  Orthogonal polynomials with respect to the weight function
$w_{\beta,\gamma}(t) = t^\beta (1-t)^\gamma$, $\gamma > -1$, on the conic
surface $\{(x,t): \|x\| = t, \, x \in \mathbb{R}^d, \, t \le 1\}$ are studied
recently, and they are shown to be eigenfunctions of a second order
differential operator $\mathcal{D}_\gamma$ when $\beta =-1$. We extend the
setting to the Sobolev inner product, defined as the integration of the $s$-th
partial derivatives in $t$ variable with respect to $w_{\beta+s,0}$ over the
conic surface plus a sum of integrals over the rim of the cone. Our main
results provide an explicit construction of an orthogonal basis and a formula
for the orthogonal projection operators; the latter is used to exploit the
interaction of differential operators and the projection operator, which allows
us to study the convergence of the Fourier orthogonal series. The study can be
regarded as an extension of the orthogonal structure to the weight function
$w_{\beta, -s}$ for a positive integer $s$. It shows, in particular, that the
Sobolev orthogonal polynomials are eigenfunctions of $\mathcal{D}_{\gamma}$
when $\gamma = -1$.
