On the boundedness of parabolic maximal operators on product domains

Abstract

In this article, appropriate sharp 𝐿 𝑝 bounds for a certain class of rough maximal operators  Ω,𝛾 with mixed homogeneity are established. Specifically, when the function Ω belongs to 𝐿 𝑞 (𝐒 𝑚−1 × 𝐒 𝑛−1 ) with 𝑚, 𝑛 ≥ 2 and 𝑞 > 1, the boundedness of the such operators is obtained. Further, the extrapolation argument employed in [1] is applied on these gotten bounds to obtain the 𝐿 𝑝 boundedness of the aforementioned operators whenever the kernels are in the space 𝐿(log 𝐿) 2 𝛾′ (𝐒 𝑚−1 × 𝐒 𝑛−1 ) or in the block space 𝐵 (0, 2 𝛾′ −1) 𝑞 (𝐒 𝑚−1 × 𝐒 𝑛−1 ) with 1 < 𝛾 ≤ 2 and 𝑞 > 1.Our obtained results are considered substantial extensions and improvements of what was known previously.