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arXiv · 2608.20941

Higher-order derivatives of radially symmetric functions

Zdeněk Mihula·Jan Vybíral
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Abstract

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 $\mathcal{D} g(r)=g'(r)/r$, 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 $p\in [1,\infty).$ Furthermore, the pointwise nature of our approach allows us to obtain similar results also for Sobolev-type spaces built upon more general Banach lattices.

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Zdeněk Mihula, Jan Vybíral. 2026-08-21. Higher-order derivatives of radially symmetric functions. https://arxiv.org/abs/2608.20941

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