Books

Analytic Capacity, the Cauchy Transform, and Non-Homogeneous Calderon-Zygmund Theory. Birkhauser (2014).

Notes on harmonic measure (by Marti Prats and Xavier Tolsa, 2023) pdf

Some Research Papers

Non-homogeneous harmonic analysis

BMO, H¹, and Calderon-Zygmund operators for non doubling measures. Math. Ann. 319 (2001), 89-149. pdf
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. pdf
The atomic space H¹ 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 Lip₁ 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 C¹ 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. Not. IMRN (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). Mem. Amer. Math. Soc. 266 (2020), no. 1293.pdf
Analytic capacity and projections (with Alan Chang). J. Eur. Math. Soc. (JEMS) 22 (2020), no. 12, 4121-4159. pdf
On C¹-approximability of functions by solutions of second order elliptic equations on plane compact sets and C-analytic capacity (with P.V. Paramonov). Anal. Math. Phys. 9 (2019), no. 3, 1133-1161. pdf
Removable singularities for Lipschitz caloric functions in time varying domains (with J. Mateu and L. Prat). Rev. Mat. Iberoam. 38, 2 (2022), pp. 547-588. pdf

Singular integrals, square functions, 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. Not. IMRN (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. PDE 57 (2018), no. 1, Art. 16, 63 pp. pdf
Singular integrals unsuitable for the curvature method whose L²-boundedness still implies rectifiability (with P. Chunaev and J. Mateu). J. Anal. Math. 138 (2019), no.2, 741-764. pdf
The measures with an associated square function operator bounded in L² (with B. Jaye and F. Nazarov). Adv. Math., 339, 1 (2018), 60-112. pdf
Rectifiability of measures and the β_p coefficients. Publ. Mat. 63 (2019), 491-519. pdf
Failure of L² boundedness of gradients of single layer potentials for measures with zero low density (with J.M. Conde-Alonso and M. Mourgoglou). Math. Ann. 373 (2019), 253-285. pdf
A family of singular integral operators which control the Cauchy transform (with P. Chunaev and J. Mateu). Math. Z. 294 (2020), 1283-1340. pdf
L-²-boundedness of gradients of single layer potentials and uniform rectifiability (with L. Prat and C. Puliatti). Analysis & PDE 14 (2021), no. 3, 717-791. pdf
Characterization of rectifiable measures in terms of α-numbers (with J. Azzam and T. Toro). Trans. Amer. Math. Soc. 373 (2020), no. 11, 7991-8037. pdf
Jump formulas for singular integrals and layer potentials on rectifiable sets. Proc. Amer. Math. Soc. 148(11) (2020), 4755-4767. pdf
A proof of Carleson's ε²-conjecture (with B. Jaye and M. Villa). Ann. of Math. (2) 194 (2021), no. 1, 97-161. pdf
The measures with L²-bounded Riesz transform satisfying a subcritical Wolff-type energy condition (with Damian Dąbrowski). Preprint (2021). pdf
The measures with L²-bounded Riesz transform and the Painlevé problem (with Damian Dąbrowski). Preprint (2021). To appear in Memoirs Amer. Math. Soc. pdf
L²-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type (with A. Molero, M. Mourgoglou, and C. Puliatti). Arch. Ration. Mech. Anal. 247 (2023), no. 3, Paper No. 38, 59 pp. pdf
Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's ε² conjecture in higher dimensions (with I. Fleschler and M. Villa). Preprint (2023). pdf
Carleson's ε² conjecture in higher dimensions (with I. Fleschler and M. Villa). Preprint (2023). pdf

Harmonic measure, unique continuation, elliptic PDE's, and related topics

Singular sets for harmonic measure on locally flat domains with locally finite surface measure (with J. Azzam and M. Mourgoglou). Int. Math. Res. Not. IMRN 2017(12) (2017), 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). J. Reine Angew. Math. 758 (2020), 183-221. 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). Int. Math. Res. Not. IMRN, 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). Amer. J. Math. 141(5) (2019), 1259-1279. 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). Int. Math. Res. Not. IMRN. (2023), no.13, 10837–10941. pdf
A two-phase free boundary problem for harmonic measure and uniform rectifiability (with J. Azzam and M. Mourgoglou). Trans. Amer. Math. Soc., vol. 373, no. 6, 2020, 4359-4388. pdf
Harmonic measure and quantitative connectivity: geometric characterization of the L^p-solvability of the Dirichlet problem (with Part I by Hofmann and Martell, and Part II by Azzam, Mourgoglou, and Tolsa). Invent. Math. 222 (2020), no. 3, 881-993. pdf
The mutual singularity of harmonic measure and Hausdorff measure of codimension smaller than one. Int. Math. Res. Not. IMRN. Vol. 2021, No. 18, pp. 13783-13811 pdf
The two-phase problem for harmonic measure in VMO (with M. Prats). Calc. Var. PDE. (2020) 59:102. pdf, web
Unique continuation at the boundary for harmonic functions in C¹ domains and Lipschitz domains with small constant. Comm. Pure Appl. Math. 76 (2), 2023, pp. 305-336. pdf
The two-phase problem for harmonic measure in VMO and the chord-arc condition (with T. Toro). Trans. Amer. Math. Soc. Ser. B 11 (2024), 1294–1315. pdf
The regularity problem for the Laplace equation in rough domains (with M. Mourgoglou). Duke Math. J. 173 (9) (2024), 1731–1837. pdf
L^p-solvability of the Poisson-Dirichlet problem and its applications to the regularity problem (with M. Mourgoglou and B. Poggi). Preprint (2022). To appear in J. Eur. Math. Soc. (JEMS). pdf
Connectivity conditions and boundary Poincare inequalities (with Olli Tapiola). Analysis & PDE 17-5 (2024), 1831–1870. pdf
The A_∞ condition, ε-approximators, and Varopoulos extensions in uniform domains (with S. Bortz, B. Poggi, and O. Tapiola). J. Geom. Anal. 34 (2024), no. 7, Paper No. 218. pdf
The dimension of harmonic measure on some AD-regular flat sets of fractional dimension. Int. Math. Res. Not. IMRN, vol. 2024, no. 8, April 2024, pp. 6579–6605. pdf
Extrapolation of solvability of the regularity problem in rough domains (with J.M. Gallegos and M. Mourgoglou). J. Funct. Anal. 288 (2025), no. 1, Paper No. 110672. pdf
A counterexample regarding a two-phase problem for harmonic measure in VMO. To appear in Potential Analysis (2025). pdf
The dimension of planar elliptic measures arising from Lipschitz matrices in Reifenberg flat domains (with I. Guillén-Mola and M. Prats). To appear in Analysis and Mathematical Physics (2025). pdf
Solvability of the Neumann problem for elliptic equations in chord-arc domains with very pieces of good superdomains (with M. Mourgoglou). Preprint (2024). 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 C¹ 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