A class of parabolic maximal functions

In this paper, we prove L^p estimates of a class of parabolic maximal functions provided that their kernels are in L^q . Using the obtained estimates, we prove the boundedness of the maximal functions under very weak conditions on the kernel. In particular, we prove the L^p -boundedness of our maximal functions when their kernels are in Llog L^1/2 (S^n-1 ) or in the block space Bq ^0,?1/2(S^n?1 ), q>1.