  This thesis studies the extension problem for higher-order fractional powers
of the heat operator $H=\Delta-\partial_t$ in $\mathbb{R}^{n+1}$. Specifically,
given $s>0$ and indicating with $[s]$ its integral part, we study the following
degenerate partial differential equation in the thick space
$\mathbb{R}^{n+1}\times \mathbb{R}_y^+$, \begin{equation} \label{a:1}
\mathscr{H}^{[s]+1}U= \left( \partial_{yy} +\frac{a}{y}\partial_y +H
\right)^{[s]+1}U=0. \quad \quad (1) \end{equation} The connection between the
Bessel parameter $a$ in (1) and the fractional parameter $s>0$ is given by the
equation \begin{equation*} a= 1-2(s-[s]). \end{equation*} When $s\in(0,1)$ this
equation reduces to the well-known relation $a=1-2s$, and in such case (1)
becomes the famous Caffarelli-Silvestre extension problem. Generalising their
result, in this thesis we show that the nonlocal operator $(-H)^{\,s}$ can be
realised as the Dirichlet-to-Neumann map associated with the solution $U$ of
the extension equation (1). In this thesis we systematically exploit the
evolutive semigroup $\{P_{\tau}^H \}_{\tau>0}$, associated with the Cauchy
problem \begin{equation*} \begin{cases} \partial_{\tau}u-Hu=0
u((x,t),0)=f(x,t). \end{cases} \end{equation*} This approach provides a
powerful tool in analysis, and it has the twofold advantage of allowing an
independent treatment of several complex calculations involving the Fourier
transform, while at same time extending to frameworks where the Fourier
transform is not available.
