Generalized Discrete Orlicz-Morrey Spaces
Hukmashabiyya Ariq Gumilar, Al Azhary Masta, Siti Fatimah
TL;DR
This paper generalizes discrete Orlicz-Morrey spaces by replacing the Young function with an $s$-Young function to form the generalized discrete Orlicz-Morrey-s spaces $\ell_{\phi,\Phi_s}$. The authors define the inner norm with $\Phi_s$ and use the $S_{m,N}$ windows along with a weight function $\phi\in G_\phi$, and show the construction recovers the classical spaces when $s=1$. They provide an explicit example to illustrate membership and prove key structural properties: the generalized norm is a quasi-norm and the space is complete, i.e., a quasi-Banach space, with several properties of the classical spaces preserved under the $s$-convex generalization. Overall, the work broadens the discrete Morrey-type function spaces and offers a robust framework for further harmonic analysis on sequences with Morrey-type control using $s$-convex growth functions.
Abstract
The Orlicz-Morrey spaces, which were introduced through the research of Nakai in 2006, are a generalization and combination of Orlicz and Morrey spaces. There are two types of Orlicz-Morrey spaces, such as continuous Orlicz-Morrey spaces and discrete Orlicz-Morrey spaces. Some properties that apply to Orlicz-Morrey spaces have been studied correspondingly to discrete Orlicz-Morrey spaces. The objectives of the study are to construct generalized discrete Orlicz-Morrey spaces by substituting a Young function with \emph{s}-Young function. Furthermore, The purpose of this study is to see the validity of the properties of the discrete Orlicz-Morrey spaces to the generality of the discrete Orlicz-Morrey spaces. The method in this research draws on the definitions and properties of the discrete Orlicz-Morrey spaces of the previous study and applies the \emph{s}-Young function to the new Orlicz-Morrey spaces. As a result, this study concludes that generalized discrete Orlicz-Morrey spaces reduce to discrete Orlicz-Morrey spaces when \emph{s} is equal to 1. Furthermore, due to the characteristics of the \emph{s}-Young function, some properties of discrete Orlicz-Morrey spaces are preserved in generalized discrete Orlicz-Morrey spaces.