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Given Ω ⊂ Z 3 + , we discuss a necessary and sufficient condition that the triple Hilbert transform associated with any polynomial of the form (t1, t2, t3, m ∈Ω am t m ) is bounded in L p (R 4 ).

Abstract. We prove the L p (R d ) (1 < p ≤ ∞) boundedness of the maximal operators associated with a family of vector polynomials given by the form {(2 k 1 p 1 (t), . . . , 2 k d p d (t)) : t ∈ R}. Furthermore, by using the lifting argument, we extend this result to a general class of vector polynomials whose coefficients are of the form constant times 2 k .

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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 1 p . 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 , αx 1 x 2 )} in the Heisenberg group H 1 where the exceptional blow up in N occurs when α = 0.