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

Abstract

The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spacesEand in particular on Lorentz spacesΓp,w={f:∫(f**)pw<∞}for any0<p<∞and a nonnegative locally integrable weight functionw, wheref**is a maximal function of the decreasing rearrangementf*for any measurable functionfon(0,α), with0<α≤∞. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages off∈E, whereEis 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 approximantsfϵoff∈Γp,worf∈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 functionf∈Γp-1,w,1<p<∞.