1999/09/01 by Michael Christ, Alexander Nagel, Elias M. Stein +1 · 1 citation
Mathematics · #math.CA
published as Ann. of Math. (2) 150 (1999), no. 2, 489-577 · 89 pages, published version, abstract added in migration
arxiv created 1999/09/01 · arxiv updated 2016/09/07
The purpose of this paper is to prove the Lp boundedness of singular Radon transforms and their maximal analogues. These operators differ from the traditional singular integrals and maximal functions in that their definition at any point x in Rn involves integration over a k-dimensional submanifold of Rn, depending on x, with k < n. The role of the underlying geometric data which determines the submanifolds and how they depend on x, makes the analysis of these operators quite different from their standard analogues. In fact, much of our work is involved in the examination of the resulting geometric situation, and the elucidation of an attached notion of curvature (a kind of ``finite-type'' condition) which is crucial for our analysis.