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Weak-Type Weights and Normable Lorentz Spaces 1 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 <p< , we define ∞ 1/p ∞ p p p Λu(w)= f; f Λu(w) = fu∗(t) w(t)dt < . k k 0 ∞ Z If u 1, we will only write Λp(w). Classical examples are obtained by choosing ≡ (p/q) 1 p q,p w(t)=t − .InthiscaseΛ(w)=L . A classical result of G.G. Lorentz (see [Lo]) shows that Λ1(w) is a 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 850 M. J. CARRO, A. GARC´IA DEL AMO, AND J. SORIA Definition 1.1 ([CS1]). Let u and w beweightsasaboveand0<p< .We define ∞ 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 <s r< ∈ ∞ ≤ ∞ 1 r C s w(t)dt w(t) dt. r ≤ s Z0 Z0 x If we set W (x)= 0 w(t)dt,thenw B1, is equivalent to saying that W is quasi-concave (see [KPS]). We now prove∈ our main∞ theorem (which is the weak-type version of the main theoremR in [AM] and the end point case p = 1 of [Sa]). Recall 1 x 1 that Sf(x)=x− 0 f(t)dt is the Hardy operator, f ∗∗(t)=Sf∗(t)andLdec(w)is the set of all positive decreasing functions in L1(w). R Theorem 2.3. The following conditions are equivalent: (i) Λ1(w) is a Banach space. (ii) w B1, . ∈ 1∞ 1 , (iii) M :Λ (w) Λ ∞ ( w ) is bounded. −→ 1 (iv) For all r>0, rt− χ t r 1, C χ(0,r) 1 . { ≥ } L ∞(w) L (w) 1 1 , ≤ k (v) S : Ldec(w) L ∞ ( w ) . −→ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use WEAK-TYPE WEIGHTS AND NORMABLE LORENTZ SPACES 851 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). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 852 M. J. CARRO, A. GARC´IA DEL AMO, AND J. SORIA 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).
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