Exponentially weighted resolvent estimates for complex Ornstein-Uhlenbeck systems

In this paper we study differential operators of the form \begin{align*} \left$\mathcal{L}$_\infty v \right = A\triangle v(x) + \left\langle Sx,\nabla v(x) \right\rangle - Bv(x), ,x \in $\mathbb{R}$^d, ,d \geqslant 2, \end{align*} for matrices $A,B\in {\mathbb{C}}^{N,N}$, where the eigenvalues of $A$ have positive real parts. The sum $A\triangle v(x) + \left\langle Sx, \nabla v(x) \right\rangle$ is known as the Ornstein-Uhlenbeck operator with an unbounded drift term defined by a skew-symmetric matrix $S\in {\mathbb{R}}^{d,d}$. Differential operators such as ${\mathcal{L}}_{\mathrm{\infty }}$ arise as linearizations at rotating waves in time-dependent reaction diffusion systems. The results of this paper serve as foundation for proving exponential decay of such waves. Under the assumption that $A$ and $B$ can be diagonalized simultaneously we construct a heat kernel matrix $H(x,\xi ,t)$ of ${\mathcal{L}}_{\mathrm{\infty }}$ that solves the evolution equation $v_t={\mathcal{L}}_{\mathrm{\infty }}v$. In the following we study the Ornstein-Uhlenbeck semigroup \begin{align*} \left T(t)v\right = \int_{$\mathbb{R}$^d} H(x,\xi,t) v(\xi) d\xi,,x \in $\mathbb{R}$^d,, t>0, \end{align*} in exponentially weighted function spaces. This is used to derive resolvent estimates for ${\mathcal{L}}_{\mathrm{\infty }}$ in exponentially weighted $L^p$-spaces $L_{\theta }^p({\mathbb{R}}^d,{\mathbb{C}}^N)$, $1⩽p<\mathrm{\infty }$, as well as in exponentially weighted $C_{\mathrm{b}}$-spaces $C_{\mathrm{b},\theta }({\mathbb{R}}^d,{\mathbb{C}}^N)$.