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

In the paper, we analyze the Lebesgue exponents p_Φ and q_Φ, and show that for any p_Φ< p < ∞ and 1< q<q_Φ, there exists an equivalent Young function $Ψ$ with p < p_Ψ< ∞ and 1<q_Ψ< q. This type of construction is used to improve upon the inclusions Lp_Φ∩ Lq_Φ⊆ L^Φ⊆ Lp_Φ + Lq_Φ. For trace type Orlicz spaces LΦ,Φ, we find that when Φ∈ Δ₂, we have LΦ,Φ ⊆ L^Φ if and only if Φ(||f||L^Φ) ≤ C ρ_Φ(f) for all f∈ L^Φ, and the reverse inclusion is equivalent to the reversed inequality.