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 there exist Young functions $\Psi$ with $p_\Phi < p_\Psi < \infty$ and $1<q_\Psi < q_\Phi$ but $L^\Psi = L^\Phi$. This type of construction is ultimately used to improve upon the inclusions $L^{p_\Phi}\cap L^{q_\Phi}\subseteq L^\Phi \subseteq L^{p_\Phi} + L^{q_\Phi}$. Investigations into the trace type Orlicz spaces $L^{\Phi,\Phi}$ are included. We find that $L^{\Phi,\Phi} \subseteq L^\Phi$ if $\Phi$ is a submultiplicative Young function, and that reversed inclusions hold if $\Phi$ is supermultiplicative.