English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We present an example of a homogeneous, left-invariant differential operator on the Heisenberg group H3 which admits fundamental solutions but no tempered ones. This answers a question raised by Corwin in the negative. Assume that N is a connected, simply connected nilpotent Lie group with Lie algebra n, and let L E if(n) be a left-invariant differential operator on N. If N is abelian, any such L can be considered as a constant coefficient differential operator on some Euclidean space Rn and, by the theorem of Malgrange and Ehrenpreis (see [H2]), has a fundamental solution F E 0'(Rn), i.e., LF = (5, where a denotes the point measure at the identity. In fact, it was proved later by Hormander [Hi] and Lojasiewicz [L] that one can even find a tempered fundamental solution F E 9v'(RDn) . The situation becomes drastically different if N is nonabelian, since then there exist many operators in i(n) which are not even locally solvable. Assume in the sequel that N admits -a one-parameter family {(5r}r>O of automorphic dilations (see [FS]) and that L is homogeneous, i.e., L((#o ?J,r) = rm (L(p) o J5r for some m > 0 and every (0 E 9(N), r > 0. Then it is at least true that various notions of solvability coincide for L. For instance, L is locally solvable at some point of N if and only if LC??(N) = C??(N), if and only if L has a fundamental solution F E 9/(N) (see, e.g., [B, Ml]). Moreover, if LT, the transpose of L, is hypoelliptic, then the same is true of the operator LLT, as can be seen by Helffer-Nourrigat's theorem [HN], and one can make use of the homogeneity of L in order to prove that LLT, hence also L, has even a tempered fundamental solution [F, G]. So, a natural question, which seems to have been open hitherto, is whether any solvable, homogeneous, left-invariant differential operator or, more generally, every globally solvable left-invariant differential operator on a nilpotent Lie group has a tempered fundamental solution. In the latter form, this question was raised by Corwin in [C]. The purpose of this note is to present an example on the 7-dimensional Received by the editors October 31, 1991 and, in revised form, August 26, 1992. 1991 Mathematics Subject Classification. Primary 22E30, 35A08. ? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

A homogeneous, globally solvable differential operator on a nilpotent Lie group which has no tempered fundamental solution