Regularized Brascamp-Lieb inequalities and an application

TL;DR

This paper introduces a regularized form of the Brascamp-Lieb inequalities, extending their applicability and leading to a generalized multilinear Kakeya inequality, with implications for harmonic analysis.

Contribution

It proposes a novel regularization of the Brascamp-Lieb inequalities and demonstrates their application to a broader multilinear Kakeya inequality.

Findings

Regularized Brascamp-Lieb inequalities established.

Generalized multilinear Kakeya inequality derived.

Potential applications in harmonic analysis explored.

Abstract

We present a regularized version of H\"{o}lder-Brascamp-Lieb inequalities studied by Bennett, Carbery, Christ, and Tao. These inequalities lead to a generalization of the multilinear Kakeya inequality.

Equations

$${\int }_{{\mathbb{R}}^n}\prod _{j=1}^J{f}_j({\pi }_j(\mathbf{x}){)}^{p_j}d\mathbf{x}\le C\prod _{j=1}^J{\left({\int }_{{\mathbb{R}}^{n_j}}{f}_j({\mathbf{x}}_j)d{\mathbf{x}}_j\right)}^{p_j}.$$

$${\int }_{[-R,R{]}^n}\prod _{j=1}^J{f}_j({\pi }_j(\mathbf{x}){)}^{p_j}d\mathbf{x}\le C\prod _{j=1}^J{\left({\int }_{{\mathbb{R}}^{n_j}}{f}_j({\mathbf{x}}_j)d{\mathbf{x}}_j\right)}^{p_j}$$

$$\text{text@underlineBL}(\vec{\pi },\vec{p},R):=\underset{\vec{f}}{\sup }\frac{{\int }_{[-R,R{]}^n}\prod _{j=1}^J{f}_j({\pi }_j(\mathbf{x}){)}^{p_j}d\mathbf{x}}{\prod _{j=1}^J{\left({\int }_{{\mathbb{R}}^{n_j}}{f}_j({\mathbf{x}}_j)d{\mathbf{x}}_j\right)}^{p_j}}$$

$$c\underset{V\le {\mathbb{R}}^n}{\sup }{R}^{\dim V-\sum _{j=1}^J{p}_j\dim {\pi }_j(V)}\le \text{text@underlineBL}(\vec{\pi },\vec{p},R)\le C\underset{V\le {\mathbb{R}}^n}{\sup }{R}^{\dim V-\sum _{j=1}^J{p}_j\dim {\pi }_j(V)}$$

$${\int }_{\{x\in {\mathbb{R}}^n:|x|\le R\}}\prod _{j=1}^J{f}_j$$

$$\le c_n{R}^n\prod _{j=1}^J{\left(\sum _{\mathbf{v}\in {\mathbb{Z}}^{n_j}}\Vert f_j{\Vert }_{L^{\infty }(\mathbf{v}+[0,1{)}^{n_j})}\right)}^{p_j}$$

$$=c_n{R}^n\prod _{j=1}^J{\left({\int }_{{\mathbb{R}}^{n_j}}{f}_j({\mathbf{x}}_j)d{\mathbf{x}}_j\right)}^{p_j}.$$

$$\int_{\mathbb{R}^{n}}\prod_{j=1}^{n}\big{(}\sum_{T_{j}\in\mathbb{T}_{j}}\chi_{T_{j}}\big{)}^{\frac{p}{n}}\leq C\delta^{n}\prod_{j=1}^{n}\big{(}\#\mathbb{T}_{j}\big{)}^{\frac{p}{n}}$$

$$\int_{[-1,1]^{n}}\prod_{j=1}^{J}\big{(}\sum_{T_{j}\in\mathbb{T}_{j}}\chi_{T_{j}}\big{)}^{p_{j}}\leq C_{\epsilon}\delta^{-\epsilon+n}\sup_{V\leq\mathbb{R}^{n}}\delta^{-\dim V+\overset{J}{\underset{j=1}{\sum}}p_{j}\dim\pi_{j}^{0}(V)}\prod_{j=1}^{J}(\#\mathbb{T}_{j})^{p_{j}}$$

$$\text{text@underlineBL}(\vec{\pi },\vec{p},R)\le C_0{R}^{\sum _{r=1}^n\max (1-\sum _{j\in {\mathcal{J}}_r}{p}_j,0)}$$

$$c\underset{V\le {\mathbb{R}}^n}{\sup }{R}^{\dim V-\sum _{j=1}^J{p}_j\dim {\pi }_j(V)}\le \text{text@underlineBL}(\vec{\pi },\vec{p},R).$$

$$S_j=\{\mathbf{v}\in {\mathbb{Z}}^{n_j}:|P_{{\pi }_j(V)}(\mathbf{v})|\le R+\sqrt{n}\text{and}|P_{{\pi }_j(V{)}^{\perp }}(\mathbf{v})|\le 1+\sqrt{n}\}$$

$$S=\{\mathbf{x}\in V:|P_V(\mathbf{x})|\le c_{0}R\text{and}|P_{V^{\perp }}(\mathbf{x})|\le c_0\}.$$

$$|P_{{\pi }_j(V)}({\pi }_j(\mathbf{x}))|\le |{\pi }_j(\mathbf{v}))|+|P_{{\pi }_j(V)}({\pi }_j(\mathbf{w}))|\le R$$

$$|P_{{\pi }_j(V{)}^{\perp }}({\pi }_j(\mathbf{x}))|=|P_{{\pi }_j(V{)}^{\perp }}({\pi }_j(\mathbf{w}))|\le 1.$$

$$C{c}_0^n{R}^{\dim V}$$

$$\le \text{text@underlineBL}(\vec{\pi },\vec{p},R)\prod _{j=1}^J{\left({\int }_{{\mathbb{R}}^{n_j}}{f}_j({\mathbf{x}}_j)d{\mathbf{x}}_j\right)}^{p_j}$$

$$\underset{\mathbf{v}\in S_j}{\cup }(\mathbf{v}+Q_j)\subseteq \{{\mathbf{x}}_j\in {\mathbb{R}}^{n_j}:|P_{{\pi }_j(V)}({\mathbf{x}}_j)|\le R+2\sqrt{n}\text{and}|P_{{\pi }_j(V{)}^{\perp }}({\mathbf{x}}_j)|\le 1+2\sqrt{n}\},$$

$$C{c}_0^n{R}^{\dim V}\le \text{text@underlineBL}(\vec{\pi },\vec{p},R){\left({\int }_{{\mathbb{R}}^{n_j}}{f}_j({\mathbf{x}}_j)d{\mathbf{x}}_j\right)}^{p_j}\le {\tilde{C}}^{p_1+\cdots +p_J}\text{text@underlineBL}(\vec{\pi },\vec{p},R)\prod _{j=1}^J{R}^{p_j\dim {\pi }_j(V)}$$

$$\Vert f{\Vert }_{A,{\mathcal{L}}_j^0}:=\sum _{\mathbf{v}\in {\mathcal{L}}_j^0}\Vert f{\Vert }_{L^{\infty }((\mathbf{v}+A{Q}_j)\cap H_j)},$$

$${\int }_{\{\mathbf{x}\in H:|\mathbf{x}|<R\}}\prod _{j=1}^J{f}_j({\pi }_j(\mathbf{x}){)}^{p_j}d\mathbf{x}\le C\underset{V\le H}{\sup }{R}^{\dim V-\sum _{i=1}^J{p}_i\dim {\pi }_i(V)}\prod _{j=1}^J\Vert f_j{\Vert }_{A_j,{\mathcal{L}}_j^0}^{p_j}$$

$${\int }_{[-R,R{]}^n}\prod _{j=1}^J{f}_j({\pi }_j(\mathbf{x}))d\mathbf{x}\le C\underset{V\le {\mathbb{R}}^n}{\sup }{R}^{\dim V-\sum _{i=1}^J{p}_i\dim {\pi }_i(V)}\prod _{j=1}^J\Vert f_j{\Vert }_{A_j,{\mathbb{Z}}^{n_j}}^{p_j}$$

$$\Vert f_j{\Vert }_{A_j,{\mathbb{Z}}^{n_j}}=\sum _{m\in {\mathbb{Z}}^{n_j}}\underset{x\in m+A_j{Q}_j}{\text{ess sup}}{f}_j(x)$$

$$\le A_j^{n_j}\sum _{k\in {\mathbb{Z}}^{n_j}}{f}_j(k)=A_j^{n_j}{\int }_{{\mathbb{R}}^{n_j}}{f}_j({\mathbf{x}}_j)d{\mathbf{x}}_j,$$

$${\int }_{\{x\in H:|x|<R\}}\prod _{j=1}^J{f}_j({\pi }_j(x){)}^{p_j}dx$$

$$=\prod _{j=1}^J|{\pi }_j{|}^{{\delta }_j-p_j}{\left({\int }_H{f}_j(x{)}^{p_j/(p_j-{\delta }_j)}dx\right)}^{p_j-{\delta }_j}$$

$$\int_{H}f_{j}(x)^{p_{j}/p_{j}-\delta_{j}}dx\leq\sum_{m\in\mathcal{L}_{j}^{0}}\|f_{j}\|_{L^{\infty}(m+[0,1))}^{p_{j}/(p_{j}-\delta_{j})}\leq\big{(}\sum_{m\in\mathcal{L}_{j}^{0}}\|f_{j}\|_{L^{\infty}(m+[0,1))}\big{)}^{p_{j}/(p_{j}-\delta_{j})},$$

$${\int }_{\{x\in H:|x|<R\}}\prod _{j=1}^J{f}_j({\pi }_j(x){)}^{p_j}dx$$

$$\le (2R{)}^{1-\sum _{i=1}^J{p}_i}\prod _{r=1}^J|{\pi }_r{|}^{-p_r}\prod _{j=1}^J\Vert f_j{\Vert }_{1,{\mathcal{L}}_j^0}^{p_j}.$$

$$\dim W-\sum _{i=1}^J{p}_i\dim {\pi }_i(W)=\underset{V\le {\mathbb{R}}^n}{\sup }\left(\dim V-\sum _{i=1}^J{p}_i\dim {\pi }_i(V)\right).$$

$${\int }_{\{x\in H:|\mathbf{x}|<R\}}\prod _{j=1}^J{f}_j({\pi }_j(\mathbf{x}){)}^{p_j}d\mathbf{x}\le {\int }_{W^{\perp }}{\int }_{\{x\in W:|x|<R\}}\prod _{j=1}^J{f}_j({\pi }_j(x)+{\pi }_j(y){)}^{p_j}dxdy.$$

$${\int }_{\{W:|x|<R\}}\prod _{j=1}^J{f}_j$$

$$\Vert f_j(\cdot +{\pi }_j(y)){\Vert }_{A_j,{\mathcal{L}}_j}$$

Full text

By Brascamp-Lieb inequalities, we mean inequalities of the form

[TABLE]

where the $f_{j}$ are nonnegative, measurable functions, $\vec{\pi}=(\pi_{j})$ is a $J$-tuple of linear maps from $\mathbb{R}^{n}$ to $\mathbb{R}^{n_{j}}$, and $\vec{p}\in[0,1]^{J}$. A related quantity is the smallest constant $C$ which satisfies the above inequality for all nonnegative input functions $f_{j}\in L^{1}(\mathbb{R}^{n_{j}})$, denoted by $\text{BL}(\vec{\pi},\vec{p})$. Locally bounded properties of $\text{BL}(\vec{\pi},\vec{p})$ lead to multilinear Kakeya inequalities: Guth [13] first did this for a special case $(\vec{\pi}_{0},\vec{p}_{0})$ and Bennett, Bez, Flock, and Lee [3] extended Guth’s approach to data $(\vec{\pi},\vec{p})$ satisfying $\text{BL}(\vec{\pi},\vec{p})<\infty$. The motivation for this paper is to formulate a regularized version of $\text{BL}(\vec{\pi},\vec{p})$ which leads to multilinear Kakeya inequalities in the case that we do not have finite Brascamp-Lieb data (see below for further discussion of multilinear Kakeya).

Some of the most fundamental inequalities of this form are Hölder’s inequality and Young’s convolution inequality. In [7], Brascamp and Lieb determined the optimal version of Young’s convolution inequality (also proved independently by Beckner in [2]), and proved a generalized Young’s inequality for more than three functions. Barthe gave a concrete description of when $\text{BL}(\vec{\pi},\vec{p})<\infty$ in the rank one case (all $n_{j}=1$) [1]. In 2008, Bennett, Carbery, Christ, and Tao (BCCT) determined the criterion for $\text{BL}(\vec{\pi},\vec{p})<\infty$ [4]. BCCT also investigated extremal configurations and variants of the Brascamp-Lieb inequality in [4] and [5] respectively.

In this paper, we consider the following regularized version of (1):

[TABLE]

where the $f_{j}$ are nonnegative, measurable functions that are constant on cubes in the unit cube lattice of $\mathbb{R}^{n_{j}}$, meaning on all sets of the form ${\bf{v}}+[0,1)^{n_{j}}$ where ${\bf{v}}\in\mathbb{Z}^{n_{j}}$. The optimal constant $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$ for this inequality is defined by

[TABLE]

where the supremum is taken over $\vec{f}=(f_{1},\ldots,f_{J})$ with nonzero, nonnegative functions $f_{j}\in L^{1}(\mathbb{R}^{n_{j}})$ that are constant on each cube in a unit cube tiling of $\mathbb{R}^{n_{j}}$. The goal of this paper is to identify the growth rate of $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$ as a function of $R$, which we do in the following theorem.

[TABLE]

for all $R\geq 1$, where the supremum is taken over all subspaces $V$ of $\mathbb{R}^{n}$.

Since the value of $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$ does not change if we replace $\mathbb{R}^{n_{j}}$ by $\pi_{j}(\mathbb{R}^{n})$, it is no loss of generality to assume that the $\pi_{j}$ are surjective. The quantity $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$ is finite because

[TABLE]

Also note that if the supremum defining $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$ were taken over nonnegative $f_{j}\in L^{1}(\mathbb{R}^{n_{j}})$ which were not necessarily constant on the unit cube lattice of $\mathbb{R}^{n_{j}}$, then $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$ is not necessarily finite (see Theorem 2.2 in [5]).

One motivation for understanding the regularized Brascamp-Lieb inequality is to extend multilinear Kakeya results. First formulated in 2006, the multilinear Kakeya inequality is an $L^{p}$-measurement of how much tubes in different directions can overlap. Let $\mathbb{T}_{j}$ be a collection of $\delta-$neighborhoods $T_{j}$ of lines in $\mathbb{R}^{n}$ that are sufficiently parallel (independent of $\delta$) to the $e_{j}$-axis, where $e_{j}$ is the standard basis vector. The multilinear Kakeya inequality states that

[TABLE]

for $\frac{n}{n-1}\leq p<\infty$. For the endpoint exponent $p=\frac{n}{n-1}$ and the special case that the $T_{j}$ are parallel to the $e_{j}$, (4) is just the Loomis-Whitney inequality, a special Brascamp-Lieb inequality. Thus multilinear Kakeya can be regarded as a generalization of Loomis-Whitney.

Bennett, Carbery, and Tao proved (4) in 2006 without an epsilon loss away from the endpoint (which implies the endpoint with an epsilon loss by Hölder’s inequality) [6]. In 2009, Guth eliminated the $\epsilon$ loss factor using an alternative proof involving Dvir’s polynomial method approach to the analogous problem over finite fields [12, 9]. Guth reproved a weaker, truncated version of the multilinear Kakeya inequality with a simple induction on scales argument in [12].

Since the multilinear Kakeya inequality is a generalization of Loomis-Whitney, it is natural to wonder if there are analogues of the multilinear Kakeya inequality which are generalizations of Brascamp-Lieb inequalities. Bennett, Bez, Flock, and Lee [3] proved a stability property of the Brascamp-Lieb constant $\text{BL}(\vec{\pi},\vec{p})$ which combined with Guth’s induction on scales result, led to a truncated multilinear Kakeya inequality with $\epsilon-$losses for finite Brascamp-Lieb data (meaning $\text{BL}(\vec{\pi},\vec{p})<\infty$). Zhang further developed Guth’s polynomial method proof in [12] to prove that these $\epsilon-$losses could be removed [14].

In this paper, we prove a multilinear Kakeya result without any assumptions on the Brascamp-Lieb data. Instead, a factor which controls the regularized Brascamp-Lieb constant appears in the upper bound. To describe our generalized (truncated) multilinear Kakeya inequality, fix some notation. Let $\vec{\pi_{0}}$ be a $J$-tuple of orthogonal projections from $\mathbb{R}^{n}$ to subspaces $(V_{j}^{0})^{\perp}\subset\mathbb{R}^{n}$ where $V_{j}^{0}$ has dimension $n_{j}^{\prime}$. Let $n_{j}=n-n_{j}^{\prime}$. We will consider affine subspaces $V_{j}\subset\mathbb{R}^{n}$ which, modulo translations, are within a distance $\nu$ of the $V_{j}^{0}$. Here distance is the standard metric on the Grassmann manifold of $n_{j}^{\prime}$-dimensional subspaces of $\mathbb{R}^{n}$.

[TABLE]

holds for all finite collections $\mathbb{T}_{j}$ of $\delta$-neighborhoods of $n_{j}^{\prime}$-dimensional affine subspaces of $\mathbb{R}^{n}$ which, modulo translations, are within a distance $\nu$ of the fixed subspace $V_{j}^{0}$.

Using a small variation of the induction-on-scales method of Guth [13], Theorem 2 follows from Theorem 1 and the locally-bounded result Theorem 5 below, which follows from work in [3]. Theorem 5 shows that the asymptotic growth rate in $R$ of $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$ is stable under perturbation of $\vec{\pi}$. A similar stability for the implicit constant $C$ in the upper bound of Theorem 1 is less clear, but turns out not to be necessary for proving Theorem 2.

The following theorem is a locally uniform bound for $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$, but which only achieves the expected sharp growth rate in $R$ for certain exponents $p_{j}$. Let $n_{j}=\dim(V_{j}^{0})^{\perp}$ and for each $r\in\{1,\ldots,n\}$, $\mathcal{J}_{r}=\{j\in\{1,\ldots,J\}:n_{j}\geq r\}$.

[TABLE]

holds for all orthogonal projections $\pi_{j}:\mathbb{R}^{n}\to V_{j}^{\perp}$ where $V_{j}$ is an $n_{j}^{\prime}$-dimensional subspace of $\mathbb{R}^{n}$ which is within a distance $\nu$ of $V_{j}^{0}$.

We remark that the power of $R$ in Theorem 3 above matches that in Theorem 1 (with $(\vec{\pi}_{0},\vec{p})$) if $p_{j}\leq\frac{1}{J}$ for all $j$, but can be larger in general. Indeed, a helpful reviewer pointed out the example in $\mathbb{R}^{2}$ for which $\pi_{1}(x,y)=x$, $\pi_{2}(x,y)=y$, and $\pi_{3}(x,y)=(x,y)$ with exponents $\vec{p}=(1/4,1,1/2)$. In this case the power of $R$ in Theorem 1 is $R^{1/4}$ while in Theorem 3 above, we have $R^{1/2}$. Also observe that Theorem 2 above implies an $R^{\varepsilon}$-loss version of Theorem 3 with the optimal growth rate $R^{\sup_{V\leq\mathbb{R}^{n}}[\dim V-\sum_{j=1}^{J}p_{j}\dim\pi_{j}^{0}(V)]}$.

An earlier draft of this manuscript had a stronger version of Theorem 3 which was not fully justified. The proof of Theorem 2, which was used in the later work [11], was edited in the present manuscript to no longer rely on Theorem 3 at all (see §4). This adjustment was facilitated by private correspondence with the authors of [11], and I am particularly appreciative of Ruixiang Zhang for helping with the revised proof.

In §2, we describe the example given by BCCT in §5 of [5], which proves the lower bound in Theorem 1. The upper bound in Theorem 1 follows as a corollary of the more technical Proposition 4 discussed in §3. The proof of Proposition 4 follows the inductive arguments of BCCT in the proofs of Theorems 2.1 and 2.5 of [5]. In §4, we discuss the proof of Theorem 2 and Theorem 3. The author wishes to thank Larry Guth for bringing this problem to her attention and to thank Michael Christ for valuable conversations.

1. Funding

The author was supported by an National Science Foundation Graduate Research Fellowship under Grant No. [DGE 1106400].

2. Lower bound for $\text{\text@underline{{BL}}}(\vec{\pi},\vec{p},R)$

In this section, we describe the example given by BCCT in §5 from [5]. We use this example to demonstrate that there exists $c>0$ satisfying

[TABLE]

Proof of the lower bound for $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$ from Theorem 1.

Fix a vector $\vec{p}\in[0,1]^{J}$ and surjective linear maps $\pi_{j}:\mathbb{R}^{n}\to\mathbb{R}^{n_{j}}$. Let $V\leq\mathbb{R}^{n}$ be a subspace. For a subspace $W\subset\mathbb{R}^{k}$ let $P_{W}:\mathbb{R}^{k}\to W$ denote linear projection onto $W$. Define the collections

[TABLE]

and let $f_{j}$ be the indicator function of the set $\underset{{\bf{v}}\in S_{j}}{\cup}({\bf{v}}+Q_{j})$, where $Q_{j}=[0,1)^{n_{j}}$.

Let $c_{0}>0$ be a constant we will define independently of $R$. Define the set $S\subset\mathbb{R}^{n}$ by

[TABLE]

Let $|\pi_{j}|$ be the operator norm of $\pi_{j}$ so that $|\pi_{j}({\bf{x}})|\leq|\pi_{j}||{\bf{x}}|$ for all ${\bf{x}}\in\mathbb{R}^{n}$ and $j=1,\ldots,J$. Choose $c_{0}=\min(\frac{1}{2|\pi_{1}|},\ldots,\frac{1}{2|\pi_{J}|},\frac{1}{\sqrt{2}})$, which guarantees that $S\subset[-R,R]^{d}$. Now verify that if ${\bf{x}}\in S$, then $f_{j}(\pi_{j}({\bf{x}}))=1$ for all $j=1,\ldots,J$: Write ${\bf{x}}\in\mathbb{R}^{n}$ uniquely as ${\bf{x}}={\bf{v}}+{\bf{w}}$ where ${\bf{v}}\in V$ and ${\bf{w}}\in V^{\perp}$. By the definition of $S$, $|{\bf{v}}|\leq c_{0}R$ and $|{\bf{w}}|\leq c_{0}$, which implies that $|\pi_{j}({\bf{v}})|\leq|\pi_{j}||{\bf{v}}|\leq\frac{1}{2}R$ and $|\pi_{j}({\bf{w}})|\leq|\pi_{j}||{\bf{w}}|\leq\frac{1}{2}$. Then

[TABLE]

and

[TABLE]

It follows from these displayed inequalities that $\pi_{j}({\bf{x}})\in\underset{{\bf{v}}\in S_{j}}{\cup}({\bf{v}}+Q_{j})$, which holds for all $j=1,\ldots,J$. Putting everything together, we have

[TABLE]

for a constant $C$ which depends on the dimension $n$. Using the inclusion

[TABLE]

and that $f_{j}$ is the indicator function of $\underset{{\bf{v}}\in S_{j}}{\cup}({\bf{v}}+Q_{j})$, obtain the final inequality

[TABLE]

where $\tilde{C}$ is a constant depending on $n$. ∎

3. Upper bound for $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$

The upper bound from Theorem 1 follows as a corollary to Proposition 4 below. The proof of Proposition 4 proceeds in an analogous way as the proofs of Theorems 2.1 and 2.5 in [5]. We introduce some notation before we state the proposition. Let $H\subset\mathbb{R}^{k}$, $H_{j}\subset\mathbb{R}^{k_{j}}$ be subspaces and let $\pi_{j}:H\to H_{j}$ be surjective linear maps. For each $j=1,\ldots,J$, fix a subset $\mathcal{L}^{0}_{j}\subset\mathbb{Z}^{k_{j}}$ which satisfies $H_{j}\subset\underset{{\bf{v}}\in\mathcal{L}_{j}^{0}}{\cup}({\bf{v}}+Q_{j})$, where $Q_{j}=[0,1)^{k_{j}}$. For a parameter $A>0$ and a function $f\in L^{\infty}(H_{j})$, define the quantity

[TABLE]

where $AQ_{j}=[0,A)^{k_{j}}\subset\mathbb{R}^{k_{j}}$.

Proposition 4**.**

Let $\vec{p}\in[0,1]^{J}$. Suppose $H$, $\{H_{j}\}$, $\{\mathcal{L}_{j}^{0}\}$, and $\vec{\pi}$ are given as above. Then there exist parameters $A_{j}\geq 1$, for $j=1,\ldots,J$, and a constant $C>0$ such that

[TABLE]

for all $R\geq 1$ and all nonnegative, measurable functions $f_{j}:H_{j}\to[0,\infty)$.

Granting this proposition, we first prove the upper bound from Theorem 1.

Proof of the upper bound for $\text{\text@underline{\emph{BL}}}(\vec{\pi},\vec{p},R)$.

Take $H=\mathbb{R}^{n}$, $H_{j}=\mathbb{R}^{n_{j}}$, and $\mathcal{L}_{j}^{0}=\mathbb{Z}^{n_{j}}$ in Proposition 4. It follows from the proposition that there exist constants $A_{j},C$ such that

[TABLE]

for all $R\geq 1$ and all measurable functions $f_{j}:\mathbb{R}^{n_{j}}\to[0,\infty)$. Consider functions $f_{j}$ which are constant on cubes ${\bf{v}}+[0,1)^{n_{j}}$ where ${\bf{v}}\in\mathbb{Z}^{n_{j}}$. Then

[TABLE]

which finishes the proof. ∎

Proof of Proposition 4.

Without loss of generality, assume that $0<\|f_{j}\|_{1,\mathcal{L}_{j}^{0}}<\infty$ for all $j=1,\ldots,J$. If $\underset{V\leq H}{\sup}[\dim V-\sum\limits_{j=1}^{J}p_{j}\dim\pi_{j}(V)]\leq 0$, then Proposition 4 follows from Theorem 2.5 in [5], so it suffices to assume that the supremum is positive. Fix the notation $n=\dim H$ and $n_{j}=\dim H_{j}$. We proceed by induction on the dimension $n$.

Begin with the base case $n=1$. In this case, $n_{j}=1$ for all $j$. If $1\leq\sum\limits_{j=1}^{J}p_{j}$, then choose $\delta_{j}\in[0,p_{j})$ so that $1=\sum\limits_{j=1}^{J}(p_{j}-\delta_{j})$. Using the generalized Hölder’s inequality,

[TABLE]

where $|\pi_{j}|$ is from the subsitution $u=\pi_{j}(x)$. For each $1\leq j\leq J$,

[TABLE]

which finishes this case.

The other case is that $1>\sum\limits_{j=1}^{J}p_{j}$. Let $s=1-\sum\limits_{j=1}^{J}p_{j}$. Then by the generalized Hölder’s inequality,

[TABLE]

This concludes the $n=1$ base case.

Next consider the case where the ambient dimension is $n>1$. Furthermore, assume we are in the subcase where there exists a proper subspace $W\leq H$ such that

[TABLE]

Define coordinates ${\bf{x}}=(x,y)\in W\oplus W^{\perp}=H$ and write

[TABLE]

For each $j$, let $P_{\pi_{j}(W)}:H_{j}\to\pi_{j}(W)$ and $P_{\pi_{j}(W)^{\perp}}:H_{j}\to\pi_{j}(W)^{\perp}$ be the linear projections. Define the subsets $\mathcal{L}_{j}\subset\mathcal{L}_{j}^{0}$ to be the collections of ${\bf{v}}\in\mathcal{L}_{j}^{0}$ satisfying $|P_{\pi_{j}(W)^{\perp}}({\bf{v}})|\leq\sqrt{k_{j}}$. Then $\mathcal{L}_{j}$ contains the collection of ${\bf{v}}\in\mathcal{L}_{j}^{0}$ such that ${\bf{v}}+Q_{j}$ has nonempty intersection with $\pi_{j}(W)$, from which it follows that $\underset{{\bf{v}}\in\mathcal{L}_{j}}{\cup}({\bf{v}}+Q_{j})$ covers $\pi_{j}(W)$. Since $\dim W<\dim H$, by the inductive hypothesis (using the linear maps $\left.\pi_{j}\right|_{W}$), there exist $C,A_{j}>0$ (depending on the maximum of $k,k_{1},\ldots,k_{J}$, $\|\vec{\pi}\|$, and ${\vec{p}}$) such that

[TABLE]

for all $y\in W^{\perp}$. Let $L_{j}=P_{\pi_{j}(W)^{\perp}}\circ\left.\pi_{j}\right|_{W^{\perp}}:W^{\perp}\to\pi_{j}(W)^{\perp}$. The maps $L_{j}$ are clearly surjective because $\pi_{j}(W\oplus W^{\perp})=\pi_{j}(W)+\pi_{j}(W^{\perp})=H_{j}=\pi_{j}(W)\oplus\pi_{j}(W)^{\perp}$. For each $y\in W^{\perp}$, define $u_{j}(y)\in\pi_{j}(W)$ by $u_{j}(y)=P_{\pi_{j}(W)}(\pi_{j}(y))$, so $\pi_{j}(y)=u_{j}(y)+L_{j}(y)$. Now analyze the quantities $\|f_{j}(\cdot+\pi_{j}(y))\|_{A_{j},\mathcal{L}_{j}}$ on the right hand side of the displayed inequality above.

[TABLE]

where in the last line, we used the fact that since $u_{j}(y)\in\pi_{j}(W)$, $\pi_{j}(W)+u_{j}(y)=\pi_{j}(W)$. Note that for each ${\bf{v}}\in\mathcal{L}_{j}$, the number of ${\bf{w}}\in\mathcal{L}_{j}$ which satisfy $({\bf{w}}+u_{j}(y)+A_{j}Q_{j})\cap({\bf{v}}+A_{j}Q_{j})\not=\emptyset$ is controlled by a dimensional constant times $A_{j}^{k_{j}}$. This means that

[TABLE]

Putting this together with (6) and (7), we have

[TABLE]

where $\tilde{C}$ depends on dimensions, $A_{j}$, $\vec{\pi}$, and $\vec{p}$. The growth rate of $\tilde{C}$ is not a concern since the number of steps in our induction is finite. For $z\in\pi_{j}(W)^{\perp}$, define

[TABLE]

It remains to bound

[TABLE]

Fix the subset $\mathcal{L}_{j}^{\prime}\subset\mathcal{L}_{j}^{0}$ of ${\bf{v}}\in\mathcal{L}_{j}^{0}$ that satisfy $|P_{\pi_{j}(W)}({\bf{v}})|\leq\sqrt{k_{j}}$. The subset $\mathcal{L}_{j}^{\prime}$ has the property that $\underset{{\bf{v}}\in\mathcal{L}_{j}^{\prime}}{\cup}({\bf{v}}+Q_{j})$ covers $\pi_{j}(W)^{\perp}$. Using that $L_{j}:W^{\perp}\to\pi_{j}(W)^{\perp}$ is surjective (discussed above), by the inductive hypothesis, there exist constants $C^{\prime},A_{j}^{\prime}$ such that

[TABLE]

For any subspace $V\subset W^{\perp}$,

[TABLE]

and for each $i$,

[TABLE]

Using the above equalities and the inequality

[TABLE]

which holds because of the subcase we are in, conclude that $\dim V\leq\sum\limits_{i=1}^{J}p_{i}\dim L_{i}(V)$ for all $V\leq W^{\perp}$. This means that the power of $R$ that appears on the right hand side of (8) is 0, and we may let $R$ tend to infinity on the right hand side to integrate over all of $W^{\perp}$. Summarizing our results for this subcase, we have

[TABLE]

Observe that for each $j$,

[TABLE]

For each ${\bf{v}}\in\mathcal{L}_{j}^{0}$, the number of $(k,m)\in\mathcal{L}_{j}^{\prime}\times\mathcal{L}_{j}$ such that

[TABLE]

is bounded by

[TABLE]

which is controlled by a constant depending only on dimension, $A_{j}$, and $A_{j}^{\prime}$. Also use the fact that $\underset{{\bf{v}}\in\mathcal{L}_{j}^{0}}{\cup}({\bf{v}}+Q_{j})$ covers $H_{j}$ to bound the quantity in (9) by

[TABLE]

which finishes this subcase.

Now suppose that the ambient dimension $n>1$ and that all proper subspaces $W\leq H$ satisfy

[TABLE]

Consider the set $K$ of all $J$-tuples $\vec{p}=(p_{1},\ldots,p_{J})\in[0,1]^{J}$ such that

[TABLE]

for all proper subspaces $V\subset\mathbb{R}^{n}$. Equivalently, $K$ equals the intersection

[TABLE]

This is the intersection of $[0,1]^{J}$ with finitely many closed half-spaces (even though there are infinitely many subspaces $V$, there are finitely many vectors $(\dim\pi_{i}(V))_{i}$). Therefore $K$ has finitely many extreme points, and since $K$ is compact and convex, $K$ equals the convex hull of its extreme points. If we prove the result for the extreme points $\vec{p}_{k}$, an application of complex interpolation says that if $\vec{p}=\sum\limits_{k}\lambda_{k}\vec{p}_{k}$ where $\sum\limits_{k}\lambda_{k}=1$, then

[TABLE]

so it suffices to consider the extreme points.

Next we want to show that $K=\tilde{K}$ where

[TABLE]

Observe $K=[0,1]^{J}\cap\tilde{K}\subset\tilde{K}$. Consider $\vec{p}\in\tilde{K}$ and any index $r$. Then for the subspace $V=\ker\pi_{r}$, $\vec{p}$ satisfies

[TABLE]

This leads to

[TABLE]

which means $p_{r}\leq 1$ and thus $\tilde{K}\subset K$.

If $\vec{p}$ is an extreme point of $K$, then at least one of the inequalities defining $K$ must be equality. If

[TABLE]

for some proper subspace $V\leq H$, then we are in subcase 1 above. The alternative is that $p_{r}=0$ for at least one $r\in\{1,\ldots,J\}$. In that case, we are trying to prove that there exist $C,A_{j}$ so that

[TABLE]

This follows from induction on the number of factors $J$ with base case $J=1$. When $J=1$, for appropriate constants $c,C>0$ depending on $\pi_{J}$,

[TABLE]

By Hölder’s inequality, this is bounded by

[TABLE]

for an appropriate constant $\tilde{C}$ and where we used that $\underset{m\in\mathcal{L}_{j}^{0}}{\cup}(m+Q_{j})$ covers $H_{j}$ in the final inequality.

∎

4. A generalized multilinear Kakeya inequality

In order to prove Theorem 2, we first prove a stability condition in the form of Theorem 5. Theorem 2 is proved using the induction-on-scales approach of Guth [13].

Begin with Theorem 5, which records the local boundedness of the growth rate of $R$ in $\text{\text@underline{BL}}(\vec{\pi},\vec{p},R)$ as a function of $\vec{\pi}$. Theorem 5 states a version of the result in §2.1 in [3] and we reproduce their argument (and some notations) here for completeness.

Theorem 5** ([3]).**

Fix $\vec{p}\in[0,1]^{J}$ and $J$ orthogonal projections $\pi_{j}^{0}$ from $\mathbb{R}^{n}$ onto $(V_{j}^{0})^{\perp}$. Then there exists $\nu>0$ such that

[TABLE]

holds for all orthogonal projections $\pi_{j}:\mathbb{R}^{n}\to V_{j}$ where $V_{j}$ is an $n_{j}^{\prime}$-dimensional subspace of $\mathbb{R}^{n}$ which is within a distance $\nu$ of the fixed subspace $V_{j}^{0}$.

Proof.

Let $1\leq k\leq n$ and let $\mathcal{E}_{k}$ denote the compact set of all orthonormal sets ${\bf{e}}=\{e_{1},\ldots,e_{k}\}$ of vectors in $\mathbb{R}^{n}$. Fix $k$ and let $\bf{e}\in\mathcal{E}_{k}$. For each $1\leq j\leq J$, choose a subset $I_{j}\subseteq\{1,\ldots,k\}$ satisfying $|I_{j}|=\dim\langle\pi_{j}^{0}(e_{1}),\ldots,\pi_{j}^{0}(e_{k})\rangle)$,

[TABLE]

and

[TABLE]

Since $(\vec{\pi},{\bf{e}^{\prime}})\mapsto\bigwedge_{i\in I_{j}}\pi_{j}(e_{i}^{\prime})\in\Lambda^{|I_{j}|}(\mathbb{R}^{n_{j}})$ is continuous for each $j$, there exists $\varepsilon(\bf{e}),\delta(\bf{e})>0$ such that $\bigwedge_{i\in I_{j}}\pi_{j}(e_{i}^{\prime})\not=0$ for each $j$ whenever $\|\bf{e}^{\prime}-\bf{e}\|<\varepsilon(\bf{e})$ and $\|\vec{\pi}-\vec{\pi}_{0}\|<\delta(\varepsilon)$. In particular, $\dim\langle\pi_{j}(e_{1}^{\prime}),\ldots,\pi_{j}(e_{k}^{\prime})\rangle)\geq|I_{j}|$ for each $j$, so

[TABLE]

Recall that $\mathcal{E}_{k}$ is compact, so there exists a finite collection ${\bf{e}}^{1},\ldots,{\bf{e}}^{N}\in\mathcal{E}_{k}$ so that the sets

[TABLE]

with $\ell=1,\ldots,N$ cover $\mathcal{E}_{k}$. Finally, choosing $\nu=\min\{\delta({\bf{e}}^{1},\ldots,{\bf{e}}^{N}\}$, conclude that if $\|\vec{\pi}-\vec{\pi}_{0}\|<\nu$ (in the sense of Theorem 5), then for any ${\bf{e}}\in\mathcal{E}_{k}$, there is some ${\bf{e}}^{\ell}$ from compactness so that

[TABLE]

Since the number of $k$ is bounded by dimension, Theorem 5 is proved.

∎

Next, we prove a weaker version of Theorem 2 in the form of the following proposition. This proposition follows the argument of Guth in [13]. It is similar to the argument described in [3], except that we do not use any locally uniform control over the (regularized) Brascamp-Lieb constants. An analogous proof could have been used in [3] to prove their Theorem 1.2 without relying on their Theorem 1.1.

Proposition 6**.**

Fix $\vec{p}\in[0,1]^{J}$ and $J$ orthogonal projections $\pi_{j}^{0}$ from $\mathbb{R}^{n}$ onto $(V_{j}^{0})^{\perp}$. There exists $\nu>0$ so that the following is true. Let $\pi_{j}:\mathbb{R}^{n}\to V_{j}^{\perp}$ be orthogonal projection maps, where the $V_{j}$ are $n_{j}^{\prime}$-dimensional subspaces within $\nu$ of the $V_{j}^{0}$. For every $\varepsilon>0$, there exist $\nu(\varepsilon,\vec{\pi})>0$ and $C(\varepsilon,\vec{\pi})<\infty$ so that

[TABLE]

holds for all finite collections $\mathbb{T}_{j}$ of $\delta$-neighborhoods of $n_{j}^{\prime}$-dimensional affine subspaces of $\mathbb{R}^{n}$ which, modulo translations, are within a distance $\nu(\varepsilon,\vec{\pi})$ of the fixed subspace $V_{j}$.

Proof.

Let $\epsilon>0$ be given and let $\nu>0$ be given by Theorem 5. Suppose that $\pi_{j}:\mathbb{R}^{n}\to V_{j}^{\perp}$ are given as in the statement of Proposition 6.

We follow the multi-scale argument of Guth which analyzes the following quantity. Define $D(\delta,\omega)$ to be the smallest constant in the inequality

[TABLE]

where the $\mathbb{T}_{j}$ are arbitrary finite collections of $\delta$-neighborhoods of $\omega$-perturbations of $V_{j}$. The multi-scale relationship relates $D(\delta,\omega)$ to $D(\delta/\omega,\omega)$.

Partition $[-1,1]^{n}$ into axis parallel cubes $Q$ of sidelength $\delta/\omega$. Then

[TABLE]

Let $T_{j}^{\prime}$ be a $O(\delta)$-neighborhood of an affine subspace which is parallel to $V_{j}$ and so that $T_{j}\cap Q\subset T^{\prime}_{j}\cap Q$. For each $Q$,

[TABLE]

where we used that $\delta^{-1}Q$ is a cube in $\mathbb{R}^{n}$ of sidelength $\omega^{-1}$ and $\delta^{-1}T_{j}$ is a 1-neighborhood of an $n_{j}^{\prime}$-dimensional affine subspace of $\mathbb{R}^{n}$, and so Theorem 1 applies. Now let $\tilde{T}_{j}=T_{j}+B(0,c\delta/\omega)$, where $c>0$ is large enough so that $T_{j}^{\prime}\cap Q\not=\emptyset$ implies $Q\subset\tilde{T}_{j}$. Then for each $x_{Q}\in Q$, $\delta^{-n_{j}}|\pi_{j}(T_{j}^{\prime}\cap Q)|\lesssim\chi_{\tilde{T}_{j}}(x_{Q})$, so

[TABLE]

Using this in (11) and with (10) leads to

[TABLE]

This proves that for a constant $\kappa>0$ which is allowed to depend on $\vec{\pi}$,

[TABLE]

Now use the multi-scale inequality. Iterate it $\ell$ times to obtain

[TABLE]

By Theorem 1 and Theorem 5, because $\vec{\pi}$ is a $\nu$-perturbation of $\vec{\pi_{0}}$, there exists $C=C(\vec{\pi})$ so that

[TABLE]

It follows from this and (12) that

[TABLE]

Choose $\omega$ so that $\omega^{-\epsilon}=C\kappa$ and take $\ell$ so that $\delta/\omega^{\ell}\geq 1\geq\delta/\omega^{\ell-1}$. It follows using the trivial bound $D(\delta/\omega^{\ell},\omega)\leq 1$ that

[TABLE]

This proves the proposition, where $\nu(\epsilon,\vec{\pi})=(C\kappa)^{-\epsilon^{-1}}$.

∎

Finally, we prove Theorem 2 using Proposition 6.

Proof of Theorem 2.

Let $\epsilon>0$ be given and let $\nu=\frac{\nu^{\prime}}{2}$, where $\nu^{\prime}>0$ is given by Theorem 5 (which is the same $\nu^{\prime}>0$ from Proposition 6). This parameter $\nu$ depends on the fixed subspaces $V_{j}^{0}$. We finish the proof of Theorem 2 using a compactness argument.

Let $\pi_{j}^{\theta}:\mathbb{R}^{n}\to(V_{j}^{\theta})^{\perp}$ denote orthogonal projection maps, where the $V_{j}^{\theta}$ are $n_{j}^{\prime}$-dimensional subspaces within $\nu$ of the $V_{j}^{0}$. For each $\vec{\pi}_{\theta}$, let the parameters $\nu(\varepsilon,\theta)=\nu(\varepsilon,\vec{\pi}_{\theta})>0$ be given by Proposition 6 and write $R(\frac{\nu(\varepsilon,\theta)}{2},\vec{\pi}_{\theta})$ to denote the open rectangle which is the product of $\frac{\nu(\varepsilon,\theta)}{2}$-neighborhoods of $V_{j}^{\theta}$ with respect to the Grassmanian metric on $n_{j}^{\prime}$-dimensional subspaces.

Let $K_{j}$ denote the closed $\nu$-neighborhood of $V_{j}^{0}$ and consider the $J$-fold product $K_{1}\times\cdots\times K_{J}$, which is compact. The open rectangles $R(\frac{\nu(\varepsilon,\theta)}{2},\vec{\pi}_{\theta})$ cover $K_{1}\times\cdots\times K_{J}$, so we may extract a finite subcover indexed by $\theta_{l}$. Let $\nu_{\varepsilon}=\frac{1}{10}\underset{l}{\min}\,\nu(\varepsilon,\theta_{l})>0$. For each $l$, cover $R(\frac{\nu(\varepsilon,\theta_{l})}{2},\vec{\pi}_{\theta_{l}})$ by a finite set of $R(\nu_{\varepsilon},\vec{\pi}_{\theta_{l}^{m}})\subset R(\frac{\nu(\varepsilon,\theta_{l})}{2},\vec{\pi}_{\theta_{l}})$.

The number of $\theta_{l}^{m}$ needed for the $R(\nu_{\varepsilon},\vec{\pi}_{\theta_{l}^{m}})$ to cover $K_{1}\times\ldots\times K_{J}$ is determined by fixed dimensions, $\nu$, and the numbers $\nu(\epsilon,\theta)$. Since the set of $\nu(\epsilon,\theta)$ themselves depends on the fixed $V_{j}^{0}$, $\nu$, and $\epsilon$, this number of $\theta_{l}^{m}$ is an acceptable contribution to the $C_{\epsilon}$ constant in the statement of Theorem 2.

We have covered $K_{1}\times\cdots\times K_{J}$ by at most $C_{\varepsilon}$ many $\nu_{\varepsilon}$-cubes, each of which is contained in some $R(\nu(\varepsilon,\theta_{l}),\vec{\pi}_{\theta_{l}})$ (where there are at most $C_{\varepsilon}$ many $l$). Write $\mathbb{T}_{j,\theta_{l}^{m}}$ for the collection of $T_{j}\in\mathbb{T}_{j}$ which are also in the $\nu_{\varepsilon}$-neighborhood of $V_{j}^{\theta_{l}^{m}}$. Then by pigeonholing,

[TABLE]

The tuple $(V_{1}^{\theta_{l_{1}}^{m_{1}}},\ldots,V_{J}^{\theta_{l_{J}}^{m_{J}}})$ is contained in $K_{1}\times\cdots\times K_{J}$, and therefore there is some $\theta_{l_{0}}^{m_{0}}$ so that $V_{j}^{\theta_{l_{j}}^{m_{j}}}$ is within $\nu_{\varepsilon}$ of $V_{j}^{\theta_{l_{0}}^{m_{0}}}$ for each $j$. Then $\mathbb{T}_{j,\theta_{l_{j}}^{m_{j}}}$ is a collection of $\delta$-neighborhoods of affines subspaces which are within $2\nu_{\varepsilon}$ of $V_{j}^{\theta_{l_{0}}^{m_{0}}}$ for each $j$. Since the rectangle of $\nu_{\varepsilon}$-neighborhoods of $V_{j}^{\theta_{l_{0}}^{m_{0}}}$ is contained in the rectangle of $\frac{\nu(\varepsilon,\theta_{l_{0}})}{2}$-neighborhoods of $V_{j}^{\theta_{l_{0}}}$, and by definition, $\nu_{\varepsilon}<\frac{\nu(\varepsilon,\theta_{l_{0}})}{2}$, conclude that $\mathbb{T}_{j,\theta_{l_{j}}^{m_{j}}}$ is a collection of affine subspaces within $\nu(\varepsilon,\theta_{l_{0}})$ of $V_{j}^{\theta_{l_{0}}}$ for each $j$. This means that we may control the right hand side of (14) by Proposition 6. Since there are $C_{\epsilon}$ many $\theta_{l}^{m}$, we are done.

∎

5. Some stability of $\underline{\text{{BL}}}(\vec{\pi},\vec{p},R)$

In this section, we describe a concrete stability result using the factoring-through-critical subspaces argument from [4]. In order to keep track of the implicit constant, we factor through one-dimensional subspaces at a time, sacrificing the probable optimal growth rate in $R$ except for special cases of exponents $\vec{p}$. Let $H$ be a vector space of dimension $n$, $V_{j}^{0}$ be $n_{j}^{\prime}$-dimensional subspaces of $H$, and $n_{j}=n-n_{j}^{\prime}$.

Notation 7**.**

Let $V_{j}^{0}\subset\mathbb{R}^{n}$ be nontrivial $n_{j}^{\prime}-$dimensional subspaces and let $H_{j}^{0}=(V_{j}^{0})^{\perp}$, which is $n_{j}$-dimensional.

Notation 8**.**

Let $V_{j}\subset\mathbb{R}^{n}$ also be $n_{j}^{\prime}-$dimensional subspaces and write $H_{j}=V_{j}^{\perp}$, which is $n_{j}$-dimensional.

Definition 9**.**

Suppose that $\pi_{j}^{0}:\mathbb{R}^{n}\to H_{j}^{0}$ and $\pi_{j}:\mathbb{R}^{n}\to H_{j}$ are orthogonal projection maps. For $\nu>0$, $\|\vec{\pi}-\vec{\pi}_{0}\|<\nu$ means that for each $j\in\{1,\ldots,J\}$, $V_{j}$ is within a distance $\nu$ of $V_{j}^{0}$ in the usual Grassmanian metric on $n_{j}^{\prime}$-dimensional subspaces.

Definition 10**.**

For each $j=1,\ldots,J$, fix a subset $\mathcal{L}^{0}_{j}\subset\mathbb{Z}^{n}$ which satisfies $H_{j}\subset\underset{{\bf{v}}\in\mathcal{L}_{j}^{0}}{\cup}({\bf{v}}+Q)$, where $Q=[0,1)^{n}$. For a subspace $W_{j}\subset H_{j}$ and a function $f\in L^{\infty}(H_{j})$, define the quantity

[TABLE]

Notation 11**.**

For vectors $w_{1},\ldots,w_{r}\in\mathbb{R}^{n}$, let $\langle w_{1},\ldots,w_{r}\rangle$ denote the linear span of $w_{1},\ldots,w_{r}$. For each $j$, $\langle\pi_{j}^{0}(w_{1}),\ldots,\pi_{j}^{0}(w_{r})\rangle^{\perp}$ will denote the orthogonal complement of $\langle w_{1},\ldots,w_{r}\rangle$ in $H_{j}^{0}$ (and the same for $\pi_{j},H_{j}$).

Lemma 12**.**

Fix $\vec{\pi}_{0}$. There exists a basis $e_{1},\ldots,e_{n}$ of $\mathbb{R}^{n}$, a small parameter $\nu>0$, and a constant $C>0$ such that the following holds. If $\vec{\pi}$ satisfies $\|\vec{\pi}_{0}-\vec{\pi}\|<\nu$, then for each $j\in\{1,\ldots,J\}$,

[TABLE]

and

[TABLE]

for all $i\in\{1,\ldots,n-1\}$. Furthermore, for each $j\in\{1,\ldots,J\}$ and $i\in\{2,\ldots,n\}$,

[TABLE]

Proof.

We describe an algorithm for choosing the vectors $e_{1},\ldots,e_{n}$.

Choose a unit vector $e_{1}\in\mathbb{R}^{n}$ so that $e_{1}\not\in V_{j}^{0}$ for all $j=1,\ldots,J$. Then $\pi_{j}^{0}(e_{1})\not=0$ for all $j$, so for $\nu>0$ sufficiently small,

[TABLE]

whenever $\|\vec{\pi}_{0}-\vec{\pi}\|<\nu$.

Now suppose that $e_{1},\ldots,e_{r}$ have been chosen to satisfy the properties in the lemma until $i=r$. Let $J_{r}\subset\{1,\ldots,J\}$ be the set of indices $j$ for which $\text{dim}\langle\pi_{j}^{0}(e_{1}),\ldots,\pi_{j}^{0}(e_{r})\rangle<n_{j}$. First suppose that $J_{r}$ is nonempty. Choose a unit vector $e_{r+1}\in\mathbb{R}^{n}$ such that $e_{r+1}\not\in\underset{j\in J_{r}}{\cup}\text{Span}(e_{1},\ldots,e_{r},V_{j}^{0})$ (recall that $V_{j}^{0}=\ker\pi_{j}^{0}$). Then for each $j\in J_{r}$, $\pi_{j}^{0}(e_{r+1})\not\in\langle\pi_{j}^{0}(e_{1}),\ldots,\pi_{j}^{0}(e_{r})\rangle$. This is because if there were constants $c_{s}$ so that $\pi_{j}^{0}(e_{r+1})=c_{1}\pi_{j}^{0}(e_{1})+\cdots+c_{r}\pi_{j}^{0}(e_{r})$, then $e_{r+1}-(c_{1}e_{1}+\cdots+c_{r}e_{r})\in V_{j}^{0}$, which contradicts how we selected $e_{r+1}$. Thus for each $j\in J_{r}$,

[TABLE]

For sufficiently small $\nu>0$ and appropriate $C>0$, this also guarantees that

[TABLE]

whenever $\|\vec{\pi}_{0}-\vec{\pi}\|<\nu$ and $j\in J_{r}$, and verifies property (16) for $i=r+1$. If $j\not\in J_{r}$, then $\text{dim}(\pi_{j}^{0}(e_{1}),\ldots,\pi_{j}^{0}(e_{r})\rangle=n_{j}$. For sufficiently small $\nu>0$, we also have $\text{dim}(\langle\pi_{j}(e_{1}),\ldots,\pi_{j}(e_{r})\rangle=n_{j}$ whenever $\|\vec{\pi}_{0}-\vec{\pi}\|<\nu$ and $j\not\in J_{r}$. Then $\langle\pi_{j}(e_{1}),\ldots,\pi_{j}(e_{r})\rangle=V_{j}^{\perp}$, so

[TABLE]

If $J_{r}$ is empty, then choose $e_{r+1},\ldots,e_{n}$ to be any orthonormal basis of $\langle e_{1},\ldots,e_{r}\rangle^{\perp}$ in $\mathbb{R}^{n}$ and the induction ends. Since $J_{r}$ is empty, for each $j\in\{1,\ldots,J\}$, $\text{dim}\langle\pi^{0}_{j}(e_{1}),\ldots,\pi_{j}^{0}(e_{r})\rangle=n_{j}$ and for sufficiently small $\nu>0$, $\text{dim}\langle\pi_{j}(e_{1}),\ldots,\pi_{j}(e_{r})\rangle=n_{j}$ whenever $\|\vec{\pi}_{0}-\vec{\pi}\|<\nu$. Thus

[TABLE]

for each $j\in\{1,\ldots,J\}$ and $s\in\{r+1,\ldots,n\}$.

∎

Definition 13**.**

For each $r\in\{1,\ldots,J\}$, let $\mathcal{J}_{r}=\{j\in\{1,\ldots,J\}:n_{j}\geq r\}$.

Proof of Theorem 3.

In this argument, all constants are permitted to depend on $\vec{\pi}_{0}$ and $\vec{p}$, but not on $\vec{\pi}$. Consider the integral

[TABLE]

Let $e_{1},\ldots,e_{n}$ and $\nu>0$ be as in Lemma 12. Express ${\bf{x}}$ in terms of this basis as ${\bf{x}}=y_{1}+\cdots+y_{n}$, where $y_{i}\in\langle e_{i}\rangle$. Perform a change of variables and use Fubini’s theorem to bound the above integral by

[TABLE]

where the constants $c,C>0$ only depend on $\vec{\pi}_{0}$ and the specific basis $e_{1},\ldots,e_{n}$ we fixed from Lemma 12. Each set $\{|y_{i}|\leq cR\}$ is a subset of $\langle e_{i}\rangle$. For each $\underline{y}=y_{2}+\cdots+y_{n}\in\langle e_{2},\ldots,e_{n}\rangle$, consider the innermost integral

[TABLE]

Let $H_{j}^{1}=\langle\pi_{j}(e_{1})\rangle$. Apply the 1-dimensional base case in the proof of Proposition 4 to obtain the upper bound

[TABLE]

The exponent $-\gamma(p_{r})$ depends on $\vec{p}$ and is negative. Since each of the $|\pi_{r}(e_{1})|$ are bounded below by Lemma 12, there is a constant $C$ so that the previous displayed math is bounded by

[TABLE]

For each $j$, let $L_{j}^{(0)}=\pi_{j}$ and let

[TABLE]

Here the notation $(H_{j}^{1})^{\perp}$ is the orthogonal complement of $\langle\pi_{j}(e_{1})\rangle$ inside of $H_{j}$. The maps $L_{j}^{(1)}$ are clearly surjective because $\pi_{j}(\mathbb{R}^{n})=H_{j}^{1}+\pi_{j}(\langle e_{2},\ldots,e_{n}\rangle)=H_{j}$. For each $\underline{y}\in\langle e_{2},\ldots,e_{n}\rangle$, define $u_{j}^{(1)}(\underline{y})\in H_{j}^{1}$ by $u_{j}^{(1)}(\underline{y})=P_{H_{j}^{1}}(\pi_{j}(\underline{y}))$, so $\pi_{j}(\underline{y})=u_{j}^{(1)}(\underline{y})+L_{j}^{(1)}(\underline{y})$. Now analyze the quantities $\|f_{j}(\cdot+\pi_{j}(\underline{y})\|_{H_{j}^{1}}$ that appear in (19).

[TABLE]

where in the last line, we used the fact that since $u_{j}^{(1)}(\underline{y})\in H_{j}^{1}$, $H_{j}^{1}+u_{j}^{(1)}(\underline{y})=H_{j}^{1}$. Note that for each ${\bf{v}}\in\mathcal{L}_{j}^{0}$, the number of ${\bf{w}}\in\mathcal{L}_{j}^{0}$ which satisfy $({\bf{w}}+u_{j}^{(1)}(\underline{y})+Q)\cap({\bf{v}}+Q)\not=\emptyset$ is controlled by the ambient dimension $n$. This means that

[TABLE]

Putting everything together so far, we have bounded (18) by

[TABLE]

where $C$ depends on $\vec{\pi}_{0}$ and $\vec{p}$.

Now suppose that the integral (18) is bounded by a constant factor times

[TABLE]

where $H_{j}^{r}=\langle\pi_{j}(e_{1}),\ldots,\pi_{j}(e_{r})\rangle$. The maps $L_{j}^{(r)}:\langle e_{r+1},\ldots,e_{n}\rangle\to(H_{j}^{r})^{\perp}$ are defined by

[TABLE]

By construction of the $e_{1},\ldots,e_{n}$, either $L_{j}^{(i)}(e_{i+1})=0$ or $|L_{j}^{(i)}(e_{i+1})|>c$ for $c$ depending only on $\vec{\pi}_{0}$. Furthermore, if $L_{j}^{(i)}(e_{i+1})=0$, then $L_{j}^{(i)}(e_{r})=0$ for all $r=i+1,\ldots,n$. In that case, for each $\underline{y}\in\langle e_{i+1},\ldots,e_{n}\rangle$,

[TABLE]

so we may ignore this constant factor from the integral (20) above. Assume without loss of generality that $|L_{j}^{(i)}(e_{i+1})|>c$ for each $j$. Then apply the one-dimensional base case of the proof of Proposition 4 to obtain for each $\underline{y}\in\langle e_{i+2},\ldots,e_{n}\rangle$

[TABLE]

The norms on the right hand side are equal to

[TABLE]

It suffices to sum over lattice points $m\in\mathcal{L}_{j}\subset\mathcal{L}_{0}$ and $r\in\mathcal{L}_{j}^{\prime}\subset\mathcal{L}_{j}^{0}$ which are within a distance $\sqrt{n}$ of $\langle L_{j}^{(i)}(e_{i+1})\rangle$ and $H_{j}^{i}$ respectively. Similar to an analogous step in the proof of Proposition 4, (21) is bounded by

[TABLE]

where $H_{j}^{i+1}=\langle\pi_{j}(e_{1}),\ldots,\pi_{j}(e_{i+1})\rangle$. Here we used that both $H_{j}^{i}$ and $\langle L_{j}^{(i)}(e_{i+1})\rangle$ are contained in $H_{j}^{i+1}$. For each ${\bf{v}}\in\mathbb{Z}^{n}$, the number of $(r,m)\in\mathcal{L}_{j}^{\prime}\times\mathcal{L}_{j}$ such that

[TABLE]

is controlled by a constant depending only on dimension. Also use the fact that $\underset{{\bf{v}}\in\mathbb{Z}^{n}}{\cup}({\bf{v}}+Q)$ covers $H_{j}^{i+1}$ to bound the quantity in (22) by a constant multiple of

[TABLE]

This concludes the intermediate inductive step.

Finally, consider the result of this induction. We have shown that (18) is bounded by a constant multiple of

[TABLE]

The nonnegative input functions $f_{j}$ in the definition of $\underline{\text{BL}}(\vec{\pi},\vec{p},R)$ are assumed to be constant on cubes in the unit cube lattice of $\mathbb{R}^{n_{j}}$, where we identify $H_{j}$ (with the induced metric from $\mathbb{R}^{n}$) isometrically with $\mathbb{R}^{n_{j}}$. Then for a dimensional constant $C$,

[TABLE]

It remains to check that the exponent of $R$ is the desired quantity. By the construction in Lemma 12, $\dim(\langle L_{j}^{(r-1)}(e_{r})\rangle)=1$ if $n_{j}\geq r$ and $\dim(\langle L_{j}^{(r-1)}(e_{r})\rangle)=0$ if $n_{j}<r$. Define indexing sets $\mathcal{J}_{r}:=\{j\in\{1,\ldots,J\}:n_{j}\geq r\}$. Then for each $r\in\{1,\ldots,n\}$,

[TABLE]

Note in addition that if $p_{j}\leq\frac{1}{J}$ for each $j$, then for each $r$, $1\leq r\leq n$,

[TABLE]

In this case,

[TABLE]

∎