Abstract. We prove an endpoint version of the Stein-Tomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate. A similar improvement is obtained for Stein's estimate on oscillatory integrals of Carleson-Sjölin-Hörmander type and some spectral projection operators on compact manifolds, and for classes of oscillatory integral operators with one-sided fold singularities.

Abstract. The Bochner-Riesz operator T α on R n of order α is defined bywhere denotes the Fourier transform and r α + = r α if r > 0, andTo be more precise, we prove that for 0 < δ < 3/2 the estimateWe also obtain some weak-type results for T α .The Bochner-Riesz operator T α on R n , n ≥ 2, of order α is a multiplier operator defined bywhere denotes the Fourier transform and r For our present purposes the following equivalent definition is more convenient:

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