A Note on the Refined Strichartz Estimates and Maximal Extension Operator

Abstract

There are two parts for this paper. In the first part we extend some results in a recent paper by Du, Guth, Li and Zhang to a more general class of phase functions. The main methods are Bourgain–Demeter’s $l^2$ decoupling theorem and induction on scales. In the second part we prove some positive results for the maximal extension operator for hypersurfaces with positive principal curvatures. The main methods are sharp $L^2$ estimates by Du and Zhang, and the bilinear method by Wolff and Tao.

Appendices

Appendix: Proof of Wave-Packet Decomposition

In this section we present a proof for Proposition 4.4. The proof based on the framework in [15] Proposition 2.6. Let $\psi (\xi )$ be a smooth function that equals to 1 in the unit ball $B^n(0,1)$, and is supported in a bigger ball $B^n(0,2)$. Define $\psi _q(\xi ):=\psi \big (R^{1/2}(\xi -c(q))\big )$ so that $\widehat{f}_q=\psi _q\widehat{f}_q$. Consider the partial Fourier series $S_N\widehat{f_q}$ for $\widehat{f_q}$ expanding in a $2R^{-1/2}$-cube 2q:

$$\begin{aligned} S_N\widehat{f_q}(\xi )\sim \sum _{m\in {\mathbb {Z}}^n,|m|\le N}a_me^{i\pi R^{1/2}m\cdot \xi }, \end{aligned}$$

(5.1)

where

$$\begin{aligned} a_m:=R^{n/2}\int _{2q} e^{-i\pi R^{1/2}m\cdot \xi }\widehat{f_q}(\xi )\psi _q(\xi )d\xi . \end{aligned}$$

(5.2)

For $T\in {\mathbb {T}}_q$, let $P_T(x):{\mathbb {R}}^{n+1}\rightarrow {\mathbb {R}}^n$ be the projection to the subspace whose normal vector coincides to the direction of T. If $P_T(c(T))=R^{1/2}m$, we let $m=m_T$ and define

$$\begin{aligned} f_T(\xi )=\int a_m e^{ix\cdot \xi }e^{i\pi R^{1/2}m\cdot \xi }\psi _q(\xi )d\xi \end{aligned}$$

(5.3)

so that $\widehat{f_T}(\xi )=a_me^{i\pi R^{1/2}m\cdot \xi }\psi _q(\xi )$. Clearly, Property (1) is true.

Next we take a look on ${\mathcal {E}}_{\Phi } f_T$. Plug in the definition of $f_T$ to have

$$\begin{aligned} {\mathcal {E}}_{\Phi } f_T(x,t)= & {} \int _{{\mathbb {R}}^n}e^{ix\cdot \xi }e^{it\Phi (\xi )}a_me^{i\pi R^{1/2}m\cdot \xi }\psi _q(\xi )d\xi \\= & {} \int _{{\mathbb {R}}^n}e^{ix\cdot \xi }e^{it\Phi (\xi )}a_me^{i\pi R^{1/2}m\cdot \xi }\psi (R^{1/2}(\xi -c(q)))d\xi . \end{aligned}$$

After a change of variable, we use Taylor’s expansion so that

$$\begin{aligned} |{\mathcal {E}}_{\Phi } f_T(x,t)|= \Big |a_m\int _{{\mathbb {R}}^n}e^{i(x+\pi R^{1/2}m+\nabla \Phi (c(q)))\cdot \xi }e^{iO(1)}\psi (R^{1/2}\xi )d\xi \Big |. \end{aligned}$$

(5.4)

When $0<|t|<R$ and $|x+\pi m+\nabla \Phi (c(q))| > rsim R^{1/2}R^\delta$, the integrand in (5.4) admits fast decay. Thus $| {\mathcal {E}}_{\Phi } f_T|\lesssim \mathrm {RapDec}(R)\Vert f\Vert _2$, as $|a_m|\le R^{n/4}\Vert f\Vert _2$. This gives the proof of Property (2).

We now prove Property (3). Recall that the partial sum $S_N\widehat{f_q}$ converges to $\widehat{f_q}$ in $L^2$. Thus, there is a positive number $N_q>0$ such that

$$\begin{aligned} \Vert S_{N_q}\widehat{f_q}-\widehat{f_q}\Vert _{L^2(2q)}\le \mathrm {RapDec}(R)\Vert f\Vert _2. \end{aligned}$$

(5.5)

Let ${{\bar{{\mathbb {T}}}}}_q$ be the collection of $T\in {\mathbb {T}}_q$ such that $|P_T(c(T))|\le N_q$. Then

$$\begin{aligned}&\Big |{\mathcal {E}}_{\Phi }f_q-\sum _{T\in {\mathbb {T}}_q}{\mathcal {E}}_{\Phi }f_T\Big | = \Big |\int _{{\mathbb {R}}^n} e^{ix\cdot \xi }e^{it\Phi (\xi )}(\widehat{f_q}-\sum _{T\in {\mathbb {T}}_q} \widehat{f_T})d\xi \Big |\nonumber \\&\quad \le \Big |\int _{{\mathbb {R}}^n} e^{ix\cdot \xi }e^{it\Phi (\xi )}(\widehat{f_q}-\sum _{T\in {{\bar{{\mathbb {T}}}}}_q} \widehat{f_T})d\xi \Big |+\Big |\int _{{\mathbb {R}}^n} e^{ix\cdot \xi }e^{it\Phi (\xi )}\Big (\sum _{T\in ({{\bar{{\mathbb {T}}}}}_q\setminus {\mathbb {T}}_q)} \widehat{f_T}\Big )d\xi \Big |. \end{aligned}$$

(5.6)

From (5.5) and the fact $\widehat{f}_q=\psi _q\widehat{f}_q$, the first part of (5.6) is bounded above by $\mathrm {RapDec}(R)\Vert f\Vert _2$. Since $(x,t)\in B^{n+1}_R$, for each $T\in {{\bar{{\mathbb {T}}}}}_q\setminus {\mathbb {T}}_q$, we set $m=P_T(C(T))$ and use the standard (non) stationary phase method to get

$$\begin{aligned} \Big |\int _{{\mathbb {R}}^n} e^{ix\cdot \xi }e^{it\Phi (\xi )} \widehat{f_T}(\xi )d\xi \Big |\lesssim \frac{a_m}{(1+|x|+|m|)^{-M}}\lesssim |m|^{-2n}\mathrm {RapDec}(R)\Vert f\Vert _2\nonumber \\ \end{aligned}$$

(5.7)

for some M large enough. Summing up all the $T\in {{\bar{{\mathbb {T}}}}}_q\setminus {\mathbb {T}}_q$ we have the second part of (5.6) is bounded by $\mathrm {RapDec}(R)\Vert f\Vert _2$. Thus, (5.6) is bounded by $\mathrm {RapDec}(R)\Vert f\Vert _2$ and we finish the proof for Property (3) by summing up all the q.

Property (4), the first part of Property (5) and Property (6) follow directly from Plancherel. For the second part of Property (5), by Plancherel we have

$$\begin{aligned} \langle {\mathcal {E}}_{\Phi } f_T,{\mathcal {E}}_{\Phi } f_{T'}\rangle =\int \langle \widehat{f_T},\widehat{f}_{T'}\rangle dt\le R|\langle \widehat{f_T},\widehat{f}_{T'}\rangle |, \end{aligned}$$

(5.8)

which is further bounded by

$$\begin{aligned} CR^{n/2}|a_{m_T}a_{m_{T'}}\widehat{\psi }(m_T-m_{T'})|\lesssim \mathrm {RapDec}(R), \end{aligned}$$

(5.9)

as $|m_T-m_{T'}|\sim R^{-1/2}$dist($T,T'$)$> rsim R^{\delta }$. $\square$

Appendix: An Epsilon Removal Lemma

In this section we prove the following lemma

Lemma 6.1

Suppose $p>2$, $\varepsilon >0$ and (4.1). Then

$$\begin{aligned} \big \Vert \sup _{t\in {\mathbb {R}}}|{\mathcal {E}}_{\Phi }f|\big \Vert _{L^{p_0}({\mathbb {R}}^n)}\lesssim \Vert f\Vert _2 \end{aligned}$$

(6.1)

for $f\in L^2({\mathbb {R}}^n)$ and

$$\begin{aligned} \frac{1}{p_0}<\frac{1}{p}-\frac{C}{\log \frac{1}{\varepsilon }}. \end{aligned}$$

(6.2)

In particular, Lemma 4.1 implies Theorem 1.7 by letting $\varepsilon \rightarrow 0$. Our proof for Lemma 6.1 is similar to the argument in [5]. See also [21].

For some technical issues, we assume $\widehat{f}$ is supported in $B^n_{1/4}$ rather then the unit ball in Theorem 1.7. Since $\widehat{f}\subset B^n_{1/4}$, $|{\mathcal {E}}_{\Phi }f|$ is essentially constant in every 1-ball in ${\mathbb {R}}^{n+1}$. Inspired by this observation, we have the following lemma:

Lemma 6.2

There exists a function $\varphi (x,t):{\mathbb {R}}^{n+1}\rightarrow {\mathbb {C}}$ such that

$$\begin{aligned} \big \Vert \sup _{t\in {\mathbb {R}}}|{\mathcal {E}}_{\Phi }f|\big \Vert _{L^p({\mathbb {R}}^n)}\sim \big \Vert ({\mathcal {E}}_{\Phi }f)\varphi \big \Vert _{L^p({\mathbb {R}}^{n+1})}, \end{aligned}$$

(6.3)

where the function $\varphi$ satisfies the following properties:

• (1) $|\varphi |\lesssim 1$, $\mathrm{supp}(\widehat{\varphi })\subset B^{n+1}_{1/4}$ .

• (2) Uniformly for any $x\in {\mathbb {R}}^n$, $\Vert \varphi (x,\cdot )\Vert _{L^1_t}=O(1)$.

• (3) For any small factor $\beta >0$, there exists $R^\beta$ many horizontally sparse sets $\{X_j\}$ (See Definition 4.3) such that $\varphi (x,t)=\mathrm {RapDec}(R)$ when $(x,t)\in {\mathbb {R}}^{n+1}\setminus (\cup _j X_j)$.

We remark that the implicit constant in $\mathrm {RapDec}(R)$ depends on $\beta$

Roughly speaking, the weight $\varphi$ is an averaging method to help us realize the linearization $\sup _t|{\mathcal {E}}_\Phi f(x,t)|=|{\mathcal {E}}_\Phi f(x,t(x))|$. We remark that $\varphi$ is essentially supported in a set satisfying Property 4.3.

Proof

Let $\{U\}$ be the lattice 1-cubes in ${\mathbb {R}}^n$ and $\{I\}$ be the lattice 1-cubes in ${\mathbb {R}}$. Let $\psi _U(x)$, $\psi _I(t)$ be two smooth functions on such that $\mathrm{supp}(\widehat{\psi }_U)\subset B^n_{1/8}$, $\mathrm{supp}(\widehat{\psi }_I)\subset [-1/8,1/8]$, $|\psi _U(x)|\sim 1$ for $x\in U$, $|\psi _I(t)|\sim 1$ for $t\in I$ and $\psi _U$, $\psi _I$ admit fast decay outside U, I, respectively. Thus, by Hausdorff-Young inequality,

$$\begin{aligned} \int _U\sup _{t\in {\mathbb {R}}}|{\mathcal {E}}_{\Phi } f|^p\lesssim \sup _{I}\sup _{x\in {\mathbb {R}}^n}\sup _{t\in I}\big |{\mathcal {E}}_{\Phi } f(\psi _U\psi _I)^3\big |^p\le \sup _I\big \Vert \big ({\mathcal {E}}_{\Phi } f(\psi _U\psi _I)^3\big )^\wedge \big \Vert _1^p.\nonumber \\ \end{aligned}$$

(6.4)

From the constructions of $\psi _U$ and $\psi _I$, we see the Fourier transform of $(\psi _U\psi _I)^3$ is compactly supported. Combining the fact that $\widehat{f}$ is compactly supported, we have that $\big ({\mathcal {E}}_{\Phi } f(\psi _U\psi _I)^3\big )^\wedge$ is compactly supported. Hence

$$\begin{aligned} \sup _I\big \Vert \big ({\mathcal {E}}_{\Phi } f(\psi _U\psi _I)^3\big )^\wedge \big \Vert _1^p\lesssim \sup _I\big \Vert \big ({\mathcal {E}}_{\Phi } f(\psi _U\psi _I)^3\big )^\wedge \big \Vert _2^p. \end{aligned}$$

(6.5)

Invoking Plancherel and Hölder’s inequality, we obtain that for $3<p<4$,

$$\begin{aligned}&\sup _I\big \Vert \big ({\mathcal {E}}_{\Phi } f(\psi _U\psi _I)^3\big )^\wedge \big \Vert _2^p=\sup _I\big \Vert {\mathcal {E}}_{\Phi } f(\psi _U\psi _I)^3\big \Vert _2^p\nonumber \\&\quad \le \sup _I\Big (\int _{{\mathbb {R}}^{n+1}}|{\mathcal {E}}_{\Phi } f|^p|\psi _U\psi _I|^{2p}\Big )\Big (\int _{{\mathbb {R}}^{n+1}}|\psi _U\psi _I|^{2p/(p-2)}\Big )^{(p-2)/2}\nonumber \\&\quad \lesssim \sup _I\int _{{\mathbb {R}}^{n+1}}|{\mathcal {E}}_{\Phi } f|^p|\psi _U\psi _I|^{2p}. \end{aligned}$$

(6.6)

We pick one $I_U$ such that

$$\begin{aligned} \sup _I\int _{{\mathbb {R}}^{n+1}}|{\mathcal {E}}_{\Phi } f|^p|\psi _U\psi _I|^{2p}\le 2\int _{{\mathbb {R}}^{n+1}}|{\mathcal {E}}_{\Phi } f|^p|\psi _U\psi _{I_U}|^{2p} \end{aligned}$$

(6.7)

and define

$$\begin{aligned} \varphi :=\sum _U|\psi _U\psi _{I_U}|^2. \end{aligned}$$

(6.8)

Note that $\psi _U(x)\psi _{I_U}(t)=\mathrm {RapDec}(R)$ when $\mathrm {dist}((x,t),U\times I_U)\ge R^\beta$ for any $\beta >0$. One can check directly that the function $\varphi$ satisfies all properties mentioned in the statement of the Lemma. Combining (6.4), (6.5), (6.6), (6.7) and summing all the $U\subset {\mathbb {R}}^n$, we have

$$\begin{aligned} \int _{{\mathbb {R}}^n}\sup _{t\in {\mathbb {R}}}|{\mathcal {E}}_{\Phi } f|^p\lesssim \sum _U|\int _{{\mathbb {R}}^{n+1}}|{\mathcal {E}}_{\Phi } f|^p|\psi _U\psi _{I_U}|^{2p}\le \int _{{\mathbb {R}}^{n+1}}|{\mathcal {E}}_{\Phi }\varphi |^p. \end{aligned}$$

(6.9)

Take pth root to both sides we get one direction for (6.3). For the other direction of the estimate above, one just need to use property (2) of the function $\varphi$. $\square$

Applying Lemma 6.2, it suffices to show the dual estimate

$$\begin{aligned} \Vert {\mathcal {E}}^*(g\varphi )\Vert _{L^2(B^{n}_{1/4})}\lesssim \Vert g\Vert _{p_0'} \end{aligned}$$

(6.10)

where

$$\begin{aligned} {\mathcal {E}}^*g(\xi )=\int _{{\mathbb {R}}^{n+1}} e^{-ix\cdot \xi }e^{-it\Phi (\xi )}g(x,t)dxdt. \end{aligned}$$

(6.11)

Let us follow a similar argument as in the proof of Lemma 6.2. Assuming (4.1), we have that for any R-ball $V\subset {\mathbb {R}}^{n+1}$,

$$\begin{aligned} \big \Vert ({\mathcal {E}}_{\Phi }f)\varphi \big \Vert _{L^p(V)}\le C_\varepsilon R^\varepsilon \Vert f\Vert _2 \end{aligned}$$

(6.12)

and the dual estimate

$$\begin{aligned} \Vert {\mathcal {E}}^*(g\mathbf{1}_{V}\varphi )\Vert _{L^2(B^{n}_{1/4})}\le C_\varepsilon R^\varepsilon \Vert g\Vert _{p'}. \end{aligned}$$

(6.13)

Let $\phi (x)$ be a smooth function in ${\mathbb {R}}^n$ that $\widehat{\phi }(\xi )=1$ when $\xi \in B^n_{1/4}$ and $\mathrm{supp}(\widehat{\phi })\subset B^n_{1/2}$. The classic result of restriction theorem (see [16] Sect. 7, [19] Proposition 6) tells us that ${\mathcal {E}}_{\Phi }\phi (x,t)$ is bounded above by the decay function $C(1+|x|+|t|)^{-n/2}$. Therefore, in order to make full use of the local estimate (6.13), we are motivated to consider a sparse collection of R-ball in ${\mathbb {R}}^{n+1}$.

Let $\{V_j\}_{j=1}^N$ be a collection of R-balls in ${\mathbb {R}}^{n+1}$ such that for any $j\not =j',j,j'\in \{1,2,\ldots ,N\}$, $\mathrm{dist}\big (c(V_j),c(V_{j'})\big ) > rsim R^{2C}N^{2C}$. Here C is a large absolute constant that will be determined later. For a measurable function $g:{\mathbb {R}}^{n+1}\rightarrow {\mathbb {C}}$, we let $G=\sum _{j}g\mathbf{1}_{V_j}$. Consider $\Vert {\mathcal {E}}^*(G\varphi )\Vert _2$:

$$\begin{aligned}&\big \Vert {\mathcal {E}}^*G\varphi \big \Vert _{L^2(B^{n}_{1/4})}^2\le \int _{{\mathbb {R}}^n} \Big |\int _{{\mathbb {R}}^{n+1}} e^{-ix\cdot \xi }e^{-it\Phi (\xi )}(G\varphi )(x,t)dxdt\Big |^2\phi (\xi )d\xi \nonumber \\&\quad = \int _{{\mathbb {R}}^{n+1}}\int _{{\mathbb {R}}^{n+1}} \Big (\int _{{\mathbb {R}}^n} e^{i(y-c)\cdot \xi }e^{i(s-t)\Phi (\xi )}\phi (\xi )d\xi \Big )(G\varphi )(x,t)\overline{(G\varphi )}(y,s)dxdtdyds. \end{aligned}$$

(6.14)

Extract the kernel

$$\begin{aligned} K(x-y,t-s)=\int _{{\mathbb {R}}^n} e^{i(y-x)\cdot \xi }e^{i(s-t)\Phi (\xi )}\phi (\xi )d\xi \end{aligned}$$

(6.15)

and expand $G=\sum _{V_j}g\mathbf{1}_{V_j}$ in the right hand side of (6.14) so that

$$\begin{aligned} \big \Vert {\mathcal {E}}^*(G\varphi )\big \Vert _{L^2(B^{n}_{1/4})}^2\le & {} \int _{{\mathbb {R}}^{2n+2}} K(x-y,t-s)\sum _{j,j'}(g\mathbf{1}_{V_j}\varphi )(x,t)(\overline{g\mathbf{1}_{V_{j'}}\varphi })(y,s)\nonumber \\= & {} \sum _{j,j'}\int _{{\mathbb {R}}^{2n+2}} K(x-y,t-s)(g\mathbf{1}_{V_j}\varphi )(\overline{g\mathbf{1}_{V_{'}}\varphi }). \end{aligned}$$

(6.16)

For $j=j'$, we use (6.13) to have

$$\begin{aligned} \int K(x-y,t-s)(g\mathbf{1}_{V_j}\varphi )(\overline{g\mathbf{1}_{V_{j'}}\varphi })\le \big \Vert {\mathcal {E}}^*(g\mathbf{1}_{V_j}\varphi )\big \Vert _{L^2(B^n_{1/2})}^2\le C_\varepsilon R^\varepsilon \Vert g\mathbf{1}_{V_j}\Vert _{p'}^2. \end{aligned}$$

(6.17)

For $j\not =j'$, we apply Hölder inequality to get

$$\begin{aligned} \int _{{\mathbb {R}}^{2n+2}} K\cdot (g\mathbf{1}_{V_j}\varphi )(\overline{g\mathbf{1}_{V_{j'}}\varphi }) \le \Vert g\mathbf{1}_{V_j}\Vert _{p'}\Big \Vert \int \widetilde{K}(x,t,y,s)(\overline{g\mathbf{1}_{V_j}})(y,s)dyds\Big \Vert _p \end{aligned}$$

(6.18)

where

$$\begin{aligned} \widetilde{K}(x,t,y,s)=\varphi (x,t)\mathbf{1}_{V_j}(x,t)K(x-y,t-s)\overline{\varphi (y,s)}\mathbf{1}_{V_{j'}}(y,s). \end{aligned}$$

(6.19)

Notice that $K(x-y,t-s)={\mathcal {E}}_{\Phi }\phi (x-y,t-s)$ and $|{\mathcal {E}}_{\Phi }\phi (x,t)|\lesssim (1+|x|+|t|)^{-n/2}$. Thus, together with $|\varphi |\lesssim 1$ and generalized Young’s inequality we have

$$\begin{aligned} \Big \Vert \int \widetilde{K}\cdot (\overline{g\mathbf{1}_{V_j}})dyds\Big \Vert _p\lesssim \Vert K\mathbf{1}_{\{V_j-V_{j'}\}}\Vert _{p/2}\Vert (g\mathbf{1}_{V_j})\Vert _{p'}\lesssim (NR)^{-C}\Vert (g\mathbf{1}_{V_j})\Vert _{p'}. \end{aligned}$$

(6.20)

Combining (6.16), (6.17), (6.18) and (6.20), we get

$$\begin{aligned} \big \Vert {\mathcal {E}}^*G\big \Vert _{L^2(B_{1/4}^n)}^2\lesssim & {} C_\varepsilon R^\varepsilon \sum _{V_j}\Vert g\mathbf{1}_{V_j}\Vert _{p'}^2+\sum _{j\not =j'}N^{-C}R^{-C}\Vert g\mathbf{1}_{V_j}\Vert _{p'}\Vert g\mathbf{1}_{V_{j'}}\Vert _{p'}\nonumber \\\lesssim & {} C_\varepsilon R^\varepsilon \Big (\big \Vert \sum _{V_j}g\mathbf{1}_{V_j}\big \Vert _{p'}^{p'}\Big )^{2/p'}\le C_\varepsilon ' R^\varepsilon \Vert G\Vert _{p'}^2. \end{aligned}$$

(6.21)

We say a set $E=\cup _{j=1}^N B^{n+1}(x_j,r)$ is sparse if $\mathrm{dist}(x_j,x_{j'}) > rsim r^{2C}N^{2C}$. Therefore, what we have proved above is the following lemma:

Lemma 6.3

Let $\{V_j\}_{j=1}^N$ be a collection of sparse R-balls in ${\mathbb {R}}^{n+1}$ and $G=\sum _{j}g\mathbf{1}_{V_j}$, then

$$\begin{aligned} \big \Vert {\mathcal {E}}^*G\big \Vert _{L^2(B_{1/4}^n)}\le C_\varepsilon R^\varepsilon \Vert G\Vert _{p'}^2. \end{aligned}$$

(6.22)

We will use (6.22) to prove (6.10). Since $\widehat{\varphi }$ is supported on $B^{n+1}_{1/2}$, without loss of generality, we assume $\widehat{g}$ is supported in $B^{n+1}_{7/8}$. Let $\widehat{\chi }$ is a smooth function supported on $B^n_1$ and $\widehat{\chi }(\xi )=1$ on $B^{n+1}_{7/8}$, then $g=\chi *g=(\chi *\chi )*g$. We also assume g is a Schwartz function so that its Fourier series converges. We expand $\widehat{g}$ on $B^{n+1}_1$ to have

$$\begin{aligned} \widehat{g}(\xi ,\tau )=\widehat{\chi }^2(\xi ,\tau )\widehat{g}(\xi ,\tau )\sim \sum _{(m,n)\in {\mathbb {Z}}^{n+1}}a_{m,n}e^{i\pi m\cdot \xi }e^{i\pi n\tau }\widehat{\chi }(\xi ,\tau ) \end{aligned}$$

(6.23)

with

$$\begin{aligned} a_{m,n}=\int _{{\mathbb {R}}^{n+1}} e^{-i\pi m\cdot \xi }e^{-i\pi n\tau }\widehat{g}(\xi ,\tau )\widehat{\chi }(\xi ,\tau )d\xi d\tau \sim \chi *g(-m,-n). \end{aligned}$$

(6.24)

Next, we assume $\sum _{m,n}|a_{m,n}|^{p_0}=1$, and sort $|a_{m,n}|$ dyadically. Let

$$\begin{aligned} g_k(x,t)=\sum _{(m,n)\in \Lambda _k}a_{m,n}\chi (x+m,t+n) \end{aligned}$$

(6.25)

where $\Lambda _k=\{(m,n):|a_{m,n}|\sim 2^{-k}\}$ so that $|\Lambda _k|\lesssim 2^{kp_0'}$. Since

• (1) $\Vert g_k\Vert _1\le \sum |a_{m,n}|\sim 2^{-k}|\Lambda _k|$,

• (2) $\Vert g_k\Vert _2\sim (\sum |a_{m,n}|^2)^{1/2}\sim 2^{-k}|\Lambda _k|^{1/2}$,

• (3) $\Vert g_k\Vert _\infty \le \sup _{m,n}|a_{m,n}|\lesssim 2^{-k}$,

applying Hölder twice, for $1<p'<2$, we have

$$\begin{aligned} \Vert g_k\Vert _{p'}\sim 2^{-k}|\Lambda _k|^{1/p'}. \end{aligned}$$

(6.26)

We need a covering lemma in [21].

Lemma 6.4

Suppose E is a union of 1-cubes. Then there exist $O(N|E|^{1/N})$ collections of sparse set that cover E, such that the balls in each sparse set have radius at most $O(|E|^{C^N})$.

Now we are in a position to prove (6.10). Since $g=\sum _{k\ge 0}g_k$, by triangle inequality

$$\begin{aligned} \Vert {\mathcal {E}}^*(g\varphi )\Vert _{L^2(B^n_{1/4})}\le \sum _{k\ge 0}\Vert {\mathcal {E}}^*(g_k\varphi )\Vert _{L^2(B^n_{1/4})}. \end{aligned}$$

(6.27)

For each $k\ge 0$, since the function $\chi$ admits fast decay, $g_k$ is essentially supported on a set E, where E is a union of 1-cubes, and $|E|\lesssim |\Lambda _k|\lesssim 2^{kp_0'}$. We apply Lemma 6.4 so that we can obtain a collection of sparse sets $\mathbf{E}=\{E_j\}$ such that $E\subset \cup E_j$ and $|\mathbf{E}|\lesssim N|E|^{1/N}$. By triangle inequality and (6.22),

$$\begin{aligned} \big \Vert {\mathcal {E}}^*(g_k\varphi )\big \Vert _{L^2(B^n_{1/4})}\lesssim \sum _j\big \Vert {\mathcal {E}}^*(2^{-k}\mathbf{1}_{E_j\cap E}\varphi )\big \Vert _{L^2(B^n_{1/4})}\le 2^{-k} C_\varepsilon |E|^{\varepsilon C^N}N|E|^{\frac{1}{N}}|E|^{\frac{1}{p'}}, \end{aligned}$$

which is further bounded by

$$\begin{aligned} 2^{-k}C_\varepsilon \frac{\log (1/\varepsilon )}{C}|E|^{\varepsilon ^{1-\frac{\log C}{C}}+\frac{C}{\log (1/\varepsilon )}+\frac{1}{p'}}, \end{aligned}$$

(6.28)

if we let $N=\log (1/\varepsilon )/C$. Since $|E|\lesssim 2^{kp_0'}$, plug it in we have

$$\begin{aligned} (6.28) \lesssim C_\varepsilon \frac{\log (1/\varepsilon )}{C}2^{k(p_0'\varepsilon ^{1-\frac{\log C}{C}}+\frac{p_0'C}{\log (1/\varepsilon )}+\frac{p_0'}{p'}-1)}. \end{aligned}$$

(6.29)

Thus, if first choose C big enough then $\varepsilon$ small enough and let $p_0$ be in (6.2), we get that for some small positive number $\varepsilon '$,

$$\begin{aligned} (6.28)\lesssim C_{\varepsilon '}2^{-k\varepsilon '}. \end{aligned}$$

(6.30)

Summing up all the k we therefore can conclude

$$\begin{aligned} \Vert {\mathcal {E}}^*(g\varphi )\Vert _{L^2(B^n_{1/4})}\lesssim _{p_0}1=\Big (\sum _{m,n}|a_{m,n}|^{p_0}\Big )^{1/p_0}. \end{aligned}$$

(6.31)

Noticing that (6.31) is also true with $a_{m,n}$ replaced by $a_{m+x,n+t}$ where $(x,t)\in B^{n+1}_1$ and

$$\begin{aligned} a_{m+x,n+t}=\int _{{\mathbb {R}}^{n+1}} e^{-i(m+x)\cdot \xi }e^{-i(n+t)\tau }\widehat{g}(\xi ,\tau )\widehat{\chi }(\xi ,\tau )d\xi d\tau \sim \chi *g(-m-x,-n-t), \end{aligned}$$

we can average over the translated ${\mathbb {Z}}^{n+1}$-lattices so that

$$\begin{aligned}&\Vert {\mathcal {E}}^*(g\varphi )\Vert _{L^2(B^n_{1/4})}=\int _{B^{n+1}_1}\Vert {\mathcal {E}}^*(g\varphi )\Vert _{L^2(B^n_{1/4})}dxdt \nonumber \\&\quad \lesssim \int _{B^{n+1}_1}\Big (\sum _{m,n}|a_{m+x,n+t}|^{p_0}\Big )^{1/p_0'}dxdt, \end{aligned}$$

(6.32)

which is

$$\begin{aligned} \sim \int _{B^{n+1}_1}\Big (\sum _{m,n}|\chi *g(-m-x,-n-t)|^{p_0}\Big )^{1/p_0'}dxdt \end{aligned}$$

(6.33)

and is further bounded by

$$\begin{aligned} \Big (\int _{B^{n+1}_1}\sum _{m,n}|\int _{{\mathbb {R}}^{n+1}}\chi (-x-m-y,-t-n-s)g(y,s)dyds|^{p_0'}dxdt\Big )^{1/p_0'}. \end{aligned}$$

(6.34)

Finally, since $\chi *g=g$ and (6.34) is nothing but $\Vert \chi *g\Vert _{p_0'}$, we get (6.10) and hence finish the proof of Lemma 6.1. $\square$