Two versions of the Nikodym maximal function on the Heisenberg group

Abstract

The classical Nikodym maximal function on the Euclidean plane R 2 is defined as the supremum over averages over rectangles of eccentricity N; its operator norm in L 2 ( R 2 ) is known to be O ( log N ) . We consider two variants, one on the standard Heisenberg group H 1 and the other on the polarized Heisenberg group H p 1 . The latter has logarithmic L 2 operator norm, while the former has the L 2 operator norm which grows essentially of order O ( N 1 / 4 ) . We shall imbed these two maximal operators in the family of operators associated to the hypersurfaces { ( x 1 , x 2 , α x 1 x 2 ) } in the Heisenberg group H 1 where the exceptional blow up in N occurs when α = 0 .