Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w
✍ Scribed by Maciej Ciesielski; Anna Kamińska; Ryszard Płuciennik
- Publisher
- John Wiley and Sons
- Year
- 2009
- Tongue
- English
- Weight
- 256 KB
- Volume
- 282
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γ~p,w~ = {f: ∫~0~^α^ (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 Γ~p,w~ , and the necessary and sufficient conditions for which a spherical element of Γ~p,w~ is a smooth point of the unit ball in Γ~p,w~ . We show that strict convexity of the Lorentz spaces Γ~p,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 ∈ Γ~p,w~ from any convex set K ⊂ Γ~p,w~ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)