On the boundedness of parabolic maximal operators on product domains

Abstract

In this article, appropriate sharp Lp bounds for a certain class of rough maximal operators MΩ,γ with mixed homogeneity are established. Specifically, when the function Ω belongs to Lq(Sm−1×Sn−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 Lp boundedness of the aforementioned operators whenever the kernels are in the space L(log⁡L)2γ′(Sm−1×Sn−1) or in the block space Bq(0,2γ′−1)(Sm−1×Sn−1) with 11. Our obtained results are considered substantial extensions and improvements of what was known previously.

Keywords

• Lp boundedness
• Parabolic operators
• Maximal operators
• Rough kernels
• Extrapolation