We prove reducibility of a class of quasi-periodically forced linear equations of the form \[ \partial_tu-\partial_x\circ (1+a(\omega t, x))u+\mathcal{Q}(\omega t)u=0,\quad x\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}, \] where $u=u(t,x)$, $a$ is a small, $C^{\infty}$ function, $\mathcal{Q}$ is a pseudo differential operator of order $-1$, provided that $\omega\in\mathbb{R}^{\nu}$ satisfies appropriate non-resonance conditions. Such PDEs arise by linearizing the Degasperis-Procesi (DP) equation at a small amplitude quasi-periodic function. Our work provides a first fundamental step in developing a KAM theory for perturbations of the DP equation on the circle. Following \cite{Airy}, our approach is based on two main points: first a reduction in orders based on an Egorov type theorem then a KAM diagonalization scheme. In both steps the key difficulites arise from the asymptotically linear dispersion law. In view of the application to the nonlinear context we prove sharp \emph{tame} bounds on the diagonalizing change of variables. We remark that the strategy and the techniques proposed are applicable for proving reducibility of more general classes of linear pseudo differential first order operators.

中文翻译：

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由 Degasperis-Procesi 方程产生的一类弱色散线性算子的可约性

我们证明了一类形如 \[ \partial_tu-\partial_x\circ (1+a(\omega t, x))u+\mathcal{Q}(\omega t)u= 0,\quad x\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}, \] 其中 $u=u(t,x)$, $a$ 为小， $C^{\infty}$ 函数，$\mathcal{Q}$ 是 $-1$ 阶伪微分算子，前提是 $\omega\in\mathbb{R}^{\nu}$ 满足适当的非-共振条件。这种偏微分方程是通过在小幅度准周期函数上线性化 Degasperis-Procesi (DP) 方程而产生的。我们的工作为开发用于圆上 DP 方程扰动的 KAM 理论提供了第一个基本步骤。在 \cite{Airy} 之后，我们的方法基于两个要点：首先是基于 Egorov 类型定理的阶数减少，然后是 KAM 对角化方案。在这两个步骤中，关键困难来自渐近线性色散定律。鉴于在非线性上下文中的应用，我们证明了变量对角化变化的尖锐 \emph{tame} 边界。我们注意到所提出的策略和技术适用于证明更一般类别的线性伪微分一阶算子的可约性。