Asymptotics of periodic subelliptic operators (Q1917958)

Presently the small time behaviour of the Gaussian bounds for the heat kernel of an elliptic pure second order operator in divergence form with bounded real symmetric measurable coefficients is reasonably well understood. The Gaussian bounds are in terms of a distance which is naturally associated with the coefficients of the operator. The authors consider pure second order subelliptic operators \(H= - \sum_{i,j=1}^{d_1} A_i\) \(c_{ij}\) \(A_j\) with bounded real symmetric measurable periodic coefficients \(c_{ij}\) on a stratified Lie group \(G\), where the \(A_i\) are left derivatives in a direction \(a_i\) and \(a_1, \dots, a_{d_1}\) is a basis for the first layer in the stratification of the Lie algebra of \(G\). Let \(\widehat C= (\widehat c_{ij})\) be the homogenization of the coefficients \((c_{ij})\) in the sense of \textit{A. Bensoussan}, \textit{J.-L. Lions} and \textit{G. C. Papanicolaou} [Asymptotic analysis for periodic structures, Studies in mathematics and its applications, vol. 5, Amsterdam: North-Holland Publ. Comp. (1978; Zbl 0404.35001)]. Then \(\widehat C\) is a strictly positive symmetric constant matrix. Let \(\widehat H\) be the associated subelliptic operator and \(S\) and \(\widehat S\) the semigroups generated by \(H\) and \(\widehat H\), with kernels \(K\) and \(\widehat K\). It is shown that \[ \lim_{t\to\infty} t^{D/(2\delta)} |S_t- \widehat S_t |_{p\to r} = 0 \] for all \(1\leq p\leq r \leq \infty\), where \(\delta = 1/p- 1/r\) and \(D\) is the homogeneous dimension of \(G\). Moreover, \(\lim_{t\to\infty} t^{D/(2q)} ||K_t - \widehat K_t ||_p= 0\) for all \(p \in [1,\infty]\), where \(1/p + 1/q = 1\) and \[ ||f ||_p= \text{ess sup}_{x \in G} \bigl(\int_G dg |f(x;y) |^p \bigr)^{1/p}. \] Thus the large time behaviour of the kernel is determined by the distance associated with the homogenized coefficients \((\widehat c_{ij})\). The last statement was proved on \(\mathbb{R}\) and conjectured on \(\mathbb{R}^d\) by \textit{E. B. Davies} [Q. J. Math., Oxf. II. Ser. 44, 283-299 (1993; Zbl 0830.34019)].