Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz Spaces Γp,w - Maciej Ciesielski; Anna Kamińska (Journal of Function Spaces and Applications)
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,w={f:∫(f**)pw<∞} for any 0<p<∞ 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<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 fϵ of f∈Γp,w or f∈Γp-1,w, 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 f∈Γp-1,w, 1<p<∞.