Variational inequalities for singular integral operators
Journées équations aux dérivées partielles (2012), article no. 7, 14 p.

In these notes we survey some new results concerning the ρ-variation for singular integral operators defined on Lipschitz graphs. Moreover, we investigate the relationship between variational inequalities for singular integrals on AD regular measures and geometric properties of these measures. An overview of the main results and applications, as well as some ideas of the proofs, are given.

DOI: 10.5802/jedp.90
Classification: 42B20, 42B25
Keywords: $\rho $-variation, singular integral operators, uniform rectifiability.
Mas, Albert 1

1 Departamento de Matemáticas, Universidad del País Vasco – Euskal Herriko Unibertsitatea, 48080 Leioa, Bizkaia (Spain)
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Mas, Albert. Variational inequalities for singular integral operators. Journées équations aux dérivées partielles (2012), article  no. 7, 14 p. doi : 10.5802/jedp.90. http://www.numdam.org/articles/10.5802/jedp.90/

[1] J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math. 69 (1989), pp. 5–45. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[4] G. David and S. Semmes, Singular integrals and rectifiable sets in R n : au-delà des graphes lipschitziens, Astérisque No. 193 (1991). | Numdam | MR | Zbl

[5] G. David and S. Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, 38, American Mathematical Society, Providence, RI (1993). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[12] M. Lacey and E. Terwilleger, A Wiener-Wintner theorem for the Hilbert transform, Ark. Mat. 46 (2008), 2, pp. 315–336. | MR | Zbl

[13] J. C. Léger, Menger curvature and rectifiability, Ann. of Math., 149 (1999), pp. 831–869. | MR | Zbl

[14] D. Lépingle, La variation d’ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 36 (1976), pp. 295–316. | MR | Zbl

[15] A. Mas, Variation for singular integrals on Lipschitz graphs: L p and endpoint estimates, to appear in Trans. Amer. Math. Soc. (2012). | MR

[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). | MR | Zbl

[17] A. Mas and X. Tolsa, Variation for the Riesz transform and uniform rectifiability, to appear in J. Eur. Math. Soc. (2012). | MR

[18] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, Cambridge (1995). | MR | Zbl

[19] P. Mattila, Cauchy singular integrals and rectifiability of measures in the plane, Adv. Math. 115 (1995), pp. 1–34. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[22] P. Mattila and D. Preiss, Rectifiable measures in n and existence of principal values for singular integrals, J. London Math. Soc. (2) 52 (1995), no. 3, pp. 482–496. | MR | Zbl

[23] P. Mattila and J. Verdera Convergence of singular integrals with general measures, J. Eur. Math. Soc. 11 (2009), pp. 257–271. | MR | Zbl

[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. | MR | Zbl

[26] H. Pajot, Analytic capacity, rectifiability, Menger curvature and the Cauchy integral, Lecture Notes in Math. 1799, Springer (2002). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[29] X. Tolsa. Principal values for the Cauchy integral and rectifiability, Proc. Amer. Math. Soc. 128(7) (2000), pp. 2111–2119. | MR | Zbl

[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. | MR | Zbl

[31] X. Tolsa, Principal values for Riesz transforms and rectifiability, J. Funct. Anal., vol. 254(7) (2008), pp. 1811–1863. | MR | Zbl

[32] X. Tolsa, Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality, Proc. London Math. Soc. 98(2) (2009), pp. 393–426. | MR | Zbl

[33] M. Vihtilä, The boundedness of Riesz s-transforms of measures in n , Proc. Amer. Math. Soc. 124 (1996), no. 12, pp 3739–3804. | MR | Zbl

[34] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58 (2003). American Math. Soc., Providence RI. | MR | Zbl

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