BOOK |
Analytic Capacity, the Cauchy Transform, and Non-Homogeneous Calderon-Zygmund Theory.
Birkhauser (2014). Some reviews: In Mathematical Reviews by T.P. Hytonen. In Zentralblatt by Dachun Yang. In Jahresber Dtsch Math-Ver by H. von der Mosel. |
SOME RESEARCH PAPERS |
NON-HOMOGENEOUS HARMONIC ANALYSIS |
BMO, H^{1}, and Calderon-Zygmund operators for non doubling measures.
Math. Ann. 319 (2001), 89-149. dvi A proof of the weak (1,1) inequality for singular integrals with non doubling measures based on a Calderon-Zygmund decomposition. Publ. Mat. 45 (2001), 163-174. dvi The atomic space H^{1} for non doubling measures in terms of a maximal operator. Trans. Amer. Math. Soc. 355 (2003), 315-348. pdf (it includes some corrections and appendix). Weighted norm inequalities for Calderon-Zygmund operators without doubling conditions. Publ. Mat. 51:2 (2007), 397-456. pdf Strong and weak type estimates for singular integrals with respect to measures separated by AD-regular boundaries, (with V. Chousionis). Inter. Math. Res. Not. 2014(23) (2014), 6497-6522. pdf Improved Cotlar's inequality in the context of local Tb theorems (with H. Martikainen and M. Mourgoglou). J. Funct. Anal. 274 (2018), no. 5, 1255-1275. pdf |
ANALYTIC CAPACITY AND OTHER RELATED CAPACITIES |
On the analytic capacity γ_{+}. Indiana Univ. Math. J. 51 (2002), 317-343. pdf The planar Cantor sets of zero analytic capacity and the local T(b) theorem (with J. Mateu and J. Verdera). J. Amer. Math. Soc. 16 (2003), 19-28. pdf Painleve's problem and the semiadditivity of analytic capacity. Acta Math. 190:1 (2003), 105-149. pdf The semiadditivity of continuous analytic capacity and the inner boundary conjecture. Amer. J. Math. 126 (2004), 523-567. pdf Riesz transforms and harmonic Lip_{1} capacity in Cantor sets (with J. Mateu). Proc. London Math. Soc. 89(3) (2004), 676-696. pdf Bilipschitz maps, analytic capacity, and the Cauchy integral. Ann. of Math. 162:3 (2005), 1241-1302. pdf Estimate of the Cauchy integral over Ahlfors regular curves (with M. Melnikov). In "Selected Topics in Complex Analysis", Operator Theory: Advances and Applications, Vol. 158, Birkhauser Verlag, 2005, pp. 159-176. pdf Characterization and semiadditivity of the C^{1} harmonic capacity (with A. Ruiz de Villa). Trans. Amer. Math. Soc. 362 (2010) 3641-3675. pdf Calderon-Zygmund capacities and Wolff potentials on Cantor sets. J. Geom. Anal. 21(1) (2011), 195-223. pdf Capacities associated with Calderon-Zygmund kernels (with V. Chousionis, J. Mateu, and L. Prat). Potential Anal. 38 (2013), no. 3, 913-949. pdf Riesz transforms of non-integer homogeneity on uniformly disconnected sets (with M.C. Reguera). Trans. Amer. Math. Soc. 368 (2016), no. 10, 7045-7095. pdf Square functions of fractional homogeneity and Wolff potentials (with V. Chousionis and L. Prat). Int. Math. Res. Notices (2016) Vol. 2016, 2295--2319. pdf The Riesz transform of codimension smaller than one and the Wolff energy (with B. Jaye, F. Nazarov, and M.C. Reguera). Preprint (2016). To appear in Mem. Amer. Math. Soc. pdf Analytic capacity and projections (with Alan Chang). Preprint (2017). pdf On C^{1}-approximability of functions by solutions of second order elliptic equations on plane compact sets and C-analytic capacity (with P.V. Paramonov). Preprint (2018). To appear in An. Math. Phys. pdf |
SINGULAR INTEGRALS AND RECTIFIABILITY |
Growth estimates for Cauchy integrals of measures and rectifiability. GAFA vol. 17 (2007), 605-643. ps Uniform rectifiability, Calderon-Zygmund operators with odd kernel, and quasiorthogonality. Proc. London Math. Soc. 98(2) (2009), 393-426. pdf On the smoothness of Holder doubling measures (with D. Preiss and T. Toro). Calc. Var. Partial Differential Equations 35(3) (2009), 339-363. pdf Principal values for Riesz transforms and rectifiability. J. Funct. Anal., vol. 254(7) 2008, 1811-1863. pdf Non existence of principal values of signed Riesz transforms of non integer dimension (with A. Ruiz de Villa). Indiana Univ. Math. J. 59:1 (2010), 115-130. pdf Mass transport and uniform rectifiability. Geom. Funct. Anal. 22 (2012), no. 2, 478-527. pdf Variation and oscillation for singular integrals with odd kernel on Lipschitz graphs (with A. Mas). Proc. London Math. Soc. 105(1) (2012), 49-86. pdf Variation for Riesz transforms and uniform rectifiability, (with A. Mas). J. Eur. Math. Soc. 16(11) (2014), 2267--2321. pdf Calderon-Zygmund kernels and rectifiability in the plane (with Chousionis, Prat and Mateu). Adv. Math. 231:1 (2012), 535-568. pdf On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1 (with Nazarov and Volberg). Acta Math. 213:2 (2014), 237-321. pdf The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions (with Nazarov and Volberg, 2012). Publ. Mat. 58:2 (2014), 517-532. pdf Uniform measures and uniform rectifiability. J. Lond. Math. Soc. (2) 92 (2015), no. 1, 1-18. pdf Square functions and uniform rectifiability (with Chousionis, Garnett and Le). Trans. Amer. Math. Soc. 368 (2016), no. 9, 6063-6102. pdf Rectifiability via a square function and Preiss' theorem (with T. Toro). Int Math Res Notices (2015), Vol. 2015, 4638-4662. pdf Rectifiable measures, square functions involving densities, and the Cauchy transform. Mem. Amer. Math. Soc. 245 (2017), no. 1158 pdf L^{p}-estimates for the variation for singular integrals on uniformly rectifiable sets (with A. Mas). Trans. Amer. Math. Soc. 369, no. 11 (2017), 8239-8275. pdf Non-existence of reflectionless measures for the s-Riesz transform (with L. Prat). Ann. Acad. Scient. Fenn. Math., vol. 40 (2015), 957-968. pdf Characterization of n-rectifiability in terms of Jones' square function: Part I. Calc. Var. PDE. (2015), no. 4, 3643-3665. pdf Characterization of n-rectifiability in terms of Jones' square function: Part II (with J. Azzam). Geom. Funct. Anal. (GAFA) 25 (2015), no. 5, 1371-1412. pdf The Riesz transform and quantitative rectifiability for general Radon measures (with D. Girela-Sarrion). Calc. Var. Partial Differential Equations 57 (2018), no. 1, Art. 16, 63 pp. pdf Singular integrals unsuitable for the curvature method whose L^{2}-boundedness still implies rectifiability (with P. Chunaev and J. Mateu). Preprint (2016). To appear in J. Anal. Math. pdf The measures with an associated square function operator bounded in L^{2} (with B. Jaye and F. Nazarov). Adv. Math., 339, 1 (2018), 60-112. pdf Rectifiability of measures and the β_{p} coefficients. Preprint (2017). To appear in Publ. Mat. pdf Failure of L^{2} boundedness of gradients of single layer potentials for measures with zero low density (with J.M. Conde-Alonso and M. Mourgoglou). Preprint (2018). To appear in Math. Ann. pdf A family of singular integral operators which control the Cauchy transform (with P. Chunaev and J. Mateu). Preprint (2018). pdf Characterization of rectifiable measures in terms of α-numbers (with J. Azzam and T. Toro). Preprint (2018). pdf Jump formulas for singular integrals and layer potentials on rectifiable sets. Preprint (2018). pdf |
HARMONIC MEASURE |
Singular sets for harmonic measure on locally flat domains with locally finite surface measure (with J. Azzam and M. Mourgoglou). Int Math Res Notices (2017) 2017 (12): 3751-3773. pdf Rectifiability of harmonic measure in domains with porous boundaries (with J. Azzam and M. Mourgoglou). Preprint (2015). pdf Absolute continuity between the surface measure and harmonic measure implies rectifiability (with Hofmann, Martell, Mayboroda, and Volberg). C. R. Math. Acad. Sci. Paris 354 (2016), no. 4, 351-355. pdf Rectifiability of harmonic measure (with Azzam, Hofmann, Martell, Mayboroda, Mourgoglou, and Volberg). Geom. Funct. Anal. (GAFA), 26(3) (2016), 703-728. pdf Harmonic measure and Riesz transform in uniform and general domains (with M. Mourgoglou). Preprint (2015). To appear in J. Reine Ang. Math. pdf Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability (with J. Azzam and M. Mourgoglou). Comm. Pure Appl. Math. Vol. LXX (2017), 2121-2163. pdf The one-phase problem for harmonic measure in two-sided NTA domains (with J. Azzam and M. Mourgoglou). Analysis & PDE 10-3 (2017), 559--588. pdf On Tsirelson's theorem about triple points for harmonic measure (with A. Volberg). International Mathematics Research Notices, Vol. 2018 (2018), No. 12, pp. 3671-3683. pdf On a two-phase problem for harmonic measure in general domains (with J. Azzam, M. Mourgoglou and A. Volberg). Preprint (2016). To appear in Amer. J. Math. pdf Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions (with J. Garnett and M. Mourgoglou). Duke Math. J. Vol. 167 (2018), No. 8, 1473-1524. pdf Uniform rectifiability, elliptic measure, square functions, and ε-approximability via an ACF monotonicity formula (with J. Azzam, J. Garnett and M. Mourgoglou). Preprint (2016). pdf A two-phase free boundary problem for harmonic measure and uniform rectifiability (with J. Azzam and M. Mourgoglou). Preprint (2017). pdf Harmonic measure and quantitative connectivity: geometric characterization of the L^{p } solvability of the Dirichlet problem. Part II (with J. Azzam and M. Mourgoglou). Preprint (2018). pdf |
QUASICONFORMAL MAPPINGS, SOBOLEV SPACES, AND RELATED TOPICS |
Analytic capacity and quasiconformal mappings with W^{1,2} Beltrami coefficient, (with Albert Clop). Math. Res. Lett. 15 (2008), no. 4, 779-793. pdf Quasiconformal maps, analytic capacity, and non linear potentials (with I. Uriarte-Tuero). Duke Math. J. 162 (2013), no. 8, 1503-1566. pdf Quasiconformal distortion of Hausdorff measures. Preprint (2009). pdf Hausdorff measure of quasicircles (with I. Prause and I. Uriarte-Tuero). Adv. Math. 229:2 (2012), 1313-1328 pdf Quasiconformal distortion of Riesz capacities and Hausdorff measures in the plane, (with Astala, Clop, Verdera and Uriarte-Tuero). Amer. J. Math. 135 (2013), no. 1, 17-52. pdf Smoothness of the Beurling transform in Lipschitz domains (with V. Cruz). J. Funct. Anal. 262(10) (2012), 4423-4457. pdf Regularity of C^{1} and Lipschitz domains in terms of the Beurling transform. J. Math. Pures Appl. (9) 100 (2013), no. 2, 137-165. pdf A T(P) theorem for Sobolev spaces on domains (with M. Prats). J. Funct. Anal. 268 (2015), no. 10, 2946--2989. pdf |
SURVEYS AND EXPOSITORY PAPERS |
On the semiadditivity of analytic capacity and planar Cantor sets, (with J. Mateu and J. Verdera) Contemp. Math. 320 (2003), 259-278. pdf Analytic capacity and Calderon-Zygmund theory with non doubling measures, Lecture notes of a course given at the Universidad de Sevilla in December 2003. pdf Singularitats de funcions analitiques, integrals singulars i conjunts fractals, Butl. Soc. Catalana Mat. 17 (2002), no. 2, 75-90 (in Catalan). pdf Painleve's poblem, analytic capacity and curvature of measures, Proceedings of the Fourth European Congress, 2004. pdf Painleve's poblem and analytic capacity, Lecture notes of a minicourse given at El Escorial, 2004. pdf Analytic capacity, rectifiability, and the Cauchy integral, Proceedings of the ICM 2006, Madrid. pdf The T1 theorem. Notes of a short PhD course on the classical T1 theorem of David and Journe, given in 2012 at Barcelona and typed by V. Chousionis. pdf About the Jones-Wolff Theorem on the Hausdorff dimension of harmonic measure (with Cufí and Verdera). Lecture Notes of a series of reading seminars held at the UAB in 2017. pdf |