I'm trying to show directly by differentiation that the solution $u(x, t)$ to the 3D wave equation (as given by kirchoff's formula), for
$$\begin{cases} u_{tt} = c^2 \nabla u \ \ \ \ \ \ \ \ \ \ \ \ \ \ (x, t) \in \mathbb{R^3} \times [0, \infty)\\ u(x, 0) = g(x) \ \ \ \ \ \ \ x \in \mathbb{R^3} \\ u_t(x, 0) = h(x) \ \ \ \ \ \ \ x \in \mathbb{R} \end{cases}$$ with $g \in C^3$, $h \in C^2$, is $C^2$
The strategy I was thinking to use would be to differentiate with respect to $x_i$ and $x_j$ in succession. We would start with the regular formula
$$ u(x, t) = \frac{3}{4\pi c^2t^2}\int_{\partial B(x, ct)}g(y) + \nabla g(y) \cdot (y-x) + th(y) dS_y$$
But then shift the domain of integration to the unit sphere, to avoid the dependence on $t$ in the boundary. This should give
$$ \frac{3}{4 \pi } \int_{\partial B(0, 1)}g(x + tw) + \nabla g(x+tw) \cdot (tw) + th(x + tw) dS_w.$$
The plan would be to take successive derivatives of $t$ and of $x_i$, which we can do because the integrand is a smooth function of $x$ or of $t$ on a compact set (surface of unit sphere). However, I am unsure about the chain rule implications--I would eventually have to write down what
$$ \frac{\partial}{\partial x_i} \left( g(x + tw) \right)$$
is ($x$ is a vector, and so is $w$). I'm embarrassed to say, but I'm actually not sure what this partial derivative is. I am tempted to say
$$ \frac{\partial}{\partial x_i} \left( g(x + tw) \right) = \nabla g(x + tw) \cdot \frac{\partial}{\partial x_i}(x + tw) = \nabla g(x + tw) \cdot \vec{e}_i = g_{x_i}(x + tw),$$
but I'm not actually sure about that. In general, I have trouble a little bit with partial derivatives of vector functions, and I don't know if I'm even doing the chain rule right here, or even if this strategy will allow me to finish the problem of showing $u(x, t)$ is $C^2$.