Small cap square function estimates

We briefly introduce the notion of dual rectangle and local orthogonality.

From our definition, we see that if $R_{2}$ is a translated copy of $R_{1}$, then $R_{1}^{*}=R_{2}^{*}$. The motivation for defining dual rectangle is the following result.

This lemma is very standard, so we omit the proof. The next result is the local orthogonality property.

There is another notion called comparable. Given two rectangles $R_{1},R_{2}$, we say $R_{1}$ is essentially contained in $R_{2}$, if there exists a universal constant $C$ (say $C=100$) such that

$$R_{1}\subset CR_{2}.$$

We say $R_{1}$ and $R_{2}$ are comparable if $R_{1}$ is essentially contained in $R_{2}$ and vice versa, i.e.,

$$\frac{1}{C}R_{1}\subset R_{2}\subset CR_{1}.$$

Throughout this paper, we will just ignore the unimportant constant $C$, and just write $R_{1}\subset R_{2}$ to denote that $R_{1}$ is essentially contained in $R_{2}$.

There are two types of examples: concentrated example and flat example.

We introduce the concentrated example. Choose $f$ such that $\widehat{f}(\xi)=\psi_{N_{R^{-1}}(\mathcal{P})}(\xi)$, where $\psi_{N_{R^{-1}}(\mathcal{P})}$ is a smooth bump function supported in $N_{R^{-1}}(\mathcal{P})$. We see that $f(0)=\int\widehat{f}(\xi)d\xi\sim R^{-1}$. Since $\widehat{f}$ is supported in the unit ball centered at the origin, $f$ is locally constant in $B(0,1)$. Therefore,

$$\|f\|_{p}\geq\|f\|_{L^{p}(B(0,1))}\gtrsim R^{-1}.$$

We consider the right hand side of (1). By definition, for each $\gamma\in\Gamma_{\alpha}(R^{-1})$, $\widehat{f}_{\gamma}$ is roughly a bump function supported in $2\gamma$. Let $\gamma^{*}$ be the dual rectangle of $\gamma$ which has dimensions $R^{\alpha}\times R$ and is centered at the origin. By an application of integration by parts and by ignoring the tails, we assume

$$f_{\gamma}=\frac{1}{|\gamma^{*}|}{\bf 1}_{\gamma^{*}}.$$

Here, “$\approx$” means up to a $C_{\epsilon}R^{\epsilon}$ factor for any $\epsilon>0$. We will use the same notation throughout the paper.

We see that

$$\Big{\|}(\sum\limits_{\gamma\in\Gamma_{\alpha}(R^{-1})}|f_{\gamma}|^{2})^{1/2}\Big{\|}_{L^{p}(\mathbb{R}^{2})}^{p}\sim R^{-(1+\alpha)p}\int_{B(0,R)}(\sum_{\gamma}{\bf 1}_{\gamma^{*}})^{p/2}.$$

We evaluate the integral above. There are two extreme regions: $B(0,R^{\alpha})$ where all the $\{\gamma^{*}\}$ overlap; $B(0,R)\setminus B(0,R/2)$ where $\{\gamma^{*}\}$ is $O(R^{2\alpha-1})$-overlapping. For the intermediate region $B(0,r)\setminus B(0,r/2)$ ($R^{\alpha}\leq r\leq R$), we see that $\{\gamma^{*}\}$ is $O(r^{-1}R^{2\alpha})$-overlapping. We may find a dyadic radius $r$ such that

$\displaystyle\int(\sum_{\gamma}{\bf 1}_{\gamma^{*}})^{p/2}$

$\displaystyle\approx\int_{B(0,r)\setminus B(0,r/2)}(\sum_{\gamma}{\bf 1}_{\gamma^{*}})^{p/2}\lesssim(r^{-1}R^{2\alpha})^{p/2}|B(0,r)|\sim r^{2-\frac{p}{2}}R^{\alpha p}.$

Since $p\geq 4\alpha+2\geq 4$, the expression above is maximized when $r=R^{\alpha}$. Plugging in, we obtain

$$\int(\sum_{\gamma}{\bf 1}_{\gamma^{*}})^{p/2}\lessapprox R^{\alpha(2+\frac{p}{2})}.$$

Plugging into (1), we have

$$R^{-1}\lessapprox C_{\alpha,p}(R)R^{-(1+\alpha)}R^{\alpha(\frac{2}{p}+\frac{1}{2})},$$

which gives

$$C_{\alpha,p}(R)\gtrapprox R^{\alpha(\frac{1}{2}-\frac{2}{p})}.$$

We introduce the flat example. Let $\theta\subset N_{R^{-1}}(\mathcal{P})$ be a $R^{-1/2}\times R^{-1}$-cap. Choose $f$ such that $\widehat{f}(\xi)=\psi_{\theta}(\xi)$, where $\psi_{\theta}$ is a smooth bump function supported in $N_{R^{-1}}(\mathcal{P})$. Let $\theta^{*}$ be the dual rectangle of $\theta$ which has dimensions $R^{1/2}\times R$ and is centered at the origin. By the locally constant property, $f$ is an $L^{1}$ normalized function essentially supported in $\theta^{*}$. By ignoring the tails, we assume

$$f=\frac{1}{|\theta^{*}|}{\bf 1}_{\theta^{*}}.$$

We see that

$$\|f\|_{p}\sim R^{-\frac{3}{2}}R^{\frac{3}{2p}}.$$

We consider the right hand side of (1). By the same reasoning as in Case 1, for each $\gamma\in\Gamma_{\alpha}(R^{-1})$ with $\gamma\subset\theta$, we know that $\widehat{f}_{\gamma}$ is roughly a bump function supported in $2\gamma$. Therefore, we can assume

$$f_{\gamma}=\frac{1}{|\gamma^{*}|}{\bf 1}_{\gamma^{*}}.$$

We also note that $\gamma_{1}^{*}$ and $\gamma_{2}^{*}$ are comparable when $\gamma_{1},\gamma_{2}\subset\theta$. We have

	$\displaystyle\Big{\|}(\sum\limits_{\gamma\in\Gamma_{\alpha}(R^{-1})}|f_{\gamma}|^{2})^{1/2}\Big{\|}_{L^{p}(\mathbb{R}^{2})}$	$\displaystyle\sim R^{-(1+\alpha)}\bigg{(}\int(\sum_{\gamma\subset\theta}{\bf 1}_{\gamma^{*}})^{p/2}\bigg{)}^{1/p}\sim R^{-(1+\alpha)}\#\{\gamma\subset\theta\}^{1/2}|\gamma^{*}|^{1/p}$
		$\displaystyle\sim R^{-(1+\alpha)}R^{\frac{1}{2}(\alpha-\frac{1}{2})}R^{\frac{1+\alpha}{p}}.$

Plugging into (1), we have

$$R^{-\frac{3}{2}}R^{\frac{3}{2p}}\lessapprox C_{\alpha,p}(R)R^{-(1+\alpha)}R^{\frac{1}{2}(\alpha-\frac{1}{2})}R^{\frac{1+\alpha}{p}},$$

which gives

$$C_{\alpha,p}(R)\gtrapprox R^{(\alpha-\frac{1}{2})(\frac{1}{2}-\frac{1}{p})}.$$

By the standard localization argument, it suffices to prove

$$\|f\|_{L^{p}(B_{R})}\lessapprox_{\epsilon}(R^{\alpha(\frac{1}{2}-\frac{2}{p})}+R^{(\alpha-\frac{1}{2})(\frac{1}{2}-\frac{1}{p})})\Big{\|}(\sum\limits_{\gamma\in\Gamma_{\alpha}(R^{-1})}|f_{\gamma}|^{2})^{1/2}\Big{\|}_{p}.$$

We introduce some notations. Throughout the proof, we use $\gamma$ to denote caps of dimensions $R^{-\alpha}\times R^{-1}$. For $R^{-1/2}\leq\Delta\leq 1$, we will consider caps $\tau$ of length $\Delta$ and thickness $R^{-1}$. We write $|\tau|=\Delta$ to indicate the length of $\tau$. We will also partition the region $B_{R}$ into rectangles of dimensions $R^{\alpha}\times R$. For simplicity, we denote these rectangles by $B_{R^{\alpha}\times R}.$ The longest direction of $B_{R^{\alpha}\times R}$ will be specified in the proof.

Let $K\sim\log R$ and let $m\in\mathbb{N}$ be such that $K^{m}=R^{1/2}$. By doing the broad-narrow reduction as in [2, Section 5.1] , we have

(6)		$\displaystyle\|f\|_{L^{p}(B_{R})}^{p}$	$\displaystyle\lesssim C^{m}\sum_{|\theta|=R^{-1/2}}\|f_{\theta}\|_{L^{p}(B_{R})}^{p}$
(7)			$\displaystyle+C^{m}K^{C}\sum_{\begin{subarray}{c}R^{-1/2}\leq\Delta\leq 1\\ \Delta\text{~{}dyadic}\end{subarray}}\sum_{|\tau|\sim\Delta}\max_{\begin{subarray}{c}\tau_{1},\tau_{2}\subset\tau\\ |\tau_{1}|=|\tau_{2}|=K^{-1}\Delta\\ \textup{dist}(\tau_{1},\tau_{2})\geq(10K)^{-1}\Delta\end{subarray}}\|(f_{\tau_{1}}f_{\tau_{2}})^{1/2}\|_{L^{p}(B_{R})}^{p}.$

Note that $C^{m}K^{C}\lesssim_{\epsilon}R^{\epsilon}$, for each $\epsilon>0$.

We first estimate the right hand side of (6).

By Lemma 3, the right hand side of (6) is bounded by

$$R^{\epsilon}R^{(\alpha-\frac{1}{2})(\frac{1}{2}-\frac{1}{p})}\bigg{(}\sum_{\theta}\|(\sum_{\gamma\subset\theta}|f_{\gamma}|^{2})^{1/2}\|_{p}^{p}\bigg{)}^{1/p}\leq C_{\alpha,p}(R)\|(\sum_{\gamma}|f_{\gamma}|^{2})^{1/2}\|_{p}.$$

Next, we estimate (7). For any summand in (7), we will show that

(9)

$$\|(f_{\tau_{1}}f_{\tau_{2}})^{1/2}\|_{L^{p}(B_{R})}\lesssim C_{\alpha,p}(R)\|(\sum_{\gamma\subset\tau}|f_{\gamma}|^{2})^{1/2}\|_{p}.$$

This will imply (7)${}^{\frac{1}{p}}$ $\lessapprox C_{\alpha,p}(R)\|(\sum\limits_{\gamma}|f_{\gamma}|^{2})^{1/2}\|_{p}$, and then finishes the proof of Theorem 1. It remains to prove (9).

Fix a $\Delta\in[R^{-1/2},1]$ and a $\tau$ with $|\tau|=\Delta$. We first consider $\bigcap_{\gamma\subset\tau}\gamma^{*}$. It is easy to see $\bigcap_{\gamma\subset\tau}\gamma^{*}$ is an $R^{\alpha}\times R^{\alpha}\Delta^{-1}$-rectangle when $\Delta\geq R^{\alpha-1}$; $\bigcap_{\gamma\subset\tau}\gamma^{*}$ is an $R^{\alpha}\times R$-rectangle when $\Delta\leq R^{\alpha-1}$. We consider these two cases separately.

We choose a partition $B_{R}=\bigsqcup B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}}$, where each $B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}}$ is a translation of $\bigcap_{\gamma\subset\tau}\gamma^{*}$. We just need to show

(10)

$$\|(f_{\tau_{1}}f_{\tau_{2}})^{1/2}\|_{L^{p}(B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}})}\lesssim C_{\alpha,p}(R)\|(\sum_{\gamma\subset\tau}|f_{\gamma}|^{2})^{1/2}\|_{L^{p}(B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}})}.$$

Since each $|f_{\gamma}|$ is locally constant on $B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}}$ when $\gamma\subset\tau$, we have

$$\|(\sum_{\gamma\subset\tau}|f_{\gamma}|^{2})^{1/2}\|_{L^{p}(B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}})}\sim(R^{2\alpha}\Delta^{-1})^{-\frac{1}{2}+\frac{1}{p}}\|(\sum_{\gamma\subset\tau}|f_{\gamma}|^{2})^{1/2}\|_{L^{2}(B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}})}.$$

Since $\{f_{\gamma}\}_{\gamma\subset\tau}$ are locally orthogonal on $B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}}$, we have

$$\|(\sum_{\gamma\subset\tau}|f_{\gamma}|^{2})^{1/2}\|_{L^{2}(B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}})}\sim\|f_{\tau}\|_{L^{2}(B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}})}.$$

Therefore, (10) is reduced to

(11)

$$\|(f_{\tau_{1}}f_{\tau_{2}})^{1/2}\|_{L^{p}(B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}})}\lesssim C_{\alpha,p}(R)(R^{2\alpha}\Delta^{-1})^{-\frac{1}{2}+\frac{1}{p}}\|f_{\tau}\|_{L^{2}(B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}})}.$$

Next, we apply the parabolic rescaling. Recall that $\tau$ is a cap of length $\Delta$. We dilate by factor $\Delta^{-1}$ in the tangent direction of $\tau$ and dilate by factor $\Delta^{-2}$ in the normal direction of $\tau$. Under the rescaling, we see that: $\tau$ becomes the $R^{-1}\Delta^{-2}$-neighborhood of $\mathcal{P}$; $\tau_{1}$ and $\tau_{2}$ become $K^{-1}$-separated caps with length $K^{-1}$ and thickness $R^{-1}\Delta^{-2}$; the rectangle $B_{R^{\alpha}\times R^{\alpha}\Delta^{-1}}$ in the physical space becomes $B_{R^{\alpha}\Delta}$. Let $g,g_{1},g_{2}$ be the rescaled version of $f_{\tau},f_{\tau_{1}},f_{\tau_{2}}$ respectively. The inequality (11) becomes

(12)

$$\|(g_{1}g_{2})^{1/2}\|_{L^{p}(B_{R^{\alpha}\Delta})}\lesssim C_{\alpha,p}(R)(R^{2\alpha}\Delta^{-1})^{-\frac{1}{2}+\frac{1}{p}}\Delta^{3(-\frac{1}{2}+\frac{1}{p})}\|g\|_{L^{2}(B_{R^{\alpha}\Delta})}.$$

We recall the following bilinear restriction estimate (see for example in [9]).

We return to (12). Noting that $R^{\alpha}\Delta\geq(R\Delta^{2})^{1/2}$, we apply the lemma above to bound the left hand side of (12) by $(R^{\alpha}\Delta)^{\frac{2}{p}-1}\|g\|_{L^{2}(B_{R^{\alpha}\Delta})}$. It suffices to prove

(14)

$$(R^{\alpha}\Delta)^{\frac{2}{p}-1}\lesssim C_{\alpha,p}(R)(R^{2\alpha}\Delta^{-1})^{-\frac{1}{2}+\frac{1}{p}}\Delta^{3(-\frac{1}{2}+\frac{1}{p})}.$$

When $p\geq 4$, we use $C_{\alpha,p}(R)\gtrsim R^{\alpha(\frac{1}{2}-\frac{2}{p})}$. Then (14) boils down to

(15)

$$(R^{\alpha}\Delta)^{\frac{2}{p}-1}\lesssim R^{\alpha(\frac{1}{2}-\frac{2}{p})}(R^{2\alpha}\Delta^{-1})^{-\frac{1}{2}+\frac{1}{p}}\Delta^{3(-\frac{1}{2}+\frac{1}{p})},$$

which is equivalent to

$$R^{\alpha(\frac{1}{2}-\frac{2}{p})}\gtrsim 1,$$

which is true since $R\geq 1$.

When $p\leq 4$, we use $C_{\alpha,p}(R)\gtrsim R^{(\alpha-\frac{1}{2})(\frac{1}{2}-\frac{1}{p})}$. Then (14) boils down to

(16)

$$(R^{\alpha}\Delta)^{\frac{2}{p}-1}\lesssim R^{(\alpha-\frac{1}{2})(\frac{1}{2}-\frac{1}{p})}(R^{2\alpha}\Delta^{-1})^{-\frac{1}{2}+\frac{1}{p}}\Delta^{3(-\frac{1}{2}+\frac{1}{p})},$$

which is equivalent to

$$R^{(\alpha-\frac{1}{2})(\frac{1}{2}-\frac{1}{p})}\gtrsim 1,$$

which is true since $\alpha\geq 1/2.$