Embeddings of concave functions and duals of Lorentz spaces

Publ. Mat. 46 (2002), 489–515 EMBEDDINGS OF CONCAVE FUNCTIONS AND DUALS OF LORENTZ SPACES Gord Sinnamon Abstract A simple expression is presented that is equivalent to the norm of the Lp v→Lq uembedding of the cone of quasi-concave functions in the case 0 <q<p<∞. The result is extended to more general cones and the case q= 1 is used to prove a reduction principle which shows that questions of boundedness of operators on these cones may be reduced to the boundedness of related operators on whole spaces. An equivalent norm for the dual of the Lorentz space Γp(v)=f:∞ 0 (f∗∗ )pv1/p <∞ is also given. The expression is simple and concrete. An application is made to describe the weights for which the Hardy Littlewood Maximal Function is bounded on these Lorentz spaces. 1. Introduction The behaviour of the collection of non-negative, non-increasing functions in weighted Lebesgue spaces is well understood. Since [6] and [9] in the early 50’s, techniques involving properties of monotone functions have been used effectively to address a wide variety of questions in weighted norm inequalities, interpolation theory, and function space theory. For a few of the many see [1], [3], [7], [8], [13], [14], [15], [16], [17]. The study of the collection of concave functions has also had its successes. See [4], [5], [10], [11] and references there. Concave functions arise naturally in interpolation theory and much of the recent work 2000 Mathematics Subject Classification. Primary: 26D15; Secondary: 46E30, 42B25. Key words. Inequalities, weights, Lorentz space, quasi-concave functions, duality, maximal function. Support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. 490 G. Sinnamon shows that they are of equal importance in weighted norm inequalities and function spaces. Rather than working with the collection of non-increasing, concave functions, it is common to study the cone of quasi-concave functions. This is the set of non-negative functions fdefined on (0,∞) such that f(x) is non-decreasing and f(x)/x is non-increasing. Passing between the two collections is routine and the latter is more convenient for various reasons. The embedding question for this cone is a key to effectively using properties of concave functions: For which indices pand qand which weights uand vare the quasi-concave functions in Lp valso in Lq u? Various partial answers to this question are available. The case 0 < p≤q<∞in particular has been simply characterized and in [10], [11] very tight bounds on the norm of the embedding have been given. For the case 0 <q=1<p<∞sufficient conditions which are similar but not identical to the necessary ones were obtained in [17]. A complete answer to the embedding question was given in [5] but the conditions given are complicated and difficult to apply. Our object here is to give simple necessary and sufficient weight conditions that characterize the embedding of the cone of quasi-concave functions from Lp vto Lq u.We also give explicit upper and lower bounds on the norm of the embedding. This is accomplished in Theorem 2.6 and the embedding question for more general cones is answered in Theorem 2.7. In Section 3, the results are applied to give a reduction principle for operators acting on such cones. This shows the equivalence of the boundedness of an operator on the cone with the boundedness of two related operators on related spaces. The dual of the Lorentz space Γp(v) is characterized in Section 4. Theorem 4.1 gives a simple expression that is equivalent to the norm in the associate space, the K¨othe dual. As an application, in Section 5 we give weight conditions to characterize the boundedness of the HardyLittlewood Maximal Function between Lorentz spaces. To study quasi-concave functions we need an operator on non-negative functions whose images are quasi-concave functions. Although the generalized Stieltjes transformation h→ ∞ 0 x x+th(t)dt is used for this purpose by some authors, we will adopt the equivalent operator h→ ∞ 0 min(1, x/t)h(t)dt which is also popular. The lack of smoothness in the kernel min(1, x/t) will not bother us. It is important to note that the results we obtain Concave Functions and Lorentz Duals 491 can easily be re-cast in term of generalized Stieltjes transformations if desired. The weighted Lebesgue spaces already referred to are defined as follows. If vis a non-negative, Lebesgue measurable function (a weight) on (0,∞) then the weighted Lebesgue space Lp vis the collection of Lebesgue measurable functions fon (0,∞) for which fp,v ≡∞ 0|f|pv1/p ,0<p<∞ ess sup{x:v(x)>0}|f(x)|,p=∞ is finite. If v≡1 we drop the weight and write Lpand fp. Throughout the paper, products of the form 0 ·∞ are taken to be zero. For an index pwe define pby 1/p +1/p= 1. We say that the expressions Cand Aare equivalent and write C≈Aprovided there are positive constants kand Ksuch that kA ≤C≤KA. The constants depend only on the indices pand q. We keep track of the constants in the statements of theorems but will often avoid such details in the proofs, preferring to focus on essential features. In particular the extended Minkowski inequality for 0 <s<∞, min(1,21/s−1)(f1s+f2s)≤f1+f2s ≤max(1,21/s−1)(f1s+f2s) (1.1) will be used repeatedly in the form f1+f2s≈f1s+f2s.(1.2) 2. Hardy inequalities and concave functions In this section we give necessary and sufficient conditions on indices p, qand weights u,vfor the cone of quasi-concave functions in Lp vto be embedded in Lq uwhen 0 <q<p<∞. We also give upper and lower bounds for the norm of this embedding. This result is in Theorem 2.6 while an analogue for more general cones may be found in Theorem 2.7. See also Theorem 3.1. Corresponding known results for the case 0 <p≤ q<∞are stated in Proposition 2.8. We begin by looking at the embedding into Lq uof a smaller cone in L1 v. Known weighted Hardy inequalities are used to give a weight characterization in this situation. From there we expand the cone to include all quasi-concave functions and then use an invariance property of the cone of quasi-concave functions to pass from L1 vto Lp v. Let L+denote the collection of non-negative, measurable functions on (0,∞). We say f∈L+is quasi-concave and write f∈Ω0,1provided 492 G. Sinnamon f(x) is non-decreasing and f(x)/x is non-increasing. More generally, if α+β>0 we write f∈Ωα,β provided xαf(x) is non-decreasing and x−βf(x) is non-increasing. As mentioned we begin with weighted Hardy inequalities. Define the Hardy and dual Hardy operators Hαand Hβby Hαh(x)=x−αx 0 tαh(t)dt and Hβh(x)=xβ∞ x t−βh(t)dt. The sum of the two will arise frequently so for α+β>0 we introduce the operator Hβ αh(x)=Hαh(x)+Hβh(x) =∞ 0 min((t/x)α,(x/t)β)h(t)dt, h ∈L+. (2.1) Since we always suppose that α+β>0, the second form for Hβ α makes it clear that xαHβ αh(x) is non-decreasing and x−βHβ ah(x) is nonincreasing whenever h∈L+. That is, Hβ αL+⊆Ωα,β. It also makes it easy to check that ∞ 0 (Hβ αh1)h2=∞ 0 h1(Hα βh2),h 1,h 2∈L+.(2.2) Proposition 2.1. Suppose 0<q<1and U, V ∈L+.IfVis nonincreasing and C0is the least Cfor which ∞ 0x 0 hq U(x)dx1/q ≤C∞ 0 hV, h ∈L+, then (1−q)(1−q)/qC0≤∞ 0 Vq/(q−1)(H0U)q/(1−q)U(1−q)/q ≤C0/(q(1−q)). If Vis non-decreasing and C∞is the least Cfor which ∞ 0∞ x hq U(x)dx1/q ≤C∞ 0 hV, h ∈L+, then (1−q)(1−q)/qC∞≤∞ 0 Vq/(q−1)(H0U)q/(1−q)U(1−q)/q ≤C∞/(q(1−q)). Proof: The estimate for C0is from [16, Theorem 3.3] and the one for C∞ follows from the first by inversion (x→1/x) on the half line. Concave Functions and Lorentz Duals 493 These weighted Hardy inequalities can be combined to give a weight characterization for the boundedness of the L1 v→Lq uembedding of a sub-cone of the quasi-concave functions. This sub-cone is the image L+ under the map H1 0. Note that H1 0h(x)=∞ 0 min(1, x/t)h(x)dx =x 0∞ y h(t)dt tdy is non-decreasing and concave for all h∈L+. In particular, H1 0his quasi-concave. Theorem 2.2. If 0<q<1and u, v ∈L+then sup f∈H1 0L+ fq,u f1,v ≈∞ 0 (H0 1v)q/(q−1)(H0 qu)q/(1−q)v(1−q)/q .(2.3) More precisely, if the above equivalence is C≈Athen m(q)A≤C≤ M(q)Awhere m(q) = min(2−1,21−1/q)q(1 −q)and M(q) = max(21/q−1,2)(1 −q)1−1/q. (2.4) Proof: We prove only the equivalence and leave the careful tracking of constants to the interested reader. The supremum in (2.3) above is the least constant Cfor which ∞ 0 (H1 0h)qu1/q ≤C∞ 0 (H1 0h)v, h ∈L+.(2.5) Since ∞ 0 (H1 0h)v=∞ 0 h(H0 1v) the inequality (2.5) may be rewritten as ∞ 0x 0 h(t)dt +x∞ x h(t)dt tq u(x)dx1/q ≤C∞ 0 h(t)H0 1v(t)dt. By (1.2), C≈C0+C∞ (2.6) where C0and C∞are the least constants for which ∞ 0t 0 h(t)dtq u(x)dx1/q ≤C0∞ 0 h(t)H0 1v(t)dt, h∈L+,(2.7) 494 G. Sinnamon and ∞ 0x ∞ x h(t)dt tq u(x)dx1/q ≤C∞∞ 0 h(t)H0 1v(t)dt, h∈L+,(2.8) hold, respectively. Since H0 1vis non-increasing, the first part of Proposition 2.1, with V=H0 1vand U=u, applied to (2.7) shows that C0≈∞ 0 (H0 1v)q/(q−1)(H0u)q/(1−q)u(1−q)/q . To estimate C∞we replace h(t)/t by h(t) in (2.8) and apply the second part of Proposition 2.1, with V(t)=tH0 1v(t) and U(x)=xqu(x). Note that tH0 1v(t) is non-decreasing. We get C∞≈∞ 0 (H0 1v)q/(q−1)(Hqu)q/(1−q)u(1−q)/q . Adding the last two estimates and appealing to (2.6) yields C≈∞ 0 (H0 1v)q/(q−1)(H0 qu)q/(1−q)u(1−q)/q which completes the proof. The connection between the cone of quasi-concave functions and the sub-cone H1 0L+is well understood. The next lemma sets out the features of this relationship that we require here. Lemma 2.3. Let fbe a quasi-concave function and let ˜ fbe the least concave majorant of f. Then 1 2˜ f≤f≤˜ fand ˜ fis the pointwise limit of an increasing sequence of functions in H1 0L+. Proof: The definition of quasi-concave in [2, Definition 2.5.6] is slightly stronger than the one we give here, requiring that falso satisfy f(x)=0 if and only if x= 0. However, it is easy to see that only the zero function is lost by this restriction. Thus, [2, Proposition 2.5.10] applies and we see that a quasi-concave function fsatisfies 1 2˜ f≤f≤˜ f. Since ˜ fis non-negative and concave, we see that a= limx→0f(x) and b= limx→∞ f(x)/x exist and are non-negative. We may therefore write ˜ f(x)=a+bx+g(x) where gis a non-negative, concave function satisfying limx→0g(x) = limx→∞ g(x)/x = 0. If we take hn(t)=anχ(0,1/n)(t) then H1 0hn(x) is a non-decreasing sequence which converges pointwise to the constant function aas n→∞. If we take hn(t)=btχ(n,n+1)(t) then H1 0hn(x) is a non-decreasing sequence which converges pointwise to the function bx as n→∞. To complete the proof it remains to Concave Functions and Lorentz Duals 495 show that gis also the pointwise limit of a non-decreasing sequence of functions in H1 0L+. The concave function g(x) has a derivative for almost every x,g(x) is non-increasing and since limx→0g(x) = limx→∞ g(x)/x = 0 we have g(x)=x 0g(t)dt and limx→∞ g(x) = 0. Set hn(t)=(g(t)−g((n+1)t/n))/log((n+1)/n) and check that ∞ y hn(t)dt t=(n+1)y/n y g(t)dt t(n+1)y/n y dt t. These averages of gform a non-decreasing sequence indexed by nwhich converges to g(y) for almost every y. It follows that the functions H1 0hn(x)=x 0∞ y hn(t)dt tdy form a non-decreasing sequence in H1 0L+which, by the Monotone Convergence Theorem, converges to x 0 g(y)dy =g(x). This completes the proof. With this, Theorem 2.2 extends to the quasi-concave functions. Corollary 2.4. Suppose 0<q<1and u, v ∈L+. sup f∈Ω0,1 fq,u f1,v ≈∞ 0 (H0 1v)q/(q−1)(H0 qu)q/(1−q)u(1−q)/q . More precisely, if the above equivalence is C≈Athen m(q)A≤C≤ 2M(q)Awhere mand Mare given by (2.4). Proof: The lower bound requires only the observation that H1 0L+⊆ Ω0,1. For the upper bound we apply Lemma 2.3 to choose a nondecreasing sequence fnof functions in H1 0L+which converges pointwise to the least concave majorant ˜ fof f. By Theorem 2.2 and the Monotone Convergence Theorem, fq,u ≤˜ fq,u = lim n→∞ fnq,u ≈lim n→∞ fn1,v =˜ f1,v ≤2f1,v. The main advantage of working with Ω0,1rather than H1 0L+is this simple observation: Suppose p>0. If f(x)p=g(xp) then f∈Ω0,1if and only if g∈Ω0,1.(2.9) 496 G. Sinnamon This gives us the means of introducing Lp-norms into the denominator. Lemma 2.5. Suppose p, q ∈(0,∞)and u, v ∈L+. Then sup f∈Ω0,1 fq,u fp,v =sup g∈Ω0,1 gq/p,U g1,V 1/p where Vand Uare defined by V(xp)dxp=v(x)dx and U(xp)dxp=u(x)dx.(2.10) Proof: The substitution in (2.9) yields the equivalence. We note that U and Vhave been defined so that a change of variable yields fp q,u = gq/p,U and fp p,v =g1,V . Now we are ready to give our estimate of the norm of the Lp v→Lq u embedding of the cone of quasi-concave functions. Theorem 2.6. Suppose that 0<q<p<∞,1/r =1/q −1/p, and u, v ∈L+. Then sup f∈Ω0,1 fq,u fp,v ≈∞ 0 (H0 pv)−r/p(H0 qu)r/pu1/r .(2.11) More precisely, if the equivalence is C≈Athen m(q/p)1/pA≤C≤ (2M(q/p))1/pAwhere mand Mare defined by (2.4). Proof: Lemma 2.5 reduces the proof to an application of Corollary 2.4 with qreplaced by q/p and uand vreplaced by the weights Uand V from (2.10). That is, sup f∈Ω0,1 fq,u fp,v =sup g∈Ω0,1 gq/p,U g1,V 1/p ≈∞ 0 H0 1V(t)−r/pH0 q/pU(t)r/pU(t)dt1/r . Note that (q/p)/(1 −q/p)=r/p. We simplify this by making the substitution t→tpand using (2.10) to obtain ∞ 0 H0 1V(tp)−r/pH0 q/pU(tp)−r/pu(t)dt1/r .(2.12) Concave Functions and Lorentz Duals 497 Now we make the substitution x→xpin the integral forms of H0 1Vand H0 q/pUand use (2.10) again to get H0 1V(tp)=∞ 0 min(x/tp,1)V(x)dx =∞ 0 min((x/t)p,1)v(x)dx =H0 pv(t) and H0 q/pU(tp)=∞ 0 min((x/tp)q/p,1)U(x)dx =∞ 0 min((x/t)q,1)u(x)dx =H0 qu(t). Replacing these in (2.12) completes the proof of equivalence and we omit the tracking of constants. Theorem 2.6 is readily extended to a result for more general cones than the quasi-concave functions. Recall that Ωα,β is the collection of nonnegative functions fsuch that xαf(x) is non-decreasing and x−βf(x)is non-increasing. Theorem 2.7. Suppose that 0<q<p<∞,1/r =1/q −1/p, and u, v ∈L+.Ifα+β>0and Hβ αL+⊆F⊆Ωα,β then sup f∈F fq,u fp,v ≈∞ 0 (Hpα pβ v)−r/p(Hqα qβ u)r/pu1/r .(2.13) More precisely, if the above equivalence is C≈Athen (1/2)m(q/p)1/pA≤ C≤(2M(q/p))1/pAwhere mand Mare defined by (2.4). Proof: Set ρ=1/(α+β) and for each f∈F define gfby gf(x)=xαρf(xρ). Set F0,1={gf:f∈F}and note that for each f∈F,gf(x) is nondecreasing and gf(x)/x is non-decreasing. Thus F0,1⊆Ω0,1. Also, if f=Hβ αhfor some h∈L+then the change of variable t→tρyields gf(x)=∞ 0 min(1, x/t)[tαρh(tρ)ρtρ−1]dt so gf∈H1 0L+.ThusH1 0L+⊆F 0,1⊆Ω0,1. 504 G. Sinnamon Remark. It is not difficult to see that V0is non-zero if and only if L1 λ⊆ Γp,λ(v) and V∞is non-zero if and only if L∞ λ⊆Γp,λ(v). This explains the appearance of the terms involving g∗∞and g∗1and shows that, despite their appearance as technical byproducts of integration by parts in Theorem 3.1, they are an essential feature of the theory. Proof: Proving Theorem 4.1 will occupy us for the rest of this section. There are four steps in the proof: 1. Reduction to the case that λis Lebesgue measure on (0,∞). 2. Proof in the case that g∗is an integral. 3. Proof in the case that the associate norm of gis finite. 4. Elimination of the remaining case. The first step is readily accomplished by appealing to the Luxemburg Representation Theorem. Observe that Γp(v) represents the norm Γp,λ(v) in the sense of [2, Theorem 2.4.10]. That is, fΓp,λ(v)=f∗Γp(v)for all f∈Γp,λ(v). It follows that the associate norm is represented in the same way so gΓp,λ(v)=g∗Γp(v)for all g∈Γp,λ(v). In view of this is it enough to prove Theorem 4.1 in the case that λis Lebesgue measure on (0,∞). The second step is to prove the theorem in the case that g∗is an integral, specifically that g∗(t)=∞ t u(x)dx x for some u∈L+. In this case we have gΓp(v)= sup f∈Γp(v)∞ 0fg fΓp(v) = sup f∈L+∞ 0f∗g∗ f∗∗p,v = sup f∈L+∞ 0f∗∗u f∗∗p,v = sup F∈F ∞ 0Fu Fp,v where F={f∗∗ :f∈L+}. Since xf∗∗(x)=x 0f∗is non-decreasing and f∗∗(x) is non-increasing we see that F⊆Ω1,0. On the other hand, let h∈L+and set f(y)=∞ yhto see that H0 1h(x)=∞ 0 min(t/x, 1)h(t)dt =1 xx 0∞ y h(t)dt dy =f∗∗(x). Concave Functions and Lorentz Duals 505 It follows that H0 1L+⊆F⊆Ω1,0so we may apply Theorem 3.1 with q=1,r=p,α= 1, and β= 0 to get gΓp(v)≈∞ 0 H1u(t)pHp 0v(t)−pH0v(t)dt t1/p +x−11,u x−1p,v +∞ 0 H0u(t)pHp 0v(t)−pHpv(t)dt t1/p +11,u 1p,v . The terms above involving ucan all be written in terms of g∗. x−11,u =∞ 0 u(x)dx x=g∗(0) = g∗∞. 11,u =∞ 0 u(x)dx =∞ 0∞ t u(x)dx xdt =∞ 0 g∗(t)dt =g∗1. H1u(t)=t∞ t u(x)dx x=tg∗(t). H0u(t)=t 0 u(x)dx =t 0t y u(x)dx xdy =t 0 g∗(y)−g∗(t)dy =t(g∗∗(t)−g∗(t)). These substitutions give the desired result in the case that g∗is an integral. The second step is complete. We now pass to the third step and assume that gΓp(v)<∞. The first thing to establish is that limt→∞ g∗(t) = 0. For each positive integer nset fn=1 nχ(0,n)and note that f∗∗ n(t) = min(1/n, 1/t). By (4.1) and the Dominated Convergence Theorem, fnΓp(v)→0asn→∞. Since ghas finite Γp(v)-norm we see that 1 nn 0g∗=∞ 0f∗ ng∗also tends to zero as n→∞. Because g∗is monotone this implies that limt→∞ g∗(t) = 0 as desired. Now for γ>1 define uγ(x)=(g∗(x)−g∗(γx))/log(γ) and gγ(t)=∞ t uγ(x)dx x. Note that g∗ γ=gγ. The results of Step 2 apply so we have gγΓp(v)≈g∗ γp,v0+g∗∗ γ−g∗ γp,v∞+V0g∗ γ∞+V∞g∗ γ1.(4.4) 506 G. Sinnamon Using the fact that limt→∞ g∗(t) = 0 we can express gγas a moving average of g∗: gγ(t)= 1 log(γ)∞ t g∗(x)−g∗(γx)dx x=γt t g∗(x)dx xγt t dx x. It follows that for each t,gγ(t) is non-decreasing as γdecreases to 1 and that gγ(t) converges to g∗(t) for almost every t. By the Monotone Convergence Theorem, we have lim γ↓1g∗ γp,v0+V0g∗ γ∞+V∞g∗ γ1=g∗p,v0+V0g∗∞+V∞g∗1. Because Γp(v)is a Banach Function Space we also have lim γ↓1gγΓp(v)=g∗Γp(v)=gΓp(v). In order to conclude that (4.2) holds we still need to show that lim γ↓1g∗∗ γ−g∗ γp,v∞=g∗∗ −g∗p,v∞.(4.5) It is evident that the pointwise limit of g∗∗ γ−g∗ γis g∗∗ −g∗. By the Dominated Convergence Theorem, (4.5) will follow once we show that 2 log(2)(g∗∗ 2−g∗ 2)isinLp v∞and dominates g∗∗ γ−g∗ γfor 1 <γ≤2. Since g∗ 2≤g∗and g∗Γp(v)<∞we have g∗ 2Γp(v)<∞because Γp(v)is a Banach Function Space. In view of (4.4) this implies that g∗∗ 2−g∗ 2p,v∞<∞and hence 2 log(2)(g∗∗ 2−g∗ 2)isinLp v∞. To see that 2 log(2)(g∗∗ 2−g∗ 2) dominates g∗∗ γ−g∗ γwe calculate as follows: log(γ)(g∗∗ γ(t)−g∗ γ(t)) =1 tt 0γy y g∗(x)dx xdy −γt t g∗(x)dx x =1 tt 0 g∗(x)x x/γ dy dx x+1 tγt t g∗(x)t x/γ dy dx x−γt t g∗(x)dx x =(1−1/γ)g∗∗(t)−1 γt γt t g∗(x)dx =(1−1/γ)g∗∗(t)−1 γt −tγt t g∗. Concave Functions and Lorentz Duals 507 If 1 <γ≤2 then 1 −1/γ ≤log(γ). Also, for each tthe moving average 1 γt−tγt tg∗is a non-increasing function of γ.Thus g∗∗ γ(t)−g∗ γ(t)=1−1/γ log(γ)g∗∗(t)−1 γt −tγt t g∗ ≤g∗∗(t)−1 2t−t2t t g∗ = 2 log(2)(g∗∗ 2(t)−g∗ 2(t)). This completes Step 3, showing that (4.2) holds whenever its left hand side is finite. If both sides are infinite then (4.2) holds trivially. Step 4 of the proof is to eliminate the remaining case by showing that if the right hand side of (4.2) is finite then so is the left hand side. For each positive integer n, define gn= min(nχ(0,n),g∗) and note that g∗ n=gn. The sequence g∗ n is non-decreasing and converges pointwise to g∗as n→∞so gn→g in the Banach Function Space Γp(v). To show that gΓp(v)<∞we show that the norms gnΓp(v)are bounded independently of n.Todo this we note that (4.1) implies that gnΓp(v)≤nχ(0,n)Γp(v)<∞ so the results of Step 3 apply and we have gnΓp(v)≈g∗ np,v0+g∗∗ n−g∗ np,v∞+V0g∗ n∞+V∞g∗ n1. Again it is easy to handle three of the terms. Since g∗ n≤g∗we have g∗ np,v0≤g∗p,v0,g∗ n∞≤g∗∞, and g∗ n1≤g∗1. Therefore, the sum of these three terms is bounded independently of nby the right hand side of (4.2) which is assumed to be finite. The fourth term, g∗∗ n−g∗ np,v∞, is also bounded by a multiple of the right hand side of (4.2) but a little more work is required to demonstrate this. The function g∗ nis non-increasing and bounded by n. Therefore it takes the value non an interval of the form (0,t n) for some tn≥0. When 0 <t<t nwe have g∗∗ n(t)−g∗ n(t) = 0. When tn<t<nwe have g∗∗ n(t)−g∗ n(t)=g∗∗ n(t)−g∗(t)≤g∗∗(t)−g∗(t). When t>nwe have g∗∗ n(t)−g∗ n(t)=g∗∗ n(t)=1 tt 0 g∗ n≤1 tn 0 g∗=n tg∗∗(n). Thus g∗∗ n−g∗ np,v∞≤g∗∗ −g∗p,v∞+ng∗∗(n)∞ n v∞(t)dt tp1/p 508 G. Sinnamon and our object is to show that the last two summands are bounded by the right hand side of (4.2). The first is trivially so and we write the second as n(g∗∗(n)−g∗(n))∞ n v∞(t)dt tp1/p +ng∗(n)∞ n v∞(t)dt tp1/p .(4.6) Observe that t(g∗∗(t)−g∗(t)) = t 0g∗∗(y)−g∗(t)dy is non-decreasing so n(g∗∗(n)−g∗(n)) ∞ n v∞(t)dt tp1/p ≤∞ n (g∗∗(t)−g∗(t))pv∞(t)dt1/p ≤g∗∗ −g∗p,v∞. The second term in (4.6) requires some integration using (4.3). np∞ n v∞(t)dt tp=−np pt 0 v(x)dx +tp∞ t v(x)dx xp1−p ∞ n ≤1 p1 npn 0 v(x)dx +∞ n v(x)dx xp1−p =n 0 v0(t)dt +1 p∞ 0 v(x)dx xp1−p . Therefore, ng∗(n)∞ n v∞(t)dt tp1/p ≤n 0 g∗(t)pv0(t)dt +1 pg∗p ∞∞ 0 v(x)dx xp1−p1/p ≤g∗p p,v0+1 pg∗p ∞Vp 01/p . Which is bounded by (a multiple of) the right hand side of (4.2). This completes Step 4 and the proof of Theorem 4.1. Concave Functions and Lorentz Duals 509 We remark that the term g∗∗ −g∗p,v∞in (4.2) may be replaced by sup h∗≤g∗ h∗∗ −h∗p,v∞. Although this new term may be substantially larger than g∗∗ −g∗p,v∞ for example when g∗is constant, the equivalence (4.2) is not affected due to the presence of the other terms. Indeed, the proof of Theorem 4.1 is simpler with the new term in place. 5. The Hardy-Littlewood Maximal Function The reduction principle in Theorem 3.2 can be used to give criteria to determine whether or not the Hardy-Littlewood Maximal Function is bounded between Lorentz spaces. If fis a locally integrable function on Rnwe define Mf to be Mf(x) = sup 1 µn(Q)Q |f|dµn where the supremum is taken over all cubes Qcontaining xwhose sides are parallel to the axes. Here µndenote Lebesgue measure on Rn. Theorem 5.1. Suppose p, q ∈(1,∞)and u, v ∈L+. Define Vby V(t)= 1 tpt 0 v(x)dx +∞ t v(x)dx xp. Then M:Γ p,µn(v)→Γq,µn(u)if and only if: Either 1<p≤q<∞, and all of sup y>0∞ y u(x)dx xq1/q y 0 (log(y/t))p−1V(t)1−pdt t1/p , sup y>0∞ y (log(x/y))qu(x)dx xq1/q V(y)−1/p, sup y>0∞ y u(x)dx xq1/q py 0 V(t)1−pdt t−V(y)1−p1/p ,and sup y>0y 0 u(x)dx1/q (ypV(y))−1/p , 510 G. Sinnamon are finite; or 1<q<p<∞,1/r =1/q −1/p, and all of ∞ 0∞ y u(x)dx xqr/p y 0 (log(y/t))p−1V(t)1−pdt tr/p u(y)dy yq, ∞ 0∞ y (log(x/y))qu(x)dx xqr/q V(y)−r/q d(−V(y)), ∞ 0∞ y u(x)dx xqr/qp y 0 V(t)1−pdt t−V(y)1−pr/q d(ypV(y)) ypV(y)p,and ∞ 0y 0 u(x)dxr/p (ypV(y))−r/p u(y)dy are finite. Proof: We cite [2, Theorem 3.8] for the well known equivalence (Mf)∗≈ f∗∗. It implies that M:Γ p,µn(v)→Γq,µn(u) if and only if sup f∈L+∞ 01 xx 0f∗∗qu(x)dx1/q ∞ 0(f∗∗)pv1/p <∞. That is, T:F∩Lp v→Lq u (5.1) where Tis the operator TF(x)=1 xx 0Fand F={f∗∗ :f∈L+}.As we observed in Part 2 of the proof of Theorem 4.1, H0 1L+⊆F⊆Ω1,0. Thus, we can apply Theorem 3.2 with α=1,β= 0, and Ls w=Lq uto see that (5.1) holds if and only if TH1:Lp v1→Lq u,(5.2) TH0:Lp v2→Lq u,(5.3) if x−1∈Lp vthen T(x−1)∈Lq u,and if 1 ∈Lp vthen T(1) ∈Lq u. Concave Functions and Lorentz Duals 511 Since T(x−1)≡∞and T(1) ≡1 the latter two conditions reduce to u≡0or ∞ 0 v(x)dx xp=∞,(5.4) and ∞ 0 v<∞=⇒∞ 0 u<∞.(5.5) The conditions (5.2) and (5.3) reduce to weighted norm inequalities for which necessary and sufficient conditions are known. Our task now is to simplify the known conditions using the definitions of v1and v2from Theorem 3.2. We have v1(t)=tp−1t 0 v(x)dx +tp∞ t v(x)dx xppt 0 v(x)dx1−p and v2(t)=tp−1t 0 v(x)dx +tp∞ t v(x)dx xpptp∞ t v(x)dx xp1−p . In terms of Vthese become ptpv1(t)1−p=V(t)−pd dt(−V(t)) and(5.6) pv2(t)1−p=(tpV(t))−pd dt(tpV(t)).(5.7) The operator in (5.2) is TH1f(x)= 1 xx 0 1 yy 0 tf(t)dt dy =1 xx 0 log(x/t)tf(t)dt so, with g(t)=tf(t), we see that (5.2) holds if and only if the inequality ∞ 0x 0 log(x/t)g(t)dtq u(x)dx xq1/q ≤C∞ 0 g(t)pv1(t)dt tp1/p (5.8) holds for some C>0 and all g∈L+.By[18, Theorems 1 and 2], (5.8) holds if and only if: Either 1 <p≤q<∞, sup y>0∞ y u(x)dx xq1/qy 0 (log(y/t))ptpv1(t)1−pdt1/p <∞,(5.9) and sup y>0∞ y (log(x/y))qu(x)dx xq1/qy 0 tpv1(t)1−pdt1/p <∞;(5.10) 512 G. Sinnamon or 1 <q<p<∞,1/r =1/q −1/p, ∞ 0∞ y u(x)dx xqr/p y 0 (log(y/t))ptpv1(t)1−pdtr/p u(y)dy yq<∞, (5.11) and ∞ 0∞ y (log(x/y))qu(x)dx xqr/q y 0 tp v1(t)1−pdtr/q yp v1(y)1−pdy<∞. (5.12) The operator in (5.3) is TH0f(x)= 1 xx 0∞ y f(t)dt dy =1 xx 0 tf(t)dt +∞ x f(t)dt, a sum of two Hardy operators. Thus (5.3) holds if and only if the two weighted Hardy inequalities ∞ 0x 0 g(t)dtq u(x)dx xq1/q ≤C∞ 0 g(t)pv2(t)dt tp1/p and ∞ 0∞ x f(t)dtq u(x)dx1/q ≤C∞ 0 f(t)pv2(t)dt1/p hold for some constant C>0 and all g∈L+and f∈L+respectively. The conditions (see [12]) under which these hold are: Either 1 <p≤ q<∞, sup y>0∞ y u(x)dx xq1/q y 0 tpv2(t)1−pdt1/p <∞,(5.13) and sup y>0y 0 u(x)dx1/q ∞ y v2(t)1−pdt1/p <∞;(5.14) or 1 <q<p<∞,1/r =1/q −1/p, ∞ 0∞ y u(x)dx xqr/q y 0 tp v2(t)1−pdtr/q yp v2(y)1−pdy<∞,(5.15) Concave Functions and Lorentz Duals 513 and ∞ 0y 0 u(x)dxr/p ∞ y v2(t)1−pdtr/p u(y)dy<∞.(5.16) Using the properties (5.4) and (5.5) and the substitutions (5.6) and (5.7) to eliminate v1and v2, (5.9), (5.10), (5.13), and (5.14) can be simplified to yield the four weight conditions given in the case 1 <p≤ q<∞. Similarly, (5.11), (5.12), (5.15), and (5.16) simplify to yield the four weight conditions given in the case 1 <q<p<∞. We have shown that the weight conditions given in the statement of the theorem, together with (5.4) and (5.5), are necessary and sufficient for the boundedness of M. All that remains is to show that (5.4) and (5.5) are consequences of the weight conditions. Write V(t)=∞ 0max(t, x)−pv(x)dx to see that V(t)≤V(0) = ∞ 0v(x)dx/xp.IfV(0) <∞then for any y>0, y 0 log(y/t)p−1V(t)1−pdt t≥V(0)1−py 0 log(y/t)p−1dt t=∞. In view of this, the first weight condition in either the case 1 <p≤q<∞ or the case 1 <q<p<∞can hold only if uis almost everywhere 0. Thus (5.4) holds. If ∞ 0v<∞it follows that ypV(y) is bounded above and hence the fourth weight condition in either the case 1 <p≤q<∞or the case 1 <q<p<∞would fail unless ∞ 0u<∞. Thus (5.5) also holds. This completes the proof. We would like to thank the referee for pointing out that the weight conditions (5.4) and (5.5) follow from the others in Theorem 5.1. Since (Mf)∗≈f∗∗ the boundedness of M:Γ p,µn(v)→Λq,µn(u) reduces to a straightforward application of Theorems 2.7 and 2.8 with α= 1 and β= 0. Here Λq,µn(u)={f:f∗q,u <∞}. Theorem 5.2. Let p, q ∈(1,∞)and u, v ∈L+. Then M:Γ p,µn(v)→ Λq,µn(u)if and only if: Either 1<p≤q<∞and sup y>0y 0 u(x)dx+yq∞ y u(x)dx xq1/qy 0 v(x)dx+yp∞ y v(x)dx xp−1/p is finite; or 1<q<p<∞,1/r =1/q −1/p, and ∞ 0y 0 u(x)dx+yq∞ y u(x)dx xqr/p y 0 v(x)dx+yp ∞ y v(x)dx xp−r/p u(y)dy is finite.