My research areas are probability, analysis and PDEs. I study probabilistic, analytic and geometric aspects of Markov semigroups, stochastic processes and random fields on spaces with smooth or rough structures. I also work on regularity problems related to elliptic operators.

My  research is partially supported by Simons Collaboration Grants for Mathematicians
.

Publications and Preprints
​Most of my papers are available at arXiv.
  1. (with F. Baudoin) Sobolev spaces and Poincaré inequalities on the Vicsek fractal,  Anna. Fenn. Math. 48, no. 1, 3-26, 2023. arXiv:2207.02949
  2. (with F. Baudoin) Dirichlet fractional Gaussian fields on the Sierpinski gasket and their discrete graph approximations, in revision for Stochastic Process. Appl. arXiv:2201.03970
  3. (with P. Auscher, J.M. Martell, C. Prisuelos-Arribas) The regularity problem for degenerate elliptic operators in weighted spaces, accepted by  Rev. Mat. Iberoam. arXiv:2106.14422 
  4. (with F. Baudoin) $L^p$-Poincaré inequalities on nested fractals, in revision for Potential Anal. arXiv:2012.03090
  5. (with F. Baudoin) A note on second order Riesz transforms in 3-dimensional Lie groups, Arch. Math. (Basel). 118, 291–304, 2022. arXiv:2003.09931
  6. (with R. Bañuelos, F. Baudoin, Y. Sire) Multiplier theorems via martingale transforms,  J. Funct. Anal. 281, no. 9, 109188, 2021. arXiv:2003.02077
  7. (with P. Alonso-Ruiz, F. Baudoin, L. Rogers, N. Shanmugalingam, A. Teplyaev) BV functions and fractional Laplacians on Dirichlet spaces, preprint. arXiv:1910.13330
  8. (with J.M. Martell, C. Prisuelos-Arribas) The regularity problem for uniformly elliptic operators in weighted spaces, accepted by Potential Analysis. arXiv:1908.03328​
  9. (with F. Baudoin, M. Bonnefont) Convergence to equilibrium for hypoelliptic non-symmetric Ornstein-Uhlenbeck type operators, preprint. arXiv:1906.10828​
  10. (with P. Alonso-Ruiz, F. Baudoin, L. Rogers, N. Shanmugalingam, A. Teplyaev) Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates, Calc. Var. Partial Differential Equations 60, no. 5, 170, 2021. arXiv:1903.10078
  11. A note on Sobolev type inequalities on graphs with polynomial volume growth, Arch. Math. (Basel), 113, no. 3, 313–323, 2019. arXiv:1902.02440
  12. (with P. Alonso-Ruiz, F. Baudoin, L. Rogers, N. Shanmugalingam, A. Teplyaev) Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates, Calc. Var. Partial Differential Equations 59, no. 3, 103,  2020. arXiv:1811.11010
  13. (with P. Alonso-Ruiz, F. Baudoin, L. Rogers, N. Shanmugalingam, A. Teplyaev) Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities, J. Funct. Anal. 278, no. 11, 108459, 2020. arXiv:1811.04267​
  14. (with R. Bañuelos, F. Baudoin) Gundy-Varopoulos martingale transforms and their projection operators on manifolds and vector bundles, Math. Ann. 378, no.1-2, 359–388, 2020. arXiv:1802.02410, Slides, Video
  15. (with T. Coulhon, B. Hua) Riesz transforms for bounded Laplacians on graphs, Math. Z. 294, no. 1-2, 397-417, 2020. arXiv:1708.05476
  16. (with J.M. Martell, C. Prisuelos-Arribas) Conical square functions for degenerate elliptic operators, Adv. Calc. Var. 13, no. 1, 75-113, 2020. arXiv:1610.05952
  17. Hardy spaces on metric measure spaces with generalized sub-Gaussian heat kernel estimates, J. Aust. Math. Soc. 104, no. 2, 162–194, 2018. arXiv:1603.05379
  18. (with T. Coulhon, J. Feneuil, E. Russ) Riesz transform for \(1\le p\le 2\) without Gaussian heat kernel bound, J. Geom. Anal. 27, no. 2, 1489–1514, 2017. arXiv:1510.08275, Slides
  19. Sub-Gaussian heat kernel estimates and quasi Riesz transforms for \(1\le p\le 2\), Publ. Mat. 59, no. 2, 313–338, 2015. arXiv:1401.2279​
  20. (with J. Zhao) Weyl transform and generalized spectrogram associated with quaternion Heisenberg group, Bull. Sci. Math. 136, no. 2, 127–143, 2012.​​

Other writings

  1. A regularization property of heat semigroups and its applications, Heat Kernels, Stochastic Processes and Functional Inequalities,  Oberwolfach Report, 2019.
  2. (with P. Alonso-Ruiz, F. Baudoin, L. Rogers, N. Shanmugalingam, A. Teplyaev) BV functions and Besov spaces associated with Dirichlet spaces, book draft (not submitted). arXiv:1806.03428
  3. Quasi Riesz transforms, Hardy spaces and generalized sub-Gaussian heat kernel estimates, Ph.D. thesis, with the abstract in Bull. Aust. Math. Soc. 92, no. 3, 508–510, 2015.