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 $\sum_{n_1^2+\cdots+ n_k^2\le x} F(n_1,\ldots,n_k)$, 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 $\sum_{n_1^2+\cdots+ n_k^2\le x} f((n_1,\ldots,n_k))$ and $\sum_{n_1^2+\cdots+ n_k^2\le x} f([n_1,\ldots,n_k])$, involving the GCD and LCM of the integers $n_1,\ldots,n_k$, where $f:{\Bbb N}\to {\Bbb C}$ belongs to certain classes of functions.