Self-improving boundedness of the maximal operator on quasi-Banach lattices over spaces of homogeneous type

We prove the self-improvement property of the Hardy--Littlewood maximal operator on quasi-Banach lattices with the Fatou property in the setting of spaces of homogeneous type. Our result is a generalization of the boundedness criterion obtained in 2010 by Lerner and Ombrosi for maximal operators on quasi-Banach function spaces over Euclidean spaces. The specialty of the proof for spaces of homogeneous type lies in using adjacent grids of Hyt\"onen--Kairema dyadic cubes and studying the maximal operator alongside its dyadic version. Then we apply the obtained result to variable Lebesgue spaces over spaces of homogeneous type.