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I'm trying to check that certain examples of Young functions in the harmonic analysis literature are actually Young functions, and in doing so need to prove the following convexity-like inequality for ...

It is a classical result that $L^p(\Omega) \subset L^q(\Omega)$ when $q<p$ and $|\Omega| < \infty$. I'd like to know if there is an Orlicz version of this fact. In other words, let $L^{G_1}$ and ...