Estimates for $k$-dimensional spherical summations of arithmetic functions of the GCD and LCM

Let $k\ge 2$ be a fixed integer. We consider sums of type ${\sum }_{n_1^2+\cdots +n_k^2\le x}F(n_1,\dots ,n_k)$, taken over the $k$-dimensional spherical region ${(n_1,\ldots,n_k)\in {\Bbb Z}^k: n_1^2+\cdots+ n_k^2\le x}$,where$F:{\Bbb Z}^k\to {\Bbb C}$ is a given function. In particular, we deduce asymptotic formulas with remainder terms for the spherical summations ${\sum }_{n_1^2+\cdots +n_k^2\le x}f((n_1,\dots ,n_k))$ and ${\sum }_{n_1^2+\cdots +n_k^2\le x}f([n_1,\dots ,n_k])$, involving the GCD and LCM of the integers $n_1,\dots ,n_k$, where $f:\mathbb{N}\to \mathbb{C}$ belongs to certain classes of functions.