Characteristic variety

Behaviours, Modules, and Duality

Published in Krzysztof Gałkowski, Jeff David Wood, Multidimensional Signals, Circuits and Systems, 2001

Jeffrey Wood, Eric Rogers, David H. Owens

The characteristic variety of B, written V(ann В), is the set of all points g € С" at which every point in the annihilator of В vanishes (Wood et al. 1999a); this concept comes from the theory of PDEs. Now if В contains a non-zero exponential trajectory w with frequency q, then ann В Ç ann w = (zj — Oi,..., z„ — an), and so a is a point in the characteristic variety. The converse also holds.

Estimates of fundamental solution for Kohn Laplacian in Besov and Triebel-Lizorkin spaces

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Journal Information

Published in Applicable Analysis, 2023

Tongtong Qin, Der-Chen Chang, Yongsheng Han, Xinfeng Wu

Inspired by the works of Beals-Greiner [15], Beals-Gaveau-Greiner [16–19] and Greiner-Li [20] we shall use the complex and real Hamiltonian and Lagrangian formalism developed in the sub-Riemannian manifolds to look for the fundamental solution of , i.e. a distribution in which satisfies Since the operator is translation invariant along the t-direction, hence, we may assume that . We shall look for a in the form Here, the functions g, E and have significant geometric meanings and we will give a brief discussion here. More precisely, let be the Hamilton function with the variables dual to . The bicharacteristic curves are complex, a consequence of nonstandard boundary conditions. It is known that the function g is given by an action integral in terms of these bicharacteristics. The format (5) has a simple geometric interpretation. The operator of (1) has a characteristic variety in the cotangent bundle given by .

Characteristic variety

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Behaviours, Modules, and Duality

Estimates of fundamental solution for Kohn Laplacian in Besov and Triebel-Lizorkin spaces