Higher-order derivatives of radially symmetric functions

We prove a surprisingly simple pointwise formula for the Frobenius norm of the tensor of $n$-th order partial derivatives of a radially symmetric function $f(x)=g(r(x))$. Using the iterations of the differential operator $\mathcal{D}g(r)=g^{\mathrm{\prime }}(r)\mathrm{/}r$, we avoid technical difficulties usually caused by higher-order radial derivatives. As a consequence, we obtain a complete characterization of the subspace of radially symmetric functions in both inhomogeneous and homogeneous Sobolev spaces of arbitrarily high order and all integrability parameters $p\in [1,\mathrm{\infty })\mathrm{.}$ Furthermore, the pointwise nature of our approach allows us to obtain similar results also for Sobolev-type spaces built upon more general Banach lattices.