Maximal functions in variable exponent spaces: limiting cases of the exponent. (English) Zbl 1180.42010

Let \(\Omega\subset\mathbb{R}^n\) be a bounded open set and let the function \(1/p:\Omega\to\mathbb{R}\) be globally log-Hölder continuous with \(1\leq p^-\leq p^+\leq\infty\). Let \(M\) denote the maximal function operator. The main problem consists in finding a function \(\Phi(x,t)\) generating an Orlicz-Musielak space \(L_{\Phi(x,t)}(\Omega)\) such that for every \(f\in L_{\Phi(x,t)}(\Omega)\) there holds the inequality \(\| Mf\|_{L^{p(\cdot)}}(\Omega)\leq\| f\|_{L_{\Phi(-,t)}}\) and, moreover, for any ball \(B\) \(Mf\in L^{p(\cdot)}(B)\) if and only if \(f\in L_{\Phi(-,t)}(B)\). It is proved that the function \(\Phi\) is of the form \(\Phi(x,t)= |t|^{p(x)}\psi_{p(x)}(t)\), where \(\psi_p(t)= \log(e+|t|)\) for \(|t|< e^{p'}- e\) and \(\psi_p(t)= 2p'- p'e^{p'}(e+|t|)^{-1}\) for \(|t|\geq e^{p'}- e\), with \(p'= p/(p- 1)\). There are also shown some consequences of this result.


42B25 Maximal functions, Littlewood-Paley theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)