The edge-of-the-wedge theorem in several complex variables gives the analytic continuation of functions defined on the poly upper half plane and the poly lower half plane, the set of points in $\mathbb{C}^{n}$ with all coordinates in the upper and lower half planes respectively, through a set in real space, $\mathbb{R}^{n}$. The geometry of the set in the real space can force the function to analytically continue within the boundary itself, which is qualified in our wedge-of-the-edge theorem. For example, if a function extends to the union of two cubes in $\mathbb{R}^{n}$ that are positively oriented with some small overlap, the functions must analytically continue to a neighborhood of that overlap of a fixed size not depending of the size of the overlap.

We give some new characterizations for compactness of weighted composition operators $u{{C}_{\varphi }}$ acting on Bloch-type spaces in terms of the power of the components of $\varphi$, where $\varphi$ is a holomorphic self-map of the polydisk ${{\mathbb{D}}^{n}}$, thus generalizing the results obtained by Hyvärinen and Lindström in 2012.

Let $f\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be ${{C}^{\infty }}$ and let $h\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be positive and continuous. For any unbounded nondecreasing sequence $\{{{c}_{k}}\}$ of nonnegative real numbers and for any sequence without accumulation points $\{{{x}_{m}}\}$ in ${{\mathbb{R}}^{n}}$, there exists an entire function $g\,:\,{{\mathbb{C}}^{n}}\,\to \,\mathbb{C}$ taking real values on ${{\mathbb{R}}^{n}}$ such that

$$\left| {{g}^{\left( \alpha \right)}}\left( x \right)-{{f}^{\left( \alpha \right)}}\left( x \right) \right|\text{ }<h\left( x \right),\left| x \right|\ge {{c}_{k}},\left| \alpha \right|\le k,k=0,1,2,...,$$

$${{g}^{\left( \alpha \right)}}\left( {{x}_{m}} \right)\,=\,{{f}^{\left( \alpha \right)}}\left( {{x}_{m}} \right),\,\,\,\left| {{x}_{m}} \right|\,\ge \,{{c}_{k}},\,\left| \alpha \right|\,\le \,k,\,m,\,k\,=\,0,\,1,\,2,\,.\,.\,.\,.$$

This is a version for functions of several variables of the case $n\,=\,1$ due to $L$. Hoischen.