2022/09/02 by Rodney, Scott, MacDonald, Sullivan F.
#28A25 #46E30 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2209.01149
Given a σ-finite measure space (X,μ), a Young function Φ, and a one-parameter family of Young functions \Ψq\, we find necessary and sufficient conditions for the associated Orlicz norms of any function f∈ LΦ(X,μ) to satisfy limq→ ∞‖f‖LΨq(X,μ)=C‖f‖L^∞(X,μ). The constant C is independent of f and depends only on the family \Ψq\. Several examples of one-parameter families of Young functions satisfying our conditions are given, along with counterexamples when our conditions fail.