Riesz spaces with generalized Orlicz growth

TL;DR

This paper introduces a generalized Riesz variation based on a $ ext{Phi}$-function, establishing a quasi-Banach space structure, explicit modular formulas, and applications to image restoration, extending prior variable exponent and Orlicz results.

Contribution

It generalizes the Riesz variation to a $ ext{Phi}$-function setting, providing new structural and explicit formulas, and addresses an open question in the field.

Findings

Generated a quasi-Banach space from the generalized Riesz $ ext{Phi}$-variation.

Derived explicit formulas for the modular in the bounded variation case.

Connected the new framework to image restoration models.

Abstract

We consider a Riesz $\phi$-variation for functions $f$ defined on the real line when $\varphi:\Omega\times[0,\infty)\to[0,\infty)$ is a generalized $\Phi$-function. We show that it generates a quasi-Banach space and derive an explicit formula for the modular when the function $f$ has bounded variation. The resulting $BV$-type energy has previously appeared in image restoration models. We generalize and improve previous results in the variable exponent and Orlicz cases and answer a question regarding the Riesz--Medvedev variation by Appell, Bana\'s and Merentes [\emph{Bounded Variation and Around}, Studies in Nonlinear Analysis and Applications, Vol. 17, De Gruyter, Berlin/Boston, 2014].