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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 $$(DC)^s_2.$$

##### MSC:
 58J10 Differential complexes 35F35 Systems of linear first-order PDEs
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