AI 中文总结
该研究推导了径向对称函数n阶偏导数张量Frobenius范数的简洁逐点公式，规避高阶径向导数的技术困难，刻画了径向对称函数在高阶Sobolev空间中的子空间，并推广到Banach格上的Sobolev型空间。
AI 中文摘要
我们证明了一个极其简洁的逐点公式，用于计算径向对称函数f(x)=g(r(x))的n阶偏导数张量的Frobenius范数。通过微分算子D g(r)=g'(r)/r的迭代，我们规避了高阶径向导数通常带来的技术困难。由此，我们对任意高阶非齐次和齐次Sobolev空间中径向对称函数的子空间，以及所有可积性参数p∈[1,∞)，得到了完整刻画。此外，我们方法的逐点特性还使我们能在更一般的Banach格上建立的Sobolev型空间中得到类似结果。
英文摘要
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.
Comments17 pages
