Trace type Orlicz spaces and analysis of Orlicz spaces by Lebesgue exponents

TL;DR

This paper investigates the structure of Orlicz spaces through Lebesgue exponents, constructing equivalent Young functions to refine inclusions and characterizing trace type spaces with conditions on Young functions.

Contribution

It introduces a method to construct equivalent Young functions to improve inclusion relations in Orlicz spaces and characterizes trace type spaces under certain conditions.

Findings

Existence of equivalent Young functions with specific Lebesgue exponents

Refinement of inclusion relations in Orlicz spaces

Characterization of trace type Orlicz spaces with inequalities

Abstract

In the paper, we analyze the Lebesgue exponents $p_\Phi$ and $q_\Phi$, and show that for any $p_\Phi< p < \infty$ and $1< q<q_\Phi$, there exists an equivalent Young function $\Psi$ with $p < p_\Psi < \infty$ and $1<q_\Psi < q$. This type of construction is used to improve upon the inclusions $L^{p_\Phi}\cap L^{q_\Phi}\subseteq L^\Phi \subseteq L^{p_\Phi} + L^{q_\Phi}$. For trace type Orlicz spaces $L^{\Phi,\Phi}$, we find that when $\Phi \in \Delta_2$, we have $L^{\Phi,\Phi} \subseteq L^\Phi$ if and only if $\Phi(||f||_{L^\Phi}) \le C \rho_\Phi(f)$ for all $f\in L^\Phi$, and the reverse inclusion is equivalent to the reversed inequality.