A Note on Inclusion Properties of Weighted Orlicz Spaces

Al Azhary Masta,Ifronika ,Muhammad Taqiyuddin

doi:10.22342/jims.26.1.816.128-136

Abstract

In this paper we present sufficient and necessary conditions for the inclusion relationbetween two weighted Orlicz spaces which complete the Osan\c{c}liol result in 2014.One of the keys to prove our results is to use the norm of the characteristic functionsof the balls in $\mathbb{R}^n$.

Highlights

Orlicz spaces are generalization of Lebesgue spaces which were firstly introduced by Z
We need to estimate the norm of the characteristic function of an open ball in relation between LuΦ1 (Rn) as in the following lemma
We come to the inclusion relation between LuΦ1 (Rn) and LuΦ2 (Rn) with respect to Young functions Φ1, Φ2

Summary

INTRODUCTION

Orlicz spaces are generalization of Lebesgue spaces which were firstly introduced by Z. The weighted Orlicz space LuΦ(Rn) is the set of all measurable functions f : Rn → R such that uf ∈ LΦ(Rn). Obtained sufficient and necessary conditions for the inclusion relation between two Orlicz spaces and between two weak Orlicz spaces by using a different technique from Maligranda. Orlicz spaces are comparable with respect to Young functions for any measurable set, the Lebesgue space Lp are not comparable with respect to the number p. Osancliol [13] has obtained sufficient and necessary conditions for the inclusion relation between two weighted Orlicz spaces, as in the following theorem. In connection with Theorem 1.1, we shall prove the inclusion relation between weighted Orlicz spaces with respect to Young functions Φ1, Φ2 and weights u1, u2. The letter C will be used for constants whose values may change from line to line, while constants with subscripts, such as C1, C2, do not change their values

RESULTS

AN ADDITIONAL CASE

CONCLUDING REMARKS