Stephen Wainger

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Part 2. Geometric theory 8. Curvature: Introduction 8.1. Three notions of curvature 8.2. Theorems 8.3. Examples 9. Curvature: Some details 9.1. The exponential representation 9.2. Diffeomorphism invariance 9.3. Curvature condition (CY ) 9.4. Two lemmas 9.5. Double fibration formulation 10. Equivalence of curvature conditions 10.1. Invariant submanifolds and(More)
X \R(1 r\\ f /OO # / ( * ) --> e-*0 \B(e9 X)\ JB(t,x) for certain types of «-dimensional sets B(e9 x) shrinking to x as e -* 0. (\B(e9 x)\ is of course the Lebesgue measure of B(e9 x).) Some standard examples of sets B(e9 x) are balls with center x and radius e, and cubes with center x and diameter e. The problem of considering other sets besides balls and(More)
This paper studies the discrete analogues of singular Radon transforms. We prove the `2 boundedness for those operators that are “quasi-translation-invariant.” The approach used is related to the “circle-method” of Hardy and Littlewood, and requires multi-dimensional extensions of Weyl sums and Gauss sums, as well as variants that replace scalar sums by(More)
where P(s, t) is a polynomial in s and t with P(0,0)= 0, and ∇P(0,0)= 0. We call H the (local) double Hilbert transform along the surface (s, t,P (s, t)). The operator may be precisely defined for a Schwartz function f by integrating where ≤ |s| ≤ 1 and η ≤ |t | ≤ 1, and then taking the limit as ,η→ 0. The corresponding 1-parameter problem has been(More)