On the boundedness of parabolic maximal operators on product domains.

In this article, appropriate sharp bounds for a certain class of rough maximal operators with mixed homogeneity are established. Specifically, when the function Ω belongs to with and , the boundedness of the such operators is obtained. Further, the extrapolation argument employed in [1] is applied on these gotten bounds to obtain the boundedness of the aforementioned operators whenever the kernels are in the space or in the block space with and . Our obtained results are considered substantial extensions and improvements of what was known previously.