2
$\begingroup$

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 $K_{p}(\delta) \lesssim_{\epsilon} \delta^{-\epsilon}$.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.