Alternative proof of the higher-order radial derivative norm formula

Establish whether the Lyons–Zumbrun formula for arbitrary partial derivatives of a radial function can be exploited to provide an alternative proof of the closed-form expression for the Frobenius norm of the tensor of higher-order derivatives of a radial function.

Background

The paper proves a closed-form pointwise formula for the squared Frobenius norm of the tensor of nth-order partial derivatives of a smooth radial function, expressed through iterates of the operator Dg(r)=g'(r)/r. The authors note that Lyons and Zumbrun previously obtained an elegant formula for each individual partial derivative of a radial function.

Although the Lyons–Zumbrun formula was used in earlier work to characterize radial subspaces of Sobolev spaces, the paper does not use it to derive its principal theorem. The authors explicitly leave open whether that formula can yield an alternative proof of their higher-order Frobenius-norm identity.