an:02188283
Zbl 1077.35004
Bergamasco, Adalberto P.; Zani, S??rgio Lu??s
Prescribing analytic singularities for solutions of a class of vector fields on the torus
EN
Trans. Am. Math. Soc. 357, No. 10, 4159-4174 (2005).
00117981
2005
j
35A20 35H10
global analytic hypoellipticity
This paper deals with the analytic singularities of the operator
\[
L= \partial_t+ (a(t)+ ib(t))\partial_x
\]
acting on distributions on the torus \(\mathbb{T}^2_{t,x}\). The functions \(a\), \(b\) are real valued, real analytic on \(\mathbb{T}^1_t\). We shall formulate some of the main results of the paper. Assume that \(b\) changes sign and \(\Sigma\) is any subset of the set of the local extrema of the local primitives of \(b\). Then there exists a solution \(u\in D'(\mathbb{T}^2)\) of \(Lu= f\in C^\omega(\mathbb{T}^2)\) such that the \(t\)-projection of its analytic singular support is \(\Sigma\). Moreover, for any \(\tau\in\Sigma\) and any closed \(F\subset\mathbb{T}^1_x\) one can find \(u\in D'(\mathbb{T}^2)\) such that \(Lu\in C^\omega(\mathbb{T}^2)\) and \(\text{sing\,supp}_A(u)= \{\tau\}\times F\). The results here proposed are sharp, i.e. if \(t\) is neither a local minimum nor a local maximum, then every \(u\in D'(\mathbb{T}^2)\), for which \(Lu\in C^\omega(\mathbb{T}^2)\), is real analytic in \((t,x)\).
Petar Popivanov (Sofia)