Results in Mathematics | 2021
Fractional Maximal Operator on Musielak–Orlicz Spaces Over Unbounded Quasi-Metric Measure Spaces
Abstract
The main target of this article is the boundedness of the fractional maximal operator , on Musielak–Orlicz spaces $$L^{\\Phi }(X)$$\n over unbounded quasi-metric measure spaces as an extension of recent results by Cruz-Uribe and Shukla (Studia Math 242(2):109–139, 2018) and the authors (2019), where $$\\eta $$\n is the order of the fractional maximal operator and $$\\lambda $$\n is its modification rate. Our results are new even for the Hardy–Littlewood maximal operator $$M_{\\lambda }$$\n or for the Orlicz spaces $$L^{p(\\cdot )}(\\log L)^{q(\\cdot )}(X)$$\n . Usually, for the proof of the boundedness, the three-line theorem is employed. This new technique of using the three-line theorem enables us to extend the function spaces with ease. An example explains why we can not remove the modification parameter $$\\lambda $$\n .