Suppose $f\in C^{\infty}(\mathbb{R}^n \backslash \{0\})$ satisfies $f(\lambda x)= f(x)$ for $\lambda >0$, and it has zero average on $\mathbb{S}^{n-1}$.
My question is: how can we show that its n-th partial derivative $\frac{\partial^n f}{\partial x_{i}^n}$ also has zero average on the unit sphere?
This is just a sentence in a proof regarding the kernel of some certain type of singular integrals, but it’s not obvious.