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A Gevrey differential complex on the torus. (English) Zbl 1434.58008
Let \(\mathbb{T}^{m+n}=\mathbb{T}^{m}\times \mathbb{T}^{n}\) be the \(m+n\) dimensional torus. We recall that a function \(f:\mathbb{T}^{m+n}\rightarrow \mathbb{R}\) is \(s\)-Gevrey (with \(s\geq 1\)) if \(f\in C^{\infty}(\mathbb{T}^{m+n})\) and there exist positive constants \(C\) and \(R\) such that for every \(\alpha \in \mathbb{Z}_+^{m+n}\) and for all \((x,t)\in \mathbb{T}^{m+n}\) we have \(|\partial^{\alpha} f(x,t)|\leq CR^{|\alpha|}(\alpha!)^s\). The space of \(s\)-Gevrey functions on \(\mathbb{T}^{m+n}\) is denoted by \(G^s(\mathbb{T}^{m+n})\). In particular for \(s=1\) \(G^1(\mathbb{T}^{m+n})\) equals \(C^{\omega}(\mathbb{T}^{m+n})\), the space of real analytic functions. Consider now a system of closed one forms \(\Omega=\{\omega^1,\dots,\omega^n\}\) on \(\mathbb{T}^n\) with \(\mathbb{R}\)-valued coefficients in the Gevrey spaces \(G^s(\mathbb{T}^m)\) with \(s\geq 1\). The authors associate to \(\Omega\) the differential complex \(\mathbb{L}_{\Omega}\) defined on the torus \(\mathbb{T}^{m+n} = \mathbb{T}^m\times \mathbb{T}^n\) by \[ \mathbb{L}_{\Omega}=d_t+\sum_{j=1}^n\omega^j\wedge \frac{\partial}{\partial x_j} \] where \(t = (t_1, \dots , t_m)\) and \(x = (x_1, \dots , x_n)\) are the angular coordinates on the tori \(\mathbb{T}^m\) and \(\mathbb{T}^n\) respectively. This paper has a twofold purpose: to understand the induced cohomology in the Gevrey category: \[ H^*_{\Omega,s}(\mathbb{T}^{m+n})=\frac{(\Omega,s)\text{-closed forms}}{(\Omega,s)\text{-exact forms}} \] and also to understand the hypoellipticity of the operator \(\mathbb{L}_{\Omega}\).
In order to attain these goals the authors associate to \(\Omega\) the following matrix \[ \mathbf{A}=\left(\int_{\gamma_i}\omega^j\right)_{1\leq i\leq m,1\leq j\leq n} \] where \(\gamma_1,\dots\gamma_m\) are loops in \(\mathbb{T}^m\) that generate \(H^1(\mathbb{T}^m)\). Moreover they introduce the following Diophantine conditions \((DC)^1_s\) and \((DC)^2_s\) defined as follows: given a matrix \(\mathbf{A}\in \mathcal{M}_{n\times m}(\mathbb{R})\) the condition \((DC)^1_s\) holds if for every \(\epsilon>0\), there exists \(C_{\epsilon}>0\) such that \[\|K + AJ\|\geq C_{\epsilon}\exp\left(-\epsilon(|K| + |J |)^{\frac{1}{s}}\right)\] for all \((K,J) \in \mathbb{Z}^m \times \mathbb{Z}^n\) satisfying \(K + \mathbf{A}J\neq 0\), where \(\|K + AJ\| =\max_{1\leq l\leq m} |k_l + a_l \cdot J |\) with \(a_l\), the \(l\)-th row of the matrix \(\mathbf{A}\) and \(K = (k_1, \dots , k_m).\) The other conditions, \((DC)^2_s\), holds if for every \(\epsilon > 0\), there exists \(C_{\epsilon} > 0\) such that \[ \|K + AJ\| \geq C_{\epsilon}\exp\left( -\epsilon(|K| + |J |)^{\frac{1}{s}}\right) \] for all \((K, J ) \in (\mathbb{Z}^m\times \mathbb{Z}^n)\setminus {0}\), where \(\|K + AJ\| = \max_{1\leq l \leq m} |kl +a_l \cdot J |\) with \(a_l\), the \(l\)-th row of the matrix \(A\) and \(K = (k_1, \dots , k_m)\).
In their first main result the authors prove that if \(\mathbf{A}\) satisfies \((DC)_1^s\) then \[H^q_{\Omega,s}(\mathbb{T}^{m+n})\cong G^s(\mathbb{T}^r)\times \cdot \cdot \cdot \times G^s(\mathbb{T}^r)\] where on the right hand side we have \(\binom{m}{q}\) factors. In their second main result the authors show that \(\mathbb{L}_{\Omega}\) is never globally hypoelliptic at the level of \(q\)-forms when \(q\geq 1\) whereas at the level of functions \((q = 0)\) \(\mathbb{L}_{\Omega}\) is \(s\)-globally hypoelliptic if and only if the matrix \(\mathbf{A}\) satisfies condition \((DC)^s_2.\)

58J10 Differential complexes
35F35 Systems of linear first-order PDEs
Full Text: DOI
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