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 ).

We study double Hilbert transforms and maximal functions along surfaces of the form (t 1 , t 2 , γ 1 (t 1 )γ 2 (t 2 )). The L p (R 3 ) boundedness of the maximal operator is obtained if each γ i is a convex increasing and γ i (0) = 0. The double Hilbert transform is bounded in L p (R 3 ) if both γ i 's above are extended as even functions. If γ 1 is odd, then we need an additional comparability condition on γ 2 . This result is extended to higher dimensions and the general hyper-surfaces of the form (t 1 , . . . , t n , Γ(t 1 , . . . , t n )) on R n+1 .