Computing Infinity Harmonic Operator of a function

Question

Consider the cone function $ C(x) := a |x-z| + b $ and the paraboloid $ P(x) := -\gamma | x- z|^2 $ for given constants $ a,b $ and $ \gamma $ where $ a >0 $. I'd like to compute $ \Delta_\infty (C+P) $. At the page 474 of the article http://users.jyu.fi/~peanju/preprints/tour.pdf is stated that \begin{equation}\label{key} \Delta_\infty (C+P)(x) = \Delta_\infty G (|x-z|) = G''(|x-Z)G'(|x-z|)^2 = -2\gamma (2 \gamma |x-z| - a)^2. \end{equation} I can see the second inequality. However, I can't see the others. I know that $$ (C+P)_{x_i} (x) = G'(|x-z|)\dfrac{(x_i- z_i)}{|x-z|} $$ \begin{eqnarray*} (C+P)_{x_ix_j} (x) &=& G''(|x-z|)^2\dfrac{(x_i- z_i)}{|x-z|} \dfrac{(x_j- z_j)}{|x-z|} \\ &+& G'(|x-z|)\left [ \dfrac{\delta_{ij}}{|x-z|} - (x_i- z_i)(x_j- z_j)\dfrac{1}{|x-z|^3}\right ] \end{eqnarray*}

Answer 1

For the first/second equality, let's just show that $\Delta_\infty(C+P)(x) = G''(\lvert x - z\rvert) G'(\lvert x - z \rvert)^2$. We compute, as you found (up to rearrangement and a typo),

\begin{align*} (C+P)_{x_i}(x) &= G'(\lvert x - z \rvert) \frac{x_i - z_i}{\lvert x - z \rvert}, \\ (C+P)_{x_i x_j}(x) &= G''(\lvert x - z \rvert) \frac{(x_i - z_i)(x_j - z_j)}{\lvert x - z \rvert^2} + G'(\lvert x - z \rvert) \left( \frac{\delta_{ij}}{\lvert x - z \rvert} - \frac{(x_i - z_i)(x_j - z_j)}{\lvert x - z \rvert^3} \right). \end{align*}

Computing the sum defining $\Delta_\infty$, the key is to recognize that the various sums of terms involving the components of $x-z$ are just powers of $\lvert x - z \rvert$:

\begin{align*} \Delta_\infty(C+P)(x) &= \sum_{i,j} (C+P)_{x_i}(x) \, (C+P)_{x_j}(x) \, (C+P)_{x_i x_j}(x) \\ &= G''(\lvert x - z \rvert) \, G'(\lvert x - z \rvert)^2 \frac{\sum_{i,j} (x_i - z_i)^2 (x_j - z_j)^2}{\lvert x - z \rvert^4} \\ &\phantom{=} \phantom{X} + G'(\lvert x - z \rvert)^3 \left( \frac{ \sum_{i,j} \delta_{ij} (x_i - z_i)(x_j - z_j)}{\lvert x - z \rvert^3} - \frac{\sum_{i,j} (x_i - z_i)^2 (x_j - z_j)^2}{\lvert x - z \rvert^5} \right) \\ &= G''(\lvert x - z \rvert) \, G'(\lvert x - z \rvert)^2 \frac{\lvert x - z \rvert^4}{\lvert x - z \rvert^4} + G'(\lvert x - z \rvert)^3 \left( \frac{\lvert x - z \rvert^2}{\lvert x - z \rvert^3} - \frac{\lvert x - z \rvert^4}{\lvert x - z \rvert^5} \right) \\ &= G''(\lvert x - z \rvert) \, G'(\lvert x - z \rvert)^2. \end{align*}

As for the third equality, just differentiate $G(s) = as - \gamma s^2 + b$.