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In this paper, we study decay estimates for a two-dimensional scalar oscillatory integral with degenerate real-analytic phase and amplitude. Integrals such as these form a model for certain higher-dimensional degenerate oscillatory integrals, for which it is known that many of the two-dimensional results fail. We define an analogue of the Newton distance in the weighted case, and prove that this gives the optimal rate of decay for the weighted oscillatory integral under certain generic hypotheses. When these hypotheses fail, we provide counterexamples to show that the optimal rate of decay may be faster in general. We have obtained bounds for the rate of decay in some of these exceptional cases.

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Abstract. In this paper, we give L p − L q estimates and the L p regularizing estimate of Radon transforms associated to real analytic functions, and we also give estimates of the decay rate of the L p operator norm of corresponding oscillatory integral operators. For L p −L q estimates and estimates of the decay rate of the L p operator norm we obtain sharp results except for extreme points; however, for L p regularity we allow some restrictions on the phase function.

We establish sharp weak-type estimates for the maximal operators T λ * associated with cylindric Riesz means for functions on H p (R 3 ) when 4/5 < p < 1 and λ = 3/ p − 5/2, and when p = 4/5 and λ > 3/ p − 5/2.

Abstract. We give a necessary and sufficient condition for the double Hilbert transform on R d+2 to be bounded on L p , 1 < p < ∞. This generalizes a result of Carbery, Wainger and Wright [1] for d = 1.