Abstract
We find the best constants in inequalities relating the standard norm, the dual norm, and the norm∥x∥(p,s):=inf{∑k∥x(k)∥p,s}, where the infimum is taken over all finite representationsx=∑kx(k)in the classical Lorentz sequence spaces. A crucial point in this analysis is the concept of level sequence, which we introduce and discuss. As an application, we derive the best constant in the triangle inequality for such spaces.
Highlights
The study of Lorentz spaces goes back to the work of Lorentz 1, 2 see 3, 4 for more recent results concerning functional properties of Lorentz spaces
Lorentz proved that · p,s is a norm if and only if 1 ≤ s ≤ p < ∞ and that the space p,s is always normable i.e., there exists a norm which is equivalent to · p,s when 1 < p < s ≤ ∞ for the remaining cases, it is known that p,s cannot be endowed with an equivalent norm
In this paper, we introduce the notion of level sequence, which corresponds to the notion of level function introduced by Halperin in 11 and Lorentz in
Summary
The study of Lorentz spaces goes back to the work of Lorentz 1, 2 see 3, 4 for more recent results concerning functional properties of Lorentz spaces. One of the main applications of our results is that we obtain the best constant Cps in the triangle inequality 1.4 This constant was found by a different approach in 13 where the best constants in q-convexity and q-concavity inequalities were found for Lorentz and Marcinkiewicz spaces of functions and sequences. In Theorem 3.7 of Section 3, we prove optimal estimates between the quasinorm in p,s of x xn n and of its level sequence x◦ xn◦ n. The number p stands for the conjugate index of p; that is, 1/p 1/p 1 and N, N∗ stand for the sets of nonnegative, respectively, positive integers
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