Two versions of the Nikodym maximal function on the Heisenberg group

The classical Nikodym maximal function on the Euclidean plane R2 is defined as the supremum over averages over rectangles of eccentricity N; its operator norm in L2(R2) is known to be O(logN). We consider two variants, one on the standard Heisenberg group H1 and the other on the polarized Heisenberg group . The latter has logarithmic L2 operator norm, while the former has the L2 operator norm which grows essentially of order O(N1/4). We shall imbed these two maximal operators in the family of operators associated to the hypersurfaces {(x1,x2,αx1x2)} in the Heisenberg group H1 where the exceptional blow up in N occurs when α=0.