Lebesgue\u27s differentiation theorems in R.I. quasi-Banach spaces and Lorentz spaces Γ p,w

The paper is devoted to investigation of new Lebesgue\u27s type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spaces E and in particular on Lorentz spaces Γ p, w = {f : ∫(f**) p w \u3c ∞} for any 0 \u3c p \u3c ∞ and a nonnegative locally integrable weight function w, where f** is a maximal function of the decreasing rearrangement f* for any measurable function f on (0, α), with 0 \u3c α ≤ ∞. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages of f ∈ E, 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 f ∈ of f ∈ Γ p, w or f ∈ Γ p-1,w, 1 \u3c p \u3c ∞, respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any function f ∈ Γ p-1,w, 1 \u3c p \u3c ∞. Copyright © 2012 Maciej Ciesielski and Anna Kamiska