\section{Funding}

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






\section{Lower bound for $\text{\underline{{\normalfont BL}}}(\vec{\pi},\vec{p},R)$ \label{lower}}
\vspace*{.03in}

In this section, we describe the example given by BCCT in \textsection 5 from \cite{BCCT2}. We use this example to demonstrate that there exists $c>0$ satisfying
\[ c\underset{V\le \mathbb R^n}{\sup}R^{\dim V-\sum\limits_{j=1}^Jp_j\dim\pi_j(V)} \le \text{\underline{BL}}(\vec{\pi},\vec{p},R) . \]

\begin{proof}[Proof of the lower bound for $\text{\underline{\normalfont BL}}(\vec{\pi},\vec{p},R)$ from Theorem \ref{main}]



Fix a vector $\vec{p}\in[0,1]^J$ and surjective linear maps $\pi_j:\mathbb R^n\to\mathbb R^{n_j}$. Let $V\le \mathbb R^n$ 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 
\[ S_{j}=\{\v\in\mathbb Z^{n_j}:|P_{\pi_j(V)}(\v)|\le R+\sqrt{n}\quad\text{and}\quad |P_{\pi_j(V)^\perp}(\v)|\le 1+\sqrt{n}\} \]
and let $f_j$ be the indicator function of the set $\underset{\v \in S_j}{\cup} (\v+Q_j)$, 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 
\[ S=\{{\bf{x}}\in V:|P_V({\bf{x}})|\le c_0R\quad\text{and}\quad |P_{V^\perp}({\bf{x}})|\le c_0\}. \]
Let $|\pi_j|$ be the operator norm of $\pi_j$ so that $|\pi_j({\bf{x}})|\le |\pi_j||{\bf{x}}|$ 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 $S$, $|{\bf{v}}|\le c_0R$ and $|{\bf{w}}|\le c_0$, which implies that $|\pi_j({\bf{v}})|\le|\pi_j||{\bf{v}}|\le \frac{1}{2}R$ and $|\pi_j({\bf{w}})|\le |\pi_j||{\bf{w}}|\le \frac{1}{2}$. Then \[|P_{\pi_j(V)}(\pi_j({\bf{x}}))|\le |\pi_j(\v))|+|P_{\pi_j(V)}(\pi_j({\bf{w}}))|\le R \]
and 
\[|P_{\pi_j(V)^\perp}(\pi_j({\bf{x}}))|= |P_{\pi_j(V)^\perp}(\pi_j({\bf{w}}))|\le 1. \]
It follows from these displayed inequalities that  $\pi_j({\bf{x}})\in\underset{\v\in S_j}{\cup}(\v+Q_j)$, which 
holds for all $j=1,\ldots,J$. Putting everything together, we have
\begin{align*}
    Cc_0^nR^{\dim V}&\le  \int_{S}\prod_{j=1}^J(1)^{p_j}d{\bf{x}}\le  \int_{[-R,R]^n}\prod_{j=1}^Jf_j(\pi_j({\bf{x}}))^{p_j}d{\bf{x}}\\
    &\le \text{\underline{BL}}(\vec{\pi},\vec{p},R)\prod_{j=1}^J\left(\int_{\mathbb R^{n_j}}f_j({\bf{x}}_j)d{\bf{x}}_j\right)^{p_j} 
\end{align*}   
for a constant $C$ which depends on the dimension $n$. Using the inclusion 
\[ \underset{\v\in S_j}{\cup}(\v+Q_j) \subseteq\{{\bf{x}}_j\in\mathbb R^{n_j}:|P_{\pi_j(V)}({\bf{x}}_j)|\le R+2\sqrt{n}\quad\text{and}\quad|P_{\pi_j(V)^\perp}({\bf{x}}_j)|\le 1+2\sqrt{n}\},\]
and that $f_j$ is the indicator function of $\underset{\v\in S_j}{\cup}(\v+Q_j)$, obtain the final inequality
\[ Cc_0^{n}R^{\dim V}\le \text{\underline{BL}}(\vec{\pi},\vec{p},R)\left(\int_{\mathbb R^{n_j}}f_j({\bf{x}}_j)d{\bf{x}}_j\right)^{p_j}\le  \tilde{C}^{p_1+\cdots+p_J}\text{\underline{BL}}(\vec{\pi},\vec{p},R) \prod_{j=1}^J R^{p_j\dim \pi_j(V)} \]
where $\tilde{C}$ is a constant depending on $n$. 
\end{proof}






\section{Upper bound for $\text{\underline{BL}}(\vec{\pi},\vec{p},R)$ \label{upper} }


The upper bound from Theorem \ref{main} follows as a corollary to Proposition \ref{tech} below. The proof of Proposition \ref{tech} proceeds in an analogous way as the proofs of Theorems 2.1 and 2.5 in \cite{BCCT2}. 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 $H_j\subset\underset{\v\in\mathcal{L}_j^0}{\cup}(\v+Q_j)$, where $Q_j=[0,1)^{k_j}$.  For a parameter $A>0$ and a function $f\in L^\infty(H_j)$, define the quantity
\begin{equation}  \|f\|_{A,\mathcal{L}_j^0}:=\sum_{\v\in\mathcal{L}_j^0}\|f\|_{L^\infty((\v+AQ_j)\cap H_j)},\end{equation}
where $AQ_j=[0,A)^{k_j}\subset\mathbb R^{k_j}$. 



\begin{prop}\label{tech} 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 $A_j\ge 1$, for $j=1,\ldots,J$, and a constant $C>0$ such that
\[ \int_{\{{\bf{x}}\in H:|{\bf{x}}|<R\}}\prod_{j=1}^Jf_j(\pi_j({\bf{x}}))^{p_j}d{\bf{x}}\le C\sup_{V\le H}R^{\dim V-\sum\limits_{i=1}^Jp_i\dim\pi_i(V)}\prod_{j=1}^J\|f_j\|_{A_j,\mathcal{L}_j^0}^{p_j} \]
for all  $R\ge 1$ and all nonnegative, measurable functions $f_j:H_j\to[0,\infty)$. 
\end{prop}

Granting this proposition, we first prove the upper bound from Theorem \ref{main}. 
\begin{proof}[Proof of the upper bound for $\text{\underline{\emph{BL}}}(\vec{\pi},\vec{p},R)$] 
Take $H=\mathbb R^n$, $H_j=\mathbb R^{n_j}$, and $\mathcal{L}_j^0=\mathbb Z^{n_j}$ in Proposition \ref{tech}. It follows from the proposition that there exist constants $A_j,C$ such that
\[ \int_{[-R,R]^n}\prod_{j=1}^Jf_j(\pi_j({\bf{x}}))d{\bf{x}}\le C\sup_{V\le \mathbb R^n}R^{\dim V-\sum\limits_{i=1}^Jp_i\dim\pi_i(V)}\prod_{j=1}^J\|f_j\|_{A_j,\mathbb Z^{n_j}}^{p_j}\]
for all $R\ge 1$ and all measurable functions $f_j:\mathbb R^{n_j}\to [0,\infty)$. Consider functions $f_j$ which are constant on cubes $\v+[0,1)^{n_j}$ where $\v\in\mathbb Z^{n_j}$. Then 
\begin{align*}
    \|f_j\|_{A_j,\mathbb Z^{n_j}}=\sum_{m\in\mathbb Z^{n_j}}\underset{x\in m+A_jQ_j}{\text{ess sup}}f_j(x) &\le \sum_{m\in\mathbb Z^{n_j}}\sum_{\substack{k\in\mathbb Z^{n_j}\\ k\in m+A_jQ_j}}f_j(k)\\
    &\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({\bf{x}}_j)d{\bf{x}}_j , 
\end{align*}
which finishes the proof.
\end{proof}


\begin{proof}[Proof of Proposition \ref{tech}] Without loss of generality, assume that $0<\|f_j\|_{1,\mathcal{L}_j^0}<\infty$ for all $j=1,\ldots,J$. If $\underset{V\le H}{\sup}[\dim V-\sum\limits_{j=1}^Jp_j\dim\pi_j(V)]\le 0$, then Proposition \ref{tech} follows from Theorem 2.5 in \cite{BCCT2}, 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 $1\le \sum\limits_{j=1}^Jp_j$, then choose $\delta_j\in[0,p_j)$ so that $1=\sum\limits_{j=1}^J(p_j-\delta_j)$. 
Using the generalized H\"{o}lder's inequality,
\begin{align} 
\int_{\{x\in H:|x|<R\}}\prod_{j=1}^Jf_j(\pi_j(x))^{p_j}dx&\le\prod_{j=1}^J\left(\int_{H}f_j(\pi_j(x))^{p_j/(p_j-\delta_j)}dx\right)^{p_j-\delta_j} \nonumber \\
&= \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} \nonumber 
\end{align} 
where $|\pi_j|$ is from the subsitution $u=\pi_j(x)$. For each $1\le j\le J$, 
\[ \int_Hf_j(x)^{p_j/p_j-\delta_j}dx\le \sum_{m\in\mathcal{L}_j^0}\|f_j\|_{L^\infty(m+[0,1))}^{p_j/(p_j-\delta_j)}\le \big(\sum_{m\in\mathcal{L}_j^0}\|f_j\|_{L^\infty(m+[0,1))}\big)^{p_j/(p_j-\delta_j)},\]
which finishes this case. 

The other case is that  $1>\sum\limits_{j=1}^Jp_j$. Let $s=1-\sum\limits_{j=1}^Jp_j$. Then by the generalized H\"{o}lder's inequality, 
\begin{align*} 
\int_{\{x\in H:|x|<R\}}\prod_{j=1}^Jf_j(\pi_j(x))^{p_j}dx&\le \left(\int_{\{x\in H:|x|<R\}}dx\right)^s\prod_{j=1}^J\left(\int_{H}f_j(\pi_j(x))dx \right)^{p_j} \\
&\le  (2R)^{1-\sum\limits_{i=1}^Jp_i}\prod_{r=1}^J|\pi_r|^{-p_r}\prod_{j=1}^J\|f_j\|_{1,\mathcal{L}_j^0}^{p_j}. 
\end{align*} 
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 $W\le H$ such that
\[ \dim W-\sum_{i=1}^Jp_i\dim\pi_i(W)=\sup_{V\le \mathbb R^n}\left(\dim V-\sum_{i=1}^Jp_i\dim\pi_i(V)\right). \]
Define coordinates ${\bf{x}}=(x,y)\in W\oplus W^\perp=H$ and write
\begin{align}\label{2} \int_{\{x\in H:|{\bf{x}}|<R\}}\prod_{j=1}^Jf_j(\pi_j({\bf{x}}))^{p_j}d{\bf{x}}\le \int_{W^\perp}\int_{\{x\in W:|x|<R\}}\prod_{j=1}^Jf_j(\pi_j(x)+\pi_j(y))^{p_j}dxdy . \end{align}
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 $\v\in\mathcal{L}_j^0$ satisfying $|P_{\pi_j(W)^\perp}(\v)|\le \sqrt{k_j}$. Then $\mathcal{L}_j$ contains the collection of $\v\in\mathcal{L}_j^0$ such that $\v+Q_j$ has nonempty intersection with $\pi_j(W)$, from which it follows that $\underset{\v\in\mathcal{L}_j}{\cup}(\v+Q_j)$ 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
\begin{align}\label{3}  
\int_{\{W:|x|<R\}}\prod_{j=1}^Jf_j&(\pi_j(x)+\pi_j(y))^{p_j}dx \le CR^{\dim W-\sum\limits_{i=1}^Jp_i\dim \pi_i(W)}\prod_{j=1}^J\|f_j(\cdot+\pi_j(y))\|_{A_j,\mathcal{L}_j}^{p_j}    
\end{align} 
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. 

\begin{align*}
    \|f_j(\cdot+\pi_j(y))\|_{A_j,\mathcal{L}_j}&=\|f_j(\cdot+u_j(y)+L_j(y))\|_{A_j,\mathcal{L}_j}\\
    &= \sum_{\v\in\mathcal{L}_j}\underset{{\bf{x}}_j\in (\v+A_jQ_j)\cap\pi_j(W)}{\text{ess sup}}f_j({\bf{x}}_j+u_j(y)+L_j(y)) \\
    &= \sum_{\v\in\mathcal{L}_j+u_j(y)}\underset{{\bf{x}}_j\in (\v+A_jQ_j)\cap\pi_j(W)}{\text{ess sup}}f_j({\bf{x}}_j+L_j(y)) 
\end{align*}
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 $\v\in\mathcal{L}_j$, the number of ${\bf{w}}\in\mathcal{L}_j$ which satisfy $({\bf{w}}+u_j(y)+A_jQ_j)\cap(\v+A_jQ_j)\not= \emptyset$ is controlled by a dimensional constant times $A_j^{k_j}$. This means that
\[ \sum_{\v\in\mathcal{L}_j+u_j(y)}\underset{{\bf{x}}_j\in (\v+A_jQ_j)\cap\pi_j(W)}{\text{ess sup}}f_j({\bf{x}}_j+L_j(y)) \le CA_j^{k_j}\|f_j(\cdot+L_j(y))\|_{A_j,\mathcal{L}_j}. \]
Putting this together with (\ref{2}) and (\ref{3}), we have
\[ \int_{\{{\bf{x}}\in H:|{\bf{x}}|<R\}}\prod_{j=1}^Jf_j(\pi_j({\bf{x}}))^{p_j}d{\bf{x}}\le \tilde{C}R^{\dim W-\sum\limits_{i=1}^Jp_i\pi_i(W)}\int_{W^\perp}\prod_{j=1}^J\|f_j(\cdot+L_j(y))\|_{A_j,\mathcal{L}_j}^{p_j} dy ,\]
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
\begin{align*} 
F_j(z):=\|f_j(\cdot+z)\|_{A_j,\mathcal{L}_j}.
\end{align*} 
It remains to bound
\[ \int_{W^\perp}\prod_{j=1}^JF_j(L_j(y))^{p_j}dy.  \]
Fix the subset $\mathcal{L}_j'\subset\mathcal{L}_j^0$ of $\v\in\mathcal{L}_j^0$ that satisfy $|P_{\pi_j(W)}(\v)|\le \sqrt{k_j}$. The subset $\mathcal{L}_j'$ has the property that $\underset{\v\in\mathcal{L}_j'}{\cup}(\v+Q_j)$ 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 $C',A_j'$ such that
\begin{align} \label{noR} \int_{\{y\in W^\perp:|y|<R\}}\prod_{j=1}^JF_j(L_j(y))^{p_j}dy\le C'\sup_{V\le W^\perp} R^{\dim V-\sum\limits_{i=1}^Jp_i\dim L_i(V)}\prod_{j=1}^J\|F_j\|_{A_j',\mathcal{L}_j'}^{p_j}. \end{align}
For any subspace $V\subset W^\perp$, 
\[ \dim(V+W)=\dim(V)+\dim(W),\]
and for each $i$,
\[ \dim\pi_i(V+W)=\dim L_i(V)+\dim\pi_i(W). \]
Using the above equalities and the inequality
\[ \dim (V+W)-\sum_{i=1}^Jp_i\dim\pi_i(V+W)\le \dim W-\sum_{i=1}^Jp_i\dim\pi_i(W),\]
which holds because of the subcase we are in, conclude that $\dim V\le \sum\limits_{i=1}^Jp_i\dim L_i(V)$ for all $V\le W^\perp$. This means that the power of $R$ that appears on the right hand side of (\ref{noR}) 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
\[ \int_{\{{\bf{x}}\in H: |{\bf{x}}|<R\}}\prod_{j=1}^Jf_j(\pi_j({\bf{x}}))^{p_j}d{\bf{x}}\le\tilde{C}R^{\dim W-\sum\limits_{i=1}^Jp_i\dim\pi_i(W)}\prod_{j=1}^J\|F_j\|_{A_j',\mathcal{L}_j'}^{p_j}. \]
Observe that for each $j$,
\begin{align}
\sum_{k\in\mathcal{L}_j'} \underset{y\in (k+A_j'Q_j)\cap\pi_j(W)^\perp}{\text{ess sup}}F_j(y)&=\sum_{k\in\mathcal{L}_j'}\underset{y\in (k+A_j'Q_j)\cap\pi_j(W)^\perp}{\text{ess sup}}\sum_{m\in\mathcal{L}_j}\underset{x\in(m+A_jQ_j)\cap\pi_j(W)}{\text{ess sup}}f_j(x+y)\nonumber\\
    &\le \sum_{\substack{k\in\mathcal{L}_j'\\textfrak{m}\in\mathcal{L}_j}}\underset{x+y\in(k+m+(A_j+A_j')Q_j)\cap H_j}{\text{ess sup}}f_j(x+y). \label{4}
\end{align}
For each $\v\in\mathcal{L}_j^0$, the number of $(k,m)\in\mathcal{L}_j'\times\mathcal{L}_j$ such that \[(k+m+(A_j+A_j')Q_j)\cap(\v+(A_j+A_j')Q_j)\cap H_j\not=\emptyset\]
is bounded by 
\[ \#\{(k,m):\max(|P_{\pi_j(W)^\perp}(\v)-k|,|P_{\pi_j(W)}(\v)-m|)<2\sqrt{k_j}(1+A_j+A_j')\},\]
which is controlled by a constant depending only on dimension, $A_j$, and $A_j'$. Also use the fact that $\underset{\v\in\mathcal{L}_j^0}{\cup}(\v+Q_j)$ covers $H_j$ to bound the quantity in (\ref{4}) by 
\[ C\sum_{\v\in\mathcal{L}_j^0}  \underset{{\bf{x}}_j\in(\v+(A_j+A_j')Q_j)\cap H_j}{\text{ess sup}}f_j({\bf{x}}_j)=C\|f_j\|_{A_j+A_j',\mathcal{L}_j^0},  \]
which finishes this subcase. 



Now suppose that the ambient dimension $n>1$ and that all proper subspaces $W\le H$ satisfy 
\[ n-\sum_{i=1}p_in_i>\dim W-\sum_{i=1}^Jp_i\dim\pi_i(W). \]
Consider the set $K$ of all $J$-tuples $\vec{p}=(p_1,\ldots,p_J)\in [0,1]^J$ such that 
\[ n-\sum_{i=1}^Jp_in_i\ge \dim V- \sum_{i=1}^Jp_i\dim\pi_i(V)\]
for all proper subspaces $V\subset\mathbb R^n$. Equivalently, $K$ equals the intersection 
\[ K=[0,1]^J\cap \left(\underset{V\le\mathbb R^n}{\bigcap}\{\vec{p}\in\mathbb R^J:n-\sum_{i=1}^Jp_in_i\ge \dim V- \sum_{i=1}^Jp_i\dim\pi_i(V)\} \right).\]
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
\begin{align*}  
\int_{H\cap|{\bf{x}}|<R}\prod_{j=1}^Jf_j(\pi_j({\bf{x}}))^{p_j}d{\bf{x}}&\le C\prod_{k}\sup_{V\le H}R^{\lambda_k(\dim V-\sum\limits_{i=1}^Jp_{k,i}\dim\pi_{i}(V))}\prod_{j=1}^J\|f_j\|_{A_j,\mathcal{L}_j^0}^{\lambda_kp_{k,j}} \\
&=  C\sup_{V\le H}R^{\dim V-\sum\limits_{i=1}^Jp_{i}\dim\pi_{i}(V)}\prod_{j=1}^J\|f_j\|_{A_j,\mathcal{L}_j^0}^{p_j},
\end{align*} 
so it suffices to consider the extreme points. 

Next we want to show that $K=\tilde{K}$ where 
\[\tilde{K}=[0,\infty)^J\cap \left(\underset{V\le\mathbb R^n}{\bigcap}\{\vec{p}\in\mathbb R^J:n-\sum_{i=1}^Jp_in_i\ge \dim V- \sum_{i=1}^Jp_i\dim\pi_i(V)\} \right). \]
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
\[ n-\sum_{i=1}^Jp_in_i\ge \dim\ker\pi_r-\sum_{i=1}^Jp_i\dim\pi_i(\ker\pi_r).\]
This leads to
\[ n_r=n-\dim\ker\pi_r\ge \sum_{i:i\not=r}p_i(n_i-\dim\pi_i(\ker\pi_r))+p_rn_r\ge p_rn_r ,\]
which means $p_r\le 1$ 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 
\[ n-\sum_{i=1}^Jp_in_i=\dim V-\sum_{i=1}^Jp_i\dim \pi_i(V)\]
for some proper subspace $V\le 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
\[ \int_{\{{\bf{x}}\in H:|{\bf{x}}|<R\}}\prod_{j:j\not=r}f_j(\pi_j({\bf{x}}))^{p_j}d{\bf{x}}\le C\sup_{V\le H}R^{\dim V-\sum\limits_{i:i\not=r}p_i\dim \pi_i(V)}\prod_{j:j\not=r}\|f_j\|_{A_j,\mathcal{L}_j^0}^{p_j} .\]
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 $\pi_J$, 
\[ \int_{\{{\bf{x}}\in H :|{\bf{x}}|\le R\}}f_J(\pi_J({\bf{x}}))^{p_J}d{\bf{x}} =  CR^{n-n_J}\int_{\{{\bf{x}}_J\in H_J:|{\bf{x}}_J|\le c R\}}f_J({\bf{x}}_J)^{p_J}d{\bf{x}}_J. \]
By H\"{o}lder's inequality, this is bounded by
\[ \tilde{C}R^{n-n_J}R^{n_J(1-p_J)}\left(\int_{H_J}f_J({\bf{x}}_J)d{\bf{x}}_J\right)^{p_j}\le\tilde{C}R^{n-p_Jn_J}\|f_J\|_{1,\mathcal{L}_J^0}^{p_J} \]
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. 

\end{proof}





\section{ A generalized multilinear Kakeya inequality \label{K}}

In order to prove Theorem \ref{genKak}, we first prove a stability condition in the form of Theorem \ref{techK}. Theorem \ref{genKak} is proved using the induction-on-scales approach of Guth \cite{guth}. 


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

\begin{theorem}[\cite{stabBL}]\label{techK} 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 
\[  \sup_{V\le \mathbb R^n}\big[\dim V-\sum_{j=1}^Jp_j\dim\pi_j(V)\big]\le\sup_{W\le \mathbb R^n}\big[\dim W-\sum_{j=1}^Jp_j\dim\pi_j^0(W)\big] \]
holds for all orthogonal projections $\pi_j:\mathbb R^n\to V_j$ where $V_j$ is an $n_j'$-dimensional subspace of $\mathbb R^n$ which is within a distance $\nu$ of the fixed subspace $V_j^0$.

\end{theorem}

\begin{proof} Let $1\le k\le n$ 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 $1\le j\le J$, choose a subset $I_j\subseteq \{1,\ldots,k\}$ satisfying $|I_j|=\dim\langle \pi_j^0(e_1),\ldots,\pi_j^0(e_k)\rangle)$, 
\[ k-\sum_{j=1}^Jp_j\dim\pi_j^0(\langle e_1,\ldots,e_k\rangle)=k-\sum_{j=1}^Jp_j|I_j|, \]
and 
\[ \bigwedge_{i\in I_j}\pi_j^0(e_i)\not= 0. \]
Since $(\vec{\pi},{\bf{e}'})\mapsto\bigwedge_{i\in I_j}\pi_j(e_i')\in\Lambda^{|I_j|}(\mathbb R^{n_j})$ is continuous for each $j$, there exists $\varepsilon(\bf{e}),\delta(\bf{e})>0$ such that $\bigwedge_{i\in I_j}\pi_j(e_i')\not=0$ for each $j$ whenever $\|\bf{e}'-\bf{e}\|<\varepsilon(\bf{e})$ and $\|\vec{\pi}-\vec{\pi}_0\|<\delta(\varepsilon)$. In particular, $\dim\langle \pi_j(e_1'),\ldots,\pi_j(e_k')\rangle)\ge |I_j|$ for each $j$, so 
\[ k-\sum_{j=1}^Jp_j\dim\pi_j(\langle e_1',\ldots,e_k'\rangle)\le k-\sum_{j=1}^Jp_j\dim\pi_j^0(\langle e_1,\ldots,e_k\rangle). \]
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 
\[ \{{\bf{e}}'\in\mathcal{E}_k:\|{\bf{e}}'-{\bf{e}}^{\ell}\|<\varepsilon({\bf{e}}^{\ell})\}\]
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 \ref{techK}), then for any ${\bf{e}}\in\mathcal{E}_k$, there is some ${\bf{e}}^{\ell}$ from compactness so that
\[ k-\sum_{j=1}^Jp_j\dim\pi_j(\langle e_1,\ldots,e_k\rangle)\le k-\sum_{j=1}^Jp_j\dim\pi_j^0(\langle e_1^\ell,\ldots,e_k^\ell\rangle). \]
Since the number of $k$ is bounded by dimension, Theorem \ref{techK} is proved. 


\end{proof} 

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

\begin{prop} \label{Kakstep1} 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 $n_j'$-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
\[ \int_{[-1,1]^n}\prod_{j=1}^J\big(\sum_{T_j\in\mathbb T_j}\chi_{T_j}\big)^{p_j}\le C(\varepsilon,\vec{\pi})\delta^{-\epsilon+n}\sup_{V\le \mathbb R^n}\delta^{-\dim V+\overset{J}{\underset{j=1}{\sum}} p_j\dim\pi_j^0(V)}\prod_{j=1}^J(\#\mathbb T_j)^{p_j} \]
holds for all finite collections $\mathbb T_j$ of $\delta$-neighborhoods of $n_j'$-dimensional affine subspaces of $\mathbb R^n$ which, modulo translations, are within a distance $\nu(\varepsilon,\vec{\pi})$ of the fixed subspace $V_j$. 
\end{prop}

\begin{proof} Let $\epsilon>0$ be given and let $\nu>0$ be given by Theorem \ref{techK}. Suppose that $\pi_j:\mathbb R^n\to V_j^\perp$ are given as in the statement of Proposition \ref{Kakstep1}. 

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 
\begin{equation} \nonumber \int_{[-1,1]^n}\prod_{j=1}^J\big(\sum_{T_j\in\mathbb T_j}\chi_{T_j}\big)^{p_j}\le D_{}(\delta,\omega)\prod_{j=1}^J(\delta^{n_j}\#\mathbb T_j)^{p_j} \end{equation}
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
\begin{align}  
\int_{[-1,1]^n}\prod_{j=1}^J\big(\sum_{T_j\in\mathbb T_j}\chi_{T_j}\big)^{p_j}&=\sum_Q\int_Q\prod_{j=1}^J\big(\sum_{T_j\in\mathbb T_j}\chi_{T_j\cap Q}\big)^{p_j}\label{ok2} . 
\end{align} 
Let $T_j'$ be a $O(\delta)$-neighborhood of an affine subspace which is parallel to $V_j^{}$ and so that $T_j\cap Q\subset T'_j\cap Q$. For each $Q$,
\begin{align}
\int_{Q}\prod_{j=1}^J\big(\sum_{\substack{T_j\in\mathbb T_j\\mathbb T_j\cap Q\not=\emptyset}}\chi_{T_j'\cap Q}\big)^{p_j}&\le  \int_{Q}\prod_{j=1}^J\big(\sum_{\substack{T_j\in\mathbb T_j\\mathbb T_j\cap Q\not=\emptyset}}\chi_{\pi_j^{}(T_j'\cap Q)}\circ\pi_j^{} \big)^{p_j}\nonumber\\
&\le \delta^n \text{\underline{BL}}(\vec{\pi_{}},\vec{p},\omega^{-1})\prod_{j=1}^J\big(\sum_{\substack{T_j\in\mathbb T_j\\mathbb T_j\cap Q\not=\emptyset}}\delta^{-n_j}|\pi_j^{}(T_j'\cap Q)|\big)^{p_j}. \label{ok}
\end{align}
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 $n_j'$-dimensional affine subspace of $\mathbb R^n$, and so Theorem \ref{main} applies. Now let $\tilde{T}_j=T_j+B(0,c\delta/\omega)$, where $c>0$ is large enough so that $T_j'\cap Q\not=\emptyset$ implies $Q\subset\tilde{T}_j$. Then for each $x_Q\in Q$, $\delta^{-n_j}|\pi_j^{}(T_j'\cap Q)|\lesssim \chi_{\tilde{T}_j}(x_Q)$, so
\[ \prod_{j=1}^J\big(\sum_{\substack{T_j\in\mathbb T_j\\mathbb T_j\cap Q\not=\emptyset}}\delta^{-n_j}|\pi_j^{}(T_j'\cap Q)|\big)^{p_j}\lesssim |Q|^{-1}\int_Q \prod_{j=1}^J\big(\sum_{\substack{T_j\in\mathbb T_j\\mathbb T_j\cap Q\not=\emptyset}}\chi_{\tilde{T}_j}\big)^{p_j} .    \]
Using this in (\ref{ok}) and with (\ref{ok2}) leads to 
\begin{align*}
    \int_{[-1,1]^n}\prod_{j=1}^J&\big(\sum_{T_j\in\mathbb T_j}\chi_{T_j}\big)^{p_j}\lesssim \delta^n\text{\underline{BL}}(\vec{\pi_{}},\vec{p},\omega^{-1})\sum_Q(\delta/\omega)^{-n}\int_Q\prod_{j=1}^J\big(\sum_{\substack{T_j\in\mathbb T_j}}\chi_{\tilde{T}_j}\big)^{p_j}    \\
    &= \omega^{n}\text{\underline{BL}}(\vec{\pi_{}},\vec{p},\omega^{-1})\int_{[-1,1]^n}\prod_{j=1}^J\big(\sum_{\substack{T_j\in\mathbb T_j}}\chi_{\tilde{T}_j}\big)^{p_j} \\
&\le \omega^{n}\text{\underline{BL}}(\vec{\pi_{}},\vec{p},\omega^{-1})D_{}(\delta/\omega,\omega)\prod_{j=1}^J\big((\delta/\omega)^{n_j}\#\mathbb T_j)\big)^{p_j} 
\end{align*}
This proves that for a constant $\kappa>0$ which is allowed to depend on $\vec{\pi}$, 
\[ D(\delta,\omega)\le \kappa \omega^{n-\sum p_jn_j}\text{\underline{BL}}(\vec{\pi_{}},\vec{p},\omega^{-1})D_{}(\delta/\omega,\omega). \]



Now use the multi-scale inequality. Iterate it $\ell$ times to obtain 
\begin{equation} \label{iteration} D_{}(\delta,\omega)\le \kappa^\ell (\omega^\ell)^{n-\sum p_jn_j}\text{\underline{BL}}(\vec{\pi_{}},\vec{p},\omega^{-1})^\ell D_{}(\delta/\omega^\ell,\omega).\end{equation}
By Theorem \ref{main} and Theorem \ref{techK}, because $\vec{\pi_{}}$ is a $\nu$-perturbation of $\vec{\pi_0}$, there exists $C=C(\vec{\pi})$ so that 
\[ \text{\underline{BL}}(\vec{\pi_{}},\vec{p},\omega^{-1})\le C_{}\sup_{V\le\mathbb R^n}\omega^{-\dim V+\sum p_j\dim\pi_j^0(V)}.  \]
It follows from this and (\ref{iteration}) that
\begin{equation} \label{iteration2} 
D_{}(\delta,\omega)\le (C_{}\kappa)^\ell (\omega^\ell)^{n-\sum p_jn_j}\sup_{V\le\mathbb R^n}(\omega^{-\ell})^{\dim V-\sum p_j\dim\pi_j^0(V)} D_{}(\delta/\omega^\ell,\omega).\end{equation}
Choose $\omega$ so that $\omega^{-\epsilon}=C_{}\kappa$ and take $\ell$ so that $\delta/\omega^\ell\ge 1\ge \delta/\omega^{\ell-1}$. It follows using the trivial bound $D(\delta/\omega^\ell,\omega)\le1 $ that
\begin{align*} 
D(\delta,\omega)&\le \omega^{-\varepsilon\ell} (\omega^{\ell})^{n-\sum p_jn_j}\sup_{V\le\mathbb R^n}(\omega^{-\ell})^{\dim V-\sum p_j\dim\pi_j(V)} \\
&\le \delta^{-\epsilon+n-\sum p_jn_j}\sup_{V\le\mathbb R^n}\delta^{-\dim V+\sum p_j\dim\pi_j^0(V)}.  
\end{align*} 
This proves the proposition, where $\nu(\epsilon,\vec{\pi})=(C\kappa)^{-\epsilon^{-1}}$. 





\end{proof} 


Finally, we prove Theorem \ref{genKak} using Proposition \ref{Kakstep1}. 

\begin{proof}[Proof of Theorem \ref{genKak}] Let $\epsilon>0$ be given and let $\nu=\frac{\nu'}{2}$, where $\nu'>0$ is given by Theorem \ref{techK} (which is the same $\nu'>0$ from Proposition \ref{Kakstep1}). This parameter $\nu$ depends on the fixed subspaces $V_j^0$. We finish the proof of Theorem \ref{genKak} 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 $n_j'$-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 \ref{Kakstep1} 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 $n_j'$-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 \ref{genKak}. 

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,
\begin{align} 
\int_{[-1,1]^n}\prod_{j=1}^J\big(\sum_{T_j\in\mathbb T_j}\chi_{T_j}\big)^{p_j}&\lesssim_\epsilon \int_{[-1,1]^n}\prod_{j=1}^J\big(\sum_{T_j\in\mathbb T_{j,\theta_{l_j}^{m_j}}}\chi_{T_j}\big)^{p_j}.  \label{final}
\end{align} 
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 \eqref{final} by Proposition \ref{Kakstep1}. Since there are $C_\epsilon$ many $\theta_l^m$, we are done. 






\end{proof} 



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

In this section, we describe a concrete stability result using the factoring-through-critical subspaces argument from \cite{BCCT}. 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 $n_j'$-dimensional subspaces of $H$, and $n_j=n-n_j'$. 

\begin{notation} Let $V_j^0\subset\mathbb R^{n}$ be  nontrivial $n_j'-$dimensional subspaces and let $H_j^0=(V_j^0)^\perp$, which is $n_j$-dimensional.  
\end{notation}


\begin{notation} Let $V_j\subset\mathbb R^{n}$ also be $n_j'-$dimensional subspaces and write $H_j=V_j^\perp$, which is $n_j$-dimensional.
\end{notation}

\begin{definition}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 $n_j'$-dimensional subspaces. 
\end{definition}


\begin{definition}
For each $j=1,\ldots,J$, fix a subset $\mathcal{L}^0_j\subset \mathbb Z^{n}$ which satisfies $H_j\subset\underset{\v\in\mathcal{L}_j^0}{\cup}(\v+Q)$, 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
\begin{equation} \label{norm} \|f\|_{W_j}:=\sum_{\v\in\mathcal{L}_j^0}\|f\|_{L^\infty((\v+Q)\cap W_j)}.\end{equation}

\end{definition}


\begin{notation} 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 $\pi_j,H_j$).

\end{notation}


\begin{lemma}\label{algo} 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 $j\in\{1,\ldots,J\}$,
\[|\pi_j(e_1)|\ge C\]
and
\[ P_{\langle \pi_j(e_1),\ldots,\pi_j(e_i)\rangle^\perp}(\pi_j(e_{i+1}))=0\quad\text{or}\quad |P_{\langle \pi_j(e_1),\ldots,\pi_j(e_i)\rangle^\perp}(\pi_j(e_{i+1}))|\ge C \]
for all $i\in\{1,\ldots,n-1\}$. Furthermore, for each $j\in\{1,\ldots,J\}$ and $i\in\{2,\ldots,n\}$,
\begin{equation}\label{imp} \pi_j^0(e_i)\not\in\langle \pi_j^0(e_1),\ldots,\pi_j^0(e_{i-1})\rangle \implies \pi_j(e_i)\not\in\langle\pi_j(e_1),\ldots,\pi_j(e_{i-1})\rangle. \end{equation}

\end{lemma}

\begin{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,
\[ |\pi_j(e_1)|\ge \frac{1}{2}\min_i|\pi_i^0(e_1)|>0 \]
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 $e_{r+1}-(c_1e_1+\cdots+c_re_r)\in V_j^0$, which contradicts how we selected $e_{r+1}$. Thus for each $j\in J_r$,
\[ P_{\langle \pi_j^0(e_1),\ldots,\pi_j^0(e_r)\rangle^\perp}(\pi_j^0(e_{r+1}))\not=0. \]
For sufficiently small $\nu>0$ and appropriate $C>0$, this also guarantees that
\[ P_{\langle \pi_j(e_1),\ldots,\pi_j(e_r)\rangle^\perp}(\pi_j(e_{r+1}))|\ge C\]
whenever $\|\vec{\pi}_0-\vec{\pi}\|<\nu$ and $j\in J_r$, and verifies property \eqref{imp} 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
\[ P_{\langle \pi_j(e_1),\ldots,\pi_j(e_r)\rangle^\perp}(\pi_j(e_{r+1}))=0 .\]



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 
\[ P_{\langle \pi_j(e_1),\ldots,\pi_j(e_{s-1})\rangle^\perp}(\pi_j(e_s))=0\]
for each $j\in\{1,\ldots,J\}$ and $s\in\{r+1,\ldots,n\}$. 



\end{proof}

\begin{definition}  For each $r\in\{1,\ldots,J\}$, let $\mathcal{J}_r=\{j\in\{1,\ldots,J\}:n_j\ge r\}$. 
\end{definition}

\begin{proof}[Proof of Theorem \ref{locbd}] In this argument, all constants are permitted to depend on $\vec{\pi}_0$ and $\vec{p}$, but not on $\vec{\pi}$. Consider the integral
\begin{equation}\label{startingpt} \int_{[-R,R]^n}\prod_{j=1}^Jf_j(\pi_j({\bf{x}}))^{p_j}d{\bf{x}}.   \end{equation}
Let $e_1,\ldots,e_n$ and $\nu>0$ be as in Lemma \ref{algo}. 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 
\begin{equation}\label{step0'}C\int_{\{|y_n|\le cR\}}\cdots\int_{\{|y_1|\le cR\}}\prod_{j=1}^Jf_j(\pi_j(y_1)+\cdots+\pi_j(y_n))^{p_j}dy_1\cdots dy_n     \end{equation}
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 \ref{algo}. Each set $\{|y_i|\le cR\}$ 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
\begin{equation*} \int_{\{|y_1|\le cR\}}\prod_{j=1}^Jf_j(\pi_j(y_1)+\pi_j(\underline{y}))^{p_j}dy_1. \end{equation*}
Let $H_j^1=\langle\pi_j(e_1)\rangle$. Apply the 1-dimensional base case in the proof of Proposition \ref{tech} to obtain the upper bound 
\[ 2(cR)^{\max(1-\sum\limits_{j=1}^Jp_j,0)}\prod_{r=1}^J|\pi_r(e_1)|^{-\gamma(p_r)}\prod_{j=1}^J\|f_j(\cdot+\pi_j(\underline{y})\|_{H_j^1}^{p_j}. \]
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 \ref{algo}, there is a constant $C$ so that the previous displayed math is bounded by
\begin{equation}\label{step1'} CR^{\max(1-\sum\limits_{j=1}^Jp_j,0)}\prod_{j=1}^J\|f_j(\cdot+\pi_j(\underline{y})\|_{H_j^1}^{p_j} . \end{equation}

For each $j$, let $L_j^{(0)}=\pi_j$ and let
\[ L_j^{(1)}=P_{(H_j^1)^\perp}\circ\left.\pi_j\right|_{\langle e_2,\ldots,e_n\rangle}:\langle e_2,\ldots,e_n\rangle \to (H_j^1)^\perp. \] 
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 \eqref{step1'}. 
\begin{align*}
    \|f_j(\cdot+\pi_j(\underline{y}))\|_{H_j^1}&=\|f_j(\cdot+u_j^{(1)}(\underline{y})+L_j^{(1)}(\underline{y}))\|_{H_j^1}\\
    &= \sum_{\v\in\mathcal{L}_j^0}\underset{x\in (\v+Q)\cap H_j^1}{\text{ess sup}}f_j(x+u_j^{(1)}(\underline{y})+L_j^{(1)}(\underline{y})) \\
    &= \sum_{\v\in\mathcal{L}_j^0+u_j^{(1)}(\underline{y})}\underset{x\in (\v+Q)\cap H_j^1}{\text{ess sup}}f_j(x+L_j^{(1)}(\underline{y})) 
\end{align*}
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 $\v\in\mathcal{L}_j^0$, the number of ${\bf{w}}\in\mathcal{L}_j^0$ which satisfy $({\bf{w}}+u_j^{(1)}(\underline{y})+Q)\cap(\v+Q)\not= \emptyset$ is controlled by the ambient dimension $n$. This means that
\[ \sum_{\v\in\mathcal{L}_j+u_j^{(1)}(\underline{y})}\underset{x\in (\v+Q)\cap H_j^1}{\text{ess sup}}f_j({\bf{x}}_j+L_j^{(1)}(y)) \le C\|f_j(\cdot+L_j^{(1)}(y))\|_{H_j^1}. \]
Putting everything together so far, we have bounded \eqref{step0'} by 
\[ CR^{\max(1-\sum\limits_{j=1}^Jp_j,0)}\int_{\{|y_n|\le cR\}}\cdots\int_{\{|y_2|\le cR\}}\prod_{j=1}^J\|f_j(\cdot+L_j^{(1)}(\underline{y}))\|_{H_j^1}^{p_j} d\underline{y} ,\]
where $C$ depends on $\vec{\pi}_0$ and $\vec{p}$. 


Now suppose that the integral \eqref{step0'} is bounded by a constant factor times
\begin{align}\label{compcase} 
R^{\sum\limits_{r=1}^{i}\max(1-\sum\limits_{j=1}^Jp_j\text{dim}(\langle L_j^{(r-1)}(e_{r})\rangle),\,0)}\int_{\{|y_n|\le cR\}}\cdots\int_{\{|y_{i+1}|\le cR\}}\prod_{j=1}^J\|f_j(\cdot+L_j^{(i)}(y))\|_{H_j^i}^{p_j} dy ,
\end{align} 
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 
\[ L_j^{(r)}=P_{(H_j^r)^\perp}\circ \left.\pi_j\right|_{\langle e_{r+1},\ldots,e_n\rangle} . \] 
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 $\underline{y}\in\langle e_{i+1},\ldots,e_n\rangle$,
\[ \|f_j(\cdot+L_j^{(i)}(\underline{y}))\|_{H_j^i}=\|f_j\|_{H_j^i} ,\]
so we may ignore this constant factor from the integral (\ref{compcase}) 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 \ref{tech} to obtain for each $\underline{y}\in\langle e_{i+2},\ldots,e_n\rangle$
\begin{align*} 
\int_{\{|y_{i+1}|<cR\}}\prod_{j=1}^J&\|f_j(\cdot+L_j^{(i)}(y_{i+1})+L_j^{(i)}(\underline{y}))\|_{H_j^i} ^{p_j}dy_{i+1} \\
&\le CR^{\max(1-\sum\limits_{j}p_j\text{dim}(\langle L_j^{(i)}(e_{i+1})\rangle),0)}\prod_{j=1}^J\|\|f_j(s+t+L_j^{(i)}(\underline{y})\|_{H_j^i,s}\|_{\langle L_j^{(i)}(e_{i+1})\rangle,t}^{p_j}. \end{align*} 
The norms on the right hand side are equal to 
\begin{align} \label{fra}
\sum_{m\in\mathcal{L}_j^0}\underset{t\in(m+Q)\cap\langle L_j^{(i)}(e_{i+1})\rangle}{\text{ess sup}}\sum_{r\in\mathcal{L}_j^0}\underset{s\in (r+AQ)\cap H_j^i}{\text{ess sup}}f_j(s+t+L_j^{(i)}(\underline{y}))
\end{align}
It suffices to sum over lattice points $m\in \mathcal{L}_j\subset\mathcal{L}_0$ and $r\in \mathcal{L}_j'\subset\mathcal{L}_j^0$ 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 \ref{tech}, (\ref{fra}) is bounded by 
\begin{align}   \sum_{\substack{r\in\mathcal{L}_j'\\textfrak{m}\in\mathcal{L}_j}}\underset{s+t\in(r+m+2Q)\cap H_j^{i+1}}{\text{ess sup}}f_j(s+t+L_j^{(i)}(\underline{y})) \label{4'}
\end{align}
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 $\v\in\mathbb Z^n$, the number of $(r,m)\in\mathcal{L}_j'\times\mathcal{L}_j$ such that \[(r+m+2Q)\cap(\v+Q)\cap H_j^{i+1}\not=\emptyset\]
is controlled by a constant depending only on dimension.
Also use the fact that $\underset{\v\in\mathbb Z^n}{\cup}(\v+Q)$ covers $H_j^{i+1}$ to bound the quantity in (\ref{4'}) by a constant multiple of 
\[ \sum_{\v\in\mathbb Z^n}  \underset{x\in(\v+Q)\cap H_j^{i+1}}{\text{ess sup}}f_j(x+L_j^{(i)}(\underline{y}))=\|f_j(\cdot+L_j^{i}(\underline{y}))\|_{H_j^{i+1}}.   \]
This concludes the intermediate inductive step. 

Finally, consider the result of this induction. We have shown that \eqref{step0'} is bounded by a constant multiple of 
\begin{align*}
R^{\sum\limits_{r=1}^{n}\max(1-\sum\limits_{j=1}^Jp_j\text{dim}(\langle L_j^{(r-1)}(e_r)\rangle),\,0)}\prod_{j=1}^J\|f_j\|_{H_j}^{p_j}  .
\end{align*}
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 $C$,
\[ \|f_j\|_{H_j}\le C\int_{H_j}f_j. \]


It remains to check that the exponent of $R$ is the desired quantity. By the construction in Lemma \ref{algo}, $\dim (\langle L_j^{(r-1)}(e_r)\rangle)=1$ if $n_j\ge r$ and $\dim (\langle L_j^{(r-1)}(e_r)\rangle)=0$ if $n_j<r$. Define indexing sets $\mathcal{J}_r:=\{j\in\{1,\ldots,J\}:n_j\ge r\}$. Then  for each $r\in\{1,\ldots,n\}$,
\[ \sum_{j=1}^Jp_j\dim (\langle L_j^{(r-1)}(e_r)\rangle )=\sum_{j\in\mathcal{J}_r}p_j.\]





Note in addition that if $p_j\le \frac{1}{J}$ for each $j$, then for each $r$, $1\le r\le n$,
\[ 1-\sum_{j=1}^Jp_j\dim L_j^{(r-1)}(e_r)\ge 0. \]
In this case, 
\begin{align*} 
n-\sum_{j=1}^Jp_j \dim n_j&=1-\sum_{j=1}^Jp_j\dim\pi_j(e_1)+(n-1)-\sum_{j=1}^Jp_j \dim L^1(\langle e_2,\ldots,e_n\rangle)\\
&=\cdots=\sum_{r=1}^n(1-\sum_{j=1}^Jp_j\dim L_j^{(r-1)}(e_r)).
\end{align*} 


\end{proof}










