, We needed to replace X T j by E( X|F T j ) because we are dealing with submartingales and unlike in the martingale case

D. Applebaum and R. Bañuelos, Probabilistic approach to fractional integrals and the Hardy-Littlewood-Sobolev inequality, Analytic methods in interdisciplinary applications, vol.116, pp.17-40, 2015.

N. Arcozzi, Riesz transforms on compact Lie groups, spheres and Gauss space, Ark. Mat, vol.36, issue.2, pp.201-231, 1998.

K. Astala, T. Iwaniec, and E. Saksman, Beltrami operators in the plane, Duke Math. J, vol.107, issue.1, pp.27-56, 2001.

P. Auscher and J. M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators, Contemporary Mathematics, pp.61-83, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00331495

R. Ba, The foundational inequalities of D. L. Burkholder and some of their ramifications, Illinois J. Math, vol.54, pp.789-868, 2010.

R. Bañuelos and F. Baudoin, Martingale transforms and their projection operators on manifolds, Potential Anal, vol.38, issue.4, pp.1071-1089, 2013.

R. Bañuelos and P. Janakiraman, L p -bounds for the Beurling-Ahlfors transform, Trans. Amer. Math. Soc, vol.360, issue.7, pp.3603-3612, 2008.

R. Bañuelos and A. Osekowski, Sharp martingale inequalities and applications to Riesz transforms on manifolds, Lie groups and Gauss space, J. Funct. Anal, vol.269, issue.6, pp.1652-1713, 2015.

R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J, vol.80, issue.3, pp.575-600, 1995.

D. Bakry, Transformations de Riesz pour les semi-groupes symétriques. II. étude sous la condition ? 2 ? 0, Séminaire de probabilités, XIX, vol.84, pp.145-174, 1983.

D. Bakry, Un critère de non-explosion pour certaines diffusions sur une variété riemannienne complète, Comptes rendus de l'Académie des Sciences, vol.303, pp.23-26, 1986.

D. Bakry, Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, vol.1247, 1987.

D. Bakry and M. Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, vol.84, pp.177-206, 1983.

F. Bernicot, D. Frey, and S. Petermichl, Sharp weighted norm estimates beyond Calderon-Zygmund theory, Anal. PDE, vol.9, pp.1079-1113, 2016.

S. Buckley, Summation condition on weights, Michigan Math. J, vol.40, pp.153-170, 1993.

A. Carbonaro and O. Dragicevic, Bellman function and linear dimension-free estimates in a theorem of Bakry, J. Funct. Anal, vol.265, issue.7, pp.1085-1104, 2013.

D. V. Cruz-uribe, J. M. Martell, and C. Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol.215, 2011.

K. Dahmani, Sharp dimension free bound for the Bakry-Riesz vector, 2016.

K. Dahmani and . Domelevo, Dimensionless l p estimates for the Riesz vector on manifolds, 2018.

E. B. Davies, Heat kernel and spectral theory, 1989.

K. Domelevo and S. Petermichl, A sharp maximal inequality for differentially subordinate martingales under a change of law, 2016.

K. Domelevo and S. Petermichl, Differential subordination under change of law, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01977592

K. Domelevo, S. Petermichl, and J. Wittwer, A linear dimensionless bound for the weighted Riesz vector. hal-01097113, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01097113

O. Dragicevic, L. Grafakos, M. C. Pereyra, and S. Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat., ?, vol.49, pp.73-91, 2005.

O. Dragicevi? and A. Volberg, Bellman function for the estimates of Littlewood-Paley type and asymptotic estimates in the p ? 1 problem, C. R. Math. Acad. Sci, vol.340, issue.10, pp.731-734, 2005.

J. Duoandikoetxea, Fourier analysis, volume 29 of Graduate Studies in Mathematics, 1995.

J. Eells and K. D. Elworthy, Stochastic dynamical systems, Control theory and topics in functional analysis, pp.179-185, 1976.

D. Elworthy, Geometric aspects of diffusions on manifolds, École d'Été de Probabilités de Saint-Flour XV-XVII, vol.87, pp.277-425, 1985.

M. Emery, Stochastic Calculus in Manifolds, 1989.

K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol.194, 2000.

R. Fefferman, C. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math, vol.134, issue.1, pp.65-124, 1991.

S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00002870

L. Grafakos, Classical Fourier Analysis?, Graduate Texts in Mathematics, vol.249, 2008.

L. Grafakos, Classical and modern Fourier analysis, 2004.

R. F. Gundy and N. Varopoulos, Les transformations de Riesz et les intégrales stochastiques, C.R. Acad. Sci, vol.289, pp.13-16, 1979.

R. F. Gundy, Sur les transformations de Riesz pour le semi-groupe d'OrnsteinUhlenbeck, C. R. Acad. Sci. Paris Sér. I Math, vol.303, issue.19, pp.967-970, 1986.

F. Richard, M. L. Gundy, and . Silverstein, On a probabilistic interpretation for the Riesz transforms, Functional analysis in Markov processes, vol.923, pp.199-203, 1981.

H. Helson and G. Szego, A problem in prediction theory, Ann. Mat. Pura Appl, pp.107-138, 1960.

E. P. Hsu, Stochastic Analysis on Manifolds, vol.38, 2001.

S. Hukovic, S. Treil, and A. Volberg, The Bellman functions and sharp weighted inequalities for square functions, Complex analysis, operators, and related topics, vol.113, pp.97-113, 2000.

R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc, vol.176, pp.227-251, 1973.

T. Hytonen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math, vol.175, issue.3, pp.1473-1506, 2012.

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 1981.

C. J. Isham, Modern differential geometry for physicists, World Scientific Lecture Notes in Physics, vol.61, 1999.

K. Itô, The Brownian motion and tensor fields on Riemannian manifolds, Proceedings of the International Congress of Mathematics, 1962.

T. Iwaniec and G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math, vol.473, pp.25-57, 1996.

D. Kim, Martingale transforms and the Hardy-Littlewood-Sobolev inequality for semigroups, Potential Anal, vol.45, issue.4, pp.795-807, 2016.

N. Kobayashi and K. Nomizu, Foundations of differential geometry, vol.1, 1963.

M. T. Lacey, An elementary proof of the A 2 Bound, Israel J. Math, 2015.

J. M. Lee, Riemmannian Manifolds an introduction to curvature. Graduate texts in mathematics, 1997.

J. M. Lee, Introduction to smooth manifolds. Graduate texts in mathematics, 2000.

A. Lerner and F. Nazarov, Intuitive dyadic calculus: the basics, 2015.

A. K. Lerner, A simple proof of the A 2 conjecture, Int. Math. Res. Not, 2013.

X. Li, Martingale transforms and L p -norm estimates of Riesz transforms on complete Riemannian manifolds, 2008.

X. Li, On the l p -estimates for Beurling-Ahlfors and Riesz transforms on Riemannian manifolds, 2013.

X. Li, On the weak L p -Hodge decomposition and Beurling-Ahlfors transforms on complete Riemannian manifolds. Probab. Theory Related Fields, vol.150, pp.111-144, 2011.

X. Li, Erratum to: On the weak L p Hodge decomposition and BeurlingAhlfors transforms on complete Riemannian manifolds [mr2800906]. Probab. Theory Related Fields, vol.159, pp.409-411, 2014.

X. Li, Riesz transforms on forms and L p -Hodge decomposition on complete Riemanian manifolds, Rev. Mat. Iberoam, vol.30, issue.1, pp.369-370, 2014.

P. Malliavin, Formule de la moyenne, calcul de perturbations et théorèmes d'annulation pour les formes harmoniques, J. Funct. Analy, pp.274-291, 1974.

P. Meyer, Transformations de Riesz pour les lois gaussiennes, Seminar on probability, vol.1059, pp.179-193, 1984.

F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc, vol.12, 1999.

F. Nazarov, S. Treil, and A. Volberg, Bellman function in stochastic control and harmonic analysis, Systems, approximation, singular integral operators, and related topics, vol.129, pp.393-423, 2000.

F. L. Nazarov and S. R. Treil, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz, vol.8, issue.5, 1996.

E. M. Ouhabaz, Analysis of Heat Equations on Domains. LMS-31, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00283205

S. , The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic, Amer. J. Math, vol.129, issue.5, pp.1355-1375, 2007.

S. , The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A(p) characteristic, American Journal of Mathematics, vol.129, issue.5, pp.1355-1375, 2007.

S. , The sharp weighted bound for the Riesz transforms, vol.136, pp.1237-1249, 2008.

S. Petermichl and S. Pott, An estimate for weighted Hilbert transform via square functions, Trans. Amer. Math. Soc, vol.354, issue.4, pp.1699-1703

S. Petermichl and A. Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J, vol.112, issue.2, pp.281-305, 2002.

S. Petermichl and J. Wittwer, A sharp estimate for the weighted Hilbert transform via Bellman functions, Michigan Math. J, 2002.

S. K. Pichorides, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, Studia Math, vol.44, pp.165-179, 1972.

G. Pisier, Riesz transforms: a simpler analytic proof of P.-A. Meyer's inequality, Séminaire de Probabilités, XXII, vol.1321, pp.485-501, 1988.

P. E. Protter, Stochastic integration and differential equations, Stochastic Modelling and Applied Probability, vol.21, 2005.

M. Riesz, Sur les fonctions conjuguées, Math. Z, vol.27, issue.1, pp.218-244, 1928.

J. L. Rubio-de-francia, Factorization theory and A p weights, Amer. J. Math, vol.106, issue.3, pp.533-547, 1984.

P. Sjögren, Operators associated with the Hermite semigroup-a survey, Proceedings of the conference dedicated to Professor Miguel de Guzmán, vol.3, pp.813-823, 1996.

E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, issue.30, 1970.

E. M. Stein, Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Mathematics Studies, issue.63, 1970.

R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal, 1983.

G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab, vol.23, issue.2, pp.522-551, 1995.

G. Wei and W. Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom, 2009.

J. Wittwer, A sharp estimate on the norm of the martingale transform, Mathematical Research Letters, vol.7, issue.1, pp.1-12, 2000.