Equations of evolution on the Heisenberg group. I (Q2367098)

Aimed at the solvability of partial differential operators on the nilpotent Lie groups, this paper treats the Cauchy problem for equations of evolution on the Heisenberg group. The operators of higher order considered here are of the form: \(P = \partial^ m_ t + \sum^ m_{j = 1} A_ j\partial^{m-j}_ t\) where \(A_ j\) are the homogeneous right invariant differential operators of order \(pj\) on the Heisenberg group \(H^ n\), \(p \in \mathbb{N}\). According to the two families of irreducible unitary representations of \(H^ n\), the author introduces two generalized symbols of \(P\), and defines the ``parabolic conditions'' on them. The basic result of this paper is the well-posedness for the Cauchy problem. That is, if \(P\) satisfies the ``parabolic'' conditions, then for any \(T > 0\), \(l \geq 2\) the Cauchy problem \[ Pu = f\quad \text{in }(0,T) \times H^ n\quad \partial^ i_ t u|_{t = 0} = g_ j \quad \text{on }H^ n,\quad 0 \leq j \leq m - 1 \] has a unique solution \(u(x,t) \in C^{m + l - 2}([0,T];H_{(\infty)}(H^ n))\) if \(f \in C^ l([0,t];H_{(\infty)}(H^ n))\) and \(g_ j \in H_{(\infty)}(H^ n)\), \(j = 0,\dots,m - 1\).