Gâteaux derivatives and their applications to approximation in Lorentz spaces Γ_p,w

Abstract

We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γp,w = {f: ∫0α (f **)p w < ∞}, where 1 ≤ p < ∞, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, α), 0 < α ≤ ∞. We also find a general form of any supporting functional for each function from Γp,w , and the necessary and sufficient conditions for which a spherical element of Γp,w is a smooth point of the unit ball in Γp,w . We show that strict convexity of the Lorentz spaces Γp,w is equivalent to 1 < p < ∞ and to the condition ∫0∞ w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γp,w from any convex set K ⊂ Γp,w (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On geometric structure of symmetric spaces

“…The studies of the local and global properties in symmetric spaces is a key for many different types of branches of mathematics. Indeed, it has been found many applications of the monotonicity and rotundity properties of symmetric spaces in approximation theory (see [9,10,12,20,27]). It is worth mentioning that the monotonicity properties play a similar significant role in the best dominated approximation problems in Banach lattices as the respective rotundity properties do in the best approximation problems in Banach spaces.…”

Strict K-monotonicity and K-order continuity in symmetric spaces

“…The next motivating research was published in [9], where there has been established among others a connection between the best dominated approximation and the KadecKlee property for global convergence in measure in Banach function spaces. Recently, in view of the previous investigation there appeared many results [5][6][7]10,14] devoted to exploration of the global and local monotonicity and rotundity structure of Banach spaces applicable in the best approximation problems. The main inspiration for this article appeared in paper [4], where there has been introduced a new type of the best dominated approximation with respect to the Hardy-Littlewood-Pólya relation ≺.…”

The best constant approximant operators in Lorentz spaces Γp,w and their applications

“…In the case where 1 ≤ p < ∞, then it is a Banach space. For more details about the properties of Γ p,w the reader is referred to [7,4].…”

“…Definition 1.2 ( [4,12]). Let f, g ∈ Γ p,w and let τ ( f,g) , τ (g| S(g)\S( f ) ,0) be measure preserving transformations given by Definition 1.1.…”

“…Recall that Γ p,w , for 0 < p < ∞ and w a nonnegative weight function, is a set of all Lebesgue measurable functions such that α 0 ( f * * ) p w < ∞, where f * * (t) = H 1 ( f )(t) = 1 t t 0 f * is the Hardy operator and 0 < α ≤ ∞. The extension under consideration is constructed in the spirit of [14] on the basis of the results obtained in [4], where the Gâteaux derivative of the norm in Γ p,w has been thoroughly investigated. Finally, maximal inequalities for the extended best constant approximant operators and the generalization of LDT in L 0 are investigated.…”