Uniform openness of multiplication in Orlicz spaces

Abstract

Let $\Phi$ and $\Psi$ be Young functions, and let $L^{\Phi }(\Omega )$ and $L^{\Psi }(\Omega )$ be corresponding Orlicz spaces on a measure space $(\Omega ,\mu )$. Our aim in this paper is to prove that, under mild conditions on $\Phi$ and $\Psi$, the multiplication from $L^{\Phi }(\Omega )\times L^{\Psi }(\Omega )$ onto $L^1(\Omega )$ is uniformly open. This generalizes an interesting recent result due to M. Balcerzak, A. Majchrzycki, and F. Strobin in 2013.