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<∞.
Highlights
The present paper is devoted to investigation of maximal inequalities and Lebesgue’s type differentiation theorems for best local approximations in r.i. quasi-Banach spaces and Lorentz spaces Γp,w for 0 < p < ∞
The interesting exploration of LDT was initiated by Stein in 3, who introduced the maximal functions on Lp Rn, associated with integral average, and applied it to obtain differentiation theorem in the notation of the norm in Lp Rn for 1 ≤ p < ∞
The first results in this subject were obtained by Bastero et al 4 in 1999, who have investigated Hardy-Littlewood maximal functions and weak maximal inequalities in rearrangement invariant quasi-Banach function spaces
Summary
The present paper is devoted to investigation of maximal inequalities and Lebesgue’s type differentiation theorems for best local approximations in r.i. quasi-Banach spaces and Lorentz spaces Γp,w for 0 < p < ∞. 2008 5 Levis et al extended the best constant approximant operator from Orlicz-Lorentz spaces Λw,φ to the spaces Λw,φ and showed monotonicity of the extended operator In view of this result, in 2009 6 Levis established maximal inequalities for the maximal function associated with the best constant approximation and proved Lebesgue’s type differentiation theorem for best constant approximants and for integral averages expressed in terms of the modular corresponding to these spaces. The present paper is a continuation of the previous results and devoted to investigation of maximal inequalities and Lebesgue’s type differentiation theorems for local approximation in r.i. quasi-Banach space E and in particular in Γp,w. We investigate relations between maximal functions and the K-functional of Banach couple Γq,p, L∞ in the spirit of the inequalities stated in
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