  We study (approximate) null-controllability of parabolic equations in
$L_p(\mathbb{R}^d)$ and provide explicit bounds on the control cost. In
particular we consider systems of the form $\dot{x}(t) = -A_p x(t) +
\mathbf{1}_E u(t)$, $x(0) = x_0\in L_p (\mathbb{R}^d)$, with interior control
on a so-called thick set $E \subset \mathbb{R}^d$, where $p\in [1,\infty)$, and
where $A$ is an elliptic operator of order $m \in \mathbb{N}$ in
$L_p(\mathbb{R}^d)$. We prove null-controllability of this system via duality
and a sufficient condition for observability. This condition is given by an
uncertainty principle and a dissipation estimate. Our result unifies and
generalizes earlier results obtained in the context of Hilbert and Banach
spaces. In particular, our result applies to the case $p=1$.
