Let $\phi : [0,1]^{n-1} \rightarrow \mathbb{R}$ be a $C^{2}$ function such that $\phi(0)=\nabla\phi(0)=0$ and $0<\frac{1}{C}<\nabla^{2}\phi<C$. Consider the compact $C^{2}$ hypersurface $S \subset \hat{\mathbb{R}}^{n}$ parametrized by $(\underline{\xi},\phi(\underline{\xi}))$ for $\underline{x}\in [0,1]^{n-1}$.
For $\delta \in 4^{-\mathbb{N}}$, let $\mathrm{Part}_{\delta^{1/2}}([0,1]^{n-1})$ denote the unique partition of $[0,1]^{n-1}$ into dyadic cubes $Q$ of side length $l(Q)=\delta^{1/2}$. For a cube $Q\subset [0,1]^{n-1}$, define the $\delta$-neighborhood of $S$ over $Q$ by \begin{equation*} N_{\delta}(Q) := \{(\underline{\xi}, \phi(\underline{\xi})+t : \underline{\xi}\in Q, 0 \leq |t|\leq \delta\} \end{equation*} where $\underline{\xi} := (\xi_{1},\ldots,\xi_{n-1})$. For $f : \mathbb{R}^{n} \rightarrow \mathbb{C}$, let $f_{N_{\delta}(Q)} := (\widehat{f}1_{N_{\delta}(Q)})^{\vee}$.
In Bourgain and Demeter's paper "The $\ell^{2}$ decoupling conjecture", the authors state (in different notation) that the following theorem holds for any compact $C^{2}$ hypersurface with positive definite second fundamental form (hence, the setup above).
$\ell^{2}$ decoupling. Let $2 \leq p \leq \frac{2(n+1)}{n-1}$. There exists a constant $C_{1, (n,p,E,\epsilon)}$ such that for any $f : \mathbb{R}^{n} \rightarrow \mathbb{C}$ with Fourier support in $N_{\delta}([0,1]^{n-1})$ and any cube $B=B(c_{B},\delta^{-1})$, \begin{equation*} \|f\|_{L^{p}(w_{E,B})} \leq C_{1,(n,p,E,\epsilon)} \delta^{-\epsilon} \left(\sum_{Q\in\mathrm{Part}_{\delta^{-1/2}}([0,1]^{n-1})}\|f_{N_{\delta^{-1}}(Q)}\|_{L^{p}(w_{E,B})}^{2}\right)^{1/2} \tag{1} \end{equation*} where $w_{E,B}(x) := (1+\frac{|x-c_{B}|}{\delta^{-1}})^{-E}$, $E\geq 100n$. Or in the equivalent global form, \begin{equation} \|f\|_{L^{p}(\mathbb{R}^{n})} \leq C_{2, (n,p,\epsilon)} \delta^{-\epsilon} \left(\sum_{Q\in \mathrm{Part}_{\delta^{-1/2}}([0,1]^{n-1})} \|f_{N_{\delta^{-1}}(Q)}\|_{L^{p}(\mathbb{R}^{n})}\right)^{1/2} \tag{2} \end{equation}
The authors first prove the theorem in the model case where $S$ is the $(n-1)$-dimensional truncated paraboloid (i.e. $\phi(xi)=|\xi|^{2}$). Then (Section 7, pg. 28) they claim that the theorem for the paraboloid implies the theorem for a general surface $S$ as above using the following argument.
One extends the theorem to elliptic paraboloids of the form \begin{equation*} \{ (\underline{\xi}, \theta_{1}\xi_{1}^{2} + \cdots + \theta_{n-1}\xi_{n-1}^{2}) \in \mathbb{R}^{n} : \underline{\xi} \in [0,1]^{n-1}\} \end{equation*} with $\theta_{i} \in [1/C, C]$.
For dyadic $\delta \ll 1$, let $K_{p}(\delta)$ be the smallest constant such that for all $f : \mathbb{R}^{n} \rightarrow \mathbb{C}$ with Fourier support in $N_{\delta}$, \begin{equation*} \|f\|_{L^{p}(\mathbb{R}^{n})} \leq K_{p}(\delta) \left(\sum_{Q \in \mathrm{Part}_{\delta^{1/2}}([0,1]^{n-1})} \|f_{Q}\|_{L^{p}(\mathbb{R}^{n})}^{2} \right)^{1/2} \end{equation*} Fix such an $f$ so that \begin{equation*} \|f\|_{L^{p}(\mathbb{R}^{n})} \leq K_{p}(\delta^{2/3}) \left(\sum_{Q \in \mathrm{Part}_{\delta^{1/3}}([0,1]^{n-1})} \|f_{Q}\|_{L^{p}(\mathbb{R}^{n})}^{2} \right)^{1/2} \end{equation*} Next, the authors claim (I have modified their notation to be consistent with above) that
"our assumption on the principal curvatures of $S$ combined with Taylor's formula shows that on each [$Q\in \mathrm{Part}_{\delta^{2/3}}([0,1]^{n-1})$], $S$ is within $\delta$ from a paraboloid with similar principal curvatures."
I don't understand how one gets the $\delta$ approximation error with merely $C^{2}$ regularity. Please correct me if I am wrong, but Taylor's formula, under $C^{2}$ assumption, gives \begin{equation*} \phi(\underline{\xi})=\phi(c_{Q})+\langle{\nabla\phi(c_{Q}), \underline{\xi} - c_{Q}}\rangle + \frac{1}{2}\langle{\nabla^{2}\phi(c_{Q})(\underline{\xi}-c_{Q}),\underline{\xi}-c_{Q}}\rangle + o(|\underline{\xi}-c_{Q}|^{2}) \end{equation*} for all $\underline{\xi}\in Q$. If we strengthen our hypothesis to $\phi \in C^{3}$, with $\|\phi\|_{C^{3}}\leq A$, then we have \begin{equation*} \phi(\underline{\xi}) = \phi(c_{Q})+\langle{\nabla\phi(c_{Q}), \underline{\xi} - c_{Q}}\rangle + \frac{1}{2}\langle{\nabla^{2}\phi(c_{Q})(\underline{\xi}-c_{Q}), \underline{\xi} - c_{Q}}\rangle + R(\underline{\xi}) \end{equation*} where $|R(\underline{\xi})|\lesssim A|\underline{\xi}-c_{Q}|^{3}$. So for all $\underline{\xi} \in Q$, $|R(\underline{\xi})| \lesssim A\delta$. Note that the paper of Pramanik and Seeger to which the authors refer for the worked out argument in the conical case assumed $C^{3}$ regularity for the planar curves considered.
The rest of the argument proceeds by using parabolic rescaling and the decoupling theorem for the paraboloid associated to $Q$ to obtain \begin{align*} \|f_{Q}\|_{L^{p}(\mathbb{R}^{n})} \lesssim_{A,C} C_{2,(n,p,\epsilon)}\delta^{-\epsilon}\left(\sum_{q\in \mathrm{Part}_{\delta^{1/2}}([0,1]^{n-1}) : q \subset Q} \|f_{q}\|_{L^{p}(\mathbb{R}^{n})}^{2} \right)^{1/2} \end{align*} Therefore for every $\epsilon > 0$, there exists $C_{\epsilon}>0$, such that \begin{equation*} K_{p}(\delta) \leq C_{\epsilon} \delta^{-\epsilon} K_{p}(\delta^{2/3}), \end{equation*} from which one can conclude by iteration that $K_{p}(\delta) \lesssim_{\epsilon} \delta^{-\epsilon}$.