$\mathcal{F}_1$ satisfies (\[3.4\]) for some $l\in{\mathbb{N}}$. And then by Proposition \[Calpha\_Prop\] we can conclude that every element in $\mathcal{F}_1$ is $(C,\frac{1}{d_\nu l})$-good on some $V_\nu\subset V_{{\mathbf{x}}_0}\subset U_\nu$ together with conclusion (2) of the Corollary above. This also implies $ a_0+u+({\mathbf{a}}-{\mathbf{c}}).{\mathbf{f}}_{\nu} $ are all $(C,\frac{1}{d_\nu l})$ good on $V_\nu$ for all $(a_0,{\mathbf{a}})\in\mathcal{O}^{n+1}$ with $|a_0+u| < 2|{\mathbf{a}}-{\mathbf{c}}|$. Otherwise $$\sup_{{\mathbf{x}}\in V_{{\mathbf{x}}_0}}|a_0+u+({\mathbf{a}}-{\mathbf{c}}).{\mathbf{f}}_{\nu}|\leq 3.\inf_{{\mathbf{x}}\in V_{{\mathbf{x}}_0}}|a_0+u+({\mathbf{a}}-{\mathbf{c}}).{\mathbf{f}}_{\nu}|$$ as $|a_0+u|\geq 2|{\mathbf{a}}-{\mathbf{c}}| $ and it turns out to be a trivial case. This implies that for $C\geq 3$ and $0<\alpha\leq1$ the aforementioned functions are $(C,\alpha)$-good.

Let us recall the following Corollary from [@KT] (Corollary 2.3).

[\[product\_good\]]{} For $ j=1,\cdots,n,$ let $X_j$ be a metric space, $\mu_j$ be a measure on $X_j $. Let $ U_j\subset X_j $ be open, $C_j,\alpha_j >0 $ and let $f$ be a function on $U_1\times\cdots \times U_d$ such that for any $j=1,\cdots d$ and any $x_i\in U_i$ with $i\neq j,$ the function $${\label{fun}}
    y~~\mapsto f(x_1,\cdots,x_{j-1}, y, x_{j+1},\cdots, x_d)$$ is $(C_j,\alpha_j)$-good on $U_j$ with respect to $\mu_j$. Then $f$ is $(\widetilde{C},\widetilde{\alpha}) $ -good on $U_1\times\cdots\times U_d $ with respect to $\mu_1\times\cdots\times\mu_d,$ where $\widetilde{C}=d,\widetilde{\alpha }$ are computable in terms of $C_j,\alpha_j $. In particular, if each of the functions (\[fun\]) is $(C,\alpha)$-good on $U_j$ with respect 