For a discrete-time Markov chain $\{X(t)\}$ evolving on $\Re \ell$ with transition kernel $P$, natural, general conditions are developed under which the following are established:

1. The transition kernel $P$ has a purely discrete spectrum, when viewed as a linear operator on a weighted Sobolev space $L\infty v,1$ of functions with norm, $$ |f|{v,1} = \sup{x \in \Re^\ell} \frac{1}{v(x)} \max {|f(x)|, |\partial_1 f(x)|,\ldots,|\partial_\ell f(x)|}, $$ where $v:\Re \ell \to [1,\infty )$ is a Lyapunov function and $\partial i:=\partial /\partial xi$.
2. The Markov chain is geometrically ergodic in $L\infty v,1$: There is a unique invariant probability measure $\pi$ and constants and $\delta >0$ such that, for each $f\in L\infty v,1$, any initial condition $X(0)=x$, and all $t\ge 0$: $$\Big| \text{E}_x[f(X(t))] - \pi(f)\Big| \le Be^{-\delta t}v(x),\quad |\nabla \text{E}_x[f(X(t))] |_2 \le Be^{-\delta t} v(x), $$ where $\pi (f)=\int fd\pi$.
3. For any function $f\in L\infty v,1$ there is a function $h\in L\infty v,1$ solving Poisson's equation: [ h-Ph = f-\pi(f). ] Part of the analysis is based on an operator-theoretic treatment of the sensitivity process that appears in the theory of Lyapunov exponents.