2.1 A multilinear weighted decoupling estimate

We first recall the multilinear restriction estimate [Reference Bennett, Carbery and Tao1]. Suppose that $U_j\subset \mathbb {S}^{d-1}$ for $j=1,2,\cdots ,d$ , such that for any $v_j\in U_j$ ,

(2.1) $$ \begin{align} |v_1 \wedge v_2 \wedge \cdots \wedge v_d| \geq \nu, \end{align} $$

for some $0<\nu \leq 1$ .

Theorem 2.1 (Multilinear restriction)

For each $1\leq j\leq d$ , let $S_j \subset S$ be a cap such that the set $U_j\subset \mathbb {S}^{n-1}$ of vectors normal to $S_j$ satisfies (2.1). Suppose that the Fourier transform of $f_j$ is supported on the $R^{-1}$ -neighborhood of $S_j$ . Then

$$\begin{align*}\| \prod_{j=1}^d |f_j|^{1/d} \|_{L^{\frac{2d}{d-1}}(B_R)} \lesssim \nu^{-O(1)} R^{\epsilon} R^{-1/2} \prod_{j=1}^d \| f_j \|_{L^2}^{1/d}.\end{align*}$$

Next, we recall the refined decoupling estimate, which is stated in terms of wave packets. We recall the setup. Given $0<\epsilon \ll 1$ , let $\epsilon _1 = \epsilon ^{100}$ . For each $\theta $ , let $\mathbb {T}_\theta $ denote a collection of finitely overlapping tubes T covering $\mathbb {R}^d$ of length $\sim R^{1+\epsilon _1}$ and radius $\sim R^{(1+\epsilon _1)/2}$ with long axis orthogonal to $\theta $ . Let $\mathbb {T}(R) = \cup _\theta \mathbb {T}_\theta $ . Given $T\in \mathbb {T}(R)$ , let $\theta (T)$ denote the $\theta $ for which $T\in \mathbb {T}_\theta $ .

We say that a function $f_T$ is microlocalized to $(T,\theta )$ if $f_T$ and $\widehat {f_T}$ are essentially supported on $2T$ and $2\theta $ , respectively, in the sense that the $L^p$ norms of the restrictions of $f_T$ to $(2T)^c$ in the physical space and to $(2\theta )^c$ in the frequency space are both $O(R^{-100d} \| f_T \|_{L^p})$ . Terms involving $O(R^{-100d})$ can be absorbed into the main term of the estimate, so we will ignore these terms in the following statements and proofs. We recall that a function f whose Fourier transform is supported on the $R^{-1}$ -neighborhood of S admits a wave packet decomposition $f=\sum _\theta f_\theta =\sum _{T\in \mathbb {T}(R)} f_T$ , where $f_T$ is microlocalized to $(T,\theta (T))$ .

We can now state the refined decoupling theorem, [Reference Guth, Iosevich, Ou and Wang16, Theorem 4.2].

Theorem 2.2 (Refined decoupling)

Let $\mathbb {W} \subset \mathbb {T}(R)$ and suppose that $f=\sum _{T\in \mathbb {W}} f_T$ , where $f_T$ is microlocalized to $(T,\theta (T))$ . Let $Y\subset B_R$ be the union of a collection of $R^{1/2}$ -cubes Q, each of which is contained in $3T$ for at most $M\geq 1$ tubes $T\in \mathbb {W}$ . Then for $2\leq p\leq p_d$ ,

$$\begin{align*}\| f \|_{L^p(Y)} \lessapprox M^{\frac{1}{2}-\frac{1}{p}} \left( \sum_{T\in \mathbb{W}} \| f_T \|_{L^p}^p \right)^{1/p}. \end{align*}$$

The following is a multilinear refinement of Theorem 2.2.

Theorem 2.3. For each $j=1,2,\cdots , d$ , let $S_j \subset S$ be a cap such that the set $U_j\subset \mathbb {S}^{n-1}$ of vectors normal to $S_j$ satisfies (2.1). Let $\mathbb {W}_j \subset \mathbb {T}(R)$ such that $2\theta (T)$ is contained in the $O(R^{-1})$ -neighborhood of $S_j$ for every $T\in \mathbb {W}_j$ and let $f_j = \sum _{T\in \mathbb {W}_j} f_T$ , where $f_T$ is microlocalized to $(T,\theta (T))$ . Let $\{Q\}$ be a collection of $R^{1/2}$ -cubes contained in $B_R$ such that each cube in the collection is contained in $3T$ for at most $M_j$ tubes $T\in \mathbb {W}_j$ for each $1\leq j\leq d$ . Let Y be a subset of $\cup _Q Q$ . Then for $2 \leq p\leq p_d$ ,

$$ \begin{align*} \left\| \prod_{j=1}^d |f_j|^{\frac{1}{d}} \right\|_{L^{p}(Y)} \lessapprox \nu^{-O(1)} ( |Y| R^{-d})^{\frac{d+1}{2d} (\frac{1}{p} - \frac{1}{p_d})} \prod_{j=1}^d \left[ M_j^{\frac{1}{2}-\frac{1}{p}} \Big( \sum_{T\in \mathbb{W}_j} \| f_T \|_{L^p}^p \Big)^{\frac{1}{p}} \right]^{\frac{1}{d}}. \end{align*} $$

For the proof, we interpolate Theorem 2.1 and Theorem 2.2. We remark that the use of refined decoupling is optional here as we shall use Theorem 2.3 with $M_j \lesssim R^{\frac {d-1}{2}}$ . We note that Theorem 2.3 for the case $p=\frac {2d}{d-1}$ was essentially proved in [Reference Gan and Wu13]. Our proof is slightly more direct than the one given there in that it does not go through the multilinear Kakeya estimate.

Proof. We prove the result for $2\leq p\leq \frac {2d}{d-1}$ ; the result for the remaining p follows from interpolation with the statement for $p=p_d$ , which is a consequence of Hölder and Theorem 2.2.

We may view $f_T$ as essentially constant on T; this can be made precise by using the frequency localization property of $f_T$ ; cf. Section 3.1. By dyadic pigeonholing, we may reduce the matter to the case where, for each $j=1,\cdots ,d$ , $| f_T | \sim \gamma _j$ on T for some $\gamma _j>0$ for all $T\in \mathbb {W}_j$ . By homogeneity, we may assume that $\gamma _j=1$ . Then for $1\leq q< \infty $ ,

$$\begin{align*}\sum_{T\in \mathbb{W}_j} \| f_T \|_{L^q}^q \sim |T| |\mathbb{W}_j |.\end{align*}$$

By Hölder, we get

(2.2) $$ \begin{align} \left\| \prod_{j=1}^d |f_j|^{\frac{1}{d}} \right\|_{L^{p}(Y)} \lesssim |Y|^{\frac{1}{p}-\frac{d-1}{2d}} \left\| \prod_{j=1}^d |f_j|^{\frac{1}{d}} \right\|_{L^{\frac{2d}{d-1}}(Y)}. \end{align} $$

The multilinear restriction estimate and the $L^2$ -orthogonality imply

(2.3) $$ \begin{align} \left\| \prod_{j=1}^d |f_j|^{\frac{1}{d}} \right\|_{L^{\frac{2d}{d-1}}(B_R)} \lessapprox R^{-\frac{1}{2}} \prod_{j=1}^d \| f_j \|_{L^2}^{\frac{1}{d}} \lessapprox R^{-\frac{1}{2}} \prod_{j=1}^d (|T||\mathbb{W}_j| )^{\frac{1}{2d}}. \end{align} $$

However, by Hölder and the refined decoupling estimate, Theorem 2.2, we have

(2.4) $$ \begin{align} \left\| \prod_{j=1}^d |f_j|^{\frac{1}{d}} \right\|_{L^{\frac{2d}{d-1}}(Y)} &\lesssim |Y|^{\frac{d-1}{2d}-\frac{1}{p_d}} \left\| \prod_{j=1}^d |f_j|^{\frac{1}{d}} \right\|_{L^{p_d}(Y)} \nonumber\\ & \lessapprox |Y|^{\frac{d-1}{2d}-\frac{1}{p_d}} \prod_{j=1}^d \left[ M_j^{\frac{1}{2}-\frac{1}{p_d}} \left( \sum_{T\in \mathbb{W}_j} \| f_T \|_{L^{p_d}}^{p_d} \right)^{\frac{1}{p_d}} \right]^{\frac{1}{d}} \\ & \nonumber\lessapprox |Y|^{\frac{d-1}{2d(d+1)}} \prod_{j=1}^d \left[ M_j^{\frac{1}{d+1}} (|T| |\mathbb{W}_j|)^{\frac{1}{p_d}} \right]^{\frac{1}{d}}. \end{align} $$

We combine the $(d+1)(\frac {1}{p}-\frac {1}{p_d})$ -th power of (2.3) and $(d+1)(\frac {1}{2}-\frac {1}{p})$ -th power of (2.4). Then we get

$$\begin{align*}\left\| \prod_{j=1}^d |f_j|^{\frac{1}{d}} \right\|_{L^{\frac{2d}{d-1}}(Y)} \lessapprox R^{-\frac{d+1}{2} (\frac{1}{p}-\frac{1}{p_d})} |Y|^{\frac{d-1}{2d}(\frac{1}{2}-\frac{1}{p})} \prod_{j=1}^d \left[ M_j^{\frac{1}{2}-\frac{1}{p}} \left( \sum_{T\in \mathbb{W}_j} \| f_T \|_{L^p}^{p} \right)^{\frac{1}{p}} \right]^{\frac{1}{d}}. \end{align*}$$

Finally, we combine this estimate with (2.2).

2.2 A weighted refined decoupling estimate

Let $v(T) \in \mathbb {S}^{d-1}$ denote the direction of $T\in \mathbb {T}$ . Following [Reference Du, Ou, Ren and Zhang9], for $R^{-1/2}\leq r\leq 1$ and $1\leq m\leq d$ , we say that $f = \sum _{T\in \mathbb {W}} f_T$ has $(r,m)$ -concentrated frequencies if there exists a m-dimensional subspace $V\subset \mathbb {R}^d$ such that $\angle (v(T), V) \leq r$ for any $T \in \mathbb {W}$ .

When f has $(R^{-1/2},m)$ -concentrated frequencies, one can decouple f by $\ell ^2$ -decoupling in dimension m (cf. [Reference Guth15, Lemma 9.3]). In the paper [Reference Du, Ou, Ren and Zhang9], it was observed that this continues to hold for the refined decoupling estimate as well.

Theorem 2.4 ([Reference Du, Ou, Ren and Zhang9, Theorem 1.2(a)]Footnote ¹ )

Let $\mathbb {W} \subset \mathbb {T}(R)$ and suppose that $f=\sum _{T\in \mathbb {W}} f_T$ , where $f_T$ is microlocalized to $(T,\theta (T))$ . Let $Y\subset B_R$ be the union of a collection of $R^{1/2}$ -cubes Q, each of which is contained in $3T$ for at most $M\geq 1$ tubes $T\in \mathbb {W}$ . Suppose that f has $(R^{-1/2},m)$ concentrated frequencies for some $1\leq m\leq d$ . Then for $2\leq p\leq p_m$ ,

$$\begin{align*}\| f \|_{L^p(Y)} \lessapprox M^{\frac{1}{2}-\frac{1}{p}} \left( \sum_{T\in \mathbb{W}} \| f_T \|_{L^p}^p \right)^{1/p}. \end{align*}$$

A consequence of Theorem 2.4 is a weighted refined decoupling estimate with weights $w: Y\to [0,1]$ such that $\int _{Q} w \lesssim R^{\alpha /2}$ holds for any $R^{1/2}$ -cube Q in Y; see [Reference Du, Ou, Ren and Zhang9, Theorem 1.2(b)]). Here we state a slightly more general version, which does not require such an assumption.

Theorem 2.5 (Weighted refined decoupling)

Let $1\leq m\leq d$ and $2\leq p\leq p_m$ . Suppose that ${f= \sum _{T\in \mathbb {W}} f_T}$ , where $\mathbb {W} \subset \mathbb {T}(R)$ and each $f_T$ is microlocalized to $(T,\theta (T))$ . Assume that f has $(R^{-1/2},m)$ -concentrated frequencies. Let $\{ Q\}$ be a collection of disjoint $R^{1/2}$ -cubes in $B_R$ such that each Q is contained in $3T$ for at most $M\geq 1$ tubes $T\in \mathbb {W}$ . Let Y be a subset of $\cup _Q Q$ . Let $w: Y \to [0,1]$ and $w(T) := \int _T w$ . Then

$$\begin{align*}\| f\|_{L^p(w)} \lessapprox \left( \max_{T\in \mathbb{W}} \frac{w(3T)}{|T|} \right)^{\frac{1}{p}-\frac{1}{p_m}} M^{\frac{1}{2}-\frac{1}{p}} \left( \sum_{T\in \mathbb{W}} \| f_T\|_{L^p}^p \right)^{1/p}.\end{align*}$$

The power of $\max _{T\in \mathbb {W}} \frac {w(3T)}{|T|}$ represents a gain over the refined decoupling estimate. Theorem 2.5 essentially follows from the proof of [Reference Du, Ou, Ren and Zhang9, Theorem 1.2(b)]). We include a sketch of the proof for the sake of completeness.

Proof. By dyadic pigeonholing, we may assume that each $R^{1/2}$ -cube Q in the collection is contained in $3T$ for $\sim M$ tubes $T\in \mathbb {W}$ . By Hölder,

$$\begin{align*}\| f\|_{L^p(w)} \lesssim w(Y)^{\frac{1}{p}-\frac{1}{p_m}} \| f\|_{L^{p_m}(Y)}.\end{align*}$$

Next, we use Theorem 2.4 with exponent $p=p_m$ and then replace the $\ell ^{p_m}(L^{p_m})$ norm back to $\ell ^{p}(L^{p})$ as in the proof of Theorem 2.3, which gives

(2.5) $$ \begin{align} \| f\|_{L^p(w)} \lessapprox \left(\frac{w(Y) M}{|\mathbb{W}| |T| }\right)^{\frac{1}{p}-\frac{1}{p_m}} M^{\frac{1}{2}-\frac{1}{p}} \left( \sum_{T\in \mathbb{W}} \| f_T\|_{L^p}^p \right)^{1/p}. \end{align} $$

Then

$$\begin{align*}w(Y) M \sim \sum_{Q} \sum_{T\in \mathbb{W}: Q \subset 3T} w(Q) = \sum_{T\in \mathbb{W}} \sum_{Q:Q \subset 3T} w(Q) \leq |\mathbb{W}| \max_{T\in \mathbb{W}} w(3T). \end{align*}$$

Combining this inequality and (2.5) finishes the proof.

3.1 Broad case

In this subsection, we denote by $A \lessapprox B$ expressions of the form $A\leq K^{O(1)} B$ ; the loss of $K^{O(1)}$ is harmless as long as it is at most $R^{\epsilon /2}$ .

In the broad case, we have $\| f\|_{L^p(Y)} \lesssim \| f\|_{L^p(Y_{\operatorname {broad}})}$ . Let $B\in \mathcal {B}_{\operatorname {broad}}$ . Then there exist $\tau _1,\tau _2,\cdots ,\tau _d \in S(B)$ satisfying (3.2). We fix such a d-tuple and denote $\bar {\tau }(B) = (\tau _1,\cdots ,\tau _d)$ . Let

$$\begin{align*}\Gamma = \{ \bar{\tau}(B) : B \in \mathcal{B}_{\operatorname{broad}} \}. \end{align*}$$

Since $\# \Gamma \lesssim K^{O(1)}$ , by dyadic pigeonholing, there exist $\bar {\tau }=(\tau _1,\cdots ,\tau _d) \in \Gamma $ and a sub-collection of broad $K^2$ -cubes $\mathcal {B}_{\operatorname {broad}}^1$ such that $\bar {\tau }(B) = \bar {\tau }$ for all $B\in \mathcal {B}_{\operatorname {broad}}^1$ and

$$\begin{align*}\| f\|_{L^p(Y_{\operatorname{broad}})}\lessapprox \| f\|_{L^p(Y_{\operatorname{broad}}^1)}, \text{ where } \; Y_{\operatorname{broad}}^1 = \cup_{B\in \mathcal{B}_{\operatorname{broad}}^1} B\cap Y. \end{align*}$$

Let $B\in \mathcal {B}_{\operatorname {broad}}^1$ . Since $\tau _j \in S(B)$ for each j, we have

$$ \begin{align*} \| f \|_{L^p(B\cap Y)}^p \lessapprox \prod_{j=1}^d \| f_{\tau_j} \|_{L^p(B\cap Y)}^{\frac{p}{d}}. \end{align*} $$

Using the Fourier support property of $f_j$ , we morally have the following reverse Hölder inequality (cf. [Reference Du and Zhang10]):

(3.3) $$ \begin{align} \prod_{j=1}^d \| f_{\tau_j} \|_{L^p(B\cap Y)}^{\frac{p}{d}} \lessapprox \| \prod_{j=1}^d |f_{\tau_j}|^{1/d} \|_{L^p(B\cap Y)}^{p}. \end{align} $$

We shall first assume (3.3), and then make it precise later. Summing (3.3) over $B\in \mathcal {B}_{\operatorname {broad}}^1$ , we get

$$\begin{align*}\| f\|_{L^p(Y_{\operatorname{broad}})}\lessapprox \| \prod_{j=1}^d |f_{\tau_j}|^{1/d} \|_{L^p(Y_{\operatorname{broad}}^1)}. \end{align*}$$

By using Theorem 2.3 with the bound $M_j\lesssim R^{\frac {d-1}{2}}$ and $\mathbb {W}_{\tau _j} \subset \mathbb {W}$ , we obtain

(3.4) $$ \begin{align} \| f\|_{L^p(Y_{\operatorname{broad}})} &\lessapprox (|Y| R^{-d})^{\frac{d+1}{2d}(\frac{1}{p}-\frac{1}{p_d})} \prod_{j=1}^d \left[ M_j^{\frac{1}{2}-\frac{1}{p}} \left( \sum_{T\in \mathbb{W}_{\tau_j}} \| f_T \|_{L^p}^p \right)^{1/p} \right]^{1/d} \nonumber\\ &\lessapprox (|Y| R^{-d})^{\frac{d+1}{2d}(\frac{1}{p}-\frac{1}{p_d})} R^{\frac{d-1}{2} (\frac{1}{2}-\frac{1}{p})} \left( \sum_{T\in \mathbb{W}} \| f_T \|_{L^p}^p \right)^{1/p}, \end{align} $$

which implies the claimed estimate (3.1) as $\alpha (p) \leq \frac {d+1}{2d}(\frac {1}{p}-\frac {1}{p_d})$ .

Now we make the step (3.3) rigorous following [Reference Hickman17]. Since $f_{\tau _j}$ has compact Fourier support, we may write $f_{\tau _j} = f_{\tau _j}* \psi $ for some $\psi $ with compact Fourier support. Therefore, we have $|f_{\tau _j}| \lesssim |f_{\tau _j}|* \Psi _{1}$ , where $\Psi _{\rho }(x) = \rho (\rho +|x|)^{-(d+1)}$ . Moreover,

(3.5) $$ \begin{align} |f_{\tau_j}|^{r} \lesssim_{r} |f_{\tau_j}|^r *\Psi_{1} \end{align} $$

for all $r>0$ . This follows from Hölder for $r>1$ . The case $0<r<1$ can be proved by using Bernstein’s inequality; see [Reference Hickman17, Lemma 5.9]. We also note that $\Psi _{1}(x) = K^{2d}\Psi _{K^{2}}(K^2 x) \lessapprox \Psi _{K^2}(x)$ . Using (3.5) with $r=p/d$ , we get

(3.6) $$ \begin{align} |f_{\tau_j}|^{\frac{p}{d}} \lessapprox |f_{\tau_j}|^{\frac{p}{d}}* \Psi_{K^2}. \end{align} $$

We are going to use the following two properties of $\Psi _{K^2}$ ; its $L^1$ norm is comparable to $1$ , and it is locally constant on balls of radius $K^2$ . The latter implies that $|f_{\tau _j}|^{\frac {p}{d}}* \Psi _{K^2}(x) \sim |f_{\tau _j}|^{\frac {p}{d}}* \Psi _{K^2}(y)$ whenever $|x-y|\lesssim K^2$ . Therefore, by (3.6),

$$ \begin{align*} \prod_{j=1}^d \left( \int_{B\cap Y} |f_{\tau_j}|^{p}(x) dx\right)^{1/d} &\lessapprox \prod_{j=1}^d \left( \int_{B\cap Y} \big( |f_{\tau_j}|^{\frac{p}{d}} * \Psi_{K^2}(x) \big)^d dx\right)^{1/d} \\ &\sim |B\cap Y| \prod_{j=1}^d |f_{\tau_j}|^{\frac{p}{d}} * \Psi_{K^2}(y), \text{ for every } y\in B \\ &\sim \int_{B\cap Y} \prod_{j=1}^d |f_{\tau_j}|^{\frac{p}{d}} * \Psi_{K^2}(x) dx. \end{align*} $$

Let $f_{\tau _j,y_j} (x)= f_{\tau _j}(x-y_j)$ . Summing this estimate over $B\in \mathcal {B}_{\operatorname {broad}}^1$ , we get

$$ \begin{align*} \| f\|_{L^p(Y_{\operatorname{broad}})}^p &\lessapprox \int_{Y} \prod_{j=1}^d |f_{\tau_j}|^{\frac{p}{d}} * \Psi_{K^2}(x) dx\\ &= \int_{(\mathbb{R}^d)^d} \| \prod_{j=1}^d |f_{\tau_j,y_j}|^{1/d} \|_{L^p(Y)}^{p} \prod_{j=1}^d \Psi_{K^2}(y_j) dy_1 \cdots dy_d. \end{align*} $$

By Theorem 2.3, we obtain (3.4).

3.2 Narrow case

In the narrow case, we use induction on the scale R. In this subsection, we denote by $A \lessapprox B$ expressions of the form $A\leq (\log R )^{O(1)} B$ and keep track of powers of K.

We start by considering the base case $R\sim 1$ . Since $|\mathbb {W}|\sim 1$ , we have

$$\begin{align*}\| f\|_{L^p(Y)}\lesssim |Y|^{1/p} \max_{T\in \mathbb{W}} \|f_T\|_{L^\infty} \lesssim |Y|^{1/p} \max_{T\in \mathbb{W}} \|f_T\|_{L^p}, \end{align*}$$

which is better than the claimed estimate since $\alpha (p) \leq 1/p$ and $|Y|\lesssim 1$ .

Let $B\in \mathcal {B}_{\operatorname {narrow}}$ . Recall that $\| f\|_{L^p(B\cap Y)} \sim \| \sum _{\tau \in S(B)} f_\tau \|_{L^p(B\cap Y)}$ . For each $\tau $ , we use a wave packet decomposition

$$\begin{align*}f_\tau = \sum_{T_1 \in \mathbb{T}_{\tau}} f_{T_1}, \end{align*}$$

where $f_{T_1}$ is microlocalized to $(T_1,\tau )$ . Note that $\mathbb {T}_{\tau } \subset \mathbb {T}(K^2)$ and each $T_1\in \mathbb {T}_{\tau }$ has length $\sim K^{2(1+\epsilon _1)}$ and radius $\sim K^{(1+\epsilon _1)}$ . For dyadic $R^{-1000d} \leq \eta \leq |T_1|$ , let $\mathbb {T}_{\tau ,\eta }$ denote the collection of $T_1\in \mathbb {T}_{\tau }$ such that $|3T_1 \cap Y| \sim \eta $ . The contribution of tubes $T_1 \in \mathbb {T}_\tau $ such that $|3T_1 \cap Y| \lesssim R^{-1000d}$ is negligible; it can be absorbed into the main term by crude estimates. By dyadic pigeonholing, there exists dyadic $\eta $ such that

$$\begin{align*}\sum_{B\in \mathcal{B}_{\operatorname{narrow}}} \| f\|_{L^p(B\cap Y)}^p \lessapprox \sum_{B\in \mathcal{B}_{\operatorname{narrow}}}\| \sum_{\tau\in S(B)} \sum_{T_1 \in \mathbb{T}_{\tau,\eta}} f_{T_1} \|_{L^p(B\cap Y)}^p. \end{align*}$$

We fix this $\eta $ and let $\mathbb {T}(K^2;B)$ denote the collection of $T_1\in \cup _{\tau \in S(B)} \mathbb {T}_{\tau ,\eta }$ such that $2T_1$ intersects B. Then

$$\begin{align*}\| f\|_{L^p(Y)}^p \lessapprox \sum_{B\in \mathcal{B}_{\operatorname{narrow}}}\| \sum_{T_1 \in \mathbb{T}(K^2;B)} f_{T_1} \|_{L^p(B\cap Y)}^p. \end{align*}$$

For each $B\in \mathcal {B}_{\operatorname {narrow}}$ , $\sum _{T_1 \in \mathbb {T}(K^2;B)} f_{T_1}$ has $(K^{-1},d-1)$ -concentrated frequencies and $|S(B)| \lesssim K^{d-2}$ . Therefore, by the weighted refined decoupling Theorem 2.5,

(3.7) $$ \begin{align} \| \sum_{T_1 \in \mathbb{T}(K^2;B)} f_{T_1}\|_{L^p(B\cap Y)} \lesssim &K^{\epsilon_1} (\eta K^{-(d+1)})^{\frac{1}{p}-\frac{1}{p_{d-1}}} K^{(d-2)(\frac{1}{2}-\frac{1}{p})} \left( \sum_{T_1 \in \mathbb{T}(K^2;B)} \| f_{T_1} \|_{L^p}^p \right)^{\frac{1}{p}}. \end{align} $$

For each $\tau $ , we cover $B_R$ by parallelepiped $\square $ of dimensions $K^{-1}R \times \cdots \times K^{-1}R \times R$ with the long axis perpendicular to $\tau $ . We denote the collection of $\square $ by $\mathbb {B}_\tau $ and $\mathbb {B} = \cup _\tau \mathbb {B}_\tau $ . In addition, given $\square \in \mathbb {B}$ , we denote by $\tau (\square )$ the $\tau $ for which $\square \in \mathbb {B}_\tau $ .

Let $\mathbb {T}_\square $ denote the collection of all $T_1 \in \cup _{B\in \mathcal {B}_{\operatorname {narrow}}} \mathbb {T}(K^2;B)$ such that $2T_1$ intersects $\square $ and ${\tau (T_1) = \tau (\square )}$ . Let $Y_\square =\cup _{T_1 \in \mathbb {T}_{\square } } 2T_1$ . Let $\mathbb {W}_\square $ denote the collection of $T\in \mathbb {W}_{\tau (\square )}$ such that $2T$ intersects $Y_\square $ and define $f_\square = \sum _{T\in \mathbb {W}_\square } f_T$ . We record here that

$$\begin{align*}\sum_{\square} \sum_{T\in \mathbb{W}_\square} \| f_T \|_{L^p}^p \lesssim \sum_{T\in \mathbb{W}} \| f_T \|_{L^p}^p.\end{align*}$$

Moreover, $f_\square $ is microlocalized to $(\square ,\tau (\square ))$ , and we have

$$ \begin{align*} \sum_{B\in \mathcal{B}_{\operatorname{narrow}}} \sum_{T_1 \in \mathbb{T}(K^2;B)} \| f_{T_1} \|_{L^p}^p &\lesssim \sum_{\square} \sum_{T_1\in \mathbb{T}_\square} \| f_{T_1} \|_{L^p}^p \lesssim \sum_{\square} \| f_{\tau(\square)} \|_{L^p(Y_\square)}^p \lesssim \sum_{\square} \| f_{\square} \|_{L^p(Y_\square)}^p. \end{align*} $$

Summing the p-th power of (3.7) over narrow $B\in \mathcal {B}_{\operatorname {narrow}}$ , we get

(3.8) $$ \begin{align} \| f\|_{L^p(Y)} & \lessapprox K^{\epsilon_1} (\eta K^{-(d+1)})^{\frac{1}{p}-\frac{1}{p_{d-1}}} K^{(d-2)(\frac{1}{2}-\frac{1}{p})} \left( \sum_{\square} \| f_{\square} \|_{L^p(Y_\square)}^p \right)^{1/p}. \end{align} $$

Next, we apply the induction hypothesis to $f_{\square }$ after a parabolic rescaling. Without loss of generality, suppose that $\tau (\square )$ is contained in $B_{K^{-1}}^{d-1}(0) \times B^1_{K^{-2}}(0)$ . Let $L(x_1,\cdots ,x_d) = (Kx_1,\cdots ,Kx_{d-1},K^2 x_d)$ . Then we may apply the induction hypothesis to $f_{\square }\circ L$ over $L^{-1}(Y_\square ) \subset L^{-1}(\square )$ at the scale $R_1= R/K^2$ :

(3.9) $$ \begin{align} \| f_{\square} \|_{L^p(Y_\square)} \lesssim R_1^{\epsilon/2} (|L^{-1}(Y_\square)| R_1^{-d})^{\alpha(p)} (R_1)^{\frac{d-1}{2} (\frac{1}{2}-\frac{1}{p})} \left( \sum_{T\in \mathbb{W}_\square} \| f_T \|_{L^p}^p \right)^{1/p}. \end{align} $$

For every $\square $ ,

$$\begin{align*}|\mathbb{T}_\square| \eta \lesssim \sum_{T_1\in \mathbb{T}_\square} |3T_1\cap Y| \lesssim |Y|.\end{align*}$$

Thus, we have

$$\begin{align*}|L^{-1}(Y_\square)| = K^{-(d+1)} |Y_\square| \sim K^{-(d+1)} |T_1| |\mathbb{T}_\square| \lesssim K^{O(\epsilon_1)} \eta^{-1} |Y|. \end{align*}$$

By combining this estimate, (3.8) and (3.9), we obtain

$$ \begin{align*} \| f\|_{L^p(Y)} \lessapprox &R^{\epsilon/2} K^{O(\epsilon_1)} (\eta K^{-(d+1)})^{\frac{1}{p}-\frac{1}{p_{d-1}}} K^{(d-2)(\frac{1}{2}-\frac{1}{p})} (\eta^{-1} K^{2d})^{\alpha(p)} K^{-(d-1) (\frac{1}{2}-\frac{1}{p})} \\ &(|Y|R^{-d})^{\alpha(p)} R^{\frac{d-1}{2} (\frac{1}{2}-\frac{1}{p})} \left( \sum_{T\in \mathbb{W}} \| f_T \|_{L^p}^p \right)^{1/p} \\ \lessapprox &R^{\epsilon/2} K^{O(\epsilon_1)} (\eta K^{-(d+1)})^{\frac{1}{p}-\frac{1}{p_{d-1}} - \alpha(p) } K^{(d-1)\alpha(p) - (\frac{1}{2}-\frac{1}{p})} \\ &(|Y|R^{-d})^{\alpha(p)} R^{\frac{d-1}{2} (\frac{1}{2}-\frac{1}{p})} \left( \sum_{T\in \mathbb{W}} \| f_T \|_{L^p}^p \right)^{1/p}. \end{align*} $$

Note that

$$\begin{align*}(\eta K^{-(d+1)})^{\frac{1}{p}-\frac{1}{p_{d-1}} - \alpha(p) } =O(K^{O(\epsilon_1)}) \end{align*}$$

since $\eta \lesssim |T_1|$ and $\frac {1}{p}-\frac {1}{p_{d-1}} - \alpha (p) \geq 0$ . Moreover, $(d-1)\alpha (p) - (\frac {1}{2}-\frac {1}{p}) \leq 0$ by the choice of $\alpha (p)$ . Therefore, we have

$$\begin{align*}\| f\|_{L^p(Y)} \lesssim R^\epsilon (|Y|R^{-d})^{\alpha(p)} R^{\frac{d-1}{2} (\frac{1}{2}-\frac{1}{p})} \left( \sum_{T\in \mathbb{W}} \| f_T \|_{L^p}^p \right)^{1/p}, \end{align*}$$

and the induction closes.

4.1 Linearization

In this subsection, we reduce Theorem 1.1 to the following.

Proposition 4.2. Fix a collection of disjoint intervals $\{I_1, \cdots , I_{2^j} \}$ of length $\sim 2^{-j}$ for which $ {\cup _{l} I_l =[1/2,1]}$ . Let $c_l \in I_l$ and let $\{ F_1, \cdots , F_{2^j}\}$ be a partition of $B(0,C2^j)$ . For a type j multiplier $m_j$ , let T denote the linear operator defined as

$$\begin{align*}T f(x) = \sum_{l=1}^{2^j} 1_{F_l}(x) m_j(D/c_l) f(x). \end{align*}$$

Then

$$\begin{align*}\| T f\|_{L^{\frac{86}{57}}} \lessapprox 2^{ \frac{9}{86} j} \| f\|_{L^{\frac{86}{57}}}, \end{align*}$$

where the implicit constant is independent of the choice of $\{c_l, F_l\}$ .

We postpone the proof of Proposition 4.2 to following subsections and first look at its implications. Clearly, T in Proposition 4.2 is dominated by the maximal function $f \mapsto \sup _{t\sim 1} |m_j(D/t)f|$ . Conversely, we have the following:

Lemma 4.3. Let T be as in Proposition 4.2. Assume that for any type j multiplier $m_j$ , there exists a constant $A_j>0$ independent of $\{F_l, c_l\}_{1\leq l\leq 2^j}$ such that

$$\begin{align*}\| T f\|_{L^p} \leq A_j \| f\|_{L^p}. \end{align*}$$

Then for every type j multiplier $m_j$ ,

$$\begin{align*}\| \sup_{t\sim 1} |m_j(D/t) f|\|_{L^p(B(0,C2^j))} \lesssim A_j \| f\|_{L^p}. \end{align*}$$

Proof. Let $m_j$ be a type j multiplier. We linearlize and discretize the maximal operator. Fix a measurable function $t:B(0,CR) \to [1/2,1]$ . For each $l=1,\cdots ,2^j$ , define $F_l = \{ x\in B(0,CR) : t(x) \in I_l \}$ so that

$$\begin{align*}|m_j(D/t(x))f(x)|^p \leq \sum_{l} 1_{F_l}(x) \sup_{t\in I_l}| m_j(D/t)f(x)|^p.\end{align*}$$

Let $a_l$ denote the left endpoint of the interval $I_l$ . By the fundamental theorem of calculus and Hölder, we have

$$\begin{align*}\sup_{t\in I_l} |m_j(D/t)f(x)|^p \lesssim |m_j(D/a_l)f(x)|^p + |I_l|^{p-1} \int_{I_l} | \partial_t [m_j(D/t)f(x)]|^p dt.\end{align*}$$

For each $1\leq k\leq d$ , let $m_{j,k} (\xi ) = |I_l| \xi _k \partial _{k} m_{j}(\xi )$ . Then $m_{j,k}$ is a type j multiplier and

$$\begin{align*}\sup_{t\in I_l} |m_j(D/t)f(x)|^p \lesssim |m_j(D/a_l)f(x)|^p + |I_l|^{-1} \int_{I_l}\sum_{1\leq k\leq d} | m_{j,k}(D/t)f(x)|^p dt.\end{align*}$$

Choose $b_l\in I_l$ for which

$$\begin{align*}\int_{F_l} \sum_{1\leq k\leq d} | m_{j,k}(D/b_l)f|^p \gtrsim \sup_{t\in I_l} \int_{F_l} \sum_{1\leq k\leq d} | m_{j,k}(D/t)f|^p. \end{align*}$$

Then

$$\begin{align*}\int_{F_l}\sup_{t\in I_l} |m_j(D/t)f|^p \lesssim \int_{F_l} |m_j(D/a_l)f|^p + \int_{F_l} \sum_{1\leq k\leq d} | m_{j,k}(D/b_l)f|^p .\end{align*}$$

By summing the inequality over l and using the assumption, we get

$$\begin{align*}\int_{B(0,C2^j)} |m_j(D/t(x)) f|^p \lesssim A_j^p \| f\|_{L^p}^p. \end{align*}$$

Since this holds for any measurable function t, the claim is proved.

By Lemma 4.3 and Tao’s reduction discussed earlier, we have reduced Theorem 1.1 to Proposition 4.2, which will be proved in the following subsections.

4.2 Decompositions

In this subsection, we work on $\mathbb {R}^d$ for $d\geq 2$ . We fix $\{ c_l,F_l\}_{l=1}^{2^j}$ as in Proposition 4.2 and let

$$\begin{align*}T f(x) = \sum_{l} 1_{F_l}(x) m_j(D/c_l) f(x). \end{align*}$$

Given any exponent $1\leq p_0\leq \infty $ , by dyadic pigeonholing, there exist $\mathcal {J} \subset \{1,2,\cdots , 2^j\}$ and $2^{-100d j} \leq \gamma \leq 2^{jd}$ for which $|F_l|\sim \gamma $ for all $l\in \mathcal {J}$ Footnote ² and

(4.3) $$ \begin{align} \| T f\|_{L^{p_0}}\lessapprox \| T_{\mathcal{J}} f\|_{L^{p_0}}, \end{align} $$

where

$$\begin{align*}T_{\mathcal{J}} f(x) = \sum_{l\in \mathcal{J}} 1_{F_l}(x) m_j(D/c_l) f(x). \end{align*}$$

By real interpolation, it suffices to work with $f=1_E$ , the characteristic function of $E\subset B(0,2^j)$ . Given $0<\epsilon \ll 1$ , let $0<\epsilon _1 \ll \epsilon $ . For a technical reason, we work with the parameter $R=2^{(1-\epsilon _1)j}$ , but one may identify R and $2^j$ as $R\approx 2^j$ . The main result of this subsection is the following.

Proposition 4.4 (cf. [Reference Gan and Wu13])

Let $E\subset B(0,2^j)$ and $2\leq p \leq p_d$ . Let $R=2^{(1-\epsilon _1)j}$ . Suppose that (3.1) holds with the exponent $\alpha (p)$ . Then

(4.4) $$ \begin{align} |\{x: |T_{\mathcal{J}}1_{E}(x)|> \alpha\}| \lessapprox \alpha^{-2} R^{\frac{1}{2p-3} (\frac{d+1}{4}p - \frac{d-1}{2}) -1} (\gamma R^{-d})^{\frac{p}{2p-3}\alpha(p)} |E|^{\frac{3p-5}{2p-3}}. \end{align} $$

Proposition 4.4 follows from the argument from [Reference Gan and Wu13]. We sketch the proof for the sake of exposition. We decompose E according to the density. For a parameter $\beta _2>0$ to be chosen, let $\{ Q\}$ denote the collection of maximal dyadic cubes such that

$$\begin{align*}|E\cap Q| \geq \beta_2 l(Q), \end{align*}$$

where $l(Q)$ is the side-length of Q. Let $E_2= \cup _Q (E\cap Q)$ and $E_1 = E \setminus E_2$ . Then

$$\begin{align*}|E_1 \cap B| \leq \beta_2 l(B)\end{align*}$$

for any dyadic cube B; if otherwise, there is a dyadic cube Q in the collection containing B leading to the contradiction $|E_1 \cap B|=0$ . These sets $E_1$ and $E_2$ are the low and the high density parts of E, respectively. Note that $T_{\mathcal {J}}1_E = T_{\mathcal {J}} 1_{E_1} + T_{\mathcal {J}} 1_{E_2}$ .

For the high-density part $E_2$ , we have the following estimate from a local $L^2$ -estimate, which goes back to [Reference Tao23].

Lemma 4.6 [Reference Gan and Wu13, Lemma 3.4]

For any $\alpha>0$ ,

$$\begin{align*}| \{ x : |T_{\mathcal{J}} 1_{E_2}(x)| \geq \alpha \} \lesssim \alpha^{-2} 2^{-j} \beta_2^{-1} |E|^2.\end{align*}$$

For the low-density part $E_1$ , we use a wave-packet decomposition. Let $R=2^{j(1-\epsilon _1)}$ , so that $m_j$ is essentially supported on the $R^{-1}$ -neighborhood of $\mathbb {S}^{d-1}$ . Cover $\mathbb {S}^{d-1}$ by finitely overlapping caps $\{ \theta \}$ of diameter $R^{-1/2}$ and let $\{ \chi _{\theta } \}$ be an associated smooth partition of unity. Define $m_{j,\theta }(\xi ) = m_j(\xi ) \chi _{\theta }(\xi /|\xi |)$ so that $m_{j,\theta }(\xi /c_l)$ is essentially a bump function supported on a box $\theta _l$ of dimensions $R^{-1/2}\times \cdots R^{-1/2}\times R^{-1}$ .

Let $f_{\theta _l}$ denote $m_{j,\theta }(D/c_l) f$ . Let $\mathbb {T}_\theta $ denote the collection of tubes as in Section 2. Then $f_{\theta _l}$ is morally constant on each $T\in \mathbb {T}_{\theta }$ . Indeed, we may choose an $L^1$ -normalized non-negative function $\Psi _\theta $ such that $\Psi _\theta (x) \sim \Psi _\theta (y)$ for all $x,y\in T$ and

$$\begin{align*}|f_{\theta_l} (x)| \lesssim 2^{O(\epsilon_1 j)} |f_{\theta_l}|* \Psi_\theta (x) \end{align*}$$

up to a negligible error term (cf. Section 3.1). Therefore, $ |f_{\theta _l}|* \Psi _\theta (x) \sim |f_{\theta _l}|* \Psi _\theta (y) $ for all $x,y \in T$ . Let $f=\chi _{E_1}$ . Given $\beta _1>0$ , we let

$$\begin{align*}\mathbb{T}_{\theta_l,small} =\{ T \in \mathbb{T}_{\theta} : \| |f_{\theta_l}|* \Psi_\theta \|_{L^\infty(T)} < \beta_1 \} \end{align*}$$

and $\mathbb {T}_{\theta _l,large} = \mathbb {T}_{\theta } \setminus \mathbb {T}_{\theta _l,small}$ . Let $\{ \chi _{T} \}_{T\in \mathbb {T}_\theta }$ denote a smooth partition of unity associated with the covering $\mathbb {T}_\theta $ . Note that $f_{\theta _l} \chi _{T}$ is microlocalized to $(T, \theta _l)$ for each $T\in \mathbb {T}_\theta $ . We define

$$\begin{align*}T_{small}(f) = \sum_{l\in \mathcal{J}} 1_{F_l} \sum_{\theta} \sum_{T\in \mathbb{T}_{\theta_l,small}} f_{\theta_l} \chi_{T}. \end{align*}$$

We define $T_{large}(f)$ similarly, which gives the decomposition $T_{\mathcal {J}} f = T_{small}(f)+T_{large}(f)$ .

Lemma 4.7 [Reference Gan and Wu13, Lemmas 3.2 and 3.3]

Suppose that the weighted $\ell ^p$ -decoupling estimate (3.1) holds with the exponent $\alpha (p)$ . Then

(4.5) $$ \begin{align} \int |T_{large} (1_{E_1})| &\lessapprox \beta_1^{-1} \beta_2 R^{-1/2} |E_1|, \nonumber \\ \int |T_{small} (1_{E_1})|^p dx &\lessapprox \left[(\gamma R^{-d})^{\alpha(p)} R^{\frac{d-1}{2}(\frac{1}{2}-\frac{1}{p})} \right]^p \beta_1^{p-2} |E_1|. \end{align} $$

The assumption on the weighted $\ell ^p$ -decoupling inequality (3.1) is utilized only for the bound (4.5). For the sake of completeness, we give the proof.

Proof of (4.5)

By the weighted $\ell ^p$ -decoupling estimate (3.1),

$$ \begin{align*} \int |T_{small} (f)|^p dx &= \sum_{l\in \mathcal{J}} \int_{F_l} | \sum_{\theta} \sum_{T\in \mathbb{T}_{\theta_l,small}} f_{\theta_l} \chi_{T}|^p dx \\ &\lessapprox \left[ ( \gamma R^{-d})^{\alpha(p)} R^{\frac{d-1}{2} (\frac{1}{2}-\frac{1}{p})} \right]^p\sum_{l\in \mathcal{J}} \sum_{\theta} \sum_{T\in \mathbb{T}_{\theta_l,small}} \|f_{\theta_l} \chi_{T} \|_{L^p}^p. \end{align*} $$

Let $f= 1_{E_1}$ . Note that

$$\begin{align*}\|f_{\theta_l} \chi_{T} \|_{L^\infty} \lessapprox \||f_{\theta_l}| * \Psi_\theta \|_{L^\infty(T)} \lessapprox \beta_1. \end{align*}$$

Therefore,

$$ \begin{align*} \sum_{l\in \mathcal{J}} \sum_{\theta} \sum_{T\in \mathbb{T}_{\theta_l,small}} \|f_{\theta_l} \chi_{T} \|_{L^p}^p \lessapprox \beta_1^{p-2} \sum_{l\in \mathcal{J}} \sum_{\theta} \sum_{T\in \mathbb{T}_{\theta}} \|f_{\theta_l} \chi_{T} \|_{L^2}^2 \lessapprox \beta_1^{p-2} \| f\|_{L^2}^2 \end{align*} $$

by the $L^2$ -orthogonality, giving the claimed bound.

We may now prove Proposition 4.4.

Proof of Proposition 4.4

By Lemma 4.7,

$$ \begin{align*} |\{x: |T_{\mathcal{J}} 1_{E_1}(x)|> \alpha\}| \leq & |\{x: |T_{large}(1_{E_1})(x)| > \alpha/2\}| + |\{x: |T_{small}(1_{E_1})(x)| > \alpha/2\}| \\ \lessapprox & \alpha^{-1} \beta_1^{-1} \beta_2 R^{-1/2} |E| + \alpha^{-p} (\gamma R^{-d})^{p \alpha(p)} R^{\frac{d-1}{2}(\frac{p}{2}-1)} \beta_1^{p-2} |E|. \end{align*} $$

We take $\beta _1$ so that the two terms are equal, which gives

(4.6) $$ \begin{align} |\{x: |T_{\mathcal{J}} 1_{E_1}(x)|> \alpha\}| \lessapprox \alpha^{-2} \beta_2^{\frac{p-2}{p-1}} R^{\frac{d-1}{4}\frac{p-2}{p-1}-\frac{1}{2}+\frac{1}{2(p-1)}} (\gamma R^{-d})^{\frac{p}{p-1} \alpha(p)} |E_1|. \end{align} $$

By combining Lemma 4.6 and (4.6),

$$ \begin{align*} |\{x: |T_{\mathcal{J}} 1_{E}(x)|> \alpha\}| \leq & |\{x: |T_{\mathcal{J}} 1_{E_1}(x)| > \alpha/2\}| + |\{x: |T_{\mathcal{J}} 1_{E_2}(x)| > \alpha/2\}| \\ \lessapprox & \alpha^{-2} \beta_2^{\frac{p-2}{p-1}} R^{\frac{d-1}{4}\frac{p-2}{p-1}-\frac{1}{2}+\frac{1}{2(p-1)}} (\gamma R^{-d})^{\frac{p}{p-1} \alpha(p)} |E| + \alpha^{-2} R^{-1} \beta_2^{-1} |E|^2. \end{align*} $$

Finally, it suffices to choose $\beta _2$ so that the two terms are equal:

$$\begin{align*}\beta_2^{-1} = |E|^{-\frac{p-1}{2p-3}} R^{\frac{1}{2p-3} (\frac{d+1}{4}p - \frac{d-1}{2}) } (\gamma R^{-d})^{\frac{p}{2p-3}\alpha(p)}.\\[-38pt]\end{align*}$$

4.3 Proof of Proposition 4.2

By applying Proposition 4.4 and Theorem 3.1 with $p=\frac {14}{5}$ and $\alpha (p) = \frac {1}{7}$ , we obtain the restricted weak-type estimate

$$\begin{align*}|\{x: |T_{\mathcal{J}} 1_{E}(x)|> \alpha\}| \lessapprox \alpha^{-2} R^{-\frac{5}{13}} (\gamma R^{-2})^{\frac{2}{13}} |E|^{\frac{17}{13}}, \end{align*}$$

or equivalently,

$$\begin{align*}\| T_{\mathcal{J}} f \|_{L^{2,\infty}} \lessapprox R^{-\frac{5}{26} } (\gamma R^{-2})^{\frac{1}{13}} \| f\|_{L^{\frac{26}{17},1}}. \end{align*}$$

By Hölder’s inequality (for Lorentz spaces), we get

(4.7) $$ \begin{align} \| T_{\mathcal{J}} f \|_{L^{\frac{26}{17}}} \lessapprox2^{2(\frac{17}{26}-\frac{1}{2})j} 2^{-\frac{5}{26} j} (\gamma R^{-2})^{\frac{1}{13}} \| f\|_{L^{\frac{26}{17},1}} = 2^{\frac{3}{26} j} (\gamma R^{-2})^{\frac{1}{13}} \| f\|_{L^{\frac{26}{17},1}}. \end{align} $$

In order to use the gain of $(\gamma R^{-2})^{\frac {1}{13}}$ in (4.7), we interpolate it with a $L^{4/3}$ -estimate. We recall the $L^{4/3}$ -estimate due to Carleson and Sjörin [Reference Carleson and Sjölin6] and Córdoba [Reference Córdoba7]:

$$\begin{align*}\| m_j(D) f \|_{L^{4/3}} \lessapprox \| f\|_{L^{4/3}}. \end{align*}$$

Since the $L^{4/3}$ -operator norm of $m_j(D/t)$ is independent of $t>0$ , we have

(4.8) $$ \begin{align} \|T_{\mathcal{J}} f\|_{L^{4/3}} = \left( \sum_{l\in \mathcal{J}} \| m_j(D/c_l) f\|_{L^{4/3}(F_l)}^{4/3} \right)^{3/4} \lessapprox |\mathcal{J}|^{3/4} \|f \|_{L^{4/3}}. \end{align} $$

We interpolate (4.7) and (4.8) so that we may apply $\gamma R^{-2} |\mathcal {J}| \lessapprox 1$ . To be specific, take $\theta = \frac {39}{43}$ so that $\theta \cdot \frac {1}{13} = (1-\theta ) \frac {3}{4}$ . Then $\theta \frac {17}{26} + (1-\theta ) \frac {3}{4}=\frac {57}{86}$ . Real interpolation of (4.7) and (4.8) gives

$$\begin{align*}\| T_{\mathcal{J}} f\|_{L^{\frac{86}{57}}} \lessapprox 2^{ \frac{9}{86} j}\| f\|_{L^{\frac{86}{57}}}, \end{align*}$$

which completes the proof of Proposition 4.2 by (4.3).

Remark 4.8. Consider the example $f(x) = e^{2\pi i x_2} \psi (x_1,2^{-j/2} x_2)$ from [Reference Tao22]. In this example, if ${x_1\sim x_2 \sim 2^j}$ , then

$$\begin{align*}\sup_{t\sim 1} |m_j(D/t)f(x)|\sim |m_j(D/t(x))f(x)| \sim 2^{-j}, \end{align*}$$

where $t(x) = |x|/x_2$ . Thus, we may choose $\{ c_l,F_l\}$ so that $\gamma \sim 2^{j}$ , $|\mathcal {J}|\sim 2^j$ , $\| T_{\mathcal {J}} f\|_{L^p} \gtrsim 2^{2j(\frac {1}{p}-\frac {1}{2})}$ and $\| f\|_{L^q} \sim 2^{j\frac {1}{2q}}$ . In particular,

$$ \begin{align*} \|T_{\mathcal{J}} f\|_{L^{4/3}} &\gtrsim |\mathcal{J}|^{1/8} \|f \|_{L^{4/3}}. \end{align*} $$

This suggests that there may be significant room for improvement in the estimate (4.8). Any improvement to (4.8) would advance our knowledge of the almost everywhere convergence of the Bochner-Riesz means.

4.4 An improved bound for $d= 3$

Theorem 4.9. Let $d=3$ , $1< p< 2$ and

$$\begin{align*}\lambda>\max\left(\frac{174}{85p}-\frac{89}{85},\frac{146}{77}\left(\frac{1}{p}-\frac{1}{2}\right) \right).\end{align*}$$

Then the weak-type estimate (1.1) holds. Consequently, $\lim _{t\to \infty } S^\lambda _t f(x) = f(x)$ for almost every $x\in \mathbb {R}^3$ for any $f\in L^p(\mathbb {R}^3)$ .

Theorem 4.9 gives a small improvement to the result obtained in [Reference Gan and Wu13]; when $p=\frac {3}{2}$ , (1.1) holds for $\lambda> \frac {27}{85}=0.317\cdots $ , improving the sufficient condition $\lambda> \frac {107}{325}=0.329\cdots $ from [Reference Gan and Wu13].

Proof. Let $E\subset B(0,2^j)$ . By taking $p=\frac {14}{5}$ and $\alpha (p)=\frac {1}{14}$ in Proposition 4.4, we get

$$\begin{align*}|\{x: |T_{\mathcal{J}} 1_{E}(x)|> \alpha\}| \lessapprox \alpha^{-2} R^{-\frac{4}{13}} (\gamma R^{-3})^{\frac{1}{13}} |E|^{\frac{17}{13}}, \end{align*}$$

or equivalently,

$$\begin{align*}\| T_{\mathcal{J}} f\|_{L^{2,\infty}} \lessapprox 2^{-\frac{2}{13}j} (\gamma R^{-3})^{\frac{1}{26}} \| f\|_{L^{\frac{26}{17},1}}.\end{align*}$$

By Hölder’s inequality, we get

(4.9) $$ \begin{align} \| T_{\mathcal{J}} f\|_{L^{\frac{26}{17}}} \lessapprox 2^{\frac{4}{13}j} (\gamma R^{-3})^{\frac{1}{26}} \| f\|_{L^{\frac{26}{17},1}}. \end{align} $$

However, by the sharp $L^p$ estimate for the Bochner-Riesz means at the exponent $p=\frac {13}{9}$ due to Wu [Reference Wu24] (see also [Reference Guo, Oh, Wang, Wu and Zhang14]), there holds

$$\begin{align*}\| m_j(D) f\|_{L^{\frac{13}{9}}} \lessapprox 2^{(3(\frac{9}{13}-\frac{1}{2})-\frac{1}{2})j} \|f \|_{L^{\frac{13}{9}}}.\end{align*}$$

Therefore,

(4.10) $$ \begin{align} \|T_{\mathcal{J}} f\|_{L^{\frac{13}{9}}} = \left( \sum_{l\in \mathcal{J}} \| m_j(D/c_l) f\|_{L^{\frac{13}{9}}(F_l)}^{\frac{13}{9}} \right)^{\frac{9}{13}} \lessapprox 2^{\frac{j}{13}} |\mathcal{J}|^{\frac{9}{13}} \|f \|_{L^{\frac{13}{9}}}. \end{align} $$

We interpolate (4.9) and (4.10) so that we may apply $\gamma |\mathcal {J}| R^{-3} \lessapprox 1$ . This gives

$$\begin{align*}\| T_{\mathcal{J}} f\|_{L^{\frac{13\cdot 19}{9\cdot 18}}} \lessapprox 2^{\frac{73}{247}j} \| f\|_{L^{\frac{13\cdot 19}{9\cdot 18}}}. \end{align*}$$

This implies, by (4.3) and Lemma 4.3, that

(4.11) $$ \begin{align} \| \sup_{t\sim 1} |m_j(D/t) f|\|_{L^{\frac{247}{162}}(B(0,C2^j) )} \lessapprox 2^{\frac{73}{247}j} \| f\|_{L^{\frac{247}{162}}}. \end{align} $$

By interpolation with the standard $L^1$ and $L^2$ estimates, we get for $1\leq p \leq 2$ ,

$$ \begin{align*} \| \sup_{t\sim 1} |m_j(D/t) f|\|_{L^{p}(B(0,C2^j) )} \lessapprox \max(2^{j(\frac{174}{85p}-\frac{89}{85})}, 2^{j\frac{146}{77}(\frac{1}{p}-\frac{1}{2})} ) \| f\|_{L^p}. \end{align*} $$

By the reduction of Tao, this implies Theorem 4.9.