ABSTRACT
Let A 1 , A 2, A 3 denote a vector space basis, formed by right invariant vector fields, of the Lie algebra g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter7_89_1.tif"/> of the three-dimensional Lie group G of Euclidean motions of the plane. We demonstrate that for m ≥ 4 the semigroup kernel Kt associated with the strongly elliptic operator H = ( − 1 ) m / 2 ∑ i = 1 2 A i m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter7_89_2.tif"/> satisfies m-th order Gaussian bounds for all t ≥ 1 if, and only if, two of the Ai span the nilradical of g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter7_89_3.tif"/> . If this condition is not satisfied the kernel has an anomalous asymptotics. It behaves like an m-th order kernel in one direction and like a second-order kernel in the other two directions. No such anomaly occurs for the kernels associated with the operators H = ( − ∑ i = 1 3 A i 2 ) m / 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/inq_chapter7_89_4.tif"/> .
AMS Subject Classification: 22E25, 35B40, 35B27
