Scalar curvature rigidity of convex polytopes

Abstract

We prove a scalar curvature rigidity theorem for convex polytopes. The proof uses the Fredholm theory for Dirac operators on manifolds with boundary. A variant of a theorem of Fefferman and Phong plays a central role in our analysis.

Appendix: A variant of a theorem of Fefferman and Phong

In this section, we describe a variant of an estimate due to Fefferman and Phong [5], which plays a central role in our argument. Throughout this section, we fix an integer $n \geq 3$. We denote by $\mathcal{Q}$ the collection of all $(n-1)$-dimensional cubes of the form

$$ [2^{m} j_{1},2^{m} (j_{1}+1)] \times \ldots \times [2^{m} j_{n-1},2^{m} (j_{n-1}+1)] \times \{0\}, $$

where $m \in \mathbb{Z}$ and $j_{1},\ldots ,j_{n-1} \in \mathbb{Z}$. For abbreviation, we put

$$ \Gamma = \bigcup _{Q \in \mathcal{Q}} \partial Q. $$

If $Q_{1},Q_{2} \in \mathcal{Q}$ satisfy $Q_{1} \cap Q_{2} \setminus \Gamma \neq \emptyset$, then $Q_{1} \subset Q_{2}$ or $Q_{2} \subset Q_{1}$.

For each $(n-1)$-dimensional cube $Q \in \mathcal{Q}$, we denote by $|Q|$ the $(n-1)$-dimensional volume of $Q$.

Theorem A.1

Let us fix an integer $n \geq 3$ and a real number $\sigma \in (1,n-1)$. Suppose that $V$ is a nonnegative continuous function defined on the hyperplane $\mathbb{R}^{n-1} \times \{0\}$ with the property that

$$ \bigg( |Q|^{-1} \int _{Q} V^{\sigma }\bigg)^{\frac{1}{\sigma}} \leq \operatorname{diam}(Q)^{-1} $$

(14)

for each $(n-1)$-dimensional cube $Q \in \mathcal{Q}$. Let $F$ be a smooth function defined on the half-space $\mathbb{R}_{+}^{n} = \{x \in \mathbb{R}^{n}: x_{n} \geq 0\}$, and let $f$ denote the restriction of $F$ to the boundary $\partial \mathbb{R}_{+}^{n} = \mathbb{R}^{n-1} \times \{0\}$. Then

$$ \int _{Q} V f^{2} \leq C \int _{Q \times [0,\operatorname{diam}(Q)]} | \nabla F|^{2} + C \, \operatorname{diam}(Q)^{-1} \int _{Q} f^{2}. $$

for each $(n-1)$-dimensional cube $Q \in \mathcal{Q}$. The constant $C$ depends only on $n$ and $\sigma$.

The proof of Theorem A.1 is a straightforward adaptation of the arguments of Fefferman and Phong [5]. Let us fix an exponent $\tau \in (1,\sigma )$. Let $V: \mathbb{R}^{n-1} \times \{0\} \to \mathbb{R}$ be a nonnegative continuous function satisfying (14). We define a measurable function $W: \mathbb{R}^{n-1} \times \{0\} \to \mathbb{R}$ by

$$ W(x) = \sup _{Q \in \mathcal{Q}, x \in Q} \bigg( |Q|^{-1} \int _{Q} V^{ \sigma }\bigg)^{\frac{1}{\sigma}} $$

for each point $x \in \mathbb{R}^{n-1} \times \{0\}$. It follows from (14) that $W$ is locally bounded. Moreover, $V \leq W$ at each point in $\mathbb{R}^{n-1} \times \{0\}$.

Let $F$ be a smooth function defined on the half-space $\mathbb{R}_{+}^{n} = \{x \in \mathbb{R}^{n}: x_{n} \geq 0\}$, and let $f$ denote the restriction of $F$ to the boundary $\partial \mathbb{R}_{+}^{n} = \mathbb{R}^{n-1} \times \{0\}$. For each $(n-1)$-dimensional cube $Q \in \mathcal{Q}$, we denote by $f_{Q} = |Q|^{-1} \int _{Q} f$ the mean value of $f$ over the cube $Q$.

Lemma A.2

For each $(n-1)$-dimensional cube $Q_{0} \in \mathcal{Q}$, we have

$$ \bigg( |Q_{0}|^{-1} \int _{Q_{0}} W^{\tau }\bigg)^{\frac{1}{\tau}} \leq C \, \sup _{Q \in \mathcal{Q}, Q_{0} \subset Q} \bigg( |Q|^{-1} \int _{Q} V^{\sigma }\bigg)^{\frac{1}{\sigma}}, $$

where $C$ depends only on $n$, $\sigma$, and $\tau$.

Proof

For abbreviation, let

$$ \Lambda = \sup _{Q \in \mathcal{Q}, Q_{0} \subset Q} \bigg( |Q|^{-1} \int _{Q} V^{\sigma }\bigg)^{\frac{1}{\sigma}}. $$

It follows from (14) that $\Lambda <\infty$. We define a bounded measurable function $W_{0}: Q_{0} \to \mathbb{R}$ by

$$ W_{0}(x) = \sup _{Q \in \mathcal{Q}, x \in Q \subset Q_{0}} \bigg( |Q|^{-1} \int _{Q} V^{\sigma }\bigg)^{\frac{1}{\sigma}} $$

for each point $x \in Q_{0}$. Then

$$ W(x) = \max \{\Lambda ,W_{0}(x)\} $$

for each point $x \in Q_{0} \setminus \Gamma$. The function $W_{0}^{\sigma}$ is bounded from above by the maximal function associated with the function $V^{\sigma }\, 1_{Q_{0}}$. Hence, the weak version of the Hardy-Littlewood maximal inequality (cf. [19], Proposition 2.9 (i)) implies

$$ |Q_{0}|^{-1} \, |\{x \in Q_{0}: W_{0}(x)^{\sigma }> \alpha \}| \leq C \alpha ^{-1} \, |Q_{0}|^{-1} \int _{Q_{0}} V^{\sigma }\leq C \alpha ^{-1} \, \Lambda ^{\sigma }$$

(15)

for all $\alpha >0$. We multiply both sides of (15) by $\frac{\tau}{\sigma} \, \alpha ^{\frac{\tau}{\sigma}-1}$ and integrate over $\alpha \in (\Lambda ^{\sigma},\infty )$. Using Fubini’s theorem, we obtain

$$\begin{aligned} &|Q_{0}|^{-1} \int _{Q_{0}} \max \{W_{0}^{\tau}-\Lambda ^{\tau},0\} \\ &= |Q_{0}|^{-1} \int _{\Lambda ^{\sigma}}^{\infty } \frac{\tau}{\sigma} \, \alpha ^{\frac{\tau}{\sigma}-1} \, |\{x \in Q_{0}: W_{0}(x)^{\sigma }> \alpha \}| \, d\alpha \\ &\leq C \int _{\Lambda ^{\sigma}}^{\infty }\frac{\tau}{\sigma} \, \alpha ^{\frac{\tau}{\sigma}-2} \, \Lambda ^{\sigma }\, d\alpha \\ &= C \, \frac{\tau}{\sigma -\tau} \, \Lambda ^{\tau}. \end{aligned}$$

Putting these facts together, we conclude that

$$ |Q_{0}|^{-1} \int _{Q_{0}} W^{\tau }\leq C \, \Lambda ^{\tau}. $$

This completes the proof of Lemma A.2. □

Lemma A.3

Given a real number $\varepsilon >0$, we can find a real number $\delta >0$ (depending only on $n$, $\sigma$, and $\varepsilon$) with the property that

$$ \int _{A} W \leq \varepsilon \int _{Q_{0}} W. $$

for every $(n-1)$-dimensional cube $Q_{0} \in \mathcal{Q}$ and every measurable set $A \subset Q_{0}$ satisfying $|A| \leq \delta \, |Q_{0}|$.

Proof

Using Lemma A.2, we obtain

$$ \bigg( |Q_{0}|^{-1} \int _{Q_{0}} W^{\tau }\bigg)^{\frac{1}{\tau}} \leq C \, \sup _{Q \in \mathcal{Q}, Q_{0} \subset Q} \bigg( |Q|^{-1} \int _{Q} V^{\sigma }\bigg)^{\frac{1}{\sigma}}. $$

Moreover,

$$ \sup _{Q \in \mathcal{Q}, Q_{0} \subset Q} \bigg( |Q|^{-1} \int _{Q} V^{ \sigma }\bigg)^{\frac{1}{\sigma}} \leq \inf _{Q_{0}} W \leq |Q_{0}|^{-1} \int _{Q_{0}} W $$

by definition of $W$. Putting these facts together, we obtain

$$ \bigg( |Q_{0}|^{-1} \int _{Q_{0}} W^{\tau }\bigg)^{\frac{1}{\tau}} \leq C \, |Q_{0}|^{-1} \int _{Q_{0}} W. $$

Hence, if $A \subset Q_{0}$ is a measurable set with $|A| \leq \delta \, |Q_{0}|$, then Hölder’s inequality gives

$$ \int _{A} W \leq |A|^{\frac{\tau -1}{\tau}} \, \bigg( \int _{Q_{0}} W^{ \tau }\bigg)^{\frac{1}{\tau}} \leq \delta ^{\frac{\tau -1}{\tau}} \, |Q_{0}|^{ \frac{\tau -1}{\tau}} \, \bigg( \int _{Q_{0}} W^{\tau }\bigg)^{ \frac{1}{\tau}} \leq C \delta ^{\frac{\tau -1}{\tau}} \int _{Q_{0}} W. $$

Hence, if we choose $\delta$ to be a small multiple of $\varepsilon ^{\frac{\tau}{\tau -1}}$, then $\delta$ has the required property. This completes the proof of Lemma A.3. □

Lemma A.4

For each $(n-1)$-dimensional cube $Q_{0} \in \mathcal{Q}$, we have

$$ |Q_{0}|^{-1} \int _{Q_{0}} W \leq C \, \operatorname{diam}(Q_{0})^{-1}, $$

where $C$ depends only on $n$ and $\sigma$.

Proof

Using Lemma A.2 and Hölder’s inequality, we obtain

$$ |Q_{0}|^{-1} \int _{Q_{0}} W \leq \bigg( |Q_{0}|^{-1} \int _{Q_{0}} W^{ \tau }\bigg)^{\frac{1}{\tau}} \leq C \, \sup _{Q \in \mathcal{Q}, Q_{0} \subset Q} \bigg( |Q|^{-1} \int _{Q} V^{\sigma }\bigg)^{ \frac{1}{\sigma}}. $$

Hence, the assertion follows from (14). □

Lemma A.5

Let us fix an $(n-1)$-dimensional cube $Q_{0} \in \mathcal{Q}$. We define a bounded measurable function $g: Q_{0} \to \mathbb{R}$ by

$$ g(x) = \sup _{Q \in \mathcal{Q}, x \in Q \subset Q_{0}} |Q|^{-1} \int _{Q} |f-f_{Q}| $$

for each point $x \in Q_{0}$. Then

$$ \int _{Q_{0}} V \, |f-f_{Q_{0}}|^{2} \leq C \int _{Q_{0}} W g^{2}, $$

where $C$ depends only on $n$ and $\sigma$.

Proof

We define a bounded measurable function $h: Q_{0} \to \mathbb{R}$ by

$$ h(x) = \sup _{Q \in \mathcal{Q}, x \in Q \subset Q_{0}} |Q|^{-1} \int _{Q} |f-f_{Q_{0}}| $$

for each point $x \in Q_{0}$. Note that $V \leq W$ and $|f-f_{Q_{0}}| \leq h$ at each point in $Q_{0}$. Hence, it suffices to prove that

$$ \int _{Q_{0}} W h^{2} \leq C \int _{Q_{0}} W g^{2}. $$

(16)

In order to prove the inequality (16), we define $\alpha _{0} = |Q_{0}|^{-1} \int _{Q_{0}} |f-f_{Q_{0}}|$. For each $\alpha > \alpha _{0}$, we denote by $\mathcal{Q}_{\alpha}$ the set of all $(n-1)$-dimensional cubes $Q \in \mathcal{Q}$ with the following properties:

• $Q \subset Q_{0}$.

• $|Q|^{-1} \int _{Q} |f-f_{Q_{0}}| > \alpha$.

• If $\tilde{Q} \in \mathcal{Q}$ is an $(n-1)$-dimensional cube with $Q \subsetneq \tilde{Q}$ and $\tilde{Q} \subset Q_{0}$, then $|\tilde{Q}|^{-1} \int _{\tilde{Q}} |f-f_{Q_{0}}| \leq \alpha$.

It follows from the definition of $\alpha _{0}$ that $Q_{0} \notin \mathcal{Q}_{\alpha}$ for each $\alpha >\alpha _{0}$. It is easy to see that

$$ \alpha < |Q|^{-1} \int _{Q} |f-f_{Q_{0}}| \leq 2^{n-1} \alpha $$

(17)

for each $\alpha >\alpha _{0}$ and each $Q \in \mathcal{Q}_{\alpha}$. Moreover,

$$ \{h>\alpha \} = \bigcup _{Q \in \mathcal{Q}_{\alpha}} Q $$

(18)

for each $\alpha >\alpha _{0}$. Finally, given a real number $\alpha >\alpha _{0}$ and a point $x \notin \Gamma$, there is at most one cube $Q \in \mathcal{Q}_{\alpha}$ that contains the point $x$.

We next apply Lemma A.3 with $\varepsilon = 2^{-2n-1}$. Hence, we can find a real number $\delta \in (0,1)$ such that

$$ \int _{A} W \leq 2^{-2n-1} \int _{Q} W $$

(19)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}$ and every measurable set $A \subset Q$ satisfying $|A| \leq 2^{1-n} \, \delta \, |Q|$.

Let us consider a real number $\alpha >\alpha _{0}$ and an $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. The upper bound in (17) implies $|f_{Q}-f_{Q_{0}}| \leq 2^{n-1} \alpha$. Using the lower bound in (17), we obtain

$$ 2^{n-1} \alpha \, |\tilde{Q}| \leq \int _{\tilde{Q}} (|f-f_{Q_{0}}| - 2^{n-1} \alpha ) \leq \int _{\tilde{Q}} |f-f_{Q}| $$

for all $(n-1)$-dimensional cubes $\tilde{Q} \in \mathcal{Q}_{2^{n} \alpha}$. In the next step, we take the sum over all $(n-1)$-dimensional cubes $\tilde{Q} \in \mathcal{Q}_{2^{n} \alpha}$ with $\tilde{Q} \subset Q$. This gives

$$ 2^{n-1} \alpha \sum _{\tilde{Q} \in \mathcal{Q}_{2^{n} \alpha}, \tilde{Q} \subset Q} |\tilde{Q}| \leq \int _{Q} |f-f_{Q}|. $$

(20)

For each $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$, we have the inclusion

$$ Q \cap \{h > 2^{n} \alpha \} \setminus \Gamma \subset \bigcup _{ \tilde{Q} \in \mathcal{Q}_{2^{n} \alpha}, \tilde{Q} \subset Q} \tilde{Q}. $$

(21)

Combining (20) and (21), we conclude that

$$ 2^{n-1} \alpha \, |Q \cap \{h > 2^{n} \alpha \}| \leq \int _{Q} |f-f_{Q}| $$

(22)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. In particular,

$$ |Q \cap \{h > 2^{n} \alpha \}| \leq 2^{1-n} \, \delta \, |Q| $$

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$ satisfying $|Q|^{-1} \int _{Q} |f-f_{Q}| \leq \delta \alpha$. Applying (19) with $A = Q \cap \{h > 2^{n} \alpha \}$ gives

$$ \int _{Q \cap \{h > 2^{n} \alpha \}} W \leq 2^{-2n-1} \int _{Q} W $$

(23)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$ satisfying $|Q|^{-1} \int _{Q} |f-f_{Q}| \leq \delta \alpha$. On the other hand, if $Q \in \mathcal{Q}_{\alpha}$ is an $(n-1)$-dimensional cube satisfying $|Q|^{-1} \int _{Q} |f-f_{Q}| > \delta \alpha$, then $g > \delta \alpha$ at each point in $Q$. Therefore,

$$ \int _{Q \cap \{h > 2^{n} \alpha \}} W \leq \int _{Q \cap \{g > \delta \alpha \}} W $$

(24)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$ satisfying $|Q|^{-1} \int _{Q} |f-f_{Q}| > \delta \alpha$. Combining (23) and (24), we conclude that

$$ \int _{Q \cap \{h > 2^{n} \alpha \}} W \leq 2^{-2n-1} \int _{Q} W + \int _{Q \cap \{g > \delta \alpha \}} W $$

(25)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. Summation over all $(n-1)$-dimensional cubes $Q \in \mathcal{Q}_{\alpha}$ gives

$$ \int _{\{h > 2^{n} \alpha \}} W \leq 2^{-2n-1} \int _{\{h > \alpha \}} W + \int _{\{g > \delta \alpha \}} W $$

(26)

for each $\alpha >\alpha _{0}$.

We now multiply both sides of (26) by $2\alpha$ and integrate over $\alpha \in (2\alpha _{0},\infty )$. Using Fubini’s theorem, we obtain

$$\begin{aligned} &\int _{Q_{0}} W \, \max \{2^{-2n} h^{2} - 4\alpha _{0}^{2},0\} \\ &= \int _{2\alpha _{0}}^{\infty }2\alpha \, \bigg( \int _{\{h > 2^{n} \alpha \}} W \bigg) \, d\alpha \\ &\leq \int _{2\alpha _{0}}^{\infty }2^{-2n} \alpha \, \bigg( \int _{ \{h > \alpha \}} W \bigg) \, d\alpha + \int _{2\alpha _{0}}^{\infty }2 \alpha \, \bigg( \int _{\{g > \delta \alpha \}} W \bigg) \, d \alpha \\ &\leq 2^{-2n-1} \int _{Q_{0}} W h^{2} + \delta ^{-2} \int _{Q_{0}} W g^{2}. \end{aligned}$$

Rearranging terms gives

$$ 2^{-2n-1} \int _{Q_{0}} W h^{2} \leq 4\alpha _{0}^{2} \int _{Q_{0}} W + \delta ^{-2} \int _{Q_{0}} W g^{2}. $$

(27)

On the other hand, $g \geq |Q_{0}|^{-1} \int _{Q_{0}} |f-f_{Q_{0}}| = \alpha _{0}$ at each point in $Q_{0}$. Consequently,

$$ 2^{-2n-1} \int _{Q_{0}} W h^{2} \leq (4+\delta ^{-2}) \int _{Q_{0}} W g^{2}. $$

(28)

The inequality (16) follows immediately from (28). This completes the proof of Lemma A.5. □

Lemma A.6

Let us fix an $(n-1)$-dimensional cube $Q_{0} \in \mathcal{Q}$. We define a bounded measurable function $g: Q_{0} \to \mathbb{R}$ by

$$ g(x) = \sup _{Q \in \mathcal{Q}, x \in Q \subset Q_{0}} |Q|^{-1} \int _{Q} |f-f_{Q}| $$

for each point $x \in Q_{0}$. Then

$$ \int _{Q_{0}} W g^{2} \leq C \int _{Q_{0} \times [0,\operatorname{diam}(Q_{0})]} |\nabla F|^{2}, $$

where $C$ depends only on $n$ and $\sigma$.

Proof

Let $\alpha _{0} = |Q_{0}|^{-1} \int _{Q_{0}} |f-f_{Q_{0}}|$. For each $\alpha > \alpha _{0}$, we denote by $\mathcal{Q}_{\alpha}$ the set of all $(n-1)$-dimensional cubes $Q \in \mathcal{Q}$ with the following properties:

• $Q \subset Q_{0}$.

• $|Q|^{-1} \int _{Q} |f-f_{Q}| > \alpha$.

• If $\tilde{Q} \in \mathcal{Q}$ is an $(n-1)$-dimensional cube with $Q \subsetneq \tilde{Q}$ and $\tilde{Q} \subset Q_{0}$, then $|\tilde{Q}|^{-1} \int _{\tilde{Q}} |f-f_{\tilde{Q}}| \leq \alpha$.

It follows from the definition of $\alpha _{0}$ that $Q_{0} \notin \mathcal{Q}_{\alpha}$ for each $\alpha >\alpha _{0}$. It is easy to see that

$$ \alpha < |Q|^{-1} \int _{Q} |f-f_{Q}| \leq 2^{n} \alpha $$

(29)

for each $\alpha >\alpha _{0}$ and each $Q \in \mathcal{Q}_{\alpha}$. Moreover,

$$ \{g>\alpha \} = \bigcup _{Q \in \mathcal{Q}_{\alpha}} Q $$

(30)

for each $\alpha >\alpha _{0}$. Finally, given a real number $\alpha >\alpha _{0}$ and a point $x \notin \Gamma$, there is at most one cube $Q \in \mathcal{Q}_{\alpha}$ that contains the point $x$.

Let us consider a real number $\alpha >\alpha _{0}$ and an $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. Using the lower bound in (29), we obtain

$$ 2^{n+2} \alpha \, |\tilde{Q}| \leq \int _{\tilde{Q}} |f-f_{\tilde{Q}}| \leq 2 \int _{\tilde{Q}} |f-f_{Q}| $$

for all $(n-1)$-dimensional cubes $\tilde{Q} \in \mathcal{Q}_{2^{n+2} \alpha}$. In the next step, we take the sum over all $(n-1)$-dimensional cubes $\tilde{Q} \in \mathcal{Q}_{2^{n+2} \alpha}$ with $\tilde{Q} \subset Q$. Using the upper bound in (29), we obtain

$$ 2^{n+2} \alpha \sum _{\tilde{Q} \in \mathcal{Q}_{2^{n+2} \alpha}, \tilde{Q} \subset Q} |\tilde{Q}| \leq 2 \int _{Q} |f-f_{Q}| \leq 2^{n+1} \alpha \, |Q| $$

(31)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. For each $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$, we have the inclusion

$$ Q \cap \{g > 2^{n+2} \alpha \} \setminus \Gamma \subset \bigcup _{ \tilde{Q} \in \mathcal{Q}_{2^{n+2} \alpha}, \tilde{Q} \subset Q} \tilde{Q}. $$

(32)

Combining (31) and (32), we conclude that

$$ 2^{n+2} \alpha \, |Q \cap \{g>2^{n+2} \alpha \}| \leq 2^{n+1} \alpha \, |Q|, $$

hence

$$ |Q \cap \{g \leq 2^{n+2} \alpha \}| \geq \frac{1}{2} \, |Q| $$

(33)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$.

We define a nonnegative function $\varphi : \mathbb{R}^{n-1} \times \{0\} \to \mathbb{R}$ by

$$ \varphi (x_{1},\ldots ,x_{n-1},0) = \bigg( \int _{0}^{ \operatorname{diam}(Q_{0})} |\nabla F(x_{1},\ldots ,x_{n-1},x_{n})|^{2} \, dx_{n} \bigg)^{\frac{1}{2}}. $$

(34)

Moreover, we define a nonnegative function $\psi : Q_{0} \to \mathbb{R}$ by

$$ \psi (x) = \sup _{Q \in \mathcal{Q}, x \in Q \subset Q_{0}} |Q|^{-1} \int _{Q} \varphi $$

(35)

for each point $x \in Q_{0}$. Using the Sobolev trace theorem, we obtain

$$\begin{aligned} \alpha &\leq |Q|^{-1} \int _{Q} |f-f_{Q}| \\ &\leq 2 \, |Q|^{-1} \inf _{a \in \mathbb{R}} \int _{Q} |f-a| \\ &\leq C \, \, |Q|^{-1} \inf _{a \in \mathbb{R}} \bigg( \int _{Q \times [0,\operatorname{diam}(Q)]} |\nabla (F-a)| \\ & \hspace{30mm} + \operatorname{diam}(Q)^{-1} \int _{Q \times [0,\operatorname{diam}(Q)]} |F-a| \bigg) \end{aligned}$$

(36)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. Using (36) and the Poincaré inequality, we conclude that

$$ \alpha \leq C \, |Q|^{-1} \int _{Q \times [0,\operatorname{diam}(Q)]} | \nabla F| $$

(37)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. Using Hölder’s inequality, we deduce that

$$ \alpha \leq C \, \operatorname{diam}(Q)^{\frac{1}{2}} \, |Q|^{-1} \int _{Q} \varphi \leq C \, \operatorname{diam}(Q)^{\frac{1}{2}} \, \inf _{Q} \psi $$

(38)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. Combining (33) and (38) gives

$$\begin{aligned} \alpha ^{2} \, \operatorname{diam}(Q)^{-1} \, |Q| &\leq 2\alpha ^{2} \, \operatorname{diam}(Q)^{-1} \, |Q \cap \{g \leq 2^{n+2} \alpha \}| \\ &\leq C \int _{Q \cap \{g \leq 2^{n+2} \alpha \}} \psi ^{2} \end{aligned}$$

(39)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. Combining the estimate (39) with Lemma A.4, we obtain

$$ \alpha ^{2} \int _{Q} W \leq C \int _{Q \cap \{g \leq 2^{n+2} \alpha \}} \psi ^{2} $$

(40)

for every $(n-1)$-dimensional cube $Q \in \mathcal{Q}_{\alpha}$. Summation over all $(n-1)$-dimensional cubes $Q \in \mathcal{Q}_{\alpha}$ gives

$$ \alpha ^{2} \int _{\{g > \alpha \}} W \leq C \int _{\{\alpha < g \leq 2^{n+2} \alpha \}} \psi ^{2} $$

(41)

for each $\alpha >\alpha _{0}$.

We now multiply both sides of (41) by $2\alpha ^{-1}$ and integrate over $\alpha \in (2\alpha _{0},\infty )$. Using Fubini’s theorem, we obtain

$$\begin{aligned} \int _{Q_{0}} W \, \max \{g^{2}-4\alpha _{0}^{2},0\} &= \int _{2 \alpha _{0}}^{\infty }2\alpha \, \bigg( \int _{\{g > \alpha \}} W \bigg) \, d\alpha \\ &\leq \int _{2\alpha _{0}}^{\infty }2C \alpha ^{-1} \, \bigg( \int _{ \{\alpha < g \leq 2^{n+2} \alpha \}} \psi ^{2} \bigg) \, d\alpha \\ &\leq 2C \log (2^{n+2}) \int _{Q_{0}} \psi ^{2}. \end{aligned}$$

(42)

On the other hand, the function $\psi$ is bounded from above by the maximal function associated with the function $\varphi \, 1_{Q_{0}}$. Hence, the strong version of the Hardy-Littlewood maximal inequality (cf. [19], Proposition 2.9 (ii)) implies

$$ \int _{Q_{0}} \psi ^{2} \leq C \int _{Q_{0}} \varphi ^{2} \leq C \int _{Q_{0} \times [0,\operatorname{diam}(Q_{0})]} |\nabla F|^{2}. $$

(43)

Combining (42) and (43) gives

$$ \int _{Q_{0}} W \, \max \{g^{2}-4\alpha _{0}^{2},0\} \leq C \int _{Q_{0} \times [0,\operatorname{diam}(Q_{0})]} |\nabla F|^{2}. $$

(44)

Finally, using the Sobolev trace theorem, we obtain

$$\begin{aligned} \alpha _{0} &= |Q_{0}|^{-1} \int _{Q_{0}} |f-f_{Q_{0}}| \\ &\leq 2 \, |Q_{0}|^{-1} \inf _{a \in \mathbb{R}} \int _{Q_{0}} |f-a| \\ &\leq C \, \, |Q_{0}|^{-1} \inf _{a \in \mathbb{R}} \bigg( \int _{Q_{0} \times [0,\operatorname{diam}(Q_{0})]} |\nabla (F-a)| \\ & \hspace{30mm} + \operatorname{diam}(Q_{0})^{-1} \int _{Q_{0} \times [0,\operatorname{diam}(Q_{0})]} |F-a| \bigg). \end{aligned}$$

(45)

Using (45) and the Poincaré inequality, we conclude that

$$ \alpha _{0} \leq C \, |Q_{0}|^{-1} \int _{Q_{0} \times [0, \operatorname{diam}(Q_{0})]} |\nabla F|. $$

(46)

Using Hölder’s inequality, we deduce that

$$ \alpha _{0}^{2} \, \operatorname{diam}(Q_{0})^{-1} \, |Q_{0}| \leq C \int _{Q_{0} \times [0,\operatorname{diam}(Q_{0})]} |\nabla F|^{2}. $$

(47)

Combing the estimate (47) with Lemma A.4 gives

$$ \alpha _{0}^{2} \int _{Q_{0}} W \leq C \int _{Q_{0} \times [0, \operatorname{diam}(Q_{0})]} |\nabla F|^{2}. $$

(48)

The assertion follows by combining (44) and (48). This completes the proof of Lemma A.6. □

After these preparations, we now complete the proof of Theorem A.1. Combining Lemma A.5 and Lemma A.6, we conclude that

$$ \int _{Q_{0}} V \, |f-f_{Q_{0}}|^{2} \leq C \int _{Q_{0} \times [0, \operatorname{diam}(Q_{0})]} |\nabla F|^{2} $$

for every $(n-1)$-dimensional cube $Q_{0} \in \mathcal{Q}$. This implies

$$ \int _{Q_{0}} V f^{2} \leq C \int _{Q_{0} \times [0,\operatorname{diam}(Q_{0})]} |\nabla F|^{2} + C \, |Q_{0}|^{-2} \, \bigg( \int _{Q_{0}} V \bigg) \, \bigg( \int _{Q_{0}} |f| \bigg)^{2} $$

for each $(n-1)$-dimensional cube $Q_{0} \in \mathcal{Q}$. Moreover,

$$ |Q_{0}|^{-1} \int _{Q_{0}} V \leq \bigg( |Q_{0}|^{-1} \int _{Q_{0}} V^{ \sigma }\bigg)^{\frac{1}{\sigma}} \leq \operatorname{diam}(Q_{0})^{-1} $$

by (14). Thus, we conclude that

$$ \int _{Q_{0}} V f^{2} \leq C \int _{Q_{0} \times [0,\operatorname{diam}(Q_{0})]} |\nabla F|^{2} + C \, \operatorname{diam}(Q_{0})^{-1} \, |Q_{0}|^{-1} \, \bigg( \int _{Q_{0}} |f| \bigg)^{2} $$

for each $(n-1)$-dimensional cube $Q_{0} \in \mathcal{Q}$. This completes the proof of Theorem A.1.

Corollary A.7

Let us fix an integer $n \geq 3$ and a real number $\sigma \in (1,n-1)$. Suppose that $V$ is a nonnegative continuous function defined on the unit sphere $S^{n-1} \subset \mathbb{R}^{n}$ with the property that

$$ \bigg( r^{\sigma +1-n} \int _{S^{n-1} \cap B_{r}(p)} V^{\sigma } \bigg)^{\frac{1}{\sigma}} \leq 1 $$

(49)

for all points $p \in \mathbb{R}^{n}$ and all $0 < r \leq 1$. Let $F$ be a smooth function defined on the unit ball $B^{n} = \{x \in \mathbb{R}^{n}: |x| \leq 1\}$, and let $f$ denote the restriction of $F$ to the boundary $\partial B^{n} = S^{n-1}$. Then

$$ \int _{S^{n-1}} V f^{2} \leq C \int _{B^{n}} |\nabla F|^{2} + C \int _{S^{n-1}} f^{2}. $$

The constant $C$ depends only on $n$ and $\sigma$.