EMBEDDINGS OF CONCAVE FUNCTIONS AND DUALS OF LORENTZ SPACES

Abstract A simple expression is presented that is equivalent to the norm of the Lv → Lu embedding 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


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 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 f defined 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 p and q and which weights u and v are the quasi-concave functions in L p v also in L q 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 L p v to L q 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 Γ p (v) is characterized in Section 4. Theorem 4.1 gives a simple expression that is equivalent to the norm in the associate space, the Köthe dual.As an application, in Section 5 we give weight conditions to characterize the boundedness of the Hardy-Littlewood 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+t h(t) dt is used for this purpose by some authors, we will adopt the equivalent operator 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 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 v is a non-negative, Lebesgue measurable function (a weight) on (0, ∞) then the weighted Lebesgue space L p v is the collection of Lebesgue measurable functions f on (0, ∞) for which we drop the weight and write L p and f p .Throughout the paper, products of the form 0 • ∞ are taken to be zero.For an index p we define p by 1/p + 1/p = 1.We say that the expressions C and A are equivalent and write C ≈ A provided there are positive constants k and K such that kA ≤ C ≤ KA.The constants depend only on the indices p and 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 < ∞, will be used repeatedly in the form

Hardy inequalities and concave functions
In this section we give necessary and sufficient conditions on indices p, q and weights u, v for the cone of quasi-concave functions in L p v to be embedded in L q u when 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 L q u of a smaller cone in L 1 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 L 1 v to L p v .Let L + denote the collection of non-negative, measurable functions on (0, ∞).We say f ∈ L + is quasi-concave and write f ∈ Ω 0,1 provided f (x) is non-decreasing and f (x)/x is non-increasing.More generally, if α + β > 0 we write f ∈ Ω α,β provided x α f (x) is non-decreasing and x −β f (x) is non-increasing.
As mentioned we begin with weighted Hardy inequalities.Define the Hardy and dual Hardy operators H α and H β by The sum of the two will arise frequently so for α + β > 0 we introduce the operator (2.1) Since we always suppose that α + β > 0, the second form for Proof: The estimate for C 0 is from [16,Theorem 3.3] and the one for C ∞ follows from the first by inversion (x → 1/x) on the half line.
These weighted Hardy inequalities can be combined to give a weight characterization for the boundedness of the L 1 v → L q u embedding of a sub-cone of the quasi-concave functions.This sub-cone is the image L + under the map H 1 0 .Note that is non-decreasing and concave for all h ∈ L + .In particular,

More precisely, if the above equivalence is
and (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 C for which the inequality (2.5) may be rewritten as By (1.2), where C 0 and C ∞ are the least constants for which hold, respectively.Since H 0 1 v is non-increasing, the first part of Proposition 2.1, with V = H 0 1 v and U = u, applied to (2.7) shows that 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) = tH 0 1 v(t) and U (x) = x q u(x).Note that tH 0 1 v(t) is non-decreasing.We get Adding the last two estimates and appealing to (2.6) yields which completes the proof.
The connection between the cone of quasi-concave functions and the sub-cone H 1 0 L + is well understood.The next lemma sets out the features of this relationship that we require here.Lemma 2.3.Let f be a quasi-concave function and let f be the least concave majorant of f .Then 1  2 f ≤ f ≤ f and f is the pointwise limit of an increasing sequence of functions in H 1 0 L + .Proof: The definition of quasi-concave in [2, Definition 2.5.6] is slightly stronger than the one we give here, requiring that f also 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 f satisfies 1  2 f ≤ f ≤ f .Since f is non-negative and concave, we see that a = lim x→0 f (x) and b = lim x→∞ f (x)/x exist and are non-negative.We may therefore write f (x) = a+bx+g(x) where g is a non-negative, concave function satisfying is a non-decreasing sequence which converges pointwise to the function bx as n → ∞.To complete the proof it remains to show that g is also the pointwise limit of a non-decreasing sequence of functions in H 1 0 L + .The concave function g(x) has a derivative for almost every x, g (x) is non-increasing and since lim x→0 g(x) = lim x→∞ g(x)/x = 0 we have These averages of g form a non-decreasing sequence indexed by n which converges to g (y) for almost every y.It follows that the functions form a non-decreasing sequence in H 1 0 L + which, by the Monotone Convergence Theorem, converges to This completes the proof.
With this, Theorem 2.2 extends to the quasi-concave functions.
More precisely, if the above equivalence is C ≈ A then m(q)A ≤ C ≤ 2M (q)A where m and M are given by (2.4).

Proof:
The lower bound requires only the observation that H 1 0 L + ⊆ Ω 0,1 .For the upper bound we apply Lemma 2.3 to choose a nondecreasing sequence f n of functions in H 1 0 L + which converges pointwise to the least concave majorant f of f .By Theorem 2.2 and the Monotone Convergence Theorem, The main advantage of working with Ω 0,1 rather than This gives us the means of introducing L p -norms into the denominator.Lemma 2.5.Suppose p, q ∈ (0, ∞) and u, v ∈ L + .Then where V and U are defined by

10)
Proof: The substitution in (2.9) yields the equivalence.We note that U and V have 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 L p v → L q u embedding of the cone of quasi-concave functions.
Proof: Lemma 2.5 reduces the proof to an application of Corollary 2.4 with q replaced by q/p and u and v replaced by the weights U and V from (2.10).That is, Note that (q/p)/(1 − q/p) = r/p.We simplify this by making the substitution t → t p and using (2.10) to obtain Now we make the substitution x → x p in the integral forms of H 0 1 V and H 0 q/p U and use (2.10) again to get and 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 Ω α,β is the collection of nonnegative functions f such that x α f (x) is non-decreasing and x −β f (x) is non-increasing.Theorem 2.7.Suppose that 0 < q < p < ∞, 1/r = 1/q − 1/p, and

More precisely, if the above equivalence is
where m and M are defined by (2.4).
The change of variable x → x ρ shows that where We have where the last equivalence relies on Lemma 2.3.Thus, by Theorem 2.6, we have The definitions of U and V above and the changes of variable x → x 1/ρ followed by t → t 1/ρ show that This completes the proof.
Next we present a statement of the corresponding result in the case 0 < p ≤ q < ∞.This result is taken from [10, Theorem 3] and formulated in our notation to facilitate comparision with Theorem 2.7.
More precisely, if the above equivalence is C ≈ A then A ≤ C ≤ 2A.

A reduction principle for operators acting on cones
An operator may be unbounded as a map from L p v to L q u and yet still map a cone in L p v boundedly into L q u .Theorem 3.2 gives a result that reduces questions of boundedness on the cones Ω α,β to boundedness of related operators between whole spaces.We begin by working with the weight condition in (2.13) to give an equivalent expression in a less compact but more convenient form.
To avoid introducing additional notation, we use the expression x −α in several places as a substitute for the power function f defined by f (x) = x −α .The same applies to the expression x β .Theorem 3.1.Suppose that 0 < q < p < ∞, 1/r = 1/q − 1/p, and x β p,v .

More precisely, if the above equivalence is
Proof: By the Monotone Convergence Theorem it is enough to establish the theorem in the case that u is compactly supported in (0, ∞).
Under this assumption we apply Theorem 2.7, break the right hand side of (2.13) into two pieces and integrate by parts in each.Since The limit of as t → ∞ is zero because u is compactly supported and the limit as t → 0 is x −α r q,u / x −α r p,v by the Monotone Convergence Theorem.For (3.2) a similar argument shows that Substituting the results of these two calculations into (3.1) and (3.2) and applying (1.2) completes the proof of equivalence.As usual, we omit the tedious tracking of constants.Now we present our reduction principle.We suppose that T is an integral operator with non-negative kernel, that is, Here v 1 and v 2 are defined by Moreover, if C is the norm of the embedding T : F ∩ L p v → L s w and then C ≈ A with constants depending only on p, q, α, and β.

Proof:
The adjoint operator T is given by We Now we set up an application of Theorem 3.1.The boundedness of T : (3.9) Theorem 3.1 with u = T g, q = 1 and r = p shows that (3.9) is equivalent to sup

Lorentz spaces
The Lorentz space Γ p,λ (v) is defined to be the collection of λ-measurable functions such that and f * is the non-increasing rearrangement of f with respect to the measure λ.Refer to [2] for definitions and basic results regarding rearrangements and rearrangement-invariant spaces.We will assume that λ is a resonant measure space, that is, that λ is totally σ-finite and either non-atomic or completely atomic with all atoms having equal measure.In this case Γ p,λ (v) is a rearrangementinvariant Banach Function Space provided p ≥ 1 and v satisfies 0 The associate space, Γ p,λ (v) , consisting of all λ-measurable functions g for which is also a rearrangement-invariant Banach Function Space.In many cases the associate space may be identified with the usual Banach space dual.Precise conditions for this to occur may be found in [2].
When λ is Lebesgue measure on the half line we drop the measure and write Γ p (v) and Γ p (v) for the Lorentz space and its associate space.
Our objective here is to give a simple expression which is equivalent to the associate norm g Γ p,λ (v) .
In [5,Theorem 3.1], under the modest assumptions that , with equivalent norms.Also, in [4] and upcoming work by A. Gogatishvili and R. Kerman, a simple formula for such a w is given.Our equivalent norm for Γ p (v) is closely related but breaks g Γp(v) into two parts corresponding to the size, g * , and the smoothness g * * − g * of g.Note that the last two terms in (4.2) below are only present in the excluded cases, when (S, λ) is a resonant measure space, and v satisfies (3.1).If g is a λ-measurable function on S then where , and The constants in the equivalence (4.2) depend only on p.
This explains the appearance of the terms involving g * ∞ and g * 1 and 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 λ is Lebesgue measure on (0, ∞).
2. Proof in the case that g * is an integral.
3. Proof in the case that the associate norm of g is finite.
4. Elimination of the remaining case.
The first step is readily accomplished by appealing to the Luxemburg Representation Theorem.Observe that Γ p (v) represents the norm Γ p,λ (v) in the sense of [2,Theorem 2.4.10].That is, It follows that the associate norm is represented in the same way so In view of this is it enough to prove Theorem 4.1 in the case that λ is Lebesgue measure on (0, ∞).
The second step is to prove the theorem in the case that g * is an integral, specifically that for some u ∈ L + .In this case we have where x 0 f * is non-decreasing and f * * (x) is non-increasing we see that F ⊆ Ω 1,0 .On the other hand, let h ∈ L + and set f (y) = ∞ y h to see that It follows that H 0 1 L + ⊆ F ⊆ Ω 1,0 so we may apply Theorem 3.1 with q = 1, r = p , α = 1, and β = 0 to get The terms above involving u can all be written in terms of g * .
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 lim t→∞ g * (t) = 0.For each positive integer n set f n = 1 n χ (0,n) and note that f * * n (t) = min(1/n, 1/t).By (4.1) and the Dominated Convergence Theorem, n g * also tends to zero as n → ∞.Because g * is monotone this implies that lim t→∞ g * (t) = 0 as desired.Now for γ > 1 define Note that g * γ = g γ .The results of Step 2 apply so we have Using the fact that lim t→∞ g * (t) = 0 we can express g γ as a moving average of g * : It follows that for each t, g γ (t) is non-decreasing as γ decreases to 1 and that g γ (t) converges to g * (t) for almost every t.
By the Monotone Convergence Theorem, we have lim Because Γ p (v) is a Banach Function Space we also have In order to conclude that (4.2) holds we still need to show that It is evident that the pointwise limit of g * * γ − g * γ is g * * − g * .By the Dominated Convergence Theorem, (4.5) will follow once we show that 2 log(2)(g If 1 < γ ≤ 2 then 1 − 1/γ ≤ log(γ).Also, for each t the moving average 1 γt−t γt t g * is a non-increasing function of γ.Thus ).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 g n = min(nχ (0,n) , g * ) and note that g * n = g n .The sequence g * n is non-decreasing and converges pointwise to g * as n → ∞ so g n → g in the Banach Function Space Γ p (v) .To show that g Γp(v) < ∞ we show that the norms g n Γp(v) are bounded independently of n.To do this we note that (4.1) implies that so the results of Step 3 apply and we have . Again it is easy to handle three of the terms.Since g * n ≤ g * we have Therefore, the sum of these three terms is bounded independently of n by 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 * n is non-increasing and bounded by n.Therefore it takes the value n on an interval of the form (0, t n ) for some t n ≥ 0. When 0 < t < t n we have Thus 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 The second term in (4.6) requires some integration using (4.3).
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.
We remark that the term g * * − g * p ,v∞ in (4.2) may be replaced by sup 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.

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 f is a locally integrable function on R n we define Mf to be where the supremum is taken over all cubes Q containing x whose sides are parallel to the axes.Here µ n denote Lebesgue measure on R n .