# Towards Transferable Adversarial Attacks with Centralized Perturbation

Shangbo Wu1, Yu-an Tan1, Yajie Wang1, Ruinan Ma1, Wencong Ma2, Yuanzhang Li2
Corresponding author.

###### Abstract

Adversarial transferability enables black-box attacks on unknown victim deep
neural networks (DNNs), rendering attacks viable in real-world scenarios.
Current transferable attacks create adversarial perturbation over the entire
image, resulting in excessive noise that overfit the source model.
Concentrating perturbation to dominant image regions that are model-agnostic
is crucial to improving adversarial efficacy. However, limiting perturbation
to local regions in the spatial domain proves inadequate in augmenting
transferability. To this end, we propose a transferable adversarial attack
with fine-grained perturbation optimization in the frequency domain, creating
centralized perturbation. We devise a systematic pipeline to dynamically
constrain perturbation optimization to dominant frequency coefficients. The
constraint is optimized in parallel at each iteration, ensuring the
directional alignment of perturbation optimization with model prediction. Our
approach allows us to centralize perturbation towards sample-specific
important frequency features, which are shared by DNNs, effectively mitigating
source model overfitting. Experiments demonstrate that by dynamically
centralizing perturbation on dominating frequency coefficients, crafted
adversarial examples exhibit stronger transferability, and allowing them to
bypass various defenses.

## Introduction

DNNs demonstrate outstanding performance across various real-world
applications (He et al. 2016; Dosovitskiy et al. 2021). However, DNNs remain
susceptible to adversarial examples — malicious samples with carefully-crafted
imperceptible perturbation that disrupt DNN functionalities (Zhang et al.
2023). The transferability of adversarial examples (Liu et al. 2017; Guo et
al. 2019) allows for cross-model black-box attacks on even unknown victim
DNNs, i.e., perturbation created on one model can fool another with no
modification, posing a practical real-world threat.

Existing transferable adversarial attacks are gradient-based iterative
attacks, stemming from FGSM (Goodfellow, Shlens, and Szegedy 2015) and PGD
(Madry et al. 2018). By greedily accumulating gradient information obtained
from the white-box source model, these attacks are able to generate
transferable adversarial perturbation. However, as attacks try to search the
entirety of the input space, the perturbation created tend to overfit the
source model, producing excessive noise.

Figure 1: Adversarial examples crafted by vanilla MI-FGSM (blue arrow, left
sample) and MI-FGSM boosted by our approach (pink arrow, right sample) on
source model ResNet50 with $\varepsilon=8/255$ and $T=10$. Perturbation
normalized for visualization (bottom row).

Yao et al. (2019) and Xu et al. (2019) explored perturbation constraints
within the spatial domain, boasting an improved efficiency for white-box
attacks and effectiveness against model interpretability methods. While these
attempts fall short for transfer-based black-box attacks, the idea of
concentrating perturbation is valuable.

Ehrlich and Davis (2019) first showed intriguing properties of the frequency
features of images that can be exploited. Prior work attempts to craft low-
frequency perturbation with a fixed frequency constraint, assuming that the
fixed low-frequency region reflect higher DNN sensitivity. However, DNNs do
not respond uniformly nor unchangeably to these regions. Sensitivity to
frequency coefficients varies across images, only with lower frequencies
tending to be more impactful. Our goal is to mitigate the effects of
perturbation overfitting to the source model, to improve transferability.
Motivated by this insight, we propose to strategically optimize adversarial
perturbation by identifying sensitive frequency regions per input, per
iteration, targeting influential frequency coefficients dynamically.

In this paper, we propose to craft centralized adversarial perturbation,
encompassing a shared frequency decomposition procedure, and two major
strategies to regularize optimization. Frequency decomposition transforms data
into frequency coefficient blocks with DCT (Discrete Cosine Transform).
Perturbation centralization is performed via a quantization on these
coefficients, omitting excessive perturbation, centralizing the optimization
towards dominant regions. Our paramount contribution is a fine-grained
quantization, controlled through a subsequent optimization of the quantization
matrix, guaranteeing its direct alignment with regional sensitivity. Finally,
we design our pipeline as a plug-and-play module, enabling seamless
integration of our strategy into existing state-of-the-art gradient-based
attacks.

Depicted in Figure 1, the effect of our proposed strategy to centralize
perturbation is apparent. On the left side of the samples, perturbations
generated by vanilla MI-FGSM exhibit obvious noise. In contrast, our
centralized perturbation, while noticeably smaller in magnitude, yield
equivalent adversarial potency. Our proposed optimization strategy eradicates
model-specific perturbation, effectively avoiding source model overfitting. By
purposely centralizing perturbation optimization within high-contribution
frequency regions, improved adversarial effectiveness naturally arise. Our
approach significantly enhances adversarial transferability by 11.7% on
average, and boosts attack efficacy on adversarially defended models by 10.5%
on average.

In summary, our contributions are as follows:

  * •

We design a shared frequency decomposition process with DCT. Excessive
perturbation is eliminated through quantization on blocks of frequency
coefficients. As such, adversarial perturbation is centralized, effectively
avoiding source model overfitting.

  * •

We propose a systematic approach for precise control over the quantization
process. Quantization matrix is optimized dynamically in parallel with the
attack, ensuring a direct alignment with model prediction at each step.

  * •

Through comprehensive experiments, we substantiate the efficacy of our
proposed centralized perturbation. Our approach yields significantly enhanced
adversarial effectiveness. Under the same perturbation budget, centralized
perturbation achieve stronger transferability, and is more successful in
bypassing defenses.

## Related Work

#### Adversarial Attacks.

Gradient-based attacks are extensively employed in both white-box and black-
box transfer scenarios. The seminal work by Goodfellow, Shlens, and Szegedy
(2015) introduced the single-step FGSM that exploits model gradients. Kurakin,
Goodfellow, and Bengio (2016) improves optimization with the multi-step
iterative BIM. Numerous endeavors have been made to enhance the
transferability of iterative gradient-based attacks, such as MI-FGSM (Dong et
al. 2018), TI-FGSM (Dong et al. 2019), DI-FGSM (Xie et al. 2019), SI-NI-FGSM
(Lin et al. 2020), VMI-FGSM (Wang and He 2021), among others. We use gradient-
based iterative solutions as our attack basis.

#### Frequency Optimizations.

Ehrlich and Davis (2019) first explored benefits of frequency-domain deep
learning. Guo, Frank, and Weinberger (2019), Sharma, Ding, and Brubaker
(2019), and Deng and Karam (2020) exploit low-frequency image features to
optimize perturbation within a predefined, constant low-frequency region.
However, the assumption that DNN sensitivity maps to fixed frequency regions
is disclaimed in Maiya et al. (2021), where different frequency regions yield
varying sensitivity, with a higher sensitivity tendency towards lower
coefficients. In contrast, our contribution lies in our effort to dynamically
adjust frequency-domain perturbation optimization over back-propagation,
augmenting attacks with centralized perturbation.

## Methodology

### Overview

Given a DNN classifier
$\mathcal{F}(\cdot):\boldsymbol{x}\in\mathbb{R}^{m}\mapsto y$, where
$\boldsymbol{x}$ and $y$ is the clean sample and ground truth label
respectively. The adversary aims to create adversarial example
$\boldsymbol{x}^{adv}=\boldsymbol{x}+\boldsymbol{\delta}$ with a minimal
perturbation $\boldsymbol{\delta}$ such that
$\mathcal{F}(\boldsymbol{x}^{adv})\neq y$, where $\boldsymbol{x}^{adv}$ is
restricted by $\ell_{p}$-ball ($\ell_{\infty}$ in this work). Hence, the
attack is an optimization which can be formulated as

$\arg\underset{\boldsymbol{x}^{adv}}{\max}\
\mathcal{J}(\boldsymbol{x}^{adv},y),\ s.t.\
\|\boldsymbol{x}^{adv}-\boldsymbol{x}\|_{\infty}\leqslant\varepsilon,$ (1)

where $\mathcal{J}(\cdot,\cdot)$ is the cross-entropy loss. Various gradient-
based approaches have been proposed to solve this optimization as stated
before, which we adopt as our foundation.

To optimize and craft centralized perturbation in the frequency domain, we
devise a three-fold approach as follows:

  1. 1.

Frequency coefficient decomposition: Inspired by the JPEG codec, we design a
shared differential data processing pipeline to decompose data into the
frequency domain with DCT.

  2. 2.

Centralized perturbation optimization: Within the frequency domain, excessive
perturbation is omitted via quantizing each Y/Cb/Cr channel, centralizing
perturbation optimization.

  3. 3.

Differential quantization optimization: Quantization matrices are optimized
through back-propagation, ensuring quantization aligns with dominant frequency
regions.

We detail these approaches in the following sections.

### Frequency Decomposition

Figure 2: The frequency decomposition procedure.

We begin by addressing the shared frequency decomposition procedure in our
work. We focus on 8-bit RGB images of shape $(3,224,224)$. Illustrated in
Figure 2, let $\boldsymbol{X}$ be the data that we would like to frequency
decompose:

Figure 3: Pipeline of our strategy. Perturbation crafted with gradient-based
attacks is centralized in the first step. Quantization matrix controlling the
centralization is then optimized via back-propagation in the second step.

  1. 1.

We first convert $\boldsymbol{X}$ from RGB into the YCbCr color space,
consisting of 3 color channels: the luma channel $Y$, and the chroma channels
$Cb$ and $Cr$.

  2. 2.

After which, a channel-wise global DCT is applied to transform the data
losslessly into the frequency domain.

  3. 3.

Next, a “blockify” process is used to reshape the data into blocks of
$(8\times 8)$. The operation is applied to the last two dimensions of the data
(width $W$ and height $H$). As such, an image of shape $(B,C,W,H)$ would be
“blockified” into shape $(B,C,W\cdot H/64,8,8)$ ($B$: batch, $C$: channel).

  4. 4.

Quantization is applied channel-wise via quantization matrices $Q$s, omitting
excessive frequency coefficients.

  5. 5.

Finally, inverse operations, namely “block merge” and “IDCT” (inverse-DCT) is
performed, reconstructing coefficients back into RGB image
$\boldsymbol{X^{\prime}}$.

This procedure would decompose $\boldsymbol{X}$ into blocks of frequency
coefficients in each Y/Cb/Cr channel. The sequential operation of “blockify”
guarantees the lossless inversion of “block merge”. Doing so, we ensure that
every stage in our procedure is losslessly invertible, enabling reconstruction
without information loss (with the exception of deliberate data quantization).
The linearity of DCT and IDCT allow for the possibility of optimization within
a reduced model space dimensionality, enabling the centralized perturbation
and quantization to be optimized effectively.

This shared frequency decomposition procedure is shown in the green-bounded
area in Figure 3. For the next two sections, we address our optimization
strategy after acquiring gradient-based attack crafted perturbation
$\boldsymbol{\delta}_{t}$ at iteration $t$.

### Centralized Perturbation Optimization

Depicted in Step 1 of Figure 3, first, the frequency decomposition is applied
to $\boldsymbol{\delta}_{t}$ at iteration $t$, where it is separated into luma
and chroma channels, then decomposed into frequency coefficients via DCT and
“blockify”. Resulting components are denoted as $B_{Y},B_{Cb},B_{Cr}$, where

$\displaystyle\begin{split}B_{Y}&=\mathrm{blockify}(DCT(Y)),\\\
B_{Cb}&=\mathrm{blockify}(DCT(Cb)),\\\
B_{Cr}&=\mathrm{blockify}(DCT(Cr)).\end{split}$ (2)

Quantization matrices $Q_{Y},Q_{Cb},Q_{Cr}$ are applied over each Y/Cb/Cr
channel as

$\displaystyle\begin{split}B_{Y}^{\prime}&=B_{Y}\odot Q_{Y},\\\
B_{Cb}^{\prime}&=B_{Cb}\odot Q_{Cb},\\\ B_{Cr}^{\prime}&=B_{Cr}\odot
Q_{Cr}.\\\ \end{split}$ (3)

Lastly, inverse operations are taken as

$\displaystyle\begin{split}Y^{\prime}&=IDCT(\mathrm{block\mbox{-}merge}(B^{\prime}_{Y})),\\\
Cb^{\prime}&=IDCT(\mathrm{block\mbox{-}merge}(B^{\prime}_{Cb})),\\\
Cr^{\prime}&=IDCT(\mathrm{block\mbox{-}merge}(B^{\prime}_{Cr})),\end{split}$
(4)

and with an RGB conversion, reconstruction is complete.

Differing from the quantization matrix employed in the JPEG codec, ours is
defined as $Q=(q_{ij})\in\\{0,1\\}^{m\times m}$ where $m=8$, and initialized
as $Q_{0}=\mathds{1}$. Through the block-wise multiplication of $B\odot Q$,
less impactful frequency coefficients are systematically discarded at every
iteration. This process confines the perturbation optimization exclusively
within frequency regions of greater impact over DNN predictions, as modeled by
the pattern in $Q$.

At iteration $t$, perturbation $\boldsymbol{\delta}_{t}$ will be centralized
via optimization, yielding $\boldsymbol{\delta}_{t}^{\prime}$. We denote the
entire process as

$\boldsymbol{\delta}_{t}^{\prime}=\mathcal{K}(\boldsymbol{\delta}_{t};Q_{t}),$
(5)

where $\mathcal{K}(\cdot;Q)$ is the frequency decomposition and quantization
function with quantization matrix $Q$. $Q_{t}$ is fixed at this stage per each
iteration $t$.

### Differential Quantization Matrix Optimization

Intuitively, quantization matrix $Q$ holds pivotal significance in the process
of centralizing frequency-sensitive perturbation. In this section, we delve
into the modeling and optimization of $Q$ to ensure its direct alignment with
frequency-wise coefficient influence on DNN predictions.

Illustrated in Step 2 of Figure 3, during this stage, the optimized
$\boldsymbol{\delta}_{t}^{\prime}$ is added onto $\boldsymbol{x}$ to get
intermediate adversarial example $\boldsymbol{x}^{adv}_{t}$ at iteration $t$.
$\boldsymbol{x}^{adv}_{t}$ goes through the identical shared frequency
decomposition process, and is quantized in the same manner as Step 1, giving
us $\boldsymbol{x}^{adv}_{t}{{}^{\prime}}$. The quantized
$\boldsymbol{x}^{adv}_{t}{{}^{\prime}}$ is then fed back into source model
$\mathcal{F}(\cdot)$, such that $Q$ is updated subsequently through back-
propagation.

We formulate the update of $Q$ as an optimization problem by leveraging the
gradient changes of the source model before and after the quantization of
$\boldsymbol{x}^{adv}_{t}$. Our optimization goal is: $Q$ should quantize
$\boldsymbol{x}^{adv}_{t}$ in a way that, at each iteration, the source model
should be less convinced that $\boldsymbol{x}^{adv}_{t}$ is $y$. As such, the
loss is formulated as

$\arg\underset{Q_{t}}{\max}\
\mathcal{J}(\mathcal{K}(\boldsymbol{x}^{adv}_{t};Q_{t}),y),$ (6)

where $Q_{t}$ is the variable for optimization. $Q$ consists of
$Q_{Y},Q_{Cb},Q_{Cr}$ for each Y/Cb/Cr channel respectively, and is
collectively optimized by the Adam optimizer to solve this optimization. With
this approach, we assure that (1) $Q$ accurately reflects the impact of
frequency coefficients on model predictions per each iteration, allowing for
fine-grained control over the centralized perturbation quantization process,
and (2) gradients from the source model accumulate across successive
iterations, further boosting transferability.

Through the process of back-propagation, $Q$ undergoes iterative updates via
optimization. We denote the optimized result as matrix
$P=(p_{ij})\in\mathbb{R}^{m\times m}$. Following the update, a rounding
function $\mathcal{R}(\cdot)$ is implemented before $Q$ is applied to the
quantization process. Specifically, for a quantization ratio of $r$ where
$0\leqslant r\leqslant 1$ for each Y/Cb/Cr channel,

$Q=\mathcal{R}(P;r)=\begin{cases}1,\quad\mathrm{where}\
p_{ij}\geqslant\rho,\\\ 0,\quad\mathrm{otherwise},\\\ \end{cases}$ (7)

where $\rho$ is the $r$-th percentile of $\\{p_{ij}\\}$, i.e., the threshold
for each Y/Cb/Cr channel. While Equation 7 is represented as a staircase
(i.e., non-differentiable), $\mathcal{R}(\cdot)$ is implemented in a
differentiable manner, which we detail in the Appendix.

Algorithm 1 Centralized Adversarial Perturbation

Input: Original image $\boldsymbol{x}$, ground truth label $y$, source model
$\mathcal{F}$ with loss $\mathcal{J}$.
Parameters: Iteration $T$, size of perturbation $\varepsilon$, learning rate
$\beta$, quantization ratios $r_{Y},r_{Cb},r_{Cr}$ and corresponding
quantization matrices $Q_{Y},Q_{Cb},Q_{Cr}$ (denoted as $Q$s for brevity) each
Y/Cb/Cr channel.
Output: $\boldsymbol{x}^{adv}_{T}$.

1: Let step size $\alpha\leftarrow\varepsilon/T$, $Q_{0}=\mathds{1}$.

2: for $t=0$ to $T-1$ do

3: Acquire gradient from $\mathcal{F}$ as
$\nabla_{\boldsymbol{x}}\mathcal{J}(\boldsymbol{x}^{adv}_{t},y)$.

4: Perturbation
$\boldsymbol{\delta}_{t}=\alpha\cdot\mathrm{sign}(\nabla_{\boldsymbol{x}}\mathcal{J}(\boldsymbol{x}^{adv}_{t},y))$.

5: Optimize $\boldsymbol{\delta}_{t}$ by Equation 5 and 7 to acquire
$\boldsymbol{\delta}_{t}^{\prime}$, and clip with respect to $\varepsilon$.

6: Acquire intermediate
$\boldsymbol{x}^{adv}_{t}=\boldsymbol{x}+\boldsymbol{\delta}_{t}^{\prime}$.

7: Update $Q_{t}$s by passing the quantized $\boldsymbol{x}^{adv}_{t}$
(through the same Equation 5 and 7) to $\mathcal{F}$ and solving the
optimization via Equation 6.

8: end for

9: return
$\boldsymbol{x}^{adv}_{T}=\boldsymbol{x}+\boldsymbol{\delta}_{T}^{\prime}$.

### The Algorithm

Finally, Algorithm 1 showcase a basic attack integrated with our strategy,
crafting centralized adversarial perturbation (based on BIM). After the
conventional step of creating intermediate perturbation from the source
model’s gradients, our approach of centralizing the perturbation in a
frequency-sensitive manner, and updating the quantization matrix with respect
to model predictions, is a streamline integration.

Our strategy guarantees that the centralized perturbation through frequency
quantization is concentrated into frequency regions that hold greater
significant to the model’s judgements. Updating the quantization matrix per
iteration allows for a more fine-grained control, making sure that each
optimization aligns with the iteration-local gradients of the model. Lastly,
as a bonus, at each iteration, gradients gets carried over and accumulated to
the next through the directional optimization of the quantization matrix,
improving adversarial transferability.

## Experiments

### Setup

Figure 4: Evaluation of adversarial transferability. Transfer fooling rates
are aggregated across attacks over $T$.

#### Dataset.

The dataset from the NeurIPS 2017 Adversarial Learning Challenge (Kurakin et
al. 2018) is used, consisting of 1000 images from ImageNet with shape
$(3,224,224)$.

#### Networks.

We choose 3 source models as local surrogate models: ResNet-50 (Res-50) (He et
al. 2016), VGG-19 (Simonyan and Zisserman 2015), Inception-v3 (Inc-v3)
(Szegedy et al. 2016). All pretrained models are sourced from Wightman (2019).

#### Attacks.

Our approach is integrated and evaluated against 5 state-of-the-art gradient-
based attacks, namely MI-FGSM, DI-FGSM, TI-FGSM, VMI-FGSM, and SI-NI-FGSM, for
a better presentation of transferability. The unmodified forms of these
attacks serve as the baselines for our experiments.

#### Implementation Details.

We intentionally drop excessive perturbation, centralizing optimization
towards dominant frequency regions. As such, to enable a fair comparison
between our attack and baseline methods: for all three Y/Cb/Cr channels, each
set at a quantization ratio of $r_{Y},r_{Cb},r_{Cr}$, the cumulative
quantization rate is $(r_{Y}+r_{Cr}+r_{Cr})/3$; With a baseline attack under
an $\ell_{\infty}$ constraint of $\varepsilon_{0}$, we equate our attack to
the same perturbation budget by imposing a constraint of
$\varepsilon=\varepsilon_{0}/((r_{Y}+r_{Cr}+r_{Cr})/3)$.

#### Hyper-parameters.

Attacks are $\ell_{\infty}$-bounded. $\varepsilon=8/255$. They run for
iteration $T=10,20,50$. Learning rate of the Adam optimizer $\beta=0.1$.
Quantization ratios are set to $r_{Y}=0.9,r_{Cb}=0.05,r_{Cr}=0.05$, with a
total quantization rate of $1/3$. All other parameters are kept exact to their
original implementations.

### Transferability

We initiate our evaluation with our primary focus: adversarial
transferability. 6 normally trained models: ResNet-152 (Res-152), VGG-11,
DenseNet-121 (Dense-121) (Huang et al. 2017), Inception-v4 (Inc-v4) (Szegedy
et al. 2017), ConvNeXt (Liu et al. 2022), and ViT-B/16 (Dosovitskiy et al.
2021) are used as black-box target models.

Figure 4 presents the transfer fooling rates of the crafted adversarial
examples used to attack the black-box models. Fooling rates are averaged over
parameter: attack iteration $T$. We report a consistent improvement in
transfer fooling rates when integrating our proposed frequency perturbation
centralization strategy with vanilla gradient-based attacks. Displayed in
orange, our approach achieves an average boost in adversarial transferability
of 11.7% compared to the baseline attacks represented in blue.

We argue that the observed increase in transferability is the direct
manifestation of the benefits that centralized perturbation brings in our
method. Dominating frequency features of an image is learnt and shared by
neural networks alike, while excessive perturbation that are crafted to fit
model-specific features is what hinders the transferability of adversarial
examples. By centralizing adversarial perturbation towards the shared dominant
frequency regions, we effectively disrupt the model’s judgment in a more
generalized manner, boosting adversarial transferability.

Figure 5: Evaluation of adversarial effectiveness against adversarial
training-based defenses. Fooling rates aggregated over $T$.

### Defense Evasion

Currently two types of defenses provide an extra layer of robustness guarantee
to neural networks: (1) filter-based defenses, where perturbation is filtered
through image transformations, and (2) adversarial training, where models are
strengthened by training on adversarial examples.

Attack |  Res-152 |  VGG-11 |  Dense-121 |  Inc-v4 |  Conv-NeXt |  ViT-B/16
---|---|---|---|---|---|---
JPEG compression — quality level 75
MI-FGSM |  23.1% (±2.0) |  59.2% (±13.8) |  41.3% (±5.7) |  35.0% (±6.7) |  15.3% (±1.3) |  12.3% (±0.5)
+Ours* |  34.8% (±1.3) |  74.8% (±11.0) |  55.8% (±5.3) |  47.2% (±6.2) |  22.3% (±1.9) |  16.3% (±1.0)
VMI-FGSM |  33.1% (±6.0) |  66.3% (±15.3) |  51.7% (±6.8) |  45.7% (±7.3) |  22.0% (±2.8) |  17.0% (±1.5)
+Ours* |  48.1% (±5.7) |  81.0% (±11.8) |  70.4% (±5.0) |  62.8% (±5.9) |  31.8% (±3.5) |  23.8% (±3.7)
Bit-depth reduction — to 3 bits
MI-FGSM |  17.9% (±2.8) |  57.7% (±12.4) |  37.8% (±6.2) |  33.3% (±5.5) |  14.3% (±0.9) |  10.4% (±0.3)
+Ours* |  30.3% (±0.8) |  71.6% (±11.1) |  53.5% (±6.0) |  43.3% (±7.3) |  18.4% (±1.4) |  13.9% (±1.4)
VMI-FGSM |  28.8% (±8.1) |  63.9% (±14.4) |  47.3% (±6.7) |  42.3% (±7.4) |  20.5% (±2.8) |  13.5% (±1.5)
+Ours* |  44.7% (±8.7) |  78.0% (±12.5) |  69.0% (±4.9) |  57.8% (±6.8) |  27.0% (±3.0) |  20.3% (±2.5)

Table 1: Evaluation of adversarial effectiveness against filter-based
adversarial defenses. Fooling rates aggregated with $\mathrm{mean}$s and
$\mathrm{std}$s over attack iteration $T$. Figure 6: The effect of the
quantization ratio of luma channel $r_{Y}$ vs. chroma channels $r_{Cb},r_{Cr}$
over transferability. Figure 7: A visualization of perturbation over $T=10$
optimization iterations (from left to right) of MI-FGSM attacking ResNet50.
Correct predictions are framed in green, where incorrect ones are framed in
red.

#### Filter-based Defense.

Guo et al. (2018) first explored using JPEG compression and bit-depth
reduction as defenses against adversarial attacks. Since our approach also
utilizes similar techniques (such as drawing inspiration from the JPEG codec
for perturbation quantization), we are interested in investigating how our
method withstands these defenses.

Shown in Table 1, we report the transfer fooling rates of attacks: MI-FGSM,
VMI-FGSM and variants integrated with our approach. Highlighted in bold, we
observe that attacks integrated with our strategy all outperform their
baselines, with an increase of at least 3.5%. We posit that these defenses
fail as they intend to preserve dominant image features to maintain visual
quality, assuming that perturbation lies within unimportant regions to ensure
imperceptibility. As such, they are unable to remove our perturbation designed
to centralize around dominating regions, enabling our strategy to bypass these
defenses.

#### Adversarial Training.

5 adversarially trained black-box models: Adv-Inc-v3, Inc-Res-v2ens (Tramèr et
al. 2018), Adv-ResNet-50 (FastAT), Adv-ResNet-50 (FreeAT) (Shafahi et al.
2019), and EfficientNet-B0 (AdvProp) (Xie et al. 2020) are used for evaluating
our approach against adversarial training as a defense mechanism.

We report our results in Figure 5, where we observe a steady boost of fooling
rates by a maximum of 20% in some scenarios by our strategy. In all, results
indicate that our proposed approach can successfully evade adversarial
defenses that transfer-based attacks frequently fail against.

### The Impact of Quantization Ratio Allocation

The quantization process is crucial for centralized perturbation optimization,
controlled by quantization ratios $r$s. We explain our rationale for the
choice of ratios through evaluating the impact of luma channel $r_{Y}$ vs.
chroma channels $r_{Cb},r_{Cr}$ on adversarial effectiveness.

We maintain a cumulative quantization rate of $1/3$, iterating the luma
channel $r_{Y}$ from $0$ to $1$ with step size $0.1$, and allocating the rest
equally to the chroma channels $r_{Cb}$ and $r_{Cr}$. Figure 6 shows the
transfer fooling rates of MI-FGSM integrated with our approach. As $r_{Y}$
increases, the fooling rates show a consistent increase as well. We reason
that this occurs because the luma channel contains more structural
information, which DNNs commonly learn as more useful features compared to the
chroma channels. However, changes in the luma channel are also more noticeable
by the human visual system. As such, we choose $r_{Y}=0.9$ for a balance of
adversarial effectiveness and perturbation imperceptibility. Further
investigation is addressed in the Appendix.

### Visualization of Perturbation Optimization

In Figure 7, we visualize the perturbation optimized by MI-FGSM + our approach
over 10 iterations, giving further insight to the centralized optimization
process of our strategy.

On iteration 1 (step $t=1$), quantization is initialized as
$Q_{0}=\mathds{1}$. On $t=2$, we observe an immediate drop in perturbation,
indicating quantization is applied. Going further ($t=3$ to $10$), we find
perturbation is gradually added at each iteration, centralized on more
significant frequency regions. By ensuring a consistent optimization alignment
with model judgement, our centralized perturbation succeeds in exploiting
dominating frequency features of an image.

### Ablation Study

We conduct an ablation study on our proposed centralized perturbation
optimization process — the core of our strategy. We consider 4 other
strategies: (1) RandA: randomly choose $Q$s fixed at the start, (2) RandB:
randomly choose $Q$s at each iteration, (3) Low: preserving only low-frequency
coefficients (which is what previous literature proposes to do), and (4) High:
preserving only high-frequency coefficients. Details addressed in the
Appendix.

In Table 2, we report the fooling rates of vanilla MI-FGSM and SI-NI-FGSM, and
ones integrated with RandA, RandB, Low, and our strategy. (High is ignored as
fooling rates mostly dropped to zero when it is integrated.) Highlighted in
bold, we prove our strategy’s ability as our approach consistently boosts
transferability. In contrast, RandA and RandB both fail to achieve comparable
adversarial effectiveness even with the baselines. While Low comes close to
matching and occasionally surpasses us, our approach still maintains the upper
hand across the majority of cases. We argue that Low and our approach both
acknowledges that the low-frequency regions hold more dominating features.
However, instead of brute-forcely constraining all perturbation towards
consecutive low-frequency coefficients, our strategy’s precise control over
the centralization process yields an improved adversarial effectiveness.

Variant |  White-box |  Res-152 |  VGG-11 |  Dense-121 |  Inc-v4 |  Conv-NeXt |  ViT-B/16
---|---|---|---|---|---|---|---
MI-FGSM
Vanilla |  100% |  44.6% |  43.0% |  45.3% |  32.0% |  16.6% |  10.3%
RandA |  86.2% |  37.0% |  52.1% |  45.2% |  31.4% |  13.0% |  9.4%
RandB |  92.3% |  37.6% |  52.2% |  45.0% |  31.8% |  13.7% |  9.6%
Low |  100% |  50.3% |  63.1% |  56.2% |  39.5% |  18.9% |  11.9%
Ours* |  100% |  50.7% |  65.7% |  58.7% |  41.0% |  17.8% |  12.0%
SI-NI-FGSM
Vanilla |  99.9% |  43.2% |  35.4% |  42.9% |  24.1% |  13.4% |  8.0%
RandA |  88.4% |  36.7% |  61.5% |  49.3% |  34.8% |  14.7% |  10.1%
RandB |  76.6% |  28.2% |  51.4% |  42.0% |  29.1% |  11.6% |  8.7%
Low |  97.4% |  45.5% |  66.5% |  57.9% |  41.2% |  18.6% |  11.9%
Ours* |  98.7% |  49.9% |  68.9% |  64.1% |  42.6% |  17.6% |  11.2%

Table 2: Ablation study of the centralized perturbation optimization strategy.
Source model chosen as ResNet50.

## Conclusion

We propose a novel approach to centralize adversarial perturbation to dominant
frequency regions. Our technique involves dynamic perturbation optimization
through frequency domain quantization, resulting in centralized perturbation
that offers improved generalization. To ensure precise quantization alignment
with the model’s judgement, we incorporate a parallel optimization via back-
propagation. Experiments prove that our strategy achieves remarkable
transferability and succeeds in evading various defenses. Implications of our
work prompt further exploration of defending centralized perturbation under a
frequency context.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Grants No. 62072037 and U1936218).

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##  Appendix A Appendix

###  A.1 The Impact of Chroma Channels

We primarily evaluated the impact of luma channel’s $r_{Y}$ over adversarial
transferability in our paper. Our assumption is that the luma channel holds
more structural information than that of the chroma channels, thereby being
the more valuable channel for centralizing perturbation. This is also our
rationale for choosing the quantization ratios of
$r_{Y}=0.9,r_{Cb}=0.05,r_{Cr}=0.05$ as defaults. Experimental results prove
that centralizing more perturbation towards the luma channel yields greater
adversarial effectiveness.

In this section, we further investigate the effect of individual chroma
channels, namely the Cb channel and the Cr channel. Following our previous
evaluation settings, where the cumulative quantization rate of $1/3$ remains
constant, we iterate $r_{Cb}$ and $r_{Cr}$ each from $0$ to $1$ with a step
size of $0.1$, allocating the rest equally to the remaining two channels
respectively.

Figure 8: The effect of the quantization ratio of chroma channels $r_{Cb}$
(left) and $r_{Cr}$ (right) over adversarial transferability respectively.

Figure 8 illustrates that neither increasing $r_{Cb}$ nor $r_{Cr}$ improves
adversarial transferability. Fooling rates decrease at a similar rate as
quantization ratios for channel Cb and Cr increases respectively. Adversarial
transferability never manages to reach our default $r$ settings, falling less
than 50% at best.

Our findings indicate:

  1. 1.

There is no significant difference in the impact of the chroma channels, i.e.,
channel Cb and Cr, on adversarial effectiveness.

  2. 2.

Centralizing perturbation towards the chroma channels results in a
deteriorated adversarial transferability, consistently.

As such, our default strategy of preserving the luma channel to the largest
extent ($r_{Y}=0.9$) and allocating the remaining quantization budget evenly
to the two chroma channels ($r_{Cb}=0.05,r_{Cr}=0.05$) is a rational approach.

###  A.2 Details of Ablation Study

In our ablation study, we evaluated our approach with 4 other strategies,
namely: RandA, RandB, Low, and High. We address the details of these frequency
quantization strategies by illustrating the quantization matrices
$Q_{Y},Q_{Cb},Q_{Cr}$ of these schemes that we consider.

To begin, we acknowledge the zig-zag sequence, which, for $(8\times 8)$
frequency blocks, is formulated as

$\begin{bmatrix}0&1&5&6&14&15&27&28\\\ 2&4&7&13&16&26&29&42\\\
3&8&12&17&25&30&41&43\\\ 9&11&18&24&31&40&44&53\\\ 10&19&23&32&39&45&52&54\\\
20&22&33&38&46&51&55&60\\\ 21&34&37&47&50&56&59&61\\\
35&36&48&49&57&58&62&63\end{bmatrix}$ (8)

This is the standard for JPEG encoding, and is the sequence for frequency
coefficient blocks from low-frequency to high-frequency.

Figure 9: The 4 strategies (top 4 rows) that we evaluate against our method
(bottom row) in the ablation study. Note that RandB and Ours consists of a
sequence of matrices, which we use to illustrate that the matrices change
dynamically over iterations. In contrast, RandA, Low, and High all use fixed
matrices.

In Figure 9, we illustrate quantization schemes evaluated in our ablation
study. We plot the heatmap of demonstrations of the quantization matrices used
in each strategy. The darker colors represent $1$, i.e., frequency coefficient
blocks that are preserved, whereas the lighter colors represent $0$, i.e.,
blocks that are eliminated.

  * •

In strategy RandA and RandB (top 2 rows), blocks are eliminated at random.

  * •

In strategy Low and High (row 3 and 4), lower and higher frequency coefficient
blocks are preserved respectively, with respect to the sequence defined in
Matrix 8.

  * •

Lastly, in our strategy (bottom row), we are able to acquire the importance of
each frequency coefficient block (illustrated as different shades of colors)
at each attack iteration through back-propagation, thereby implementing our
dynamic and guided fine-grained centralization of adversarial perturbation.

We layer multiple matrices over each other in strategy RandB and Ours in
Figure 9, so as to represent the dynamicality of the process, where
quantization differs at each iteration. Where the rest of the strategies,
namely RandA, Low, and High all use fixed quantization matrices defined at the
start of the attack.

The results presented in our ablation study demonstrate that the lower
frequency coefficient blocks exert greater dominance, proving that
concentrating perturbations in these areas would enhance adversarial
effectiveness.

Interestingly, we discovered that solely preserving the higher frequency
regions leads to a catastrophic failure of the attack, resulting in fooling
rates dropping to zero in most cases.

Finally, as depicted, our strategy surpasses the brute centralization approach
of Low by allowing for dynamically adjustments to quantization, facilitating a
better alignment with model predictions.

###  A.3 Details on Differential Quantization

Listing 1 Differential Implementation of $\mathcal{R}(\cdot)$

⬇

1def quantize(x: torch.Tensor, ratio: float = 0.9) -> torch.Tensor:

2 # Calculate the threshold value of the top ‘ratio‘

3 threshold = torch.quantile(x, 1 - ratio)

4

5 # Keep all values above the threshold as 1

6 x_hat = torch.zeros_like(x)

7 x_hat[x >= threshold] = 1

8

9 # A hack to make the function differentiable

10 return x_hat - x.detach() + x

Equation 7 in our paper presents us applying rounding $\mathcal{R}(\cdot)$ and
acquiring the quantization matrix in a non-differential manner, as the
function is a staircase, hindering back-propagation based optimization. To
solve this issue, we implemented the function as in Listing 1.


}\sum_{|\tilde{\mathbf{k}}|\leq|\mathbf{k}_{2}|+3}\Big{(}\int_{\tau_{0}}^{\tau_{1}}\int_{v(R,u)}^{v}\int_{S^{2}}r^{-1+\delta}|\partial_{v}\mathfrak{D}^{\tilde{\mathbf{k}}}\psi|^{2}r^{2}d\omega
dudv^{\prime}\Big{)}^{\frac{1}{2}},$

by the Sobolev inequality on the incoming cones (C.8). Hence, if
$\tau_{1}-\tau_{0}\geq 1$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}\
}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}},$

and, if $\tau_{1}-\tau_{0}<1$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}\
}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}{}^{*}}.$

Estimate for $\partial_{v}\psi$ term: Next, for any $|\mathbf{k}|\leq k$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}\
}\Big{(}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}+\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\
\Big{)}.$ (C.12)

Indeed, if $|\mathbf{k}_{1}|\leq k-2$, then

$\displaystyle\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\qquad\lesssim\sup_{u,v,\theta}|r\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\Big{(}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-1}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-1}|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle\qquad\lesssim\big{(}1+(\tau_{1}-\tau_{0})^{-\frac{1}{2}}\big{)}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}}\Big{(}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-1}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|^{2}\Big{)}^{\frac{1}{2}},$

by the Sobolev inequality on the incoming cones (C.7). Hence, if
$\tau_{1}-\tau_{0}\geq 1$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime},$

and, if $\tau_{1}-\tau_{0}<1$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}}.$

If $|\mathbf{k}_{2}|\leq|\mathbf{k}|-2$ then one similarly estimates,

$\displaystyle\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\qquad\lesssim\int_{v(R,\tau_{0}+R)}^{v}\sup_{\begin{subarray}{c}\tau_{0}\leq
u\leq T_{1}^{\prime}\\\ \theta\in
S^{2}\end{subarray}}|r^{\frac{1}{2}}\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\left(\int_{\tau_{0}}^{T_{1}^{\prime}}\int_{S^{2}}r^{-1}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|r^{2}d\omega
du\right)^{\frac{1}{2}}\cdot\left(\int_{\tau_{0}}^{T_{1}^{\prime}}\int_{S^{2}}|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|r^{2}d\omega
du\right)^{\frac{1}{2}}dv^{\prime}$
$\displaystyle\qquad\lesssim\big{(}1+(\tau_{1}-\tau_{0})^{-\frac{1}{2}}\big{)}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\times$
$\displaystyle\qquad\qquad\quad\times\sum_{|\tilde{\mathbf{k}}|\leq|\mathbf{k}_{2}|+3}\int_{v(R,\tau_{0}+R)}^{v}\left(\int_{\tau_{0}}^{T_{1}^{\prime}}\int_{S^{2}}r^{-1-\delta}|\partial_{v}\mathfrak{D}^{\tilde{\mathbf{k}}}\psi|r^{2}d\omega
du\right)^{\frac{1}{2}}\left(\int_{\tau_{0}}^{T_{1}^{\prime}}\int_{S^{2}}r^{-1+\delta}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|r^{2}d\omega
du\right)^{\frac{1}{2}}dv^{\prime}$
$\displaystyle\qquad\lesssim\big{(}1+(\tau_{1}-\tau_{0})^{-\frac{1}{2}}\big{)}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}}\left(\int_{v(R,\tau_{0}+R)}^{v}\int_{\tau_{0}}^{T_{1}^{\prime}}\int_{S^{2}}r^{-1+\delta}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|r^{2}d\omega
dudv^{\prime}\right)^{\frac{1}{2}},$

by the Sobolev inequality on the incoming cones (C.8). Hence, if
$\tau_{1}-\tau_{0}\geq 1$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}},$

and, if $\tau_{1}-\tau_{0}<1$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}}.$

Estimate for $r^{-1}\not{\nabla}\psi$ and $r^{-1}\psi$ terms: For any
$|\mathbf{k}|\leq k$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-1}|\not{\nabla}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}\
}\Big{(}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}+\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\
\Big{)},$ (C.13)

and

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-1}|\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}\
}\Big{(}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}+\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\
\Big{)}.$ (C.14)

Indeed, (C.13) and (C.14) follow as in the proof of (C.12), using now the fact
that

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-1-\delta}|r^{-1}\not{\nabla}\psi|^{2}+r^{-1-\delta}|r^{-1}\psi|^{2}\leq\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime},$

and

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-1+\delta}|r^{-1}\not{\nabla}\psi|^{2}+r^{-1+\delta}|r^{-1}\psi|^{2}\leq\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime},\qquad\int_{v(R,\tau)}^{v}\int_{S^{2}}\left(|r^{-1}\not{\nabla}\psi|^{2}+|r^{-1}\psi|^{2}\right)r^{2}d\omega
dv^{\prime}\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}.$

Estimate for quartic term: For any $|\mathbf{k}|\leq k$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|^{2}\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\lesssim\Big{(}\
\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\
\Big{)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}+\Big{(}\
\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}\
\Big{)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}.$
(C.15)

Indeed, supposing first that $|\mathbf{k}_{1}|\leq k-3$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|^{2}\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\\\
\lesssim\sup_{u,\theta}\left(r|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\right)^{2}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-2}|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}\big{(}1+(\tau_{1}-\tau_{0})^{-\frac{1}{2}}\big{)}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-2}|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2},$

by the Sobolev inequality on the incoming cones (C.7). Hence, if
$\tau_{1}-\tau_{0}\geq 1$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|^{2}\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime},$

and, if $\tau_{1}-\tau_{0}<1$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|^{2}\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}.$

Similarly, if $|\mathbf{k}_{2}|\leq k-3$,

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|^{2}\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\lesssim\int_{v(R,\tau_{0}+R)}^{v}\sup_{u,\theta}|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\int_{\tau_{0}}^{\tau_{1}}\int_{S^{2}}|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}r^{2}d\omega
dudv^{\prime}\\\
\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}\big{(}1+(\tau_{1}-\tau_{0})^{-\frac{1}{2}}\big{)}\sum_{|\widetilde{\textbf{k}}|\leq|\textbf{k}_{2}|+3}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{-2}|\partial_{v}\mathfrak{D}^{\widetilde{\textbf{k}}}\psi|^{2},$

and so one similarly has

$\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|^{2}\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\lesssim\Big{(}\
\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}\
\Big{)}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}.$

Estimate for $r^{p-1}\partial_{v}(r\psi)$ term: Finally, for any
$|\mathbf{k}|\leq k$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\\\
\lesssim\bigg{(}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}+\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(p-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\bigg{)}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}+\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}{}^{*}}+\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(p-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\
}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(p-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}\
}.$ (C.16)

Indeed, consider first the case $\tau_{1}-\tau_{0}\geq 1$. Suppose first that
$|\mathbf{k}_{1}|\leq k-3$. Then, if $\tau_{1}-\tau_{0}\geq 1$,

$\displaystyle\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\quad\lesssim\sup_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}(r|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|)\Big{(}\int_{\mathcal{R}_{R}(\tau_{0},\tau_{1},v)}r^{p-3}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\int_{\mathcal{R}_{R}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\Big{)}^{\frac{1}{2}}\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(p-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime},$

by the Sobolev inequality on the incoming cones (C.7). Similarly, if
$\tau_{1}-\tau_{0}<1$,

$\displaystyle\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\quad\lesssim(\tau_{1}-\tau_{0})^{\frac{1}{2}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\Big{(}\int_{\mathcal{R}_{R}(\tau_{0},\tau_{1},v)}r^{p-2}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\int_{\mathcal{R}_{R}(\tau_{0},\tau_{1},v)}r^{p-2}|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\Big{)}^{\frac{1}{2}}\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}}.$

Suppose now $|\mathbf{k}_{2}|\leq k-3$. Then, if $\tau_{1}-\tau_{0}\geq 1$,

$\displaystyle\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\quad\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\int_{v(R,\tau_{0})}^{v}\sup_{u,\theta}\big{(}r^{\frac{p+1}{2}}|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\big{)}\left(\int_{\tau_{0}}^{\tau_{1}}\int_{S^{2}}r^{p-3}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|^{2}r^{2}d\omega
du\right)^{\frac{1}{2}}dv$
$\displaystyle\quad\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\left(\int_{\mathcal{R}_{R}(\tau_{0},\tau_{1},v)}r^{p-3}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|^{2}r^{2}d\omega
dudv\right)^{\frac{1}{2}}\sum_{|\widetilde{\textbf{k}}|\leq|\mathbf{k}_{2}|+3}\left(\int_{\mathcal{R}_{R}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}\mathfrak{D}^{\widetilde{\mathbf{k}}}\psi|^{2}r^{2}d\omega
dudv\right)^{\frac{1}{2}}$
$\displaystyle\quad\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(p-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\
}\sqrt{\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(p-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}^{\prime}\
},$

using again the Sobolev inequality on the incoming cones (C.8). Similarly, if
$\tau_{1}-\tau_{0}<1$,

$\displaystyle\int_{\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|\cdot|\partial_{u}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\partial_{v}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\quad\lesssim(\tau_{1}-\tau_{0})^{-\frac{1}{2}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\left(\int_{\mathcal{R}_{R}(\tau_{0},\tau_{1},v)}r^{p-2}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|^{2}\right)^{\frac{1}{2}}\sum_{|\widetilde{\textbf{k}}|\leq|\mathbf{k}_{2}|+3}\left(\int_{\mathcal{R}_{R}(\tau_{0},\tau_{1},v)}r^{p-2}|\partial_{v}\mathfrak{D}^{\widetilde{\mathbf{k}}}\psi|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\quad\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}{}^{*}}.$

∎

####  C.1.3 The proof of Proposition C.1.1 for the nonlinearity
$F=\not{g}^{AB}\not{\nabla}_{A}\psi\not{\nabla}_{B}\psi$

Consider now the nonlinearity

$F=\not{g}^{AB}\not{\nabla}_{A}\psi\not{\nabla}_{B}\psi.$

In view of the property (C.2), the proof of Proposition C.1.1 follows from the
following proposition.

######  Proposition C.1.5 (Nonlinear error estimates for
$F=\not{g}^{AB}\not{\nabla}_{A}\psi\not{\nabla}_{B}\psi$ on Minkowski space).

Under the assumptions of Proposition C.1.1, for each $|\mathbf{k}|\leq k$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\Bigg{(}\int_{\mathcal{R}(\tau_{0},\tau_{1},v)\cap\\{r\geq
R\\}}\Big{(}|\partial_{u}\mathfrak{D}^{\mathbf{k}}\psi|+|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|+\frac{1}{r}|\not{\nabla}\mathfrak{D}^{\mathbf{k}}\psi|+\frac{1}{r}|\mathfrak{D}^{\mathbf{k}}\psi|\Big{)}\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\\\
+\int_{\mathcal{R}(\tau_{0},\tau_{1},v)\cap\\{r\geq
R\\}}|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|^{2}\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\Bigg{)}\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}_{R}+\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}{}^{*}_{\frac{8R}{9}}}\Big{(}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}_{R}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}_{R}^{\prime}\
\Big{)}$

and

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\mathcal{R}(\tau_{0},\tau_{1},v)\cap\\{r\geq
R\\}}r^{p-1}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}_{R}+\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}{}^{*}_{\frac{8R}{9}}}\Big{(}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}_{R}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(p-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}_{R}^{\prime}\
\Big{)}.$

###### Proof.

The separate terms are estimated individually, and the proof follows from the
estimates (C.17)–(C.21) below.

The notation of Section C.1.1 will again be used throughout. As in the proof
of Proposition C.1.4, the cases $\tau_{1}-\tau_{0}\geq 1$ and
$\tau_{1}-\tau_{0}<1$ are typically considered separately. Similarly, the
region near the corner $\\{u=\tau_{0}\\}\cap\\{r=R\\}$ is treated separately.

Estimate in the corner region $\\{v^{\prime}\leq v(R,\tau_{0}+R)\\}$: For any
$|\tilde{\mathbf{k}}|\leq k+1$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\tilde{\mathbf{k}}|-1}\int_{\mathcal{R}_{\mathrm{Corner}}(\tau_{0},\tau_{1},v)}|\mathfrak{D}^{\tilde{\mathbf{k}}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{E}}}{}^{*}_{\frac{8R}{9}}}\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}_{R}^{\prime}.$
(C.17)

Indeed, (C.17) follows exactly as in the proof of Proposition C.1.4, using the
Sobolev inequality (C.9).

The remainder of the proof concerns the region
$\check{\mathcal{R}}_{R}(\tau_{0},\tau_{1},v)$.

Estimate for $\partial_{u}\psi$ term: First note that, for any
$|\mathbf{k}|\leq k$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}.$
(C.18)

Indeed, for $\tau_{1}-\tau_{0}\geq 1$, assuming without loss of generality
that $|\mathbf{k}_{1}|\leq k-3$,

$\displaystyle\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\qquad\qquad\lesssim\sup_{u,v,\theta}r|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\Big{(}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{-1-\delta}|\partial_{u}\mathfrak{D}^{\mathbf{k}}\psi|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{-1+\delta}|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\Big{)}^{\frac{1}{2}}\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime},$

by the Sobolev inequality on the incoming cones (C.7), since $p\geq\delta$.
For $\tau_{1}-\tau_{0}<1$,

$\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}|\partial_{u}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{F}}}{}^{*}}.$

Estimates for $\partial_{v}\psi$, $r^{-1}\not{\nabla}\psi$, and $r^{-1}\psi$
terms: Next, for any $|\mathbf{k}|\leq k$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}\big{(}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|+r^{-1}|\not{\nabla}\mathfrak{D}^{\mathbf{k}}\psi|+r^{-1}|\mathfrak{D}^{\mathbf{k}}\psi|\big{)}\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\Big{(}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\
\Big{)}.$ (C.19)

Indeed, for $\tau_{1}-\tau_{0}\geq 1$, assuming again without loss of
generality that $|\mathbf{k}_{1}|\leq k-3$,

$\displaystyle\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\qquad\lesssim\sup_{u,v,\theta}r|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\Big{(}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{-1}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{-1}|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\Big{)}^{\frac{1}{2}}\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime},$

and similarly, for $\tau_{1}-\tau_{0}<1$,

$\displaystyle\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}|\partial_{v}\mathfrak{D}^{\mathbf{k}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\
\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*},$

using again the Sobolev inequality on the incoming cones (C.7). Similarly for
the $r^{-1}\psi$ and $r^{-1}\not{\nabla}\psi$ terms, using the fact that

$\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}\big{(}r^{-1}|\not{\nabla}\mathfrak{D}^{\mathbf{k}}\psi|^{2}+r^{-3}|\mathfrak{D}^{\mathbf{k}}\psi|^{2}\big{)}\leq\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(\delta-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime},\qquad\int_{v(R,u)}^{v}\int_{S^{2}}\left(|r^{-1}\not{\nabla}\psi|^{2}+|r^{-1}\psi|^{2}\right)r^{2}d\omega
dv^{\prime}\leq\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}.$

Estimate for quartic term: For any $|\mathbf{k}|\leq k$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|^{2}\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\lesssim\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}\Big{(}\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(-1-\delta)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}\
\Big{)},$ (C.20)

which follows as an easy consequence of the Sobolev inequality (C.7).

Estimate for $r^{p-1}\partial_{v}(r\psi)$ term: Finally, for any
$|\mathbf{k}|\leq k$,

$\sum_{|\mathbf{k}_{1}|+|\mathbf{k}_{2}|\leq|\mathbf{k}|}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\Big{(}\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}+\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(p-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}\
\Big{)}.$ (C.21)

Assume again, without loss of generality, that $|\mathbf{k}_{1}|\leq k-3$.
Then, using (C.7), for $\tau_{1}-\tau_{0}\geq 1$,

$\displaystyle\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\qquad\qquad\lesssim\sup_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\Big{(}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{p-3}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{p-1}|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\Big{)}^{\frac{1}{2}}\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\int^{*}\accentset{\scalebox{0.6}{\mbox{\tiny$(p-1)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}^{\prime}.$

Similarly, for $\tau_{1}-\tau_{0}<1$,

$\displaystyle\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{p-1}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{1}}\psi|\cdot|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|$
$\displaystyle\qquad\qquad\lesssim(\tau_{1}-\tau_{0})^{-\frac{1}{2}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\Big{(}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{p-2}|\partial_{v}(r\mathfrak{D}^{\mathbf{k}}\psi)|^{2}\Big{)}^{\frac{1}{2}}\Big{(}\int_{\check{\mathcal{R}}(\tau_{0},\tau_{1},v)}r^{p-2}|\not{\nabla}\mathfrak{D}^{\mathbf{k}_{2}}\psi|^{2}\Big{)}^{\frac{1}{2}}\lesssim\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$\ll\mkern-6.0muk$}}}{\mathcal{F}}}{}^{*}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(p)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}}\sqrt{\accentset{\scalebox{0.6}{\mbox{\tiny$(0)$}}}{\underaccent{\scalebox{0.8}{\mbox{\tiny$k$}}}{\mathcal{E}}}{}^{*}}.$

∎

###  C.2 More general nonlinearities and Kerr

For more general nonlinearities on Minkowski space of the form

$F=N^{\mu\nu}(\psi,x)\partial_{\mu}\psi\partial_{\nu}\psi,$

with $N$ satisfying a null condition of the form

$\sup_{|\xi|\leq 1,r\geq R}\sum_{|\textbf{k}|+s\leq
k}\sum_{A,B=1,2}\mathfrak{D}^{\mathbf{k}}\partial_{\xi}^{s}\big{(}rN^{uu}+N^{uv}+N^{vv}+rN^{Au}+rN^{Av}+r^{2}N^{AB}\big{)}(\xi,x)\lesssim
D_{k},$ (C.22)

where the $A$, $B$ indices refer to coordinates $\vartheta,\varphi$ (recall
the $(u,v,\vartheta,\varphi)$ coordinate system of Section 2.7.1), and $D_{k}$
are arbitrary constants, the proof of Proposition C.0.1 follows exactly as
above, where the bound (4.29) is now also used to obtain pointwise bounds on
$\psi$ through an easy weighted Sobolev estimate which allow as to invoke
(C.22).

######  Remark C.2.1.

Note that, besides the nonlinearities of the classical null condition [Kla86],
our class (C.22) includes for instance also nonlinearities which for large $r$
take the form $F=(\sin r)\partial_{u}\psi\partial_{v}\psi$. It does not,
however, include even more general examples like $F=(\sin
x)\partial_{u}\psi\partial_{v}\psi$ considered recently in [AZ22], due to the
presence of the weighted vector fields $\Omega_{i}$ in our set of commutation
vector fields $\mathfrak{D}^{\mathbf{k}}$.

More generally, if $g_{0}$ is a metric with asymptotics suitably close to
those of Minkowski space, with extra terms suitably small, then the above
proof again applies.

We will consider explicitly the case that $g_{0}$ is the Kerr metric. Define
double null $(u,v,\theta^{1},\theta^{2})$ coordinates on Kerr, when $0<|a|<M$,
in terms of the Boyer–Lindquist coordinates $(t,r,\vartheta,\phi)$ of Appendix
A.1, by

$u=t-r_{*},\qquad v=t+r_{*},$

with $\theta^{1}$ defined implicitly by the relation

$F(\theta^{1},r,\vartheta)=0,\quad\text{where}\quad
F(\theta^{1},r,\vartheta)=\int_{\theta^{1}}^{\vartheta}\frac{1}{a\sqrt{\sin^{2}\theta^{1}-\sin^{2}\theta^{\prime}}}d\theta^{\prime}+\int_{r}^{\infty}\frac{1}{\sqrt{((r^{\prime})^{2}+a^{2})^{2}-a^{2}\sin^{2}\theta^{1}\Delta^{\prime}}}dr^{\prime},$
(C.23)

and $\theta^{2}$ defined by

$\theta^{2}=\phi+h(r_{*},\theta^{1}),\quad\text{where $h$
satisfies}\quad\partial_{r_{*}}h(r_{*},\theta^{1})=\frac{2Mar}{\Sigma\mathrm{R}^{2}},\quad\lim_{r_{*}\to\infty}h(r_{*},\theta^{1})=0.$

See for instance [PI98, DL17]. Here

$r_{*}(r,\vartheta)=\int\frac{r^{2}+a^{2}}{\Delta}dr+\int_{r}^{\infty}\frac{(r^{\prime})^{2}+a^{2}-\sqrt{((r^{\prime})^{2}+a^{2})^{2}-a^{2}\sin^{2}\theta^{1}\Delta^{\prime}}}{\Delta^{\prime}}dr^{\prime}+\int^{\vartheta}_{\theta^{1}}a\sqrt{\sin^{2}\theta^{1}-\sin^{2}\theta^{\prime}}d\theta^{\prime},$
(C.24)

(note that the expression (C.24) is independent of $\theta^{1}$ in view of
(C.23)), where $\int\frac{r^{2}+a^{2}}{\Delta}dr$ is a function satisfying

$\partial_{r}\int\frac{r^{2}+a^{2}}{\Delta}dr=\frac{r^{2}+a^{2}}{\Delta},$

and

$\Delta=r^{2}+a^{2}-2Mr,\qquad\Delta^{\prime}=(r^{\prime})^{2}+a^{2}-2Mr^{\prime},\qquad\Sigma=r^{2}+a^{2}\cos^{2}\vartheta,\qquad\mathrm{R}^{2}=r^{2}+a^{2}+\frac{2Ma^{2}r\sin^{2}\vartheta}{\Sigma}.$

Note in particular that $r_{*}$ is distinct from $r^{\star}$ defined in
Section A.1.

In these double null coordinates the Kerr metric takes the form

$g_{a,M}=-\Omega^{2}dudv+\not{g}_{AB}(d\theta^{A}-b^{A}dv)(d\theta^{B}-b^{B}dv),$

where

$\Omega^{2}=\frac{\Delta}{\mathrm{R}^{2}},\qquad b^{1}=0,\qquad
b^{2}=\frac{4Mar}{\Sigma\mathrm{R}^{2}},$

and

$\not{g}_{11}=\frac{a^{2}(\partial_{\theta^{1}}F)^{2}(\sin^{2}\theta^{1}-\sin^{2}\vartheta)((r^{2}+a^{2})^{2}-a^{2}\sin^{2}\theta^{1}\Delta)}{\mathrm{R}^{2}}+(\partial_{\theta^{1}}h)^{2}\mathrm{R}^{2}\sin^{2}\vartheta,$
$\not{g}_{22}=\mathrm{R}^{2}\sin^{2}\vartheta,\qquad\not{g}_{12}=\not{g}_{21}=-\mathrm{R}^{2}\sin^{2}\vartheta\partial_{\theta^{1}}h.$

In particular, one has

$L=\partial_{v}+b^{A}\partial_{\theta^{A}},\qquad\underline{L}=\partial_{u}.$

Moreover $T=\partial_{u}+\partial_{v}$ and
$\Omega_{i}={\Omega_{i}}(\theta^{1},\theta^{2})^{A}\partial_{\theta^{A}}$ for
$i=1,2,3$, for functions ${\Omega_{i}}(\theta^{1},\theta^{2})^{A}$, so that,
in particular,

$[\partial_{u},T]=[\partial_{u},\Omega_{i}]=[\partial_{v},T]=[\partial_{v},\Omega_{i}]=0.$

The function $b^{2}=b^{2}(u,v,\theta^{1},\theta^{2})$ satisfies, for any
$k\geq 0$,

$\sum_{k_{1}+k_{2}+|\textbf{k}_{3}|\leq
k}|\partial_{u}^{k_{1}}(r\partial_{v})^{k_{2}}\Omega^{\textbf{k}_{3}}b^{2}|\leq\frac{C_{k}}{r^{3}},$

for large $r$, and hence the commutation relations

$\displaystyle|\mathfrak{D}^{\mathbf{k}}L\psi|$
$\displaystyle\lesssim\sum_{|\widetilde{\bf k}|\leq|\bf
k|}|L\mathfrak{D}^{\widetilde{\bf
k}}\psi|+\frac{1}{r^{3}}\sum_{i=1}^{3}\sum_{|\widetilde{\bf k}|\leq|\bf
k|-1}|\Omega_{i}\mathfrak{D}^{\widetilde{\bf k}}\psi|,$ (C.25)
$\displaystyle|\mathfrak{D}^{\mathbf{k}}\underline{L}\psi|$
$\displaystyle\lesssim\sum_{|\widetilde{\bf k}|\leq|\bf
k|}|\underline{L}\mathfrak{D}^{\widetilde{\bf
k}}\psi|+\frac{1}{r^{3}}\sum_{i=1}^{3}\sum_{|\widetilde{\bf k}|\leq|\bf
k|-1}|\Omega_{i}\mathfrak{D}^{\widetilde{\bf k}}\psi|,$ (C.26)

hold for large $r$. Moreover (C.2) remains true. The proof of Proposition
C.0.1 in this case then follows just as in the case where $g_{0}$ was the
Minkowski metric, using now (C.25) and (C.26) in place of (C.1).

We note finally that the condition (C.22) applied to Kerr indeed includes in
particular the nonlinearities considered in [Luk13].

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