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

In the paper, we analyze the Lebesgue exponents $p_{\mathrm{\Phi }}$ and $q_{\mathrm{\Phi }}$, and show that for any $p_{\mathrm{\Phi }}<p<\mathrm{\infty }$ and $1<q<q_{\mathrm{\Phi }}$, there exists an equivalent Young function $\mathrm{\Psi }$ with $p<p_{\mathrm{\Psi }}<\mathrm{\infty }$ and $1<q_{\mathrm{\Psi }}<q$. This type of construction is used to improve upon the inclusions $L^{p_{\mathrm{\Phi }}}\cap L^{q_{\mathrm{\Phi }}}\subseteq L^{\mathrm{\Phi }}\subseteq L^{p_{\mathrm{\Phi }}}+L^{q_{\mathrm{\Phi }}}$. For trace type Orlicz spaces $L^{\mathrm{\Phi },\mathrm{\Phi }}$, we find that when $\mathrm{\Phi }\in {\mathrm{\Delta }}_2$, we have $L^{\mathrm{\Phi },\mathrm{\Phi }}\subseteq L^{\mathrm{\Phi }}$ if and only if $\mathrm{\Phi }(\mathrm{\mid }\mathrm{\mid }f\mathrm{\mid }{\mathrm{\mid }}_{L^{\mathrm{\Phi }}})\le C{\rho }_{\mathrm{\Phi }}(f)$ for all $f\in L^{\mathrm{\Phi }}$, and the reverse inclusion is equivalent to the reversed inequality.