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Journal of Function Spaces and Applications
Volume 2012, Article ID 682960, 28 pages
Research Article

Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γ𝑝,𝑀

1Institute of Mathematics, PoznaΕ„ University of Technology, Piotrowo 3a, 60-965 PoznaΕ„, Poland
2Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA

Received 10 October 2011; Accepted 20 January 2012

Academic Editor: Lech Maligranda

Copyright Β© 2012 Maciej Ciesielski and Anna KamiΕ„ska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spaces 𝐸 and in particular on Lorentz spaces Γ𝑝,π‘€βˆ«={π‘“βˆΆ(π‘“βˆ—βˆ—)𝑝𝑀<∞} for any 0<𝑝<∞ and a nonnegative locally integrable weight function 𝑀, where π‘“βˆ—βˆ— is a maximal function of the decreasing rearrangement π‘“βˆ— for any measurable function 𝑓 on (0,𝛼), with 0<π›Όβ‰€βˆž. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages of π‘“βˆˆπΈ, where 𝐸 is an order continuous r.i. quasi-Banach space. The second type of LDT establishes conditions for pointwise convergence of the best or extended best constant approximants π‘“πœ– of π‘“βˆˆΞ“π‘,𝑀 or π‘“βˆˆΞ“π‘βˆ’1,𝑀, 1<𝑝<∞, respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any function π‘“βˆˆΞ“π‘βˆ’1,𝑀, 1<𝑝<∞.