  We define the fractional powers $L^s=(-a^{ij}(x)\partial_{ij})^s$, $0 < s <
1$, of nondivergence form elliptic operators $L=-a^{ij}(x)\partial_{ij}$ in
bounded domains $\Omega\subset\mathbb{R}^n$, under minimal regularity
assumptions on the coefficients $a^{ij}(x)$ and on the boundary
$\partial\Omega$. We show that these fractional operators appear in several
applications such as fractional Monge--Amp\`ere equations, elasticity, and
finance. The solution $u$ to the nonlocal Poisson problem $$\begin{cases}
(-a^{ij}(x) \partial_{ij})^su = f&\hbox{in}~\Omega\\
u=0&\hbox{on}~\partial\Omega \end{cases}$$ is characterized by a local
degenerate/singular extension problem. We develop the method of sliding
paraboloids in the Monge--Amp\`ere geometry and prove the interior Harnack
inequality and H\"older estimates for solutions to the extension problem when
the coefficients $a^{ij}(x)$ are bounded, measurable functions. This in turn
implies the interior Harnack inequality and H\"older estimates for solutions
$u$ to the fractional problem.
