  This work mainly focuses on the spatial decay properties of solutions to the
Zakharov-Kuznetsov equation. In earlier studies for the two- and
three-dimensional cases, it was established that if the initial condition $u_0$
verifies $\langle \sigma\cdot x\rangle^{r}u_{0}\in L^{2}(\left\{\sigma\cdot
x\geq \kappa\right\}),$ for some $r\in\mathbb{N}$, $\kappa \in\mathbb{R}$,
being $\sigma$ be a suitable non-null vector in the Euclidean space, then the
corresponding solution $u(t)$ generated from this initial condition verifies
$\langle \sigma\cdot x\rangle ^{r}u(t)\in L^2\left(\left\{\sigma\cdot
x>\kappa-\nu t\right\}\right)$, for any $\nu >0$. In this regard, we first
extend such results to arbitrary dimensions, decay power $r>0$ not necessarily
an integer, and we give a detailed description of the gain of regularity
propagated by solutions in terms of the magnitude of the weight $r$. The
deduction of our results depends on a new class of pseudo-differential
operators, which is useful to quantify decay and smoothness properties on a
fractional scale. Secondly, we show that if the initial data $u_{0}$ has a
decay of exponential type on a particular half space, that is, $e^{b\,
\sigma\cdot x}u_{0}\in L^{2}(\left\{\sigma\cdot x\geq \kappa\right\}),$ then
the corresponding solution satisfies $e^{b\, \sigma\cdot x} u(t)\in
H^{p}\left(\left\{\sigma\cdot x>\kappa-\nu t\right\}\right),$ for all
$p\in\mathbb{N}$, and time $t\geq \delta,$ where $\delta>0$. To our knowledge,
this is the first study of such property. As a further consequence, we also
obtain well-posedness results in anisotropic weighted Sobolev spaces in
arbitrary dimensions.
  Finally, as a by-product of the techniques considered here, we show that our
results are also valid for solutions of the Korteweg-de Vries equation.
