Variational inequalities for singular integral operators

 Access to full text

 Full (PDF)

1. J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math. 69 (1989), pp. 5–45. Zbl0705.28008MR1019960
2. J. Campbell, R. L. Jones, K. Reinhold, and M. Wierdl, Oscillation and variation for the Hilbert transform, Duke Math. J. 105 (2000), pp. 59–83. Zbl1013.42008MR1788042
3. J. Campbell, R. L. Jones, K. Reinhold, and M. Wierdl, Oscillation and variation for singular integrals in higher dimensions, Trans. Amer. Math. Soc. 35 (2003), pp. 2115–2137. Zbl1022.42012MR1953540
4. G. David and S. Semmes, Singular integrals and rectifiable sets in $R^n$: au-delà des graphes lipschitziens, Astérisque No. 193 (1991). Zbl0743.49018MR1113517
5. G. David and S. Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, 38, American Mathematical Society, Providence, RI (1993). Zbl0832.42008MR1251061
6. H. Farag, The Riesz kernels do not give rise to higher-dimensional analogues of the Menger-Melnikov curvature, Pub. Mat. 43 (1999), no. 1, pp. 251–260. Zbl0936.42010MR1697524
7. T. A. Gillespie and J. L. Torrea, Dimension free estimates for the oscillation of Riesz transforms, Israel Journal of Math. 141 (2004), pp. 125–144. Zbl1072.42011MR2063029
8. P. Huovinen, Singular integrals and rectifiability of measures in the plane, dissertation, University of Jyväskylä, Jyväskylä (1997). Ann. Acad. Sci. Fenn. Math. Diss. No. 109 (1997), 63 pp. Zbl0883.28007MR1428746
9. R. L. Jones, R. Kaufman, J. Rosenblatt, and M. Wierdl, Oscillation in ergodic theory, Ergodic Theory and Dynam. Sys. 18 (1998), pp. 889–936. Zbl0924.28009MR1645330
10. R. L. Jones, A. Seeger, and J. Wright, Strong variational and jump inequalities in harmonic analysis, Trans. Amer. Math. Soc. 360 (2008), pp. 6711–6742. Zbl1159.42013MR2434308
11. P. W. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Harmonic analysis and partial differential equations (El Escorial, 1987), Lecture Notes in Math., 1384, Springer, Berlin, 1989, pp. 24–68. Zbl0675.30029MR1013815
12. M. Lacey and E. Terwilleger, A Wiener-Wintner theorem for the Hilbert transform, Ark. Mat. 46 (2008), 2, pp. 315–336. Zbl1214.42003MR2430729
13. J. C. Léger, Menger curvature and rectifiability, Ann. of Math., 149 (1999), pp. 831–869. Zbl0966.28003MR1709304
14. D. Lépingle, La variation d’ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 36 (1976), pp. 295–316. Zbl0325.60047MR420837
15. A. Mas, Variation for singular integrals on Lipschitz graphs: $L^p$ and endpoint estimates, to appear in Trans. Amer. Math. Soc. (2012). Zbl1281.42014MR3091264
16. A. Mas and X. Tolsa, Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs, to appear in Proc. London Math. Soc. (2012). Zbl1271.42024MR2948789
17. A. Mas and X. Tolsa, Variation for the Riesz transform and uniform rectifiability, to appear in J. Eur. Math. Soc. (2012). MR2948789
18. P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, Cambridge (1995). Zbl0819.28004MR1333890
19. P. Mattila, Cauchy singular integrals and rectifiability of measures in the plane, Adv. Math. 115 (1995), pp. 1–34. Zbl0842.30029MR1351323
20. P. Mattila and M. S. Melnikov, Existence and weak-type inequalities for Cauchy integrals of general measures on rectifiable curves and sets, Proc. Amer. Math. Soc. 120 (1994), no. 1, pp. 143–149. Zbl0838.30036MR1160305
21. P. Mattila, M. S. Melnikov and J. Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1996), pp. 127–136. Zbl0897.42007MR1405945
22. P. Mattila and D. Preiss, Rectifiable measures in ${ℛ}^n$ and existence of principal values for singular integrals, J. London Math. Soc. (2) 52 (1995), no. 3, pp. 482–496. Zbl0880.28002MR1363815
23. P. Mattila and J. Verdera Convergence of singular integrals with general measures, J. Eur. Math. Soc. 11 (2009), pp. 257–271. Zbl1163.42005MR2486933
24. F. Nazarov, X. Tolsa, and A. Volberg. Private communication. In preparation. 
25. R. Oberlin, A. Seeger, T. Tao, C. Thiele, and J. Wright A variation norm Carleson theorem, J. Eur. Math. Soc. 14 (2012), 2, pp. 421–464. Zbl1246.42016MR2881301
26. H. Pajot, Analytic capacity, rectifiability, Menger curvature and the Cauchy integral, Lecture Notes in Math. 1799, Springer (2002). Zbl1043.28002MR1952175
27. A. Ruiz de Villa and X. Tolsa, Non existence of principal values of signed Riesz transforms of non integer dimension, Indiana Univ. Math. J. 59 (2010), no. 1, pp. 115–130. Zbl1200.28004MR2666475
28. X. Tolsa. Cotlar’s inequality without the doubling condition and existence of principal values for the Cauchy integral of measures, J. Reine Angew. Math. 502 (1998), pp. 199–235. Zbl0912.42009MR1647575
29. X. Tolsa. Principal values for the Cauchy integral and rectifiability, Proc. Amer. Math. Soc. 128(7) (2000), pp. 2111–2119. Zbl0944.30022MR1654076
30. X. Tolsa. A proof of the weak $(1,1)$ inequality for singular integrals with non doubling measures based on a Calderón-Zygmund decomposition, Pub. Mat. 45(1) (2001), pp. 163–174. Zbl0980.42012MR1829582
31. X. Tolsa, Principal values for Riesz transforms and rectifiability, J. Funct. Anal., vol. 254(7) (2008), pp. 1811–1863. Zbl1153.28003MR2397876
32. X. Tolsa, Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality, Proc. London Math. Soc. 98(2) (2009), pp. 393–426. Zbl1194.28005MR2481953
33. M. Vihtilä, The boundedness of Riesz $s$-transforms of measures in ${ℛ}^n$, Proc. Amer. Math. Soc. 124 (1996), no. 12, pp 3739–3804. Zbl0876.28008MR1343727
34. C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58 (2003). American Math. Soc., Providence RI. Zbl1106.90001MR1964483

1. L. Giraud, P. Spiteri, Résolution parallèle de problèmes aux limites non linéaires