## 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.

### MSC:

 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.)