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.

32 pages. This is the second of presumably many versions