ISSN:
1432-0835
Keywords:
AMS Subject Classification (1991):35J15, 35K05, 22E30
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. Let G be a connected Lie group with Lie algebra $\mathfrak{g}$ and $a_1,\ldots,a_{d'}$ an algebraic basis of $\mathfrak{g}$ . Further let $A_i$ denote the generators of left translations, acting on the $L_p$ -spaces $L_p(G\,;dg)$ formed with left Haar measure dg, in the directions $a_i$ . We consider second-order operators \[ H=-\sum_{i,j=1}^{d'} A_i \, c_{ij} \, A_j + \sum_{i=1}^{d'} (c_i \, A_i + A_i \, c'_i) + c_0 \, I \] corresponding to a quadratic form with complex coefficients $c_{ij}$ , $c_{i}$ , $c'_{i}$ , $c_{0}\in L_{\infty}$ . The principal coefficients $c_{ij}$ are assumed to be Hölder continuous and the matrix $C=(c_{ij})$ is assumed to satisfy the (sub)ellipticity condition \[ \mathfrak{R} C = 2^{-1}\Big(C+C^*\Big)\geq \mu I〉0 \] uniformly over G. We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel. Moreover, we establish Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for operators \[ H'=-\sum_{i,j=1}^{d'} c_{ij} \, A_i \, A_j + \sum_{i=1}^{d'} c_i \, A_i + c_0 \, I \] in nondivergence form for which the principal coefficients are at least once differentiable.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s005260050129
