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Uniform openness of multiplication in Orlicz spaces. (English) Zbl 1354.46029

Summary: 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 [“Uniform openness of multiplication in Banach spaces \(L_p\)”, arXiv:1309.3433].

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47A07 Forms (bilinear, sesquilinear, multilinear)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
06B30 Topological lattices
46B25 Classical Banach spaces in the general theory