Endpoint estimates for some maximal operators associated to the circular conical surface

For ψ ∈ C 0 ∞ ( R d ) and m > 0 we consider the maximal operator given by M m f ( x , t ) = sup r > 0 | ∫ R d f ( x − y , t − | y | m ) 1 r d ψ ( y r ) d y | . It is well known that M m is a L p -bounded operator for 1 < p ⩽ ∞ . Also A. Seeger and T. Tao proved that M m is of weak-type L log log L if m ≠ 1 . In this paper we consider the case m = 1 and prove M 1 maps the standard Hardy space H 1 to weak L 1 if d ⩾ 4 .