Sub-Laplacians with drift on Lie groups of polynomial volume growth (Q2784245)

In this book some basic properties of a centered non-symmetric sub-Laplacian \(L\) on a connected Lie group \(G\) of polynomial volume growth are studied in detail. Main attention is focused, in particular, on the large time behaviour of the heat kernel \(p_t(x,y)\), i.e. of the fundamental solution of the equation \((\partial/\partial t+L)u=0\). A parabolic Harnack inequality (1969, Bony; 1993, Varopoulos, Saloff-Coste, Coulhon) for the sub-Laplacian \(L\) is proved by using ideas from homogenization theory (1978, Bensoussan, Lions, Papanicolaou; 1994, Jikov, Kozlov, Oleinik) and by adapting the method of Krylov and Safonov (1981). With the help of this inequality a Taylor formula for the heat functions \(u=u(t,x)\), satisfying the equation \((\partial/ \partial t+L)u=0\), is obtained. As a consequence of this formula, the Harnack inequalities for the space and time derivatives of the heat functions are derived. The harmonic functions which grow polynomially are also characterized. The Gaussian estimates (1992, Hebisch; 1991, Ben Arous, Leandre) for the heat kernel \(p_t(x,y)\) and estimates similar to the classical Berry-Esseen estimate (1971, Feller; 1975, Petrov) are established. Finally, the Riesz transform operators are studied (1992, Alexopoulos) by using the Berry-Esseen estimates for the spatial derivatives of the kernel \(p_t (x,q)\). It is indicated that the results obtained may also be extended to the non-centered sub-Laplacians: if \(L\) is non-centered, one can conjugate \(L\) by a convenient multiplicative function and obtain another centered sub-Laplacian \(L_c\).