Let \(\Omega\) be an open subset of \(\mathbf R^n\). The authors study the Hardy-Littlewood maximal function on the variable Lebesgue space \(L^{p(x)} (\Omega )\), the Banach space of measurable functions on \(\Omega\) such that \[ \int_{\Omega}\Big | \frac{f(x)}{\lambda}\Big | ^{p(x)}dx<\infty \] for some \(\lambda>0\), with norm \[ \| f\| _{p(x),\Omega}=\inf\Big \{\lambda>0:\int_{\Omega}\Big | \frac{f(x)}{\lambda}\Big | ^{p(x)}dx\geq 1 \Big \}. \] They prove that the maximal function is bounded on \(L^{p(x)}(\Omega )\) when \(p\,\) satisfies the following three conditions:
1. \(1<\inf\{ p(y):y\in\Omega\} \leq\sup\{ p(y):y\in\Omega\} <\infty\),
2. \(| p(x)-p(y)| \leq\frac{C}{-\log| x-y| }, \text{ for }\, x,y\in\Omega \,\text{and}\,| x-y| <\frac{1}{2}\),
3. \(| p(x)-p(y)| \leq\frac{C}{log(e+| x| )}, \text{ for }\, x,y\in\Omega\, \text{and}\,| y| \geq| x| .\)
In addition, the authors prove a weak-type inequality under the sole condition that the function \(\frac{1}{p}\) satisfies a reverse Hölder’s condition: there is a constant \(C>0\) so that \[ \frac{1}{p(x)}\leq\frac{C}{| B| }\int_{B}\frac{1}{p(y)}dy \] for almost every \(x\in B\), where \(B\) is an arbitrary ball in \(\mathbf{R}^n\).