Certain identities on derivatives of radial homogeneous and logarithmic functions

Abstract

Let $k$ be a natural number and $s$ be real. In the 1-dimensional case, the $k$-th order derivatives of the functions $|x{|}^s$ and $\log |x|$ are multiples of $|x{|}^{s-k}$ and $|x{|}^{-k}$, respectively. In the present paper, we generalize this fact to higher dimensions by introducing a suitable norm of the derivatives, and give the exact values of the multiples.

17pages Certain identities on derivatives of radial homogeneous and logarithmic functions. Commun. Math. Anal. 9 (2010), no. 2, 51-66