Global solvability and global hypoellipticity in Gevrey classes for vector fields on the torus
Let \(L={\partial\over\partial t}+ \sum^N_{j=1} (a_j+ ib_j)(t){\partial\over\partial x_j}\) be a vector field defined on \(\mathbb T^{N+1}\simeq \mathbb R^{N+2}/2\pi \mathbb Z^{N+1}\), where \(a_j\), \(b_j\) are real-valued functions and belonging to the Gevrey class \(G^s(\mathbb T^1)\), \(s> 1\), for \(j= 1,\dots, N\). The author presents a complete characterization for the \(s\)-global solvability and \(s\)-global hypoellipticity of \(L\). His results are linked to Diophantine properties of the coefficients and, also, connectedness of certain sublevel sets. For \(s>1\) introduce the following exponential Diophantine conditions for a pair \((\alpha,\beta)\in \mathbb R^N\times\mathbb R^N\): \((EDC)^s_1\): For each \(\varepsilon> 0\) there exists a positive constant \(C_\varepsilon\) such that \[ |\tau+\langle\xi,\alpha+i\beta\rangle|\geq C_\varepsilon\exp\{-\varepsilon(|\xi|+ |\tau|)^{1/s}\}, \] for all \((\xi,\tau)= (\xi_1,\dots,\xi_N,\tau)\in \mathbb Z^{N+1}\setminus\{0\}\). \((EDC)^s_2\): For each \(\varepsilon>0\) there exists a positive constant \(C_\varepsilon\) such that \[ |\tau+\langle\xi,\alpha+i\beta\rangle|\geq C_\varepsilon\exp\{-\varepsilon|\xi|+ |\tau|)^{1/s}\}, \] for all \((\xi,\tau)= (\xi_1,\dots, \xi_N,\tau)\in \mathbb Z^{N+1}\) such that \(\tau+\langle\xi,\alpha+\beta\rangle\neq 0\). Define \(\alpha= (a_1,\dots, a_N),\quad \beta=(\beta_1,\dots, \beta_N)\), \(a_{j,0}= {1\over 2\pi} \int^{2\pi}_0 a_j(t)\,dt,\quad b_{j0}= {1\over 2\pi} \int^{2\pi}_0 a_j(t)\,dt\), \(\alpha_0= (a_{10},\dots, a_{N0})\), \(\beta_0= (b_{10},\dots, b_{N0})\). The author has obtained the following two theorems. Theorem 1. Let \(L\) be given by above. Then, \(L\) is \(s\)-global solvable \((s>1)\) if and only if one of the following situations occurs: {\parindent=1cm \begin{itemize}\item[(I)] For each \(j=1,\dots,N\), \(b_j\) vanishes identically, and \((\alpha_0, 0)\) satisfies \((EDC)^s_2\). \item[(II)] At least one \(b_j\) dones not vanishes identically, \(b_{k0}=0\) for each \(k= 1,\dots, N\), \(\alpha_0\in \mathbb Z^N\), and the sublevel sets \[ \Biggl\{t\in \mathbb T^1;\,\int^t_0 (\xi,\beta(\tau))\,d\tau< r\Biggr\},\;r\in \mathbb R,\;\xi\in \mathbb Z^N, \] are connected. \item[(III)] \(b_{j0}\neq 0\) for at least one \(j\in\{1,\dots,N\}\), and the following conditions hold: \item[(III.1)] dim span\(\{b_1,\dots,b_N\}=1\); \item[(III.2)] the functions \(b_j\) do not change sign; \item[(III.3)] the pair \((\alpha_0,\beta_0)\) satisfies \((EDC)^s_2\). \end{itemize}} Theorem 2. Let \(L\) be given by above. Then, \(L\) is \(s\)-global hypoelliptic \((s>1)\) if and only if the following conditions are satisfied: {\parindent=0.7cm \begin{itemize}\item[(1)] each \(b_j\) does not change sign; \item[(2)] dim span\(\{b_1,\dots,b_N\}\leq 1\); \item[(3)] \((\alpha_0,\beta_0)\) satisfies \((EDC)^s_1\). \end{itemize}}
Diophantine properties of the coefficients
scientific article; zbMATH DE number 6824720