Rate of Convergence of the Kantorovich Operator Near \(L^1\)
Abdul Karim Munir Aszari,Denny Ivanal Hakim

Institut Teknologi Bandung

Abstract

The study of the rate of convergence of the Kantorovich operator has predominantly focused on the \(L^p\) spaces, yet the behaviour near \(L^1\) remains less understood, particularly as \(p\) approaches \(1\). To bridge this gap, we investigate the rate of convergence within the framework of the grand Lebesgue spaces \(L^{p)}[0,1]\), which encompass all \(L^p\) spaces for \(-1<p<\infty\) but remain a subset of \(L^1\).

Our approach leverages the intrinsic properties of \(L^{p)}[0,1]\) to derive new results on the convergence rate of the Kantorovich operator. Specifically, we aim to demonstrate that the Kantorovich operator exhibits a significant rate of convergence within this broader context, thereby providing insights applicable to the boundary behavior as \(p\to1\).

We will then apply these findings to Holder continuous functions to further understand the rate of convergence of the Kantorovich operator in these settings. This combined approach suggests that functions with derivatives in \(L^{p)}\) exhibit specific convergence rates under the Kantorovich operator.

Keywords: Kantorovich operator,maximal operator,Hardy-Littlewood,Lebesgue spaces,grand Lebesgue spaces

Topic: Minisymposia Fourier Analysis and Integral Operators