On universal functions
Let \({\mathcal F}\subset L^ 0(0,1)\) be a function space endowed with some topology. We call \(f\in {\mathcal F}^ a \)universal function if there exists \(\{\lambda_ n\}\subset {\mathbb{R}}\), \(\lambda_ n\to 0\), such that \((f(x+\lambda_ n)-f(x))/\lambda_ n\subset {\mathcal F}\) is dense in \({\mathcal F}.\) \textit{J. Marcinkiewicz} proved [Fundam. Math. 24, 305-308 (1935; Zbl 0011.10705)] that for \({\mathcal F}=L^ 0(0,1)\) there exists a universal function. Recently this statement was proved by I. Joó for \({\mathcal F}=L^ p(0,1),\quad 0<p<1\quad\) I. Joó asked: does there exist a universal function for \(p\geq 1\), \({\mathcal F}=L^ p(0,1)?\) We give a negative answer, proving: If \(p\geq 1\) and \({\mathcal F}=L^ p(0,1)\) then there is no universal function in \(L^ p(0,1).\) Independently, by similar method, this was proved also by Z. Buczolich. M. Horváth gave a completely different proof for this probem. His result is interesting in itself and runs as follows: Let \(f\in L^ p(\Omega)\) where \(\Omega \subset {\mathbb{R}}^ N\) is an arbitrary domain and suppose that f is not constant in any direction \(e\in {\mathbb{R}}^ N,\quad | e| =1\). Then for some \(\delta_ f>0\) and \(c_ f>0\) \[ \| f(x+\lambda)-f(x)\|_{L^ p(\Omega_{-| \lambda |})}\quad \geq c_ f| \lambda | \quad (| \lambda | <\delta_ f). \] Here \(\Omega_{-s}:=\{x\in \Omega\), \(dist(x,\partial \Omega)>s\}\).
scientific article; zbMATH DE number 4002411
scientific article