• Feng, Xiaojing ;
  • Niu, Pengcheng ;
  • Zhu, Maochun
  • Received : 2011.03.28
  • Published : 2014.09.30


We consider a class of hypoelliptic operators of the following type $$L=\sum_{i,j=1}^{p_0}a_{ij}{\partial}^2_{x_ix_j}+\sum_{i,j=1}^{N}b_{ij}x_i{\partial}_{x_j}-{\partial}_t$$, where ($a_{ij}$), ($b_{ij}$) are constant matrices and ($a_{ij}$) is symmetric positive definite on $\mathbb{R}^{p_0}$ ($p_0{\leqslant}N$). By establishing global Morrey estimates of singular integral on the homogenous space and the relation between Morrey space and weak Morrey space, we obtain the global weak Morrey estimates of the operator L on the whole space $\mathbb{R}^{N+1}$.


ultraparabolic operators;weak Morrey estimates;homogeneous type space


  1. E. Barucci, S. Polidoro, and V. Vespri, Some results on partial differential equations and Asian options, Math. Models Methods Appl. Sci. 11 (2001), no. 3, 475-497.
  2. M. Bramanti and L. Brandolini, $L^p$ estimates for uniformly hypoelliptic operators with discontinuous coefficients on homogeneous groups, Rend. Sem. Mat. Univ. Poltec. Torino 58 (2000), no. 4, 389-433.
  3. M. Bramanti, M. C. Cerutti, and M. Manfredini, $L^p$ estimates for some ultraparabolic operators with discontinuous coeficients, Sixth International Colloquium on Differential Equations, Plovdiv, 18-23, August 1995.
  4. M. Bramanti, M. C. Cerutti, and M. Manfredini, $L^p$ estimates for some ultraparabolic operators with discontinuous coefficients, J. Math. Anal. Appl. 200 (1996), no. 2, 332-354.
  5. M. Bramanti, G. Cupini, E. Lanconelli, and E. Priola, Global $L^p$ estimates for degenerate Ornstein-Uhlenbeck operators, Math. Z. 266 (2010), no. 4, 789-816.
  6. L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation, Comm. Pure Appl. Math. 54 (2001), no. 1, 1-42.<1::AID-CPA1>3.0.CO;2-Q
  7. G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), no. 2, 241-256.
  8. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv Math. 13 (1975), no. 2, 161-207.
  9. G. B. Folland and E. M. Stein, Estimates for the $\overline{\partial}_b$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), no. 4, 429-522.
  10. M. Di Francesco and A. Pascucci, On the complete model with stochastic volatility by Hobson and Rogers, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2051, 3327-3338.
  11. M. Di Francesco and S. Polidoro, Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form, Adv. Differential Equations 11 (2006), no. 11, 1261-1320.
  12. L. Hormander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), no. 1, 147-171.
  13. L. P. Kupcov, The fundamental solutions of a certain class of elliptic-parabolic second order equations, (Russian) Differencial'nye Uravnenija 8 (1972), 1649-1660; English Transl, Differential Equations 8 (1972), 1269-1278.
  14. E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Partial differential equations II (Turin, 1993), Rend. Sem. Mat. Univ. Politec. Torino 52 (1994), no. 1, 29-63.
  15. G. M. Lieberman, A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients, J. Funct. Anal. 201 (2003), no. 2, 457-479.
  16. P. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A 346 (1994), no. 1679, 191-204.
  17. M. Manfredini and S. Polidoro, Interior regularity for weak solutions of ultraparabolic equations in divergence form with discontinuous coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 3, 651-675.
  18. D. Mumford, Elastica and Computer Vision, Algebraic geometry and its applications (West Lafayette, IN, 1990), 491-506, Springer, New York, 1994.
  19. S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Matematiche (Catania) 49 (1994), no. 1, 53-105.
  20. S. Polidoro, A global lower bound for the fundamental solution of Kolmogorov-Fokker-Planck equations, Arch. Rational Mech. Anal. 137 (1997), no. 4, 321-340.
  21. L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247-320.
  22. S. Tang and P. Niu, Morrey estimates for parabolic nondivergence operators of Hormander type, Rend. Semin. Mat. Univ. Padova 123 (2010), 91-129.