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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)

- Stephen Wainger
- 2007

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)

- Alexander Nagel, Elias M. Stein, Stephen Wainger
- Proceedings of the National Academy of Sciences…
- 1978

Let {theta(j)} be a lacunary sequence going to zero. Let [Formula: see text]. Define [Formula: see text]. We prove [Formula: see text].

Endpoint estimates are proved for model cases of restricted X-ray transforms and singular fractional integral operators in R4.

- RADON TRANSFORMS, Alexandru Dan Ionescu, Stephen Wainger, Michael Christ, Alexander Nagel, Elias M. Stein
- 2006

Received by the editors February 27, 2004. 1991 Mathematics Subject Classification. Primary 11L07, 42B20.

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)

- Alexander Nagel, Nicolas Rivière, Stephen Wainger
- Proceedings of the National Academy of Sciences…
- 1976

THE AUTHORS PROVE THAT [FORMULA: see text] is bounded from L(P)(R(2)) to L(P)(R(2)) for 1 < p </= infinity.

- Alexander Nagel, Elias M. Stein, Stephen Wainger
- Proceedings of the National Academy of Sciences…
- 1981

We study several notions of distance and ball on the boundary of domains of finite type in C(n). We use these notions to develop a theory of maximal functions and area integrals for functions holomorphic in such domains.