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

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.