Research

My research areas are in the intersection of analysis, probability and PDEs. I study analytic, geometric and probabilistic 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.

Links: Google Scholar, PURE, ReseachGate

Publications and Preprints

Most of my papers are available at arXiv

1. (with F. Baudoin, M. Yang) Korevaar-Schoen and heat kernel characterizations of Sobolev and BV spaces on local trees, submitted, 2025. arXiv

2. (with A. Bendikov, L. Saloff-Coste) Riesz transform, function spaces and their applications on infinite dimensional compact groups, accepted by Potential Analysis, 2026. arXiv

3. (with F. Baudoin, C.-H. Huang,  C. Ouyang, S. Tindel, J. Wang) Weighted Besov spaces on Heisenberg groups and applications to the Parabolic Anderson model,  submitted, 2025. arXiv

4. (with F. Baudoin, C.-H. Huang,  C. Ouyang, S. Tindel, J. Wang) Parabolic Anderson model in bounded domains of recurrent metric measure spaces, Trans. Amer. Math. Soc., 379 (2026), 1799-1851. arXiv

5. (with F. Baudoin) Heat kernel gradient estimates for the Vicsek set, Math. Nachr., 297, no. 12,  2024. arXiv 

6. (with F. Baudoin) Sobolev spaces and Poincaré inequalities on the Vicsek fractal,  Anna. Fenn. Math., 48, no. 1, 3-26, 2023. arXiv

7. (with F. Baudoin) Dirichlet fractional Gaussian fields on the Sierpinski gasket and their discrete graph approximations, Stochastic Process. Appl., 162, 593-616, 2023. arXiv

8. (with P. Auscher, J.M. Martell, C. Prisuelos-Arribas) The regularity problem for degenerate elliptic operators in weighted spaces, Rev. Mat. Iberoam., 39, no. 2, 563–610, 2023. arXiv 

9. (with F. Baudoin) A note on second order Riesz transforms in 3-dimensional Lie groups, Arch. Math. (Basel), 118, 291–304, 2022. arXiv

10. (with R. Bañuelos, F. Baudoin, Y. Sire) Multiplier theorems via martingale transforms, J. Funct. Anal., 281, no. 9, 109188, 2021. arXiv

11. (with P. Alonso-Ruiz, F. Baudoin, L. Rogers, N. Shanmugalingam, A. Teplyaev) BV functions and fractional Laplacians on Dirichlet spaces, Asian J. Math., 27, no. 4, 441--466, 2023. arXiv

12. (with J.M. Martell, C. Prisuelos-Arribas) The regularity problem for uniformly elliptic operators in weighted spaces, Potential Analysis, 58, 409--439, 2023. arXiv​

13. (with F. Baudoin, M. Bonnefont) Convergence to equilibrium for hypoelliptic non-symmetric Ornstein-Uhlenbeck type operators, Contemp. Math., 809, 103--122, 2025. arXiv​

14. (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

15. A note on Sobolev type inequalities on graphs with polynomial volume growth, Arch. Math. (Basel), 113, no. 3, 313–323, 2019. arXiv

16. (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

17. (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​

18. (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, Slides, Video

19. (with T. Coulhon, B. Hua) Riesz transforms for bounded Laplacians on graphs, Math. Z., 294, no. 1-2, 397-417, 2020. arXiv

20. (with J.M. Martell, C. Prisuelos-Arribas) Conical square functions for degenerate elliptic operators, Adv. Calc. Var., 13, no. 1, 75-113, 2020. arXiv

21. Hardy spaces on metric measure spaces with generalized sub-Gaussian heat kernel estimates, J. Aust. Math. Soc., 104, no. 2, 162–194, 2018. arXiv

22. (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, Slides

23. Sub-Gaussian heat kernel estimates and quasi Riesz transforms for \(1\le p\le 2\), Publ. Mat. 59, no. 2, 313–338, 2015. arXiv​

24. (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.