Sharp constants related to the triangle inequality in Lorentz spaces

We study the Lorentz spaces $L^{p,s}(R,\mu)$ in the range $1<p<s\le \infty$, for which the standard functional $$ ||f||_{p,s}=(\int_0^\infty (t^{1/p}f^*(t))^s\frac{dt}{t})^{1/s} $$ is only a quasi-norm. We find the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm: $$ ||f||_{(p,s)}=\inf\bigg\{\sum_{k}||f_k||_{p,s}\bigg\}, $$ where the infimum is taken over all finite representations $f=\sum_{k}f_k. $ We also prove that the decomposition norm and the dual norm $$ ||f||_{p,s}'= \sup\left\{\int_R fg d\mu: ||g||_{p',s'}=1\right\} $$ agree for all values $p,s>1$.


Introduction
The study of the normability of the Lorentz spaces L p,s (R, µ) goes back to the work of G.G. Lorentz [10,11] (see also [13,3,2] for a more recent account of the normability results for the weighted Lorentz spaces). The condition defining these spaces is given in terms of the distribution function and, equivalently, the non-increasing rearrangement of f (see [1] for standard notations and basic definitions): with the usual modification if s = ∞. Lorentz proved that p,s is a norm, if and only if 1 ≤ s ≤ p < ∞, and the space L p,s (R, µ) is always normable (i.e., there exists a norm equivalent to p,s ), for the range 1 < p < s ≤ ∞ (for the remaining cases it is known that L p,s (R, µ) cannot be endowed with an equivalent norm). From now on we will only consider the range 1 < p < ∞, 1 ≤ s ≤ ∞.
Note that the spaces L p,s , with p < s, play an important role not only as dual spaces for the Banach spaces L p ′ ,s ′ (see [1,7]). For example, they arise naturally in limiting embeddings of Lipschitz spaces ( [8]).
The study of the normability for p < s was carried out by means of the maximal norm: It is easy to see that * p,s is always a norm. Moreover, one can prove that * p,s is equivalent to p,s , with the following optimal estimates: (see [14,9]; as usual, p ′ denotes the conjugate exponent, 1/p+1/p ′ = 1).
As a consequence of the fact that p,s is equivalent to a norm, it is easy to see that it is a quasi-norm satisfying the triangle inequality, uniformly on the number of terms: there exists a constant c p,s > 0 such that, for every finite collection {f k } k=1,··· ,N ⊂ L p,s (R, µ): It can readily be proved the converse result; namely, (1.2) is equivalent to the fact that p,s is normable and, even more, that an alternative equivalent norm is given by means of the following decomposition norm: where the infimum is taken over all finite representations f = k f k .
It is easy to prove that (p,s) is a norm, equivalent to p,s , that agrees with p,s if 1 ≤ s ≤ p. Moreover, the best constant in the inequality f p,s ≤ c p,s f (p,s) is the same as the optimal one in (1.2). One of the main problems studied in this paper is to find the best constant in the triangle inequality (1.2) and its continuous version, the Minkowski integral inequality (the control of these constants is sometimes very relevant for estimating different type of integral operators, where the use of the maximal norm and the inequalities (1.1) do not usually give optimal results).
For the Lorentz norms we have the following version of Hölder's inequality: if f ∈ L p,s (R, µ) and (see [1, p. 220]).
In the theory of Banach Function Spaces (L p,s (R, µ) is the canonical example in this context), and based on (1.4), it is also very natural to consider another norm defined in terms of the Köthe duality, which is denoted as the dual norm: As in the case of the decomposition norm, ′ p,s is a norm, equivalent to p,s and f ′ p,s = f p,s , if 1 ≤ s ≤ p (see (4.5)). Therefore, . The main result that we will prove in this paper shows that the decomposition and dual norm agree in the whole range of indices (Theorem 5.2), in spite of their quite different definitions. We also find the best constants in the inequalities relating either of these norms and p,s (see (4.4), Theorem 4.4, and Remark 4.3). In particular, these results give an alternative proof of the normability of L p,s (R, µ) with optimal estimates. We would like to remark that, while (1.1) follows easily from standard estimates, finding the best constants in our context requires new ideas and much more complicated constructions.
In Section 2 we prove several technical lemmas used in subsequent sections. Section 3 introduces one of the key tools used in the paper: the level function (see Theorems 3.1 and 3.2). Sections 4 and 5 are the core of the paper, dealing with both the dual and decomposition norms, and proving the main results already mentioned above. Finally, in Section 6 we obtain the best constant in both the triangle and Minkowski's integral inequalities for the Lorentz spaces.

Auxiliary propositions
In this section we consider some auxiliary results that will be used in the sequel. We begin with some general inequalities.
Corollary 2.2. Let g be a non-increasing nonnegative function on [0, 1] and let 0 < α < 1. Then Proof. We will prove that for all x, y ∈ (0, 1) Then (2.2) will follow from (2.3) if we take x = t 1/p , y = t 1/p ′ . To prove (2.3), fix y and denote We have This implies that x s = y s ′ , and hence, the function ϕ has an absolute minimum for x = y 1/(s−1) and this minimum is 0, which proves (2.3).
The following lemma gives the sharp constant in the relation between Lorentz norms with different second indices (see [14, p. 192] We consider now some auxiliary statements related to dual norm and decomposition norm.
We shall use the following properties of the decomposition norm.
Proof. It is known that there exists a measure preserving transforma- ). Since g k • σ and g k are equimeasurable, we have that This implies (2.9).
It will be proved below that for any f we have the equality in (2.9).
Proof. The equality (2.15) is immediate. We shall prove (2.16). Denote α = 1 − s ′ /p ′ and set We have To evaluate the dual norm of h, we assume that g ∈ L p ′ ,s ′ (R + ), g ≥ 0 and g p ′ ,s ′ = 1. Applying (2.1), Hölder's inequality (1.4), and (2.18), we obtain On the other hand, if, We prove now the second equality in (2.16) (in Section 4 we shall prove that the dual and the decomposition norms always agree, but the proof of this fact for a characteristic function is much simpler). Let 1 < p < s < ∞. Assume that the function ϕ in (2.17) is extended to the whole line R periodically with period 1. Set Applying Lebesgue's dominated convergence theorem and (2.18), we obtain Then Then, by (2.21), h = N k=1 h k and by (2.20) This implies that for all t > 0. Then, for every nonnegative and non-increasing function g on R + , we have that Finally, we recall the definition of the Hardy-Littlewood-Pólya relation. Let (R, µ) be a measure space and let f and g be µ−measurable and µ−a.e. finite functions on R.

The level function
The notion of a level function was first introduced by Halperin [5]. We shall use the extension of this notion given by Lorentz [12] and based on the following theorem.
Then, there exists a nonnegative function f • on R + satisfying the following conditions: This theorem is a slight modification of the results in [5] and [12, §3.6]; the proof is similar to the one given in [12, §3.6] for functions defined on [0, 1]. It is easy to show that the function f • is uniquely determined (see [5,Theorem 3.7]). It is called the level function of f with respect to ϕ.
The constants in the inequalities (3.1) are optimal.
Proof. First we assume that s < ∞. We consider the left hand side inequality in (3.1). Applying Theorem 3.1(c), we have f Since α = (s/p − 1)/(s − 1), and f • (t) s−1 t s/p−1 = λ s−1 k then, applying Hölder's inequality, we obtain This estimate and property (c) yield the first inequality in (3.1). Now, denote Let ψ(t) be the level function of ψ with respect to ϕ(t) = 1. Applying Theorem 3.1, Lemma 2.11, and the inequality (1.4), we obtain To obtain the second inequality in (3.1), it suffices to prove that where the constant c p,s is defined by (3.2). Let E = {t ∈ R + : ψ(t) = ψ(t)}. Then, up to a set of measure zero, where (a k , b k ) are bounded disjoint intervals such that Using (3.6) and applying Lemma 2.3, we obtain that We also have that we obtain (3.5). Thus, the inequalities in (3.1) are proved for s < ∞. Let now s = ∞ and hence α = 1/p. For any k, Thus, λ k ≤ ||f || p,∞ , which implies that ||f • || p,∞ ≤ ||f || p,∞ . On the other hand, for any t ∈ (a k , b k ) we have (see Theorem 3.1 (b)) This implies the second inequality in (3.1) for s = ∞.

The left hand side inequality in (3.1) becomes equality for
and we have equality ||f || p,s = c p,s ||f • || p,s . Thus, the constants in (3.1) are optimal. In other words, (3.7) holds if and only if f (t)t α decreases on R + .

The dual norm
Recall that for a function f ∈ L p,s (R, µ) (1 < p < ∞, 1 ≤ s ≤ ∞) its dual norm is defined by where the supremum is taken over all functions g ∈ L p ′ ,s ′ (R, µ) with ||g|| p ′ ,s ′ = 1.
If s ≤ p, then the function ψ is non-increasing and we have The latter two equalities imply that ||f || ′ p,s ≥ ||f || p,s . Together with (4.4) this yields (4.5). Observe also that the supremum in (4.2) is attained on the function g(t) = ψ(t)/||ψ|| p ′ ,s ′ . Now we assume that p < s ≤ ∞. Let f ∈ L p,s (R + ). If the function f * (t)t 1−s ′ /p ′ is non-increasing, then as above we have the equality (4.5). Let f be an arbitrary nonnegative function in L p,s (R + ) and let g ∈ L p ′ ,s ′ (R + ), g ≥ 0, be a nonincreasing function. By Lemma 2.11, we have that Note that in the case s ≤ p the infimum in (4.7) is equal to ||f || p,s . However, for s > p the infimum may be smaller than ||f || p,s and (4.6) may give a refinement of the inequality (4.3). It was proved by Halperin [5,Theorem 4.2] (see also [12,Theorem 3.6.5]) that equality in (4.7) holds and the infimum is attained for some h ∈ L p,s (R + ). Since the proofs given in [5] and [12] do not cover explicitly the case s = ∞, and for the sake of completeness, we show the result for all p < s ≤ ∞. Proof. In view of (4.7) and Theorem 3.1 (b), it suffices to prove that By Theorem 3.1, up to a set of measure zero, (4.10) We first assume that s < ∞. Denote ψ(t) = f • (t) s−1 t s/p−1 . As above, we have Besides, we have and thus, from which we obtain (4.9). Let now s = ∞. In this case we have We assume first that for some k we have a k = 0. Set . Then ||g|| p ′ ,1 = 1. We have This implies (4.9). Now we assume that a k = 0 for each k. Then, for any δ > 0 we have On the other hand, by Theorem 3.1 (c), for any t ∈ A Let ε > 0. By (4.11), there exists δ > 0 such that Let ξ ∈ (0, δ)∩A. Set g(t) = χ (0,ξ) /(p ′ ξ 1/p ′ ). Then ||g|| p ′ ,1 = 1. Applying (4.13) and (4.12), we get which again implies (4.9).
Remark 4.3. Let 1 < p < s ≤ ∞, and let f ∈ L p,s (R + ) be a nonnegative and non-increasing function on R + . Then, by Remark 3.3, the equality ||f || ′ p,s = ||f || p,s holds if and only if f (t)t α decreases on R + .
The following theorem gives the sharp estimate of the standard norm via the dual norm. where The constant c p,s is optimal.
This theorem follows immediately from Theorems 4.1 and 3.2. However, a direct proof can be given exactly as in Theorem 3.2. Indeed, assume that f is nonnegative and non-increasing on R + . As in the proof of Theorem 3.2, we have Applying the inequality (3.5), we obtain (4.14). Let now f = χ [0,1] . Then, by Lemma 2.10 which shows that the constant in (4.14) is optimal.

The decomposition norm
In this section we prove one of the main results of this paper -the coincidence of the dual and the decomposition norms. The following lemma plays an important role in the proof of the equality of these two norms.
Let ε > 0. For any ν ∈ N, define the function g ν in the following way.
We also have the following continuous version of the Minkowski type inequality. Theorem 6.3. Suppose that (R, µ) is a σ-finite nonatomic measure space and (Q, ν) is a σ-finite measure space. Let f be a nonnegative measurable function on (R × Q, µ × ν). Assume that 1 < p < s ≤ ∞ and that, for almost all y ∈ Q the function f y (x) = f (x, y), x ∈ R, belongs to L p,s (R, µ). Set F (x) = Q f (x, y)dν(y), x ∈ R. Then ||F || p,s ≤ c p,s Q ||f y || p,s dν(y), where the constant c p,s is defined by (6.4). Let g ∈ L p ′ ,s ′ (R, µ) and assume that ||g|| p ′ ,s ′ = 1. Applying Fubini's Theorem and Hölder's inequality, we obtain Together with (6.5), this implies (6.3). Finally, it follows from Theorem 6.1 that the constant c p,s in (6.3) cannot be replaced by a smaller one.