)= w(t)dt. r ≤ 2k 1s ≤ 2k 1s s Z0 − − Z0 To see the converse, we use Theorem 1.1 of [KPS, II] and obtain that there exists a decreasing weight v so that if V (x)= xv(t)dt,then§ W V, and it is now easy 0 ≈ to show that Λ1(v) is a Banach space and Λ1(w)=Λ1(v). To finish, we observe that R r/y 1 rt− χ t r L1, (w) =supy w(t)dt, { ≥ } ∞ y 1 r ≤ Z and hence, (iv) is equivalent to

y r/y 1 r sup w(t)dt C w(t) dt, y 1 r 0 ≤ r 0 ≤ Z Z which is equivalent to (ii). 

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Remarks 2.4. (a) The equivalence between (i) and (ii) was already known, as a consequence of the study of the p-convexity of a rearrangement invariant space (we want to thank Professor Y. Raynaud for pointing out this to us). (b) It is now clear that, by (iii), B1 B1, .ItisalsotruethatB1, Bp,for p>1. In fact, if r>0, we have just⊂ shown∞ that for all k =1,2, ∞, W⊂(2kr) C2kW (r), and hence, ··· ≤

2k+1r 2k+1r ∞ w(t) ∞ w(t) 1 ∞ 1 p dt = p dt p kp w(t) dt t k t ≤ r 2 r k=0 2 r k=0 0 Z X Z X Z r r 1 ∞ C2k+1 C ∞ 1 w(t) dt w(t) dt. ≤ rp 2kp ≤ rp 2k(p 1) k=0 0 k=0 − 0 X Z  X  Z

(c) If we set f =supy>0 f∗∗(y)W (y), then is a norm, and if w k k∗ k·k∗ ∈ B1, ,then f Cf Λ1(w). Sometimes the converse inequality is also true, ∞ kk∗ ≤ kk and hence is the equivalent norm in Λ1(w). This happens, for example, if w 1 (seek·k [Sj]∗ for related results). The equivalence is not true in general: if ≡ 1/2 1 1/2 1 w(t)=t− ,thenΛ(w) is a Banach space, but if f ∗(t)=t− ,thenf/Λ(w) 1 ∈ and f < . However, if y− W (y) is equivalent to an increasing function, then k k∗ ∞ f Λ1(w) C f . It is also easy to show that this condition is not necessary. k k(d) For≤ somek k related∗ results see [Ne, 6]. § 3. Extensions to general measures

p We want to study geometric properties of the spaces Λu(w), for general weights. For example, it was proved in [CS1] that p is a quasinorm, if and only if W k·kΛu(w) satisfies a certain doubling condition (W ∆2). Also it is an easy exercise to show that these spaces are always complete (as∈ long as W (x) > 0ifx>0). We now ask the same question we did for Λp(w): under which conditions are they Banach spaces? We observe that Theorem 2.3 does not hold in general, since for w 1, p p ≡ Λu(w)=Lu, which is a Banach space if p 1, but the Hardy-Littlewood maximal p ≥ p, operator is not always bounded M : Lu L u ∞. However, we can show that, in many cases, this property only depends on−→ the weight w: Theorem 3.1. If u/L1 (or du is an infinite non-atomic measure) and p>0, p ∈ p then Λu(w) is a Banach space, if and only if Λ (w) is a Banach space. Proof. By a symmetric argument, it suffices to consider the case Λp(w) Banach: we want to show that there exists C>0 so that for simple functions S1, ,SN, ··· N N

p (5) Sj C Sj Λu(w), p ≤ k k j=1 Λu(w) j=1 X X since the Monotone Convergence Theorem implies (5) for arbitrary functions, and it is easy to show that this implies the existence of an equivalent norm. We will only consider the case N = 2 (the case N>2 is completely analogous). So, let j j Sj = λ χ j , j =1,2, and the sets A are pairwise disjoint. Then, k k Ak k

P 1 2 1 2 S1 + S2 = λkχA1 A2 + λkχA2 A1 + (λk + λj )χA1 A2 . k\ j j k\ j j k∩ j k k k,j X S X S X

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1 2 We choose a family of disjoint sets E ,E ,Ik,j : k, j such that { k j }

E1 = u(A1 A2), | k| k \ j j [ 2 2 1 Ek = u(Ak Aj ), | | \ j [ 1 2 Ik,j = u(A A ), | | k ∩ j where, as usual, for a set A Rn, u(A)= u(x)dx. Define also, ⊂ A R 1 1 C1 = λkχE ( Ik,j ), k∪ j k X S 2 2 C2 = λkχE ( Ij,k ), k∪ j k X S so that C1∗ =(S1)u∗,C2∗ =(S2)u∗ and (C1 + C2)∗ =(S1+S2)u∗. Finally,

S1 + S2 p = C1 + C2 p C( C1 p + C2 p ) k kΛu(w) k kΛ (w) ≤ k kΛ (w) k kΛ (w) = C( S1 p + S2 p ) k kΛu(w) k kΛu(w) · 

This result allows us to extend (2.6) of [Hu]:

Corollary 3.2. Let u/L1 (or du an infinite non-atomic measure) let p>0,and suppose Λp (w) is a Banach∈ space. Then p 1. u ≥ Proof. From the previous theorem we see that it is enough to consider u 1. Assume now that 0

1 N fj Λp(w) = 1 but fk . k k N p N−→ ∞ k=1 Λ (w) →∞ X

n pk k Choose Ak R , Ak+1 Ak, such that W ( Ak )=2− .Letfk=2 χAk. Then, ⊂ ··· ⊂ ⊂ | |

p 1/p k ∞ k 1/p fk p =2 χA ∗ w(t)dt =2 W(Ak ) =1. k kΛ (w) k | | Z0  


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But, if for N fixed, we set AN+1 = , then, ∅ N N k 1 1 j fk = 2 χ(Ak Ak+1) N N \ k=1 Λp(w) k=1  j=1  Λp(w) X X X N k p 1/p 1 ∞ j ∗ = 2 χ(Ak Ak+1) w(t) dt N \ 0 k=1 j=1  Z  X  X     N k p 1/p 1 ∞ j = 2 χ[ Ak+1 , Ak ) w(t) dt N | | | | 0 k=1 j=1  Z  X  X    N k p 1/p 1 Ak = 2j | | w(t) dt N k=1 j=1 Ak+1  X  X  Z| |  N 1/p 1 k+1 p kp (k+1)p (2 2) 2− 2− ≥ N − −  k=1  X
1 1/p Cp N  ≥ N N−→ ∞· →∞

Remark 3.3. The above result is not true for general measures. In fact, if du = 1 p 1/p δ0, the Dirac delta at the origin, then f Λu(w) = f(0) ( 0 w(t) dt) . Hence, p k k | | Λu(w) R, for all p>0. ≈ R 4. Indices Given Λp(w) we define the following invariant index: Definition 4.1. Given k =1,2, set ··· W (2kt) Dk =sup , t>0 W (t) and kp p(w)=inf p>0; for some C>0,Dk C2 ,k =1,2, . ≤ ···  We now prove the fundamental property of p(w): Theorem 4.2. 1/k p(w) Dk 2 . k −→→∞

Proof. Set ak =logDk.Thenaj+k aj+akand therefore there exists l 0 ≤ ≥ such that ak/k l ,ask (this is an easy exercise, see for example [GuR]). 1/k −→ l →∞ p(w) Thus, Dk D =e. So, we need to show that D =2 .Ifp>p(w), then k−→ kp →∞ p p(w) p(w) Dk C2 and hence D 2 ; i.e., D 2 .Conversely,ifD<2 ,choose ≤ ≤ p p(w) ≤ 1/k p 0

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Remarks 4.3. (i) As a consequence of this result, it turns out that if two weights w0,w1 satisfy W0 W1,thenp(w0)=p(w1). ≈ k (ii) It is not true, in general, that Dk = D . For this, it suffices to consider the k+1 weight w(t)=χ(0,1)(t)+2χ[1, )(t). Here, Dk =2 1. ∞ − (iii) The definition of p(w) is closely related to the condition on the weights Rp in [Ne] (notice that R1 = B1, ). ∞ The following result will be needed later. Recall that B1, p>1 Bp. ∞ ⊂

Lemma 4.4. There exists w p>1 Bp B1, . T ∈ \ ∞ Proof. We choose a decreasingT sequence pk k, with limk pk =1(pk =1,k= { } 6 1,2, ). Consider also ak k, ak > 0and ak = 1. Let us define ··· { } k

pPk 1 ∞ akpkt − if 0 t 1, w(t)= k=1 ≤ ≤ 0ift>1. P It is clear that the series converges uniformly on [0, 1]. Now,

p ∞ akt k if 0 1. P 1 Thus, t− W (t) 0, as t 0, and therefore w/B1, .Toprovethatw Bp, −→ → ∈ ∞ ∈ p>1, it suffices to show that if p = pk,thereexistsCp >0 such that condition (1) holds, for 0

1 w(t) ∞ ∞ a p ∞ pk 1 p k k pk p p dt = akpk t − − dt = (1 r − ) t pk p − r k=1 r k=1 Z X Z X − and r ∞ p pk p r− w(t) dt = akr − . 0 k=1 Z X Thus, if we let Sp =sup pk/(pk p)>0, Ip =infkpk/(pk p)<0andDp = k − − card k : pk >p ,andchoosek0 so that pk

Theorem 4.5. (a) If 1 p< ,thenw Bp if and only if p(w)

Proof. (a) If w Bp then (see [AM]) there exists q

k k 1 2 t 2 t w(s) C t w(s) ds ds w(s) ds, 2kqtq ≤ sq ≤ tq Zt Zt Z0 and hence, 2kt t w(s) ds (1 + C2kq) w(s) ds, ≤ Z0 Z0

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kq kq which implies Dk C2 .Conversely,ifDk C2 ,withq0, ≤ ≤

2k+1r 2k+1r ∞ w(t) ∞ w(t) 1 ∞ 1 p dt = p dt p kp w(t) dt t k t ≤ r 2 r k=0 2 r k=0 0 Z X Z X Z r r 1 ∞ D 2qC ∞ 1 k+1 w(t) dt w(t) dt. ≤ rp 2kp ≤ rp 2k(p q) k=0 0 k=0 − 0 X Z  X  Z

(b) This is clear by definition. Now if w p>1 Bp B1, , as in Lemma 4.4, ∈ \ ∞ then p(w) 1(infactp(w)=1). ≤ T (c) By Corollary 3.2, we have that p(w) 1. If p(w) > 1, then w Bp(w) and by (a) we obtain a contradiction. Now, with≥w as in (b), we show that∈ the converse does not hold.  Remarks 4.6. (i) An equivalent result to part (a) of the previous theorem was proved by Raynaud ([Ra]), using more complicated arguments. (ii) If w(t)=tα,with 1<α 0, then p(w)=1+α,andsop(w) 1isthe best we can say in (b) of the− theorem.≤ ≤ kp (iii) It is easy to show that Dk C2 is equivalent to saying that W is a p-quasi-concave function. ≤ (iv) There are other indices (Simonenko, Matuszewska-Orlicz, etc.) for which it a a is possible to show the equivalence with p(w). In particular p(w)=βW =σW = a a BW =QW (see [Ma] for the definitions).

References

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[Sa] E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145–158. MR 91d:26026 [Sj] P. Sj¨ogren, How to recognize a discrete maximal function, Indiana Univ. Math. J. 37 (1988), 891–898. MR 90f:42020

(M. J. Carro and J. Soria) Departamento Matematica` Aplicada i Analisi,` Universidad de Barcelona, 08071 Barcelona, Spain E-mail address: [email protected] E-mail address: [email protected]

(A. Garc´ıa del Amo) Departamento Analisis´ Matematico,´ Facultad de Ciencias, Mate- maticas,´ Universidad Complutense, 28040 Madrid, Spain Current address: Departamento Matem´atica Pura y Aplicada, Facultad de Ciencias, Univer- sidad de Salamanca, 37008 Salamanca, Spain E-mail address: [email protected]

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