On the boundedness of parabolic maximal operators on product domains

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