Continuous norms and absolutely continuous norms in Banach function spaces are not the same. (English) Zbl 1032.46044

In this interesting paper, the authors construct an example of a Banach function space \(X\) such that every function has a continuous norm and only the zero function has an absolutely continuous norm. For the definition of a function with an absolutely continuous norm, see the well-known book of C. Bennett and R. Sharpley [“Interpolation of Operators” (Pure and Applied Mathematics 129, Boston, Academic Press) (1988; Zbl 0647.46057)]. We say that a function \(f\in X=(X(\Omega), \|\cdot\|)\) has continuous norm in \(X\) if \(\lim_{r\to 0+}\|f\chi_{(x-r,x+r)\cap \Omega}\|=0\) for every \(x\in \overline{\Omega}\) and \(\lim_{r\to \infty}\|f\chi_{\Omega\setminus (-r,r)}\|=0.\)


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)


Zbl 0647.46057