Two versions of the Nikodym maximal function on the Heisenberg group

摘要

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), alpha x(1)x(2))} in the Heisenberg group H-1 where the exceptional blow up in N occurs when alpha = 0.