Singular and Maximal Radon Transforms: Analysis and Geometry

Abstract

Part 3. Analytic theory 11. Statements and reduction to the free case 12. The multiple mapping Γ 13. The space L 1 δ 14. The almost orthogonal decomposition 15. The kernel of (T j T * j ) N ; the L 2 theorem 16. The L p argument; preliminaries 17. Further L 2 estimates 18. The L p estimates; conclusion 19. The maximal function 20. The smoothing property 21. Complements and remarks Part 4. Appendix 22. Proof of the lifting theorem References

$L^p$ boundedness of discrete singular Radon transforms

“…See [8], [13], [20] for some results in the translation invariant setting, as well as [14], [12], [5], [17], [6], [9] for the general case. For more references in the continuous case, we refer the reader to the recent paper of M. Christ, A. Nagel, E. M. Stein, and S. Wainger [6].…”

“…For more references in the continuous case, we refer the reader to the recent paper of M. Christ, A. Nagel, E. M. Stein, and S. Wainger [6]. We will use the boundedness of the (translation invariant) continuous singular Radon transforms in our proof of Theorem 1.1; see Lemma 6.5 in Section 6.…”

Basic properties of nonsmooth Hörmander's vector fields and Poincaré's inequality

“…We need the following finite BCH formula with a remainder: for any Ω ′ ⋐ Ω, given two positive integers k 0 , j 0 there exist r 0 > 0 and C > 0 such that, if |s| , |t| < r 0 then exp (sX) exp (tY ) (x) = exp   k+j≥1,k≤k0,j≤j0 where the constants r 0 , C only depend on a finite number of C k (Ω ′ ) norms of the coefficients of X, Y . Although the above fact is probably well known, we have not been able to find a precise reference for the last statement about the dependence of the constants r 0 and C. However, revising the proof of this formula given for instance in [3], one can check that this is actually the case.…”

Multiparameter singular integrals and maximal operators along flat surfaces

“…Their L p boundedness is determined by certain geometric conditions that are described in numerous ways. In [5], Nagel, Christ, Stein, Wainger have shown the equivalence of those finite type conditions in the very general setting. An interesting problem is to establish the L p theory for flat manifolds, which are in lack of the finite type condition.…”