Let ε(X, d, μ) be a Banach function space over a space of homogeneous type (X, d, μ). We show that if the Hardy-Littlewood maximal operator M is bounded on the space ε(X, d, μ), then its boundedness on the associate space ε'(X, d, μ) is equivalent to a certain condition A∞. This result extends a theorem by Andrei Lerner from the Euclidean setting of ℝn to the setting of spaces of homogeneous type.
- Associate space
- Banach function space
- Dyadic cubes
- Hardy-Littlewood maximal operator
- Space of homogeneous type