Sharp $\ell^{q}(L^{p})$ decoupling for paraboloids

Tongou Yang Department of Mathematics, University of California
Los Angeles, CA 90095, United States tongouyang@math.ucla.edu

Abstract.

In this short expository note, we prove the following result, which is a special case of the main theorem in [GOZZK23]. For each $n\geq 2$ and $p,q\in[2,\infty]$ , we prove upper bounds of $\ell^{q}(L^{p})$ decoupling constants for paraboloids in $\mathbb{R}^{n}$ , as well as presenting extremisers for each case. Both are sharp up to $\varepsilon$ -losses.

1. Introduction

Let $d\geq 2$ and $v_{i}\in\{-1,1\}$ for $i=1,\dots,d-1$ . Write $\xi=(\xi_{1},\dots,\xi_{d-1})$ , $x^{\prime}=(x_{1},\dots,x_{d-1})$ and $x=(x^{\prime},x_{d})$ . Define the paraboloid

(1)

\phi(\xi)=\sum_{i=1}^{d-1}v_{i}\xi_{i}^{2},

and its associated extension operator

(2)

Eg(x_{1},\dots,x_{d})=\int_{[0,1]^{d-1}}g(\xi)e(x^{\prime}\cdot\xi+x_{d}\phi(\xi))d\xi.

For $\delta\in\mathbb{N}^{-1/2}$ , let $\mathcal{P}(\delta)$ be a tiling of $[0,1]^{d-1}$ by cubes of side length $\delta^{1/2}$ .

For $p,q>0$ and each nonzero $g\in L^{1}([0,1]^{d-1})$ , denote

(3)

R(g)=R_{p,q}(g):=\sup_{B:\text{$d$-cube, }l(B)=\delta^{-1}}\frac{\lVert Eg\rVert_{L^{p}(B)}}{\lVert\lVert E(g1_{Q})\rVert_{L^{p}(w_{B})}\rVert_{l^{q}(Q\in\mathcal{P}(\delta))}},

where $w_{B}$ is a weight function adapted to $B$ , such as (see [BD17b])

w_{B}(x)=\left(1+\frac{|x-c_{B}|}{\delta^{-1}}\right)^{-100d},

where $c_{B}$ is the centre of $B$ . Let

(4)

D_{p,q}(\delta)=\sup\{R(g):0\neq g\in L^{1}([0,1]^{d-1})\}.

Denote

(5)

p_{d}=\frac{2(d+1)}{d-1},\quad N=\delta^{-\frac{d-1}{2}}.

In the case of an elliptic paraboloid, Bourgain and Demeter [BD15] proved

Theorem 1.

Let $v_{i}=1$ for $i=1,\dots,d-1$ . Then for every $p,q\in[2,\infty]$ we have

(6)

D_{p,2}(\delta)\lessapprox\max\{1,N^{\frac{1}{2}-\frac{p_{d}}{2p}}\}.

Here and throughout this article, $A(\delta)\lessapprox B(\delta)$ will always mean that for every $\varepsilon>0$ , there exists some $C_{\varepsilon}$ such that $A(\delta)\leq C_{\varepsilon}\delta^{-\varepsilon}B(\delta)$ for every $\delta\in(0,1)$ . We define $\gtrapprox$ and $\approx$ in a similar way.

For general $v_{i}=\pm 1$ , Bourgain and Demeter [BD17a] also proved

Theorem 2.

Let $p\in[2,\infty]$ . Then

(7)

D_{p,p}(\delta)\lessapprox\max\{N^{\frac{1}{2}-\frac{1}{p}},N^{1-\frac{1}{p}-\frac{p_{d}}{2p}}\}.

They also proved a version of $\ell^{2}(L^{p})$ decoupling in the case of hyperbolic paraboloids:

Theorem 3.

Let $p\in[2,\infty]$ . Let $d(v)$ denote the minimum number of positive and negative entries of $v_{i}$ , $1\leq i\leq d-1$ , and denote

(8)

p_{v}=\frac{2(d+1-d(v))}{d-1-d(v)}.

Then

(9)

D_{p,2}(\delta)\lessapprox\begin{cases}N^{\frac{d(v)}{d-1}\left(\frac{1}{2}-\frac{1}{p}\right)},&\quad\text{if}\quad 2\leq p\leq p_{v},\\ N^{\frac{1}{2}-\frac{p_{d}}{2p}},&\quad\text{if}\quad p_{v}\leq p\leq\infty.\end{cases}

Note that $p_{d}\leq p_{v}$ , with equality if and only if $d(v)=0$ , that is, we are in the case of elliptic paraboloids.

1.1. Main results

In the case of an elliptic paraboloid, we prove that

Theorem 4.

Let $v_{i}=1$ for $i=1,\dots,d-1$ . Then for all $p,q\in[2,\infty]$ , we have

(10)

D_{p,q}(\delta)\approx\max\{N^{\frac{1}{2}-\frac{1}{q}},N^{1-\frac{p_{d}}{2p}-\frac{1}{q}}\}.

Figure 1. The interpolation diagram

More generally, for hyperbolic paraboloids, we also have

Theorem 5.

In the $(1/p,1/q)$ interpolation diagram (see Figure 1), let

	$\displaystyle A_{1}=\left(0,0\right),\quad A_{2}=\left(0,\frac{1}{2}\right),\quad A_{3}=\left(\frac{1}{p_{v}},\frac{1}{2}\right),\quad A_{4}=\left(\frac{1}{p_{d}},\frac{1}{p_{d}}\right)$
	$\displaystyle A_{5}=\left(\frac{1}{p_{d}},0\right),\quad A_{6}=\left(\frac{1}{2},0\right),\quad A_{7}=\left(\frac{1}{2},\frac{1}{2}\right).$

Then the following holds:

(11)

D_{p,q}(\delta)\approx\begin{cases}&N^{\frac{1}{2}-\frac{1}{q}},\quad\text{if $(1/p,1/q)$ lies in trapezoid $A_{4}A_{5}A_{6}A_{7}$},\\ &N^{\frac{1}{2}-\frac{1}{q}+\frac{d(v)}{d-1}\left(\frac{1}{q}-\frac{1}{p}\right)},\quad\text{if $(1/p,1/q)$ lies in triangle $A_{3}A_{4}A_{7}$},\\ &N^{1-\frac{p_{d}}{2p}-\frac{1}{q}},\quad\text{if $(1/p,1/q)$ lies in pentagon $A_{1}A_{2}A_{3}A_{4}A_{5}$}.\end{cases}

Acknowledgement. The author thanks Jianhui Li for thoughtful discussions.

2. extremisers for elliptic paraboloids

2.1. The constant test function

Theorem 6.

Let $g\equiv 1$ . For every $p\in[2,\infty]$ , $q\in(0,\infty]$ we have

(12)

R(g)\gtrsim N^{1-\frac{p_{d}}{2p}-\frac{1}{q}}.

In particular, if $p\geq p_{d}$ , then the constant function is an extremizer for $D_{p,q}(\delta)$ for elliptic paraboloids.

Proof.

If $p<\infty$ , we use the equivalent neighbourhood version of decoupling. For each $Q$ , let $f_{Q}$ be a Schwartz function such that $\widehat{f_{Q}}$ is essentially the normalised indicator function of the $\delta$ -neighbourhood of the graph of $\phi$ over $Q$ . Let $f=\sum_{Q}f_{Q}$ . Let $B$ be centred at the origin and have side length $\delta^{-1}$ .

For $|x|\leq c\ll 1$ , we have $|e(x^{\prime}\cdot\xi+x_{d}\phi(\xi))|>1/2$ . Thus $|f(x)|\sim N$ over $[-c,c]^{d-1}$ , and hence

(13)

\lVert f\rVert_{L^{p}(B)}\geq\lVert f\rVert_{L^{p}([-c,c]^{d-1})}\sim N.

On the other hand, we have $f_{Q}\sim 1_{T_{Q}}$ where $T_{Q}$ is a rectangle with dimensions $(\delta^{-1/2})^{d-1}\times\delta^{-1}$ . Thus

(14)

\lVert f_{Q}\rVert_{L^{p}(w_{B})}\sim\delta^{-\frac{d-1}{2p}-\frac{1}{p}},

which holds for every $Q$ . Thus we have

\displaystyle R(g)

\displaystyle\gtrsim\frac{N}{N^{\frac{1}{q}}\delta^{-\frac{d-1}{2p}-\frac{1}{p}}}=N^{1-\frac{1}{q}-\frac{1}{p}}\delta^{\frac{1}{p}}=N^{1-\frac{p_{d}}{2p}-\frac{1}{q}}.

For $p=\infty$ , it is easy to see that $\lVert E1\rVert_{L^{\infty}(B)}=|E1(0)|\sim 1$ , and $\lVert E1_{Q}\rVert_{L^{\infty}(w_{B})}=|E1_{Q}(0)|\sim|Q|=N^{-1}$ . Thus we also have

(15)

R(g)\gtrsim\frac{1}{N^{\frac{1}{q}-1}}=N^{1-\frac{1}{q}}.

∎

2.2. The exponential sum test function

Lemma 7.

For $\delta\in\mathbb{N}^{-2}$ we define the function on $\mathbb{T}^{2}$ as

f(x,y)=\sum_{j=1}^{\delta^{-1/2}}e\left(jx+j^{2}y\right).

Then for all $2\leq p\leq\infty$ , we have

\lVert f\rVert_{L^{p}(\mathbb{T}^{2})}\approx\begin{cases}\delta^{-\frac{1}{4}}&\text{ if }2\leq p\leq 6\\ \delta^{-\frac{1}{2}+\frac{3}{2p}}&\text{ if }6\leq p\leq\infty\end{cases}.

Proof.

We first prove the upper bound by testing

g(s)=\sum_{j=1}^{\delta^{-1/2}}\Delta_{j\delta^{1/2}}(s),

where $\Delta_{a}$ is the delta-mass at $a$ . (A more rigorous argument is to take an approximation to the identity at each of the delta masses.) Then we have

Eg(x,y)=\sum_{j=1}^{\delta^{-1/2}}e\left(j\delta^{1/2}x+j^{2}\delta y\right)=f(\delta^{1/2}x,\delta y).

Also, for each $j$ we have

Eg_{j}(x,y)=e\left(j\delta^{1/2}x+j^{2}\delta y\right).

Thus $\lVert Eg_{j}\rVert_{L^{p}(w_{B})}\sim\delta^{-\frac{2}{p}}$ , and so

\lVert\lVert Eg_{j}\rVert_{L^{p}(w_{B})}\rVert_{l^{2}(j)}\sim\delta^{-\frac{1}{4}}\delta^{-\frac{2}{p}}.

But by periodicity, we have

\lVert Eg\rVert_{L^{p}(B)}=\delta^{-\frac{2}{p}}\lVert f\rVert_{L^{p}(\mathbb{T}^{2})}.

The desired upper bound then follows from Theorem 1 with $d=2$ .

The lower bound follows from [BB10]. We provide a little more detail for clarity. The case $p=\infty$ is trivial, taking $x=y=0$ . The case $6\leq p<\infty$ follows by considering $(x,y)$ near the origin; the detail can be found in Theorem 2.2 of [Yip20] (with $s=p/2$ , and the proof there is easily seen to work for all real numbers $s>0$ .) The case $p=2$ and $p=4$ follows from the first (trivial) bound of Theorem 2.3 of [Yip20].

Thus, the only case remaining is the case when $2<p<6$ and $p\neq 4$ . Assume $2<p<4$ first. Then $4\in(p,6)$ and using the log-convexity of $L^{p}$ -norms, we have

\lVert f\rVert_{L^{4}(\mathbb{T}^{2})}\leq\lVert f\rVert^{1-\theta}_{L^{p}(\mathbb{T}^{2})}\lVert f\rVert^{\theta}_{L^{6}(\mathbb{T}^{2})},

where $\frac{1-\theta}{p}+\frac{\theta}{6}=\frac{1}{4}$ . But since $\lVert f\rVert_{L^{4}(\mathbb{T}^{2})}\sim\delta^{-1/4}$ and $\lVert f\rVert_{L^{6}(\mathbb{T}^{2})}\sim\delta^{-1/4}$ , we also have $\lVert f\rVert_{L^{p}(\mathbb{T}^{2})}\gtrsim\delta^{-1/4}$ . The case $4<p<6$ is similar. ∎

Theorem 8.

Let $g=\sum_{Q}\Delta(c_{Q},\phi(c_{Q}))$ be the sum of delta-masses at $(c_{Q},\phi(c_{Q}))$ . Then for every $p\in[2,\infty]$ , $q\in(0,\infty]$ we have

(16)

R(g)\gtrsim\max\{N^{\frac{1}{2}-\frac{1}{q}},N^{1-\frac{3}{p}-\frac{1}{q}}\}.

As a result, if $2\leq p\leq p_{d}$ , then $g$ is an extremizer for $D_{p,q}(\delta)$ for elliptic paraboloids.

Remark. For $d\geq 3$ and $p>p_{d}$ , the bound $N^{1-\frac{3}{p}-\frac{1}{q}}$ does not agree with the sharp bound $N^{1-\frac{p_{d}}{2p}-\frac{1}{q}}$ . Thus to get sharp decoupling, we will need the constant test function. Thus, only the first lower bound $N^{\frac{1}{2}-\frac{1}{q}}$ will be useful in our proof later. Nevertheless, if $d=2$ , $p_{d}=6$ , then $g$ is an extremizer for every $p\geq 2$ .

Proof.

Let $B$ be centred at the origin and have side length $\delta^{-1}$ . We have $|E(g1_{Q})(x)|\equiv 1$ , and thus

(17)

\lVert\lVert E(g1_{Q})\rVert_{L^{p}(w_{B})}\rVert_{l^{q}(Q\in\mathcal{P}(\delta))}\sim N^{\frac{1}{q}}|B|^{\frac{1}{p}}=N^{\frac{1}{q}+\frac{2d}{p(d-1)}}.

For the left hand side, we have

	$\displaystyle Eg(x)$	$\displaystyle=\sum_{Q}e(x^{\prime}c_{Q}+x_{d}\phi(c_{Q}))$
		$\displaystyle=\prod_{i=1}^{d-1}\sum_{j_{i}=1}^{\delta^{-1/2}}e(j_{i}\delta^{1/2}x_{i}+v_{i}j_{i}^{2}\delta x_{d}).$

For $p<\infty$ ,

	$\displaystyle\int_{B}\|Eg(x)\|^{p}dx$
	$\displaystyle=\int_{-\delta^{-1}}^{\delta^{-1}}\left(\prod_{i=1}^{d-1}\int_{-\delta^{-1}}^{\delta^{-1}}\left\|\sum_{j_{i}=1}^{\delta^{-1/2}}e(j_{i}\delta^{1/2}x_{i}+v_{i}j_{i}^{2}\delta x_{d})\right\|^{p}dx_{i}\right)dx_{d}$
	$\displaystyle=\delta^{1+\frac{d-1}{2}}\int_{-1}^{1}\left(\prod_{i=1}^{d-1}\int_{-\delta^{-1/2}}^{\delta^{-1/2}}\left\|\sum_{j_{i}=1}^{\delta^{-1/2}}e(j_{i}x_{i}+v_{i}j_{i}^{2}x_{d})\right\|^{p}dx_{i}\right)dx_{d}$
(18)		$\displaystyle=\delta^{-d}\int_{-1}^{1}\left(\prod_{i=1}^{d-1}\int_{-1}^{1}\left\|\sum_{j_{i}=1}^{\delta^{-1/2}}e(j_{i}x_{i}+v_{i}j_{i}^{2}x_{d})\right\|^{p}dx_{i}\right)dx_{d},$

where the last line follows from periodicity. By taking complex conjugation and symmetry, it is easy to see that for each $i$ we have

\displaystyle\int_{-1}^{1}\left|\sum_{j_{i}=1}^{\delta^{-1/2}}e(j_{i}x_{i}+v_{i}j_{i}^{2}x_{d})\right|^{p}dx_{i}=\int_{-1}^{1}\left|\sum_{j=1}^{\delta^{-1/2}}e(jt+j^{2}x_{d})\right|^{p}dt:=S_{p}(x_{d}).

Thus

(19)

\int_{B}|Eg(x)|^{p}dx=\delta^{-d}\int_{-1}^{1}S_{p}(x_{d})^{d-1}dx_{d}.

In particular, this computation shows that the test function $g$ disregards the signs of $v_{i}$ , and thus it should only extremize the case of elliptic paraboloids.

We then use Jensen’s inequality:

	$\displaystyle\int_{-1}^{1}S_{p}(x_{d})^{d-1}dx_{d}$
	$\displaystyle\gtrsim\left(\int_{-1}^{1}S_{p}(x_{d})dx_{d}\right)^{d-1}$
	$\displaystyle\sim\max\{(\delta^{-\frac{1}{2}})^{\frac{p}{2}(d-1)},(\delta^{-\frac{1}{2}})^{(p-3)(d-1)}\}$
	$\displaystyle=\max\{N^{\frac{p}{2}},N^{p-3}\},$

where the $\sim$ follows from Lemma 7. This gives

(20)

\lVert Eg\rVert_{L^{p}(B)}\gtrsim\delta^{-\frac{d}{p}}\max\{N^{\frac{1}{2}},N^{1-\frac{3}{p}}\}.

If $p=\infty$ , it is also easy to see that $\lVert Eg\rVert_{L^{\infty}(B)}\geq|Eg(0)|\sim N$ , which agrees with the expression above.

As a result, we have

(21)

R(g)\gtrsim\frac{\delta^{-\frac{d}{p}}\max\{N^{\frac{1}{2}},N^{1-\frac{3}{p}}\}}{N^{\frac{1}{q}+\frac{2d}{p(d-1)}}}=\max\{N^{\frac{1}{2}-\frac{1}{q}},N^{1-\frac{3}{p}-\frac{1}{q}}\}.

∎

Remark. When $d=2$ , $p=6$ , a more refined number-theoretic argument (see [BB10] and Theorem 3.5 of [Yip20]) proves $\lVert Eg\rVert_{L^{p}(B)}\gtrsim N^{7/6}(\log N)^{1/6}$ , implying $D_{6,2}(\delta)\gtrsim|\log\delta|^{1/6}$ . So in general, we must have at least logarithmic losses in the upper bound. In fact, in [GMW24], the authors proved that when $d=2$ , $p=6$ , we can have $D_{6,2}(\delta)\lesssim|\log\delta|^{O(1)}$ .

2.3. Proof of Theorem 4

Proof.

The upper bound follows from Theorem 1 and Hölder’s inequality. The lower bound follows from Theorems 6 and 8. ∎

3. extremisers for hyperpolic paraboloids

We are now going to prove Theorem 5. By renaming the variables, we may let

(22)

\phi(\xi)=\xi_{1}^{2}+\cdots+\xi^{2}_{d(v)}-\xi^{2}_{d(v)+1}-\cdots-\xi_{d-1}^{2},

where we recall $d(v)\leq(d-1)/2$ is the minimum number of positive and negative entries of $v_{i}$ , $1\leq i\leq d-1$ .

First, if $p\geq p_{v}$ , then since $p_{v}\geq p_{d}$ , by Theorem 3 with Hölder’s inequality and Theorem 6, our claim follows. Thus it suffices to consider $2\leq p\leq p_{v}$ .

3.1. The hyperplane test function

Theorem 9.

Let $p,q\in[2,\infty]$ and $h$ be an extremizer for the decoupling for the elliptic paraboloid in dimension $d-1-2d(v)$ (set $h\equiv 1$ if $d-1-2d(v)=0$ ). Define $g(\xi)=h(\xi_{2d(v)+1},\cdots,\xi_{d-1})\prod_{j=1}^{d(v)}\Delta(\xi_{j}-\xi_{d(v)+j})$ . Then

(23)

R(g)\gtrsim N^{\frac{1}{2}-\frac{1}{q}+\frac{d(v)}{d-1}(\frac{1}{p}-\frac{1}{q})}.

Proof.

Let $B$ be centred at the origin and have side length $\delta^{-1}$ . By the separability of $Eg$ , we can easily compute that

(24)

Eg(x)=\tilde{E}h(x_{2d(v)+1},\dots,x_{d-1},x_{d})\prod_{k=1}^{d(v)}\left(\sum_{j_{k}=1}^{\delta^{-1/2}}e(\delta^{\frac{1}{2}}j_{k}(x_{k}+x_{k+d(v)}))\right),

where $\tilde{E}$ is the extension operator for the elliptic paraboloid over the unit cube in $d-1-2d(v)$ dimensions. We first compute

	$\displaystyle\int_{[-\delta^{-1},\delta^{-1}]^{2d(v)}}\prod_{k=1}^{d(v)}\left\|\sum_{j_{k}=1}^{\delta^{-1/2}}e(\delta^{\frac{1}{2}}j_{k}(x_{k}+x_{k+d(v)}))\right\|^{p}dx_{1}\cdots dx_{2d(v)}$
	$\displaystyle=\left(\int_{[-\delta^{-1},\delta^{-1}]^{2}}\left\|\sum_{j=1}^{\delta^{-1/2}}e(\delta^{\frac{1}{2}}j(x+y))\right\|^{p}dxdy\right)^{d(v)}$
	$\displaystyle\sim\left(\int_{-\delta^{-1}}^{\delta^{-1}}\delta^{-1}\left\|\sum_{j=1}^{\delta^{-1/2}}e(\delta^{\frac{1}{2}}jz)\right\|^{p}dz\right)^{d(v)}$
	$\displaystyle\sim\delta^{-\frac{(p+3)d(v)}{2}}.$

Thus

(25)

\lVert Eg\rVert_{L^{p}(B)}\sim\delta^{-\frac{(p+3)d(v)}{2p}}\lVert\tilde{E}h\rVert_{L^{p}(\tilde{B})},

where $\tilde{B}\subseteq\mathbb{R}^{d-2d(v)}$ is the cube centred at $0$ and has side length $\delta^{-1}$ .

Similarly, given each $Q:=\prod_{l=1}^{d-1}I_{l}\subseteq[0,1]^{d-1}$ of side length $\delta^{1/2}$ . Let $\tilde{Q}=\prod_{l=2d(v)+1}^{d-1}I_{l}$ , we see $E(g1_{Q})(x)$ is nonzero only if and for each $1\leq l\leq d(v)$ we have $I_{l}=I_{l+d(v)}$ . Thus

(26)

|E(g1_{Q})(x)|=|\tilde{E}_{\tilde{Q}}h(x_{2d(v)+1},\dots,x_{d-1},x_{d})|\prod_{l=1}^{d(v)}1_{I_{l}=I_{l+d(v)}}.

For such $Q$ we have

(27)

\lVert E(g1_{Q})\rVert_{L^{p}(B)}\sim\lVert\tilde{E}_{\tilde{Q}}h\rVert_{L^{p}(\tilde{B})}\delta^{-\frac{2d(v)}{p}}.

Thus

(28)

\lVert\lVert E(g1_{Q})\rVert_{L^{p}(B)}\rVert_{l^{q}(Q)}\sim\lVert\lVert\tilde{E}_{\tilde{Q}}h\rVert_{L^{p}(\tilde{B})}\rVert_{l^{q}(\tilde{Q})}\delta^{-\frac{2d(v)}{p}}\delta^{-\frac{d(v)}{2q}}.

For $\tilde{E}h$ , the critical exponent this time becomes $p_{d-2d(v)}$ instead of $p_{d}$ . Since $p\leq p_{v}\leq p_{d-2d(v)}$ , we use Theorem 8. Since $h$ is defined to be an extremizer, we have

(29)

\lVert\tilde{E}h\rVert_{L^{p}(\tilde{B})}\sim(\delta^{-\frac{d-1-2d(v)}{2}})^{\frac{1}{2}-\frac{1}{q}}\lVert\lVert\tilde{E}_{\tilde{Q}}h\rVert_{L^{p}(\tilde{B})}\rVert_{l^{q}(\tilde{Q})}

This implies that

	$\displaystyle R(g)$	$\displaystyle\gtrsim\delta^{-\frac{(p+3)d(v)}{2p}}\delta^{\frac{2d(v)}{p}}\delta^{\frac{d(v)}{2q}}(\delta^{-\frac{d-1-2d(v)}{2}})^{\frac{1}{2}-\frac{1}{q}}$
		$\displaystyle=N^{\frac{1}{2}-\frac{1}{q}}N^{\frac{d(v)}{d-1}(\frac{1}{q}-\frac{1}{p})}.$

∎

3.2. The case $p_{d}\leq p\leq p_{v}$

Combining the sharp lower bounds in Theorems 6 and 9, we get

(30)

D_{p,q}(\delta)\gtrsim\max\{N^{1-\frac{p_{d}}{2p}-\frac{1}{q}},N^{\frac{1}{2}-\frac{1}{q}}N^{\frac{d(v)}{d-1}(\frac{1}{q}-\frac{1}{p})}\}.

We need to determine which one is larger. Setting

(31)

1-\frac{p_{d}}{2p}-\frac{1}{q}=\frac{1}{2}-\frac{1}{q}+\frac{d(v)}{d-1}\left(\frac{1}{q}-\frac{1}{p}\right),

that is,

(32)

\frac{1}{2}-\frac{p_{d}}{2p}=\frac{d(v)}{d-1}\left(\frac{1}{q}-\frac{1}{p}\right),

we see this corresponds to the critical line $l_{1}$ that passes through the points $(1/p_{v},1/2)$ and $(1/p_{d},1/p_{d})$ in the $(1/p,1/q)$ interpolation diagram.

•

If $(1/p,1/q)$ lies above $l_{1}$ , then we have

(33) $D_{p,q}(\delta)\gtrsim N^{\frac{1}{2}-\frac{1}{q}+\frac{d(v)}{d-1}(\frac{1}{p}-\frac{1}{q})}.$

We claim that $\lessapprox$ also holds in this case. By interpolation, it suffices to prove upper bounds at the following three points:

(34) $\left(\frac{1}{p_{v}},\frac{1}{2}\right),\quad\left(\frac{1}{p_{d}},\frac{1}{p_{d}}\right),\quad\left(\frac{1}{p_{d}},\frac{1}{2}\right).$

The point $(\frac{1}{p_{v}},\frac{1}{2})$ has been settled by the case $p\geq p_{v}$ , since we can check by direct computation that

(35) $\frac{d(v)}{d-1}\left(\frac{1}{2}-\frac{1}{p_{v}}\right)=\frac{1}{2}-\frac{p_{d}}{p_{v}}.$

The upper bound at $(\frac{1}{p_{d}},\frac{1}{p_{d}})$ follows from Theorem 2 when $p=p_{d}$ . The upper bound at $(\frac{1}{p_{d}},\frac{1}{2})$ follows from Theorem 3 when $p=p_{d}$ .
•

If $(1/p,1/q)$ lies below $l_{1}$ , then we have

(36) $D_{p,q}(\delta)\gtrsim N^{1-\frac{p_{d}}{2p}-\frac{1}{q}}.$

We have $\lessapprox$ also holds in this case, by the upper bounds at $(\frac{1}{p_{v}},\frac{1}{2})$ and $(\frac{1}{p_{d}},\frac{1}{p_{d}})$ , together with Hölder’s inequality.

3.3. The case $2\leq p\leq p_{d}$

Combining the sharp lower bounds in Theorems 8 and 9, we get

(37)

D_{p,q}(\delta)\gtrsim\max\{N^{\frac{1}{2}-\frac{1}{q}},N^{\frac{1}{2}-\frac{1}{q}}N^{\frac{d(v)}{d-1}(\frac{1}{q}-\frac{1}{p})}\}.

It is easy to see that the critical line $l_{2}$ in this case is given by $1/p=1/q$ .

•

If $1/p\leq 1/q$ , then we have

(38) $D_{p,q}(\delta)\gtrsim N^{\frac{1}{2}-\frac{1}{q}+\frac{d(v)}{d-1}(\frac{1}{p}-\frac{1}{q})}.$

We have $\lessapprox$ also holds in this case, which follows immediately by interpolating the upper bounds we already obtained at $(\frac{1}{p_{d}},\frac{1}{p_{d}})$ and $(\frac{1}{p_{d}},\frac{1}{2})$ with $(\frac{1}{2},\frac{1}{2})$ given by Plancherel.
•

If $1/p\geq 1/q$ , then we have

(39) $D_{p,q}(\delta)\gtrsim N^{\frac{1}{2}-\frac{1}{q}}.$

We have $\lessapprox$ also holds in this case, by the upper bound at $(\frac{1}{p_{d}},\frac{1}{p_{d}})$ and Plancherel at $(\frac{1}{2},\frac{1}{2})$ , together with Hölder’s inequality.

References

[BB10] V. Blomer and J. Brüdern. The number of integer points on Vinogradov’s quadric. Monatsh. Math., 160(3):243–256, 2010.
[BD15] J. Bourgain and C. Demeter. The proof of the $l^{2}$ decoupling conjecture. Ann. of Math. (2), 182(1):351–389, 2015.
[BD17a] J. Bourgain and C. Demeter. Decouplings for curves and hypersurfaces with nonzero Gaussian curvature. J. Anal. Math., 133:279–311, 2017.
[BD17b] J. Bourgain and C. Demeter. A study guide for the $l^{2}$ decoupling theorem. Chin. Ann. Math. Ser. B, 38(1):173–200, 2017.
[GMW24] L. Guth, D. Maldague, and H. Wang. Improved decoupling for the parabola. J. Eur. Math. Soc. (JEMS), 26(3):875–917, 2024.
[GOZZK23] S. Guo, C. Oh, R. Zhang, and P. Zorin-Kranich. Decoupling inequalities for quadratic forms. Duke Math. J., 172(2):387–445, 2023.
[Yip20] C. H. Yip. Vinogradov’s mean value conjecture, 2020. https://drive.google.com/file/d/15WtAhVM0WUUXKRhvfnC0HhCWtfzOP_cL/view.

	$\displaystyle\int_{B}\|Eg(x)\|^{p}dx$
	$\displaystyle=\int_{-\delta^{-1}}^{\delta^{-1}}\left(\prod_{i=1}^{d-1}\int_{-\delta^{-1}}^{\delta^{-1}}\left\|\sum_{j_{i}=1}^{\delta^{-1/2}}e(j_{i}\delta^{1/2}x_{i}+v_{i}j_{i}^{2}\delta x_{d})\right\|^{p}dx_{i}\right)dx_{d}$
	$\displaystyle=\delta^{1+\frac{d-1}{2}}\int_{-1}^{1}\left(\prod_{i=1}^{d-1}\int_{-\delta^{-1/2}}^{\delta^{-1/2}}\left\|\sum_{j_{i}=1}^{\delta^{-1/2}}e(j_{i}x_{i}+v_{i}j_{i}^{2}x_{d})\right\|^{p}dx_{i}\right)dx_{d}$
(18)		$\displaystyle=\delta^{-d}\int_{-1}^{1}\left(\prod_{i=1}^{d-1}\int_{-1}^{1}\left\|\sum_{j_{i}=1}^{\delta^{-1/2}}e(j_{i}x_{i}+v_{i}j_{i}^{2}x_{d})\right\|^{p}dx_{i}\right)dx_{d},$

	$\displaystyle\int_{[-\delta^{-1},\delta^{-1}]^{2d(v)}}\prod_{k=1}^{d(v)}\left\|\sum_{j_{k}=1}^{\delta^{-1/2}}e(\delta^{\frac{1}{2}}j_{k}(x_{k}+x_{k+d(v)}))\right\|^{p}dx_{1}\cdots dx_{2d(v)}$
	$\displaystyle=\left(\int_{[-\delta^{-1},\delta^{-1}]^{2}}\left\|\sum_{j=1}^{\delta^{-1/2}}e(\delta^{\frac{1}{2}}j(x+y))\right\|^{p}dxdy\right)^{d(v)}$
	$\displaystyle\sim\left(\int_{-\delta^{-1}}^{\delta^{-1}}\delta^{-1}\left\|\sum_{j=1}^{\delta^{-1/2}}e(\delta^{\frac{1}{2}}jz)\right\|^{p}dz\right)^{d(v)}$
	$\displaystyle\sim\delta^{-\frac{(p+3)d(v)}{2}}.$

Sharp ℓq​(Lp)\ell^{q}(L^{p}) decoupling for paraboloids

Abstract.

1. Introduction

Theorem 1.

Theorem 2.

Theorem 3.

1.1. Main results

Theorem 4.

Theorem 5.

2. extremisers for elliptic paraboloids

2.1. The constant test function

Theorem 6.

Proof.

2.2. The exponential sum test function

Lemma 7.

Proof.

Theorem 8.

Proof.

2.3. Proof of Theorem 4

Proof.

3. extremisers for hyperpolic paraboloids

3.1. The hyperplane test function

Theorem 9.

Proof.

3.2. The case pd≤p≤pvp_{d}\leq p\leq p_{v}

3.3. The case 2≤p≤pd2\leq p\leq p_{d}

References

Sharp $\ell^{q}(L^{p})$ decoupling for paraboloids

3.2. The case $p_{d}\leq p\leq p_{v}$

3.3. The case $2\leq p\leq p_{d}$