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Analytic-probabilistic methods for borderline singular integrals (AnProb)

A five-year research project (11/2011-10/2016) in the area of Mathematics, funded through an ERC Starting Grant (1.1 M€) within the Seventh Framework Programme of the European Union.


AnProb was an extensive project aimed at advancing the understanding of singular integral operators of Harmonic Analysis in various situations on the borderline of the existing theory. This was achieved by a creative combination of techniques from Analysis and Probability.

The addressed problems fell under the following categories, with many interconnections and overlap: (i) sharp weighted inequalities; (ii) non-homogeneous singular integrals and metric spaces; (iii) local Tb theorems with borderline assumptions; (iv) rough boundary value problems; and (v) vector-valued singular integrals and time-frequency analysis.

Research team

During its 5-year span, the following researchers have been employed by AnProb:

  • principal investigator: Tuomas Hytönen
  • 4 international postdocs: Wendolin Damian, Ana Grau de la Herran, Kangwei Li, Ioannis Parissis
  • 1 PhD student continuing as postdoc: Timo Hänninen
  • 2 further PhD students Olli Tapiola, Emil Vuorinen
  • 1 research associate: Jori Merikoski

Selected achievements

  • A new approach giving best available bounds for the problem of quantitative affine approximation (with S. Li and A. Naor, Discrete Analysis (2016) and A. Naor, arXiv:1608.01915)
  • Bounded variation extensions of Lp boundary values, complementing classical results available only for p = ∞ (with A. Rosén, arXiv:1405.2153)
  • A characterization of the two-weight inequality for the Hilbert transform with general measures ( arXiv:1312.0843 )
  • A proof of the pointwise convergence of Fourier series of functions taking values in any known example of UMD spaces (with M. Lacey, Math. Ann. 357 (2013), 1329-1361)
  • A solution to Hofmann's question on the minimal integrability conditions on the test functions in the local Tb theorem (with F. Nazarov,  arXiv:1206.0907 )




  1. Tuomas Hytönen, Jan van Neerven, Mark Veraar, Lutz Weis: Analysis in Banach Spaces - Volume I: Martingales and Littlewood-Paley Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 63, Springer (2016). doi:10.1007/978-3-319-48520-1 


  1. Timo S. Hänninen: Dyadic analysis of integral operators: median oscillation decomposition and testing conditions, PhD thesis, University of Helsinki (2015). URN:ISBN:978-951-51-1393-1
  2. Olli Tapiola: Adjacent and random dyadic systems and their applications to metric, Euclidean and vector-valued analysis, PhD thesis, University of Helsinki (2016). URN:ISBN:978-951-51-2119-6 
  3. Emil Vuorinen: Two-weight inequalities and local Tb theorems: quadratic testing and big piece methods, PhD thesis, University of Helsinki (2017). URN:ISBN:978-951-51-3174-4

Research articles (by topic)

Sharp weighted inequalities

  1. Tuomas P. Hytönen, Carlos Pérez, Ezequiel Rela: Sharp Reverse Hölder property for A weights on spaces of homogeneous type, Journal of Functional Analysis 263 (2012), 3883–3899. doi:10.1016/j.jfa.2012.09.013, arXiv:1207.2394
  2. Tuomas P. Hytönen, Michael T. Lacey and Carlos Pérez: Sharp weighted bounds for the q-variation of singular integrals, Bulletin of the London Mathematical Society 45 (2013), 529-540.  doi:10.1112/blms/bds114, arXiv:1202.2229   
  3. Timo S. Hänninen, Tuomas P. Hytönen: The A2 theorem and the local oscillation decomposition for Banach space valued functions, Journal of Operator Theory 72 (2014), no. 1, 193-218. doi:10.7900/jot.2012nov21.1972, arXiv:1210.6236
  4. Tuomas P. Hytönen: The A2 theorem: Remarks and complements, Contemporary Mathematics 612 (2014), 91-106. doi:10.1090/conm/612/12226, arXiv:1212.3840
  5. Tuomas P. Hytönen, Carlos Pérez: The L(log L)ϵ  endpoint estimate for maximal singular integral operators, Journal of Mathematical Analysis and Applications 428 (2015), no. 1, 605-626. doi:10.1016/j.jmaa.2015.03.017, arXiv:1503.04008
  6. Timo S. HänninenTuomas P. Hytönen, Kangwei Li: Two-weight Lp-Lq bounds for positive dyadic operators: unified approach to p<q and p>q, Potential Analysis 45 (2016), no. 3, 579-608. doi:10.1007/s11118-016-9559-9 arXiv:1412.2593
  7. Tuomas P. Hytönen: The Holmes-Wick theorem on two-weight bounds for higher order commutators revisited. Archiv der Mathematik (Basel) 107 (2016), no. 4, 389–395. doi:10.1007/s00013-016-0956-5arXiv:1604.02244
  8. Kangwei Li: Two weight inequalities for bilinear forms. Collectanea Mathematica (2016). doi:10.1007/s13348-016-0182-2arXiv:1511.07250
  9. Tuomas P. Hytönen Representation of singular integrals by dyadic operators, and the A2 theorem. Expositiones Mathematicae (2016). doi:10.1016/j.exmath.2016.09.003arXiv:1108.5119
  10. Timo S. Hänninen: Two weight inequality for vector-valued positive dyadic operators by parallel stopping cubes. Israel Journal of Mathematics 219 (2017), no. 1, 71-114, doi:10.1007/s11856-017-1474-2, arXiv:1404.6933 
  11. Tuomas P. Hytönen, Luz Roncal, Olli Tapiola: Quantitative weighted estimates for rough homogeneous singular integrals. Israel Journal of Mathematics 218 (2017), no. 1, 133-164, doi:10.1007/s11856-017-1462-6arXiv:1510.05789
  12. Timo S. HänninenRemark on dyadic pointwise domination and median oscillation decomposition, Houston Journal of Mathematics 43 (2017), no. 1, 183-197, arXiv:1502.05942

Non-homogeneous singular integrals and metric spaces

  1. Tuomas P. Hytönen, Anna Kairema: What is a cube?, Annales Academiæ Scientiarum Fennicæ Mathematica 38 (2013), 405–412.  doi:10.5186/aasfm.2013.3838, arXiv:1209.2885
  2. Tuomas P. Hytönen, Olli Tapiola: Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes, Journal of Approximation Theory 185 (2014), 12–30. doi:10.1016/j.jat.2014.05.017 arXiv:1310.2047
  3. Tuomas P. Hytönen, Henri Martikainen: Non-homogeneous T1 theorem for bi-parameter singular integrals, Advances in Mathematics 261 (2014), 220–273. doi:10.1016/j.aim.2014.02.011, arXiv:1209.4473
  4. Theresa C. Anderson, Tuomas P. HytönenOlli Tapiola: Weak A weights and weak Reverse Hölder property in a space of homogeneous type, Journal of Geometric Analysis  (2016). doi:10.1007/s12220-015-9675-6 arXiv:1410.3608

Local Tb theorems

  1. Ana Grau de la Herrán, Jarod Hart and Lucas Oliveira: Multilinear local Tb theorem for square functions, Annales Academiæ Scientiarum Fennicæ Mathematica 38 (2013), 697–720. doi:10.5186/aasfm.2013.3841, arXiv:1210.0735   
  2. Tuomas Hytönen, Antti Vähäkangas: The local non-homogeneous Tb theorem for vector-valued functions, Glasgow Mathematical Journal 57 (2015), 17–82. doi:10.1017/S0017089514000123, arXiv:1201.0648
  3. Ana Grau de la Herrán: Comparison of T1 conditions for multiparameter operators, Proceedings of the American Mathematical Society   144 (2016), no. 6, 2437–2443. doi:10.1090/proc/12891,  arXiv:1403.7968

Vector-valued singular integrals and time-frequency analysis

  1. Tuomas P. Hytönen, Michael T. Lacey: Pointwise convergence of vector-valued Fourier series, Mathematische Annalen 357 (2013), 1329-1361. doi:10.1007/s00208-013-0935-0, arXiv:1205.0261 (star)
  2. Tuomas P. Hytönen, Michael T. Lacey, Ioannis Parissis: The vector valued quartile operator, Collectanea Mathematica 64 (2013), 427-454.  doi:10.1007/s13348-012-0070-3, arXiv:1203.5604
  3. Tuomas P. Hytönen, Assaf Naor: Pisier’s inequality revisited, Studia Mathematica 215 (2013), 221-235. doi:10.4064/sm215-3-2, arXiv:1207.5375
  4. Tuomas P. Hytönen, Michael T. Lacey, Ioannis Parissis: A variation norm Carleson theorem for vector-valued Walsh-Fourier series, Revista Matemática Iberoamericana 30 (2014), no. 3, 979-1014. doi:10.4171/RMI/804, arXiv:1209.3383
  5. Alejandro J. Castro, Tuomas P. Hytönen: Bounds for partial derivatives: necessity of UMD and sharp constants. Mathematische Zeitschrift 282 (2016), no. 3-4, 635-650. doi:10.1007/s00209-015-1556-yarXiv:1406.2832
  6. Timo S. HänninenTuomas P. Hytönen : Operator-valued dyadic shifts and the T(1) theorem. Monatshefte für Mathematik 180 (2016), no. 2, 213-253. doi:10.1007/s00605-016-0891-3 arXiv:1412.0470
  7. Tuomas Hytönen, Sean Li, Assaf Naor: Quantitative affine approximation for UMD targets, Discrete Analysis (2016), paper no. 614, 37 pp. doi:10.19086/da.614arxiv:1510.00276  (star)

Preprints (by topic)

Sharp weighted inequalities

  1. Tuomas P. Hytönen: The two-weight inequality for the Hilbert transform with general measures, submitted arXiv:1312.0843  (star)
  2. Tuomas P. HytönenKangwei Li: Weak and strong Ap-A estimates for square functions and related operators, to appear in Proceedings of the American Mathematical Society, arXiv:1509.00273
  3. Wendolín Damián, Mahdi Hormozi, Kangwei Li: New bounds for bilinear Calderón-Zygmund operators and applications, to appear in Revista Matemática IberoamericanaarXiv:1512.02400
  4. Kangwei Li: Sparse domination theorem for multiliner singular integral operators with Lr-Hörmander condition, arXiv:1606.03925
  5. Tuomas P. Hytönen, Emil Vuorinen: A two-weight inequality between Lp ( 2 )    and Lp, submitted,  arXiv:1608.07199

Non-homogeneous singular integrals and metric spaces

  1. Henri Martikainen, Mihalis Mourgoglou, Emil Vuorinen: Non-homogeneous square functions on general sets: suppression and big pieces methods, to appear in Journal of Geometric AnalysisarXiv:1606.04309

Local Tb theorems

  1. Tuomas P. Hytönen, Fedor Nazarov: The local Tb theorem with rough test functions, submitted arXiv:1206.0907 (star)
  2. Henri Martikainen, Emil Vuorinen: Dyadic-probabilistic methods in bilinear analysis, arXiv:1609.01706
  3. Tuomas P. Hytönen, Ana Grau de la Herrán: Dyadic representation and boundedness of non-homogeneous Calderón--Zygmund operators with mild kernel regularity, submittedarXiv:1612.05133

Rough boundary value problems

  1. Tuomas P. Hytönen, Andreas Rosén: Bounded variation approximation of Lp dyadic martingales and solutions to elliptic equations, to appear in Journal of the European Mathematical Society, arXiv:1405.2153 (star)

Vector-valued singular integrals and time-frequency analysis

  1. Tuomas P. Hytönen, Michael T. Lacey: Pointwise convergence of Walsh-Fourier series of vector-valued functions,  arXiv:1202.0209 
  2. Tuomas P. Hytönen, Assaf Naor: Heat flow and quantitative differentiation, to appear in Journal of the European Mathematical SocietyarXiv:1608.01915



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