A note on the convolution in Orlicz spaces. (English) Zbl 07791188
The authors obtain some results concerning the convolution product in Orlicz spaces. The introduction gives some information about the construction of the Orlicz space \(L^\Phi(G)\) where \(G\) is a locally compact group with a fixed left Haar measure. Assume that \(G\) is non-discrete. The main result states that if \(\Phi\) and \(\Psi\) are concave Orlicz functions such that \(\limsup_{t\longrightarrow \infty}\frac{\Phi(t)}{t}=0\), then for any compact symmetric neighborhood \(V\) of the neutral element of \(G\), a certain subset of \(L^\Phi(G)\times L^\Psi(G)\) related to \(V\) and the convolution product is of first category. Many consequences are derived from the main result among which there is the fact that the convolution of two functions in \(L^\Phi(G)\) exists if and only if \(G\) is discrete. Also, a well-known result due to W. Żelazko is recovered: \(L^p(G)\), \( 0<p<1\), is a convolution algebra if and only if \(G\) is discrete.
Reviewer: Yaogan Mensah (Lomé)
MSC:
| 43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 44A35 | Convolution as an integral transform |
| 54E52 | Baire category, Baire spaces |
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