# Sharp Fourier extension for functions with localized support on the circle

Lars Becker  Mathematical Institute, University of Bonn, Endenicher Allee 60,
53115, Bonn, Germany<EMAIL_ADDRESS>
###### Abstract.

A well known conjecture states that constant functions are extremizers of the
$L^{2}\to L^{6}$ Tomas-Stein extension inequality for the circle. We prove
that functions supported in a $\sqrt{6}/80$-neighbourhood of a pair of
antipodal points on $S^{1}$ satisfy the conjectured sharp inequality. In the
process, we make progress on a programm formulated in [CFO+17] to prove the
sharp inequality for all functions.

###### Key words and phrases:

Circle, Fourier restriction, sharp inequalities

###### 2020 Mathematics Subject Classification:

42B10

##  1\. Introduction

We are interested in the conjecture that constant functions are extremizers
for the Tomas-Stein Fourier extension inequality for the circle

$\|\widehat{f\sigma}\|_{L^{6}(\mathbb{R}^{2})}\leq C\|f\|_{L^{2}(\sigma)}\,.$
(1)

Here $\sigma$ is the arc length measure on the unit circle
$S^{1}\subset\mathbb{R}^{2}$ and $\hat{\mu}(\xi)=\int
e^{-ix\xi}\,\mathrm{d}\mu(\xi)$ is the Fourier transform.

The corresponding conjecture for $S^{2}$ was proven by Foschi [Fos15], and in
[CFO+17] Foschi’s argument is adapted to $S^{1}$, and the conjecture of
interest is reduced to the following.

######  Conjecture 1.

The quadratic form

$Q(f):=\int_{(S^{1})^{6}}(\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}-1)(f(\omega_{1},\omega_{2},\omega_{3})^{2}-f(\omega_{1},\omega_{2},\omega_{3})f(\omega_{4},\omega_{5},\omega_{6}))\,\mathrm{d}\Sigma(\omega)$

is positive semi-definite on the subspace $V$ of all antipodal functions in
$L^{2}((S^{1})^{3},\mathbb{R})$. Here we denote
$\mathrm{d}\Sigma(\omega)=\delta(\sum_{j=1}^{6}\omega_{k})\prod_{j=1}^{6}\mathrm{d}\sigma(\omega_{j})$,
and a function $f$ is antipodal if
$f(\pm\omega_{1},\pm\omega_{2},\pm\omega_{3})$ does not depend on the choice
of signs.

1 has been verified for all functions with Fourier modes up to degree $120$ in
[OTZ22] and [BTZ20], via a numerical computation of the eigenvalues of $Q$ on
the finite dimensional space of such functions. Further, using different
methods, in [CG22] the conjectured sharp form of inequality (1) has been
established for certain infinite dimensional subspaces of $L^{2}(\sigma)$ with
constrained Fourier support. Our main result establishes 1 for functions with
localized spatial support.

Let $C$ be the cylinder of radius $\varepsilon$ centered at the line
$\mathbb{R}(1,1,1)$, and define

$V_{\varepsilon}:=\bigg{\\{}f\in
V\,:\,\operatorname{supp}f(e^{i\theta_{1}},e^{i\theta_{2}},e^{i\theta_{3}})\subset\bigcup_{k\in\pi\mathbb{Z}^{3}}k+C\bigg{\\}}.$

######  Theorem 2.

Let $\varepsilon=1/20$. Then for all $f\in V_{\varepsilon}$ it holds that
$Q(f)\geq 0$.

Note that since constant functions are in the kernel of $Q$, the same result
holds for $V_{\varepsilon}\oplus\langle\mathbf{1}\rangle$, where $\mathbf{1}$
is the constant $1$ function.

As a corollary, functions with support sufficiently close to a pair of
antipodal points satisfy (1) with the conjectured sharp constant. Define
$\Phi(g):=\|\widehat{g\sigma}\|_{L^{6}(\mathbb{R}^{2})}/\|g\|_{L^{2}(\sigma)}$.

######  Corollary 3.

Let $\varepsilon^{\prime}=\sqrt{3/8}\varepsilon$. Suppose that $g\in
L^{2}(\sigma)$ is such that $g(e^{i\theta})$ is supported in
$(-\varepsilon^{\prime},\varepsilon^{\prime})+\pi\mathbb{Z}$. Then
$\Phi(g)\leq\Phi(\mathbf{1})$, where $\mathbf{1}$ is the constant $1$ function
on $S^{1}$.

Note that by rotation symmetry, the same holds when $g(e^{i\theta})$ is
supported in $I+\pi\mathbb{Z}$ for any interval $I$ of length
$2\varepsilon^{\prime}$.

The constants $\varepsilon$ and $\varepsilon^{\prime}$ in Theorem 2 and
Corollary 3 are not optimal. Numerical computations suggest that with our
method $\varepsilon$ can be improved up to about $0.104$ and
$\varepsilon^{\prime}$ up to about $0.063$, see Section 7.

The numerical results in [BTZ20] suggest that eigenfunctions of $Q$ on the
subspace of functions with Fourier modes up to degree $N$ corresponding to
small eigenvalues concentrate in space. Theorem 2 shows that $Q$ is positive
on all such sufficiently concentrated functions, thus it should be a useful
partial result in establishing positive semi-definiteness of $Q$ on the full
space of antipodal functions. A more precise observation by Jiaxi Cheng, a
graduate student in Bonn, is that the smallest eigenvalue is of size $\sim
N^{-2}\log(N)$, see Section 2 of [NOT22]. The existence of such an eigenvalue
is also explained by the asymptotic formula for the multiplier $m$ in Lemma 7,
which looks like $c\lvert\log\lvert x\rvert\rvert\lvert x\rvert^{2}$ near $0$.
Unfortunately, we cannot prove that this is the smallest eigenvalue.

More generally, the topic of sharp Fourier extension inequalities has
attracted a lot of interest in recent years. In the following we consider
general dimensions $d\geq 2$. Then the Tomas-Stein extension inequality states
that for every $q\geq q_{d}:=2(d+1)/(d-1)$, there exists $C(d,q)>0$ such that
for all $f\in L^{2}(S^{d-1},\sigma^{d-1})$

$\|\widehat{f\sigma}\|_{L^{q}(\mathbb{R}^{d})}\leq
C(d,q)\|f\|_{L^{2}(\sigma)}\,.$ (2)

Here $\sigma^{d-1}$ denotes the $d-1$-dimensional Hausdorff measure on
$S^{d-1}$.

It is known that extremizers for (2) exist when $q>q_{d}$, for all $d$, see
[FVV11]. At the endpoint $q=q_{d}$, existence and smoothness of extremizers
have been shown for $d=3$ in [CS12], [CS12a] and for $d=2$ in [Sha16a],
[Sha16]. For higher dimensional spheres $d\geq 4$, existence of extremizers
for $q=q_{d}$ is known conditional on the conjecture that Gaussians maximize
the corresponding extension inequality for the paraboloid, see [FLS16].

For certain specific choices of $(d,q)$, a full characterization of the
extremizers of (2) is known. Most such results grew out of the work of Foschi
[Fos15], who showed that constant functions maximize (2) for $(d,q)=(2,4)$,
and gave a full characterization of all complex valued maximizers. His method
can be adapted for some non-endpoint extension inequalities on higher
dimensional spheres, see [CO14]. Using different methods, maximizers of (2)
for some choices of $(d,q)$ with even $q>4$ are characterized in [OQ21]. In
some further cases it is known that constant functions are local maximizers.
This was shown in [CFO+17] for $(d,q)=(2,6)$, and in [GN22] for $(d,q_{d})$
with $3\leq d\leq 60$. For further background and references on sharp Fourier
extension inequalities we refer to [FO17] and [NOT22].

### Acknowledgement

I am grateful to Christoph Thiele for introducing me to this problem and for
many helpful discussions, and to Jan Holstermann for pointing out the short
proof of Lemma 15. The author was supported by the Collaborative Research
Center 1060 funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) and the Hausdorff Center for Mathematics, funded by the
DFG under Germany’s Excellence Strategy - GZ 2047/1, ProjectID 390685813.

##  2\. Proof of Corollary 3

Corollary 3 is a direct consequence of Theorem 2 and the program formulated in
[CFO+17]. We give a brief sketch of the implication here, for the details of
the program and proofs we refer the reader to [CFO+17].

###### Proof of Corollary 3.

Let $g\in L^{2}(\sigma)$ such that $g(e^{i\theta})$ is supported in
$(-\varepsilon^{\prime},\varepsilon^{\prime})+\pi\mathbb{Z}$. Define
$\tilde{g}(x)=g(-x)$ and

$g_{\\#}=\sqrt{\frac{\lvert g\rvert^{2}+\lvert\tilde{g}\rvert^{2}}{2}}\,.$

As shown in [CFO+17], Step 1 and 2, it holds that $\Phi(g)\leq\Phi(g_{\\#})$,
and $g_{\\#}$ is antipodal and $g_{\\#}(e^{i\theta})$ is supported in
$(-\varepsilon_{2},\varepsilon_{2})+\pi\mathbb{Z}$. Define
$f(\omega_{1},\omega_{2},\omega_{3}):=g_{\\#}(\omega_{1})g_{\\#}(\omega_{2})g_{\\#}(\omega_{3})$.
Then $f\in V_{\sqrt{8/3}\varepsilon^{\prime}}=V_{\varepsilon}$, hence
$Q(f)\geq 0$, by Theorem 2. This verifies Conjecture 1.4 in [CFO+17] for
$g_{\\#}$. Using Step 3, 4 and 5 in [CFO+17], we conclude that
$\Phi(g)\leq\Phi(1)$. ∎

As written, this proof depends on Theorem 1.2 in [CFO+17], which is the main
result of that paper and verifies Step 5. However, this step has a shorter
proof if one assumes that the functions have small support, as we do. The step
consists in proving the inequality

$\int
h(\omega_{1})h(\omega_{2})h(\omega_{3})(\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}-1)\,\sigma*\sigma*\sigma(\omega_{1}+\omega_{2}+\omega_{3})\,\mathrm{d}\sigma(\omega_{1})\,\mathrm{d}\sigma(\omega_{2})\,\mathrm{d}\sigma(\omega_{3})\\\
\leq\frac{\|h\|_{L^{1}(\sigma)}^{3}}{\|\mathbf{1}\|_{L^{1}(\sigma)}^{3}}\int(\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}-1)\,\sigma*\sigma*\sigma(\omega_{1}+\omega_{2}+\omega_{3})\,\mathrm{d}\sigma(\omega_{1})\,\mathrm{d}\sigma(\omega_{2})\,\mathrm{d}\sigma(\omega_{3})\,,$
(3)

valid for all antipodal $h\in L^{1}(\sigma)$, which is then applied to
$h=g^{2}$. Under the assumptions of Corollary 3, $g$ and therefore also $h$
are supported in $(-\varepsilon^{\prime},\varepsilon^{\prime})+\pi\mathbb{Z}$.
Then (3) follows from the estimate

$\max_{[-\varepsilon^{\prime},\varepsilon^{\prime}]^{3}}\frac{1}{8}\sum_{\gamma\in\\{-1,1\\}^{3}}(\lvert\gamma_{1}e^{i\theta_{1}}+\gamma_{2}e^{i\theta_{2}}+\gamma_{3}e^{i\theta_{3}}\rvert^{2}-1)\,\sigma*\sigma*\sigma(\gamma_{1}e^{i\theta_{1}}+\gamma_{2}e^{i\theta_{2}}+\gamma_{3}e^{i\theta_{3}})\\\
\leq\frac{1}{\|\mathbf{1}\|_{L^{1}(\sigma)}^{3}}\int(\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}-1)\,\sigma*\sigma*\sigma(\omega_{1}+\omega_{2}+\omega_{3})\,\mathrm{d}\sigma(\omega_{1})\,\mathrm{d}\sigma(\omega_{2})\,\mathrm{d}\sigma(\omega_{3})\,,$

which can be verified using Lemma 9.

##  3\. Proof of Theorem 2

###  3.1. Orthogonal decomposition

We consider the sesquilinear form

$B(f,g)=\int_{(S^{1})^{6}}(\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}-1)(f(\omega_{1},\omega_{2},\omega_{3})\overline{g(\omega_{1},\omega_{2},\omega_{3})}-f(\omega_{1},\omega_{2},\omega_{3})\overline{g(\omega_{4},\omega_{5},\omega_{6})})\,\mathrm{d}\Sigma(\omega)\,.$

By a change of variables, it holds that $B(f,g)=B(Rf,Rg)$, where
$Rf(\omega_{1},\omega_{2},\omega_{3})=f(e^{i}\omega_{1},e^{i}\omega_{2},e^{i}\omega_{3})$.
Define
$Z_{d}=\\{(k_{1},k_{2},k_{3})\in(2\mathbb{Z})^{3}\,:\,k_{1}+k_{2}+k_{3}=d\\}$
and

$X_{d}=\\{\sum_{k\in
Z_{d}}a_{k}\omega_{1}^{k_{1}}\omega_{2}^{k_{2}}\omega_{3}^{k_{3}}\ :\
a_{k}\in\ell^{2}(Z_{d})\\}\subset L^{2}((S^{1})^{3})\,.$

For $d\neq d^{\prime}$, the spaces $X_{d}$ and $X_{d^{\prime}}$ are
eigenspaces of $R$ with different eigenvalues, and hence are orthogonal with
respect to $B$. Note that the orthogonal projection $\pi_{d}$ onto $X_{d}$ can
be expressed as

$\pi_{d}(f)(\omega_{1},\omega_{2},\omega_{3})=\int_{0}^{1}e^{-2\pi
idt}f(e^{2\pi it}\omega_{1},e^{2\pi it}\omega_{2},e^{2\pi
it}\omega_{3})\,\mathrm{d}t\,,$

which implies that $\pi_{d}(V_{\varepsilon})\subset V_{\varepsilon}$.
Therefore, we have that

$V_{\varepsilon}=\overline{\bigoplus_{d\in\mathbb{Z}}\pi_{d}(V_{\varepsilon})}=\overline{\bigoplus_{d\in\mathbb{Z}}(V_{\varepsilon}\cap
X_{d})}\,.$

Hence, it suffices to show positive semi-definiteness of $B$ on each of the
spaces $X_{d,\varepsilon}:=V_{\varepsilon}\cap X_{d}$.

###  3.2. Reducing the dimension

We now integrate out simultaneous rotations of all $\omega_{i}$ by the same
angle. From now on, we use the convention that

$\omega_{i}=(\cos(\theta_{i}),\sin(\theta_{i}))\,,$ (4)

and abuse notation by writing $f(\omega(\theta))=f(\theta)$. We also define

$\displaystyle a(\theta_{1},\theta_{2},\theta_{3})$
$\displaystyle:=(\cos(\theta_{1})+\cos(\theta_{2})+\cos(\theta_{3}))^{2}+(\sin(\theta_{1})+\sin(\theta_{2})+\sin(\theta_{3}))^{2}=\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}\,,$

so that the weight in the bilinear form $B$ is given by $a-1$, and record the
useful identity

$a(\theta_{1},\theta_{2},\theta_{3})=3+2\cos(\theta_{1}-\theta_{2})+2\cos(\theta_{2}-\theta_{3})+2\cos(\theta_{3}-\theta_{1})\,.$
(5)

######  Lemma 4.

Let $C$ be the convex hull of the four points

$(\pi,-\pi,0)\,,\quad(-\frac{\pi}{3},-\frac{\pi}{3},\frac{2\pi}{3})\,,\quad(-\pi,\pi,0)\,,\quad(\frac{\pi}{3},\frac{\pi}{3},-\frac{2\pi}{3})\,.$

Then the prism $P$ over $C$ of height $2\pi\sqrt{3}$ is a fundamental domain
for $\mathbb{R}^{3}/(2\pi\mathbb{Z})^{3}$.

###### Proof.

For each point in $\mathbb{R}^{3}/(2\pi\mathbb{Z})^{3}$, there exists a unique
representative $(\theta_{1},\theta_{2},\theta_{3})\in\mathbb{R}^{3}$ such that
$0\leq\theta_{1}+\theta_{2}+\theta_{3}<2\pi$, and such that the orthogonal
projection of $(\theta_{1},\theta_{2},\theta_{3})$ onto the hyperplane
$\\{(x_{1},x_{2},x_{3})\ :\ x_{1}+x_{2}+x_{3}=0\\}$ is contained in a fixed
fundamental domain $C_{1}$ of the lattice generated by $(2\pi,-2\pi,0)$ and
$(-2\pi,0,2\pi)$, such as

$\displaystyle C_{1}:=\left\\{\lambda_{1}\begin{pmatrix}2\pi\\\ -2\pi\\\
0\end{pmatrix}+\lambda_{2}\begin{pmatrix}-2\pi\\\ 0\\\ 2\pi\end{pmatrix}\ :\
\lambda_{1},\lambda_{2}\in[0,1)\right\\}.$ (6)

Thus $C_{1}+\\{(t,t,t)\,:\,t\in[0,\frac{2\pi}{3})\\}$ is a fundamental domain
for $\mathbb{R}^{3}/(2\pi\mathbb{Z})^{3}$. By decomposing this fundamental
domain into finitely many pieces and rearranging them, it is easy to see that
$C+\\{(t,t,t)\ :\ t\in[0,2\pi)\\}$ is also a fundamental domain. ∎

######  Lemma 5.

For all $d\in\mathbb{Z}$ there exists a function $\lambda_{d}:C\times C\to
S^{1}$ such that for all $f\in X_{d}$

$B(f,f)=12\pi\int_{C^{2}}\delta(a(\theta)-a(\theta^{\prime}))(a(\theta)-1)(\lvert
f(\theta)\rvert^{2}-\lambda_{d}(\theta^{\prime},\theta)f(\theta)\overline{f(\theta^{\prime})})\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta)\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})\,.$

Here $\mathcal{H}^{2}_{C}$ denotes the $2$-dimensional Hausdorff measure on
$C$.

###### Proof.

By Lemma 4, we have

$B(f,f)=\int_{P\times
P}\delta(\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}-\omega_{5}-\omega_{6})(\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}-1)\\\
\times(\lvert
f(\omega_{1},\omega_{2},\omega_{3})\rvert^{2}-f(\omega_{1},\omega_{2},\omega_{3})\overline{f(\omega_{4},\omega_{5},\omega_{6})})\prod_{j=1}^{6}\mathrm{d}\theta_{j}\
$ $=2\pi\sqrt{3}\int_{C\times
P}\delta(\omega_{1}+\omega_{2}+\omega_{3}-\omega_{4}-\omega_{5}-\omega_{6})(\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}-1)\\\
\times(\lvert
f(\omega_{1},\omega_{2},\omega_{3})\rvert^{2}-f(\omega_{1},\omega_{2},\omega_{3})\overline{f(\omega_{4},\omega_{5},\omega_{6})})\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta_{1},\theta_{2},\theta_{3})\,\prod_{j=4}^{6}\mathrm{d}\theta_{j}\,.$

Here we have used that $f\in X_{d}$, to integrate out simultaneous rotations
of all $6$ points $\omega_{j}$ by the same angle. For $x,y\in\mathbb{R}^{2}$,
it holds that

$\delta(x-y)=2\delta(\lvert x\rvert^{2}-\lvert
y\rvert^{2})\delta(\arg(x)-\arg(y))\,.$

Hence, we can rewrite the last expression as

$=4\pi\sqrt{3}\int_{C\times
P}\delta(\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}-\lvert\omega_{4}+\omega_{5}+\omega_{6}\rvert^{2})\delta(\arg(\omega_{1}+\omega_{2}+\omega_{3})-\arg(\omega_{4}+\omega_{5}+\omega_{6}))\\\
\times(\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2}-1)(\lvert
f(\omega_{1},\omega_{2},\omega_{3})\rvert^{2}-f(\omega_{1},\omega_{2},\omega_{3})\overline{\omega_{4},\omega_{5},\omega_{6})})\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta_{1},\theta_{2},\theta_{3})\,\prod_{j=4}^{6}\mathrm{d}\theta_{j}$
$=12\pi\int_{C\times
C}\int_{0}^{2\pi}\delta(a(\theta_{1},\theta_{2},\theta_{3})-a(\theta_{4},\theta_{5},\theta_{6}))\delta(\arg(\omega_{1}+\omega_{2}+\omega_{3})-\arg(\omega_{4}+\omega_{5}+\omega_{6})-t)\\\
\times(a(\theta_{1},\theta_{2},\theta_{3})-1)(\lvert
f(\omega_{1},\omega_{2},\omega_{3})\rvert^{2}-f(\omega_{1},\omega_{2},\omega_{3})\overline{f(e^{it}\omega_{4},e^{it}\omega_{5},e^{it}\omega_{6})})\,\mathrm{d}t\,\mathrm{d}\mathcal{H}^{4}_{C\times
C}(\theta)\,\,.$

Since $f\in X_{d}$, we have
$f(e^{it}\omega_{4},e^{it}\omega_{5},e^{it}\omega_{6})=e^{itd}f(\omega_{4},\omega_{5},\omega_{6})$.
Thus, we can integrate out $t$ and obtain the claimed identity. ∎

###  3.3. Completing the proof

By Lemma 5, we have for all $d$ and all $f\in X_{d}$:

$\displaystyle B(f,f)$ $\displaystyle\geq 12\pi\int_{C}\lvert
f(\theta)\rvert^{2}(a(\theta)-1)\int_{C}\delta(a(\theta)-a(\theta^{\prime}))\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta)$
$\displaystyle\quad-12\pi\int_{C^{2}}\delta(a(\theta)-a(\theta^{\prime}))\lvert
a(\theta)-1\rvert\lvert f(\theta)\rvert\lvert
f(\theta^{\prime})\rvert\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta)\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})$
$\displaystyle=:12\pi(I-I\\!I)\,.$ (7)

If $f\in X_{d,\varepsilon}$, then the restriction of $f$ onto the hyperplane
$H:=\\{(\theta_{1},\theta_{2},\theta_{3})\,:\,\theta_{1}+\theta_{2}+\theta_{3}=0\\}$
is supported in $\Lambda+B_{\varepsilon}(0)$, where
$\Lambda=\mathbb{Z}v_{1}\oplus\mathbb{Z}v_{2}$ and

$v_{1}:=(\frac{2\pi}{3},-\frac{\pi}{3},-\frac{\pi}{3}),\qquad
v_{2}:=(-\frac{\pi}{3},\frac{2\pi}{3},-\frac{\pi}{3})\,.$

Furthermore, the function $\lvert f\rvert$ is periodic with respect to
$\Lambda$, since it is periodic with respect to $\pi\mathbb{Z}^{3}$ and
invariant under all translations in direction $(1,1,1)$. Thus it suffices to
show:

######  Lemma 6.

Suppose that $\varepsilon\leq 1/20$. Then for all functions $f:H\to[0,\infty)$
that are periodic with respect to $\Lambda$ and supported in
$\Lambda+B_{\varepsilon}(0)$, it holds that $I\geq I\\!I$.

###### Proof.

Note that $C$ is a fundamental domain of the lattice $2\Lambda$. The
expressions in the integrals for term $I$ and $I\\!I$ are $2\Lambda$ periodic,
so we may replace $C$ by any other fundamental domain $C^{\prime}$. Since $f$
is supported in $\Lambda+B(0,\varepsilon)$, there exists a fundamental domain
$C^{\prime}$ such that $f|_{C^{\prime}}$ is supported in

$B_{\varepsilon}(0,0,0)\cup
B_{\varepsilon}(\frac{2\pi}{3},-\frac{\pi}{3},-\frac{\pi}{3})\cup
B_{\varepsilon}(-\frac{\pi}{3},\frac{2\pi}{3},-\frac{\pi}{3})\cup
B_{\varepsilon}(-\frac{\pi}{3},-\frac{\pi}{3},\frac{2\pi}{3})=:B_{1}\cup
B_{2}\cup B_{3}\cup B_{4}\,.$

We decompose

$I=\sum_{i=1}^{4}\int_{B_{i}}\lvert
f(\theta)\rvert^{2}(a(\theta)-1)\int_{C}\delta(a(\theta)-a(\theta^{\prime}))\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta)=:\sum_{i=1}^{4}I_{i}\,,$
(8) $I\\!I=\sum_{1\leq i,j\leq 4}\int_{B_{i}\times
B_{j}}\delta(a(\theta)-a(\theta^{\prime}))\lvert a(\theta)-1\rvert\lvert
f(\theta)\rvert\lvert
f(\theta^{\prime})\rvert\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta)\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})=:\sum_{1\leq
i,j\leq 4}I\\!I_{ij}\,.$ (9)

Note that $\lvert\theta\rvert<\pi/6$ implies, by (5), that $a(\theta)\geq
3+6\cos(\pi/3)=6$, and $\lvert\theta-(2\pi/3,-\pi/3,-\pi/3)\rvert<\pi/6$
implies similarly that $a(\theta)\leq 3$. Therefore, for $j=2,3,4$ the measure
$\delta(a(\theta)-a(\theta^{\prime}))$ vanishes on $B_{1}\times B_{j}$, thus
$I_{1j}=I_{j1}=0$. Next, we record that $I\\!I_{11}\leq I_{1}$, by Cauchy-
Schwarz and since $a(\theta)\geq 6$ on $B_{1}$:

$\displaystyle I\\!I_{11}$
$\displaystyle=\int_{B_{1}^{2}}\delta(a(\theta)-a(\theta^{\prime}))\lvert
a(\theta)-1\rvert\lvert f(\theta)\rvert\lvert
f(\theta^{\prime})\rvert\mathrm{d}\mathcal{H}^{2}_{C}(\theta)\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})$
$\displaystyle\leq\frac{1}{2}\int_{B_{1}^{2}}\delta(\tilde{a}(\theta)-\tilde{a}(\theta^{\prime}))(\tilde{a}(\theta)-1)(\lvert
f(\theta)\rvert^{2}+\lvert
f(\theta^{\prime})\rvert^{2})\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta)\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})$
$\displaystyle\leq\int_{B_{1}}\lvert
f(\theta)\rvert^{2}(\tilde{a}(\theta)-1)\int_{C}\delta(\tilde{a}(\theta)-\tilde{a}(\theta^{\prime}))\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta)\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})=I_{1}\,.$

The remaining terms are estimated in the next two sections. By Lemma 7 and
Lemma 8, we have

$I_{2}+I_{3}+I_{4}\geq 30\int_{B_{1}}\lvert\theta\rvert^{2}\lvert
f(\theta)\rvert^{2}\,\mathrm{d}\mathcal{H}^{2}_{H}(\theta)>9\frac{101}{100}\pi\int_{B_{1}}\lvert\theta\rvert^{2}\lvert
f(\theta)\rvert^{2}\,\mathrm{d}\mathcal{H}^{2}_{H}(\theta)\geq\sum_{2\leq
i,j\leq 4}I\\!I_{ij}\,,$

which completes the proof. ∎

##  4\. Estimating term $I$

######  Lemma 7.

It holds that

$I_{2}+I_{3}+I_{4}=\int_{B_{1}}m(\theta)\lvert
f(\theta)\rvert^{2}\,\mathrm{d}\mathcal{H}^{2}_{H}(\theta)\,,$ (10)

where $I_{j}$ is defined in (8), and $m(\theta)\geq 30\lvert\theta\rvert^{2}$.

###### Proof.

By definition of the $I_{j}$, equation (10) holds with

$m(\theta)=\sum_{j=2}^{4}(a(\theta+c_{j})-1)\int_{C}\delta(a(\theta+c_{j})-a(\theta^{\prime}))\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})\,,$

where $c_{j}$ is the center of the ball $B_{j}$. Reversing the argument in the
proof of Lemma 5, it follows that for $x\in\mathbb{R}^{2}$

$\int_{C}\delta(\lvert
x\rvert^{2}-a(\theta^{\prime}))\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})=\frac{1}{\sqrt{3}}\int_{P}\delta(\lvert
x\rvert^{2}-\lvert\omega_{1}+\omega_{2}+\omega_{3}\rvert^{2})\delta(\arg(x)-\arg(\omega_{1}+\omega_{2}+\omega_{3}))\,\prod_{j=1}^{3}\mathrm{d}\theta^{\prime}_{j}\\\
=\frac{1}{2\sqrt{3}}\int_{(S^{1})^{3}}\delta(x-(\omega_{1}+\omega_{2}+\omega_{3}))\,\prod_{j=1}^{3}\mathrm{d}\sigma(\omega_{j})=\frac{1}{2\sqrt{3}}\sigma*\sigma*\sigma(x)\,.$

The convolution $\sigma*\sigma*\sigma$ is radial. We set
$\sigma*\sigma*\sigma(x)=\rho(\lvert x\rvert)$, giving

$m(\theta)=\frac{1}{2\sqrt{3}}\sum_{j=2}^{4}(a(\theta+c_{j})-1)\rho(\sqrt{a(\theta+c_{j})})\,.$
(11)

In polar coordinates

$\displaystyle\begin{pmatrix}\theta_{1}\\\ \theta_{2}\\\
\theta_{3}\end{pmatrix}$
$\displaystyle=s\cos(\alpha)\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\ -1\\\
0\end{pmatrix}+s\sin(\alpha)\frac{1}{\sqrt{6}}\begin{pmatrix}1\\\ 1\\\
-2\end{pmatrix}$ (12)

we compute in Lemma 14 the asymptotic expansion

$\displaystyle(a(\theta+c_{4})-1)\rho(\sqrt{a(\theta+c_{4})})$
$\displaystyle=-12s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))\log(s)$ (13)
$\displaystyle\quad-6s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))\log\lvert
3\sin^{2}(\alpha)-\cos^{2}(\alpha)\rvert$ (14) $\displaystyle\quad+18\log
2\,s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))$ (15) $\displaystyle\quad+E\,,$

with $\lvert E\rvert\leq-180s^{4}\log s+71s^{4}$ when $s\leq 1/20$. As the
function $a$ is invariant under permutation of its arguments and constant in
direction $(1,1,1)$, it is invariant under the rotation $T$ by $2\pi/3$ about
the line $\mathbb{R}(1,1,1)$. Since

$c_{2}+\theta(\alpha,s)=T(c_{4}+\theta(\alpha+4\pi/3,s))\,,\qquad
c_{3}+\theta(\alpha,s)=T^{2}(c_{4}+\theta(\alpha+2\pi/3,s))\,,$

we obtain the same asymptotic expansion for
$a(\theta+c_{j})\rho(\sqrt{a(\theta+c_{j})})$, $j=2,3$, but with $\alpha$
replaced by $\alpha+\frac{4\pi}{3}$ and $\alpha+\frac{2\pi}{3}$.

We now consider (11). The term (13) contributes $-6\sqrt{3}s^{2}\log(s)$ to
$m$ and the term (15) contributes $9\sqrt{3}\log(2)s^{2}$, since for all
$\alpha$

$\sum_{j=1}^{3}(3\sin^{2}(\alpha+\frac{2\pi j}{3})-\cos^{2}(\alpha+\frac{2\pi
j}{3}))=3\,.$

For term (14) we use the sharp estimate

$\sum_{j=1}^{3}(3\sin^{2}(\alpha+\frac{2\pi j}{3})-\cos^{2}(\alpha+\frac{2\pi
j}{3}))\log\lvert 3\sin^{2}(\alpha+\frac{2\pi
j}{3})-\cos^{2}(\alpha+\frac{2\pi j}{3})\rvert\leq 3\log(3)\,,$

which we prove in Lemma 15. Hence, for $s\leq 1/20$,

$\displaystyle m(\theta)$
$\displaystyle\geq-6\sqrt{3}s^{2}\log(s)+(9\sqrt{3}\log(2)-3\sqrt{3}\log(3))s^{2}+90\sqrt{3}s^{4}\log
s-62s^{4}$
$\displaystyle\geq(6\sqrt{3}\log(20)+9\sqrt{3}\log(2)-3\sqrt{3}\log(3)-90\sqrt{3}/400\log(20)-62/400)s^{2}\approx
34.906\ s^{2}\,,$

as claimed. ∎

##  5\. Estimating term $I\\!I$

######  Lemma 8.

For all $2\leq i,j\leq 4$ and all $f$, it holds that

$I\\!I_{ij}\leq\frac{101}{100}\pi\int_{B_{1}}\lvert\theta\rvert^{2}\lvert
f(\theta)\rvert^{2}\,\mathrm{d}\mathcal{H}^{2}_{H}(\theta)\,.$

###### Proof.

We first treat the term $I\\!I_{44}$, and later explain the changes for the
other terms. We have:

$\displaystyle I\\!I_{44}$ $\displaystyle=\int_{B_{4}\times
B_{4}}\delta(1-\frac{1-a(\theta^{\prime})}{1-a(\theta)})\lvert
f(\theta)\rvert\lvert
f(\theta^{\prime})\rvert\,\mathrm{d}\mathcal{H}^{2}_{H}(\theta)\,\mathrm{d}\mathcal{H}^{2}_{H}(\theta^{\prime})$
$\displaystyle=\int_{B_{1}\times
B_{1}}\delta(1-\frac{1-a(c_{4}+\theta^{\prime})}{1-a(c_{4}+\theta)})\lvert
f(\theta)\rvert\lvert
f(\theta^{\prime})\rvert\,\mathrm{d}\mathcal{H}^{2}_{H}(\theta)\,\mathrm{d}\mathcal{H}^{2}_{H}(\theta^{\prime})$

We introduce polar coordinates $\theta=\theta(s,\alpha)$ as in (12) and write
also $\theta^{\prime}=\theta(t,\beta)$. With the definitions
$h(s,t,\alpha,\beta):=(1-a(c_{4}+\theta^{\prime}))/(1-a(c_{4}+\theta))$ and
$g(s,\alpha):=\lvert\theta\rvert^{2}\lvert f(\theta)\rvert$, we obtain by
changing variables

$I\\!I_{44}=\int_{0}^{2\pi}\int_{0}^{2\pi}\int_{0}^{\varepsilon}\int_{0}^{\varepsilon}\delta(1-h(s,t,\alpha,\beta))g(s,\alpha)g(t,\beta)\,\frac{\mathrm{d}s}{s}\,\frac{\mathrm{d}t}{t}\,\mathrm{d}\alpha\,\mathrm{d}\beta\,.$
(16)

Doing a Taylor expansion of $1-a(c_{4}+\theta)$ at $0$ yields (see Lemma 13)

$h(s,t,\alpha,\beta)=\frac{t^{2}}{s^{2}}\frac{3\sin^{2}(\beta)-\cos^{2}(\beta)}{3\sin^{2}(\alpha)-\cos^{2}(\alpha)}\frac{1+\psi(t,\beta)}{1+\psi(s,\alpha)}\,,$
(17)

where $\psi$ is a smooth function of $s,\alpha$, and
$\psi(s,\alpha)=O(s^{2})$. If the last factor in (17) was equal to $1$, then
the inner two integrals in (16) would simplify to

$\int_{0}^{\infty}g(s,\alpha)g(c(\alpha,\beta)s,\beta)\,\frac{\mathrm{d}s}{s}\,,$

for some constant $c(\alpha,\beta)$, which is easily estimated using Cauchy-
Schwarz. The following is a perturbed version of this argument.

Fix $\alpha$, $\beta$ and write $h(s,t)=h(s,t,\alpha,\beta)$. Let $s(t)$ be
defined implicitly by $h(s(t),t)=1$ (note that $s$ also depends on $\alpha$
and $\beta$). Then

$\displaystyle\int_{0}^{\varepsilon}\int_{0}^{\varepsilon}\delta(1-h(s,t))g(s,\alpha)g(t,\beta)\,\frac{\mathrm{d}s}{s}\,\frac{\mathrm{d}t}{t}$
$\displaystyle=\int_{0}^{\varepsilon}g(t,\beta)g(s(t),\alpha)\frac{1}{\lvert\partial_{s}h(s(t),t)\rvert}\frac{1}{s(t)t}\,\mathrm{d}t$
$\displaystyle=\int_{0}^{\varepsilon}g(t,\beta)g(s(t),\alpha)\frac{1}{2+s(t)\frac{\psi^{\prime}(s(t),\alpha)}{1+\psi(s(t),\alpha)}}\frac{1}{t}\,\mathrm{d}t\,.$
(18)

Here we used that

$\partial_{s}h(s,t)=-h(s,t)(\frac{2}{s}+\frac{\psi^{\prime}(s,\alpha)}{1+\psi(s,\alpha)})$

and hence

$-\partial_{s}h(s(t),t)=\frac{2}{s(t)}+\frac{\psi^{\prime}(s(t),\alpha)}{1+\psi(s(t),\alpha)}\,.$

Applying Cauchy-Schwarz, we obtain

$\eqref{eq
g}\leq\left(\int_{0}^{\varepsilon}g(t,\beta)^{2}\frac{1}{2+s(t)\frac{\psi^{\prime}(s(t),\alpha)}{1+\psi(s(t),\alpha)}}\frac{1}{t}\,\mathrm{d}t\right)^{1/2}\left(\int_{0}^{\varepsilon}g(s(t),\alpha)^{2}\frac{1}{2+s(t)\frac{\psi^{\prime}(s(t),\alpha)}{1+\psi(s(t),\alpha)}}\frac{1}{t}\,\mathrm{d}t\right)^{1/2}\,.$
(19)

After substituting $s=s(t)$ in the second integral, its integrand becomes the
same as in the first one, but with the roles of $(s,\alpha)$ and $(t,\beta)$
interchanged. By Lemma 13, it holds for $s\leq 1/20$ that

$\lvert\psi(s,\alpha)\rvert<1/100\quad\text{and}\quad\lvert\psi^{\prime}(s,\alpha)\rvert\leq
1/10\,,$

giving

$\lvert
s\frac{\psi^{\prime}(s,\alpha)}{1+\psi(s,\alpha)}\rvert\leq\frac{1}{198}\,.$

Thus, the factor in the integrals in (19) is bounded above by
$198/395<101/200$. It follows that

$I\\!I_{44}\leq\frac{101}{200}\int_{0}^{2\pi}\int_{0}^{2\pi}\left(\int_{0}^{\varepsilon}g(t,\beta)^{2}\frac{\mathrm{d}t}{t}\right)^{1/2}\left(\int_{0}^{\varepsilon}g(s,\alpha)^{2}\frac{\mathrm{d}s}{s}\right)^{1/2}\,\mathrm{d}\alpha\,\mathrm{d}\beta\\\
\leq\frac{101}{100}\pi\int_{0}^{2\pi}\int_{0}^{\varepsilon}\lvert
g(s,\alpha)\rvert^{2}\,\frac{\mathrm{d}t}{t}\,\mathrm{d}\alpha=\frac{101}{100}\pi\int_{B_{1}}\lvert\theta\rvert^{2}\lvert
f(\theta)\rvert^{2}\,\mathrm{d}\mathcal{H}^{2}_{H}(\theta)\,.$

For the other eight integrals the same estimate holds: By the argument in the
proof of Lemma 7, changing $c_{4}$ to some other $c_{j}$ only changes the
expansion in (17) by a translation in $\alpha$ and $\beta$. Then the rest of
the argument goes through exactly as for $I\\!I_{44}$. ∎

##  6\. Technical estimates

Here we prove the computational lemmas that were used in the main argument.

We have the following explicit formula for $\rho$ (see [CFO+17], Lemma 8):

$\rho(r)=\frac{4}{r}\int_{A(r)}^{1}\frac{\mathrm{d}u}{\sqrt{1-u^{2}}\sqrt{\frac{(1-r)^{2}}{2r}+1-u}\sqrt{\frac{(3+r)(1-r)}{2r}+1+u}}$
(20)

with $A(r)=-1+\max\\{0,(3+r)(r-1)/(2r)\\}$. From this we obtain the following
asymptotic formula:

######  Lemma 9.

Let $\rho$ be defined by $\rho(\lvert x\rvert)=\sigma*\sigma*\sigma(x)$. Then
we have for all $r$ with $\lvert r-1\rvert\leq 1/10$

$\displaystyle\lvert\rho(r)+6\log\lvert 1-r\rvert-12\log 2\rvert\leq-22\lvert
r-1\rvert\log\lvert r-1\rvert+23\lvert r-1\rvert\,.$

We have not tried to optimize the error in this estimate. We give an
elementary, self-contained proof below. For an alternative proof one can use
the identity (see [Pea06] or [BSW+12])

$\rho(x)=\begin{cases}\frac{16}{\sqrt{(x+1)^{3}(3-x)}}K\left(\sqrt{\frac{16x}{(x+1)^{3}(3-x)}}\right)\quad&\text{if
$0\leq x<1$}\\\
\frac{4}{\sqrt{x}}K\left(\sqrt{\frac{(x+1)^{3}(3-x)}{16x}}\right)\quad&\text{if
$1<x\leq 3$}\\\ 0\quad&\text{if $x>3$}\end{cases}$

where

$K(k)=\int_{0}^{1}\frac{1}{\sqrt{1-x^{2}}\sqrt{1-k^{2}x^{2}}}\,\mathrm{d}x$

is the complete elliptic integral of the first kind, together with known
asymptotics for $K(k)$ as $k\nearrow 1$. Since we could not find a proof of
(6) in the literature, and its proof is indeed not short, we have decided to
give the elementary argument instead.

We first prove some auxilliary lemmas.

######  Lemma 10.

For all $\delta>0$ it holds that

$0\leq\int_{0}^{1}\frac{1}{\sqrt{u}\sqrt{u+\delta}}\,\mathrm{d}u-\log(\frac{4}{\delta})\leq\frac{1}{2}\delta\,.$

###### Proof.

We have

$\int_{0}^{1}\frac{1}{\sqrt{u}\sqrt{u+\delta}}\,\mathrm{d}u=-\log(\delta)+2\log(1+\sqrt{1+\delta})\,.$

Furthermore, by the mean value theorem, there exists
$0<\delta^{\prime}<\delta$ such that

$\displaystyle\log(1+\sqrt{1+\delta})=\log(2)+\delta g(\delta^{\prime})$

where

$0<g(\delta)=\frac{1}{2(1+\sqrt{1+\delta})\sqrt{1+\delta}}\leq\frac{1}{4}$

is the derivative of $\log(1+\sqrt{1+\delta})$. ∎

######  Lemma 11.

For all $0<a,b<1$, we have:

$\left\lvert\int_{0}^{1}\frac{1}{\sqrt{1-x^{2}}\sqrt{a+1-x}\sqrt{b+1+x}}\,\mathrm{d}x-\int_{0}^{1}\frac{1}{\sqrt{1-x^{2}}\sqrt{a+1-x}\sqrt{1+x}}\,\mathrm{d}x\right\rvert\leq\frac{b}{2}(\log(\frac{4}{a})+\frac{a}{2})\,.$

###### Proof.

By the mean value theorem, we have for all $x\geq 0$

$\lvert(b+1+x)^{-1/2}-(1+x)^{-1/2}\rvert\leq\frac{1}{2}b\,.$

Hence the left hand side of the claimed inequality is estimated by

$\frac{b}{2}\int_{0}^{1}\frac{1}{\sqrt{1-x}\sqrt{a+1-x}}\,\mathrm{d}x\leq\frac{b}{2}(\log(\frac{4}{a})+\frac{a}{2})\,,$

where we applied Lemma 10. ∎

######  Lemma 12.

For all $1>a>0$, we have

$\left\lvert\int_{0}^{1}\frac{1}{(1+x)\sqrt{1-x}\sqrt{a+1-x}}\,\mathrm{d}x-\frac{1}{2}\log(\frac{8}{a})\right\rvert\leq\frac{1}{2}a\log(1+\frac{1}{a})\,.$

###### Proof.

We have with $v=1-x$

$\int_{0}^{1}\frac{1}{(1+x)\sqrt{1-x}\sqrt{a+1-x}}\,\mathrm{d}x=\int_{0}^{1}\frac{1}{(2-v)\sqrt{v}\sqrt{a+v}}\,\mathrm{d}v$

which can be expanded to equal

$\frac{1}{2}\int_{0}^{1}\frac{1}{\sqrt{v}\sqrt{a+v}}\,\mathrm{d}v+\frac{1}{2}\int_{0}^{1}\frac{1}{2-v}\,\mathrm{d}v-\frac{a}{2}\int_{0}^{1}\frac{1}{(2-v)\sqrt{a+v}(\sqrt{v}+\sqrt{a+v})}\,\mathrm{d}v\,.$

Computing the second integral and using Lemma 10 for the first one yields the
main term $\log(8/a)/2$. For the error estimate we combine Lemma 10 and the
bound

$\int_{0}^{1}\frac{1}{(2-v)\sqrt{a+v}(\sqrt{v}+\sqrt{a+v})}\,\mathrm{d}v\leq\int_{0}^{1}\frac{1}{v+a}\,\mathrm{d}v=\log(1+\frac{1}{a})\,,$

and note that the errors have opposite signs. ∎

###### Proof of Lemma 9.

We start with the case $r=1-\varepsilon<1$. By (20), we have

$\frac{1-\varepsilon}{4}\rho(1-\varepsilon)=\int_{-1}^{1}\frac{1}{\sqrt{1-u^{2}}\sqrt{\frac{\varepsilon^{2}}{2-2\varepsilon}+1-u}\sqrt{\frac{(4-\varepsilon)\varepsilon}{2-2\varepsilon}+1+u}}\,\mathrm{d}u\,.$

Combining Lemma 11 and Lemma 12 with $a=\varepsilon^{2}/(2-2\varepsilon)$ and
$b=(4-\varepsilon)\varepsilon/(2-2\varepsilon)$, we obtain for this integral:

$=\frac{1}{2}(\log(\frac{8}{a})+\log(\frac{8}{b}))+E=3\log(2)-\frac{3}{2}\log(\varepsilon)-\log(2-2\varepsilon)+\frac{1}{2}\log(4-\varepsilon)+E$

with

$\lvert
E\rvert\leq\frac{1}{2}(b\log(\frac{4}{a})+a\log(\frac{4}{b})+ab+a\log(1+\frac{1}{a})+b\log(1+\frac{1}{b}))\,.$
(21)

It is easy to see that

$\lvert\frac{1}{2}\log(4-\varepsilon)-\log(2-2\varepsilon)\rvert\leq\frac{\varepsilon}{2}\,.$

Further, one verifies that, when $0<\varepsilon\leq 1/10$,

$a\leq\frac{1}{18}\varepsilon\,,\ b\leq\frac{19}{9}\varepsilon\,,\
\log(\frac{4}{a})\leq 3\log(2)-2\log(\varepsilon)\,,\
\log(\frac{4}{b})\leq\log(2)-\log(\varepsilon)$

and

$\log(1+\frac{1}{a})\leq\log(2)-2\log(\varepsilon)\,,\
\log(1+\frac{1}{b})\leq-\log(\varepsilon)\,.$

Using this, one can check that

$E\leq\frac{13}{4}\varepsilon\log(\frac{1}{\varepsilon})+\frac{5}{2}\varepsilon\,.$

To summarize, we have shown that

$\left\lvert\frac{1-\varepsilon}{4}\rho(1-\varepsilon)-3\log(2)+\frac{3}{2}\log(\varepsilon)\right\rvert\leq\frac{13}{4}\varepsilon\log(\frac{1}{\varepsilon})+3\varepsilon\,.$

We multiply by $4/(1-\varepsilon)$, and use that $\lvert
4/(1-\varepsilon)-4\rvert\leq 40/9\varepsilon$ to obtain

$\lvert\rho(1-\varepsilon)-12\log(2)+6\log(\varepsilon)\rvert\leq
22\varepsilon\log(\frac{1}{\varepsilon})+23\varepsilon\,.$

Now we turn to the case $r=1+\varepsilon>1$. There we have

$\rho(1+\varepsilon)=\frac{4}{1+\varepsilon}\int_{-1+\frac{(4+\varepsilon)\epsilon}{2+2\varepsilon}}^{1}\frac{1}{\sqrt{1-u^{2}}\sqrt{\frac{\varepsilon^{2}}{2+2\varepsilon}+1-u}\sqrt{-\frac{(4+\varepsilon)\varepsilon}{2+2\varepsilon}+1+u}}\,\mathrm{d}u$
$=\frac{16}{4-\varepsilon^{2}}\int_{-1}^{1}\frac{1}{\sqrt{1-v^{2}}\sqrt{\frac{2\varepsilon^{2}}{4-\varepsilon^{2}}+1-v}\sqrt{\frac{8\varepsilon}{4-\varepsilon^{2}}+1+v}}\,\mathrm{d}v\,.$

We first approximate the integral. We can argue as in the case $r<1$, now with
$a=2\varepsilon^{2}/(4-\varepsilon^{2})$ and
$b=8\varepsilon/(4-\varepsilon^{2})$. The main term is easily seen to be the
same as in the case $r<1$, and the error is bounded by

$-\log(1-\frac{\varepsilon^{2}}{4})+E\leq\frac{\varepsilon}{40}+E\,,$

with $E$ satisfying (21). Now we have

$a\leq\frac{1}{15}\varepsilon\,,\ b\leq\frac{800}{399}\varepsilon\,,\
\log(\frac{4}{a})\leq 3\log(2)-2\log(\varepsilon)\,,\
\log(\frac{4}{b})\leq\log(2)-\log(\varepsilon)$

and

$\log(1+\frac{1}{a})\leq\log(\frac{201}{100})-2\log(\varepsilon)\,,\
\log(1+\frac{1}{b})\leq-\log(\varepsilon)\,.$

Using this, we obtain

$\lvert
E\rvert+\frac{\varepsilon}{40}\leq\frac{13}{4}\varepsilon\log(\frac{1}{\varepsilon})+\frac{9}{4}\varepsilon\,.$

In other words, it holds that

$\left\lvert\frac{4-\varepsilon^{2}}{16}\rho(1+\varepsilon)-3\log(2)+\frac{3}{2}\log(\varepsilon)\right\rvert\leq\frac{13}{4}\varepsilon\log(\frac{1}{\varepsilon})+\frac{9}{4}\varepsilon\,.$

We multiply by $16/(4-\varepsilon^{2})$ and use that $\lvert
16/(4-\varepsilon^{2})-4\rvert\leq 40/399\varepsilon$ to obtain

$\lvert\rho(1+\varepsilon)-12\log(2)+6\log(\varepsilon)\rvert\leq
14\varepsilon\log(\frac{1}{\varepsilon})+9\varepsilon\,.$

This completes the proof. ∎

######  Lemma 13.

Let $\theta$ be given by (12). Then it holds that

$a(c_{4}+\theta)-1=s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))(1+\psi(s,\alpha))\,,$

where $\psi(s,\alpha)$ is a smooth function satisfying the following
estimates:

$\displaystyle\lvert\psi(s,\alpha)\rvert$
$\displaystyle\leq\frac{7}{24}s^{2}+\frac{17}{720}s^{4}+s^{6}e^{\sqrt{2}s}$
$\displaystyle\lvert\psi^{\prime}(s,\alpha)\rvert$
$\displaystyle\leq\frac{14}{24}s+\frac{17}{180}s^{3}+2s^{5}e^{\sqrt{2}s}\,.$

###### Proof.

By the definition of $h$, the trigonometric identities and the Taylor
expansion of $\cos$, we have

$\displaystyle a(c_{4}+\theta)-1$ $\displaystyle=a((0,0,\pi)+\theta)-1$
$\displaystyle=(\cos(\theta_{1})+\cos(\theta_{2})-\cos(\theta_{3}))^{2}+(\sin(\theta_{1})+\sin(\theta_{2})-\sin(\theta_{3}))^{2}-1$
$\displaystyle=2+2\cos(\theta_{1}-\theta_{2})-2\cos(\theta_{1}-\theta_{3})-2\cos(\theta_{2}-\theta_{3})$
$\displaystyle=2\sum_{k=1}^{\infty}\frac{(-1)^{k}}{(2k)!}((\theta_{1}-\theta_{2})^{2k}-(\theta_{1}-\theta_{3})^{2k}-(\theta_{2}-\theta_{3})^{2k})$
(22)
$\displaystyle=:\sum_{k=1}^{\infty}s^{2k}\frac{(-1)^{k}}{(2k)!}P_{2k}(\sin(\alpha),\cos(\alpha))\,.$

It follows from (22) that each $P_{2k}$ vanishes when $\theta_{1}=\theta_{3}$
and when $\theta_{2}=\theta_{3}$, which is equivalent to $\alpha=\pm\pi/6$, or
to $\cos(\alpha)=\pm\sqrt{3}\sin(\alpha)$. Hence, the homogeneous polynomial
$P_{2k}(X,Y)$ vanishes on the lines $\sqrt{3}X+Y=0$ and $\sqrt{3}X-Y=0$. We
conclude that for all $k$, the factor $3X^{2}-Y^{2}$ divides $P_{2k}(X,Y)$.
Define $Q_{2k}$ by

$Q_{2k}(X,Y)(3X^{2}-Y^{2})=(-1)^{k}P_{2k}(X,Y)\,.$

Then we have, using that $Q_{2}=1$:

$\displaystyle
a(c_{4}+\theta)-1=s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))(1+\psi(s,\alpha))$

where $\psi$ is defined by

$\psi(s,\alpha)=\sum_{k=2}^{\infty}s^{2k-2}\frac{1}{(2k)!}Q_{2k}(\cos(\alpha),\sin(\alpha))\,.$

Now we fix $k$ and estimate $p(\alpha):=P_{2k}(\sin(\alpha),\cos(\alpha))$ and
$q(\alpha):=Q_{2k}(\sin(\alpha),\cos(\alpha))$. By (12), we have that

$\theta_{1}-\theta_{2}=\sqrt{2}\cos(\alpha)$
$\theta_{3}-\theta_{1}=-\frac{1}{\sqrt{2}}\cos(\alpha)-\frac{\sqrt{3}}{\sqrt{2}}\sin(\alpha)=\sqrt{2}\cos(\alpha+\frac{2\pi}{3})$
$\theta_{2}-\theta_{3}=-\frac{1}{\sqrt{2}}\cos(\alpha)+\frac{\sqrt{3}}{\sqrt{2}}\sin(\alpha)=\sqrt{2}\cos(\alpha-\frac{2\pi}{3})\,.$

Thus, by (22),

$p(\alpha)=2^{k+1}(-1)^{k}(\cos(\alpha)^{2k}-\cos(\alpha+\frac{2\pi}{3})^{2k}-\cos(\alpha-\frac{2\pi}{3})^{2k})\,,$

Taking derivatives, and noting that the terms inside the brackets are each at
most $1$, we obtain:

$\lvert p(\alpha)\rvert\leq 6\cdot 2^{k}\,,\qquad\lvert
p^{\prime}(\alpha)\rvert\leq 12k2^{k}\,,\qquad\lvert
p^{\prime\prime}(\alpha)\rvert\leq 24k^{2}2^{k}\,.$

Denote

$q(\alpha):=\frac{p(\alpha)}{3\sin^{2}(\alpha)-\cos^{2}(\alpha)}=\frac{p(\alpha)}{(\sqrt{3}\sin(\alpha)-\cos(\alpha))(\sqrt{3}\sin(\alpha)+\cos(\alpha))}\,.$

If both factors $\lvert\sqrt{3}\sin(\alpha)\pm\cos(\alpha)\rvert$ are at least
$1/2$, we have that

$q(\alpha)\leq 24\cdot 2^{k}\,.$

If not, then $\lvert\alpha-\pi/6\rvert<1/5$ or $\lvert\alpha+\pi/6\rvert<1/5$.
Without loss of generality we are in the first case. Then, by Taylor’s
formula:

$\left\lvert\frac{p(\alpha)}{\alpha-\pi/6}-p^{\prime}(\pi/6)\right\rvert\leq\frac{1}{2}|\alpha-\pi/6|\sup|p^{\prime\prime}|\leq\frac{1}{10}24k^{2}2^{k}\,,$

hence

$\left\lvert\frac{p(\alpha)}{\alpha-\pi/6}\right\rvert\leq 15k^{2}2^{k}\,.$

Furthermore, since $\lvert\alpha-\pi/6\rvert\leq 1/5$,

$\left\lvert\frac{\alpha-\pi/6}{(\sqrt{3}\sin(\alpha)-\cos(\alpha))(\sqrt{3}\sin(\alpha)+\cos(\alpha))}\right\rvert\leq
2\left\lvert\frac{\alpha-\pi/6}{\sqrt{3}\sin(\alpha)-\cos(\alpha)}\right\rvert\leq\frac{1/5}{\sin(1/5)}<2\,.$

$k$ | $P_{2k}(X,Y)$ | $Q_{2k}(X,Y)$
---|---|---
$1$ | $-3X^{2}+Y^{2}$ | $1$
$2$ | $-9X^{4}-18X^{2}Y^{2}+7Y^{4}$ | $-3X^{2}-7Y^{2}$
$3$ | $-\frac{1}{2}(27X^{6}+135X^{4}Y^{2}+45X^{2}Y^{4}-31Y^{6})$ | $\frac{1}{2}(9X^{4}+48X^{2}Y^{2}+31Y^{4})$
Table 1. The polynomials $P_{2k}$, $Q_{2k}$ for small values of $k$.

Multiplying the last two estimates, we conclude that $\lvert q\rvert\leq
30k^{2}2^{k}$. We also directly compute for small $k$:

$\lvert
Q_{4}(\sin(\alpha),\cos(\alpha))\rvert=\lvert-7\cos^{2}(\alpha)-3\sin^{2}(\alpha)\rvert\leq
7$

and

$\lvert Q_{6}(\sin(\alpha),\cos(\alpha))\rvert=\frac{1}{2}\lvert
9\sin^{4}(\alpha)+48\sin^{2}(\alpha)\cos^{2}(\alpha)+31\cos^{2}(\alpha)\rvert\leq\frac{5125}{312}<17\,.$

Plugging in these estimates, we obtain

$\lvert\psi(s,\alpha)\rvert\leq\frac{7}{24}s^{2}+\frac{17}{720}s^{4}+\sum_{k=4}^{\infty}\frac{60k^{2}}{(2k)!}(\sqrt{2}s)^{2k-2}\leq\frac{7}{24}s^{2}+\frac{17}{720}s^{4}+s^{6}e^{\sqrt{2}s}\,$

and

$\lvert\psi^{\prime}(s,\alpha)\rvert\leq\frac{14}{24}s+\frac{17}{180}s^{3}+\sqrt{2}\sum_{k=4}^{\infty}\frac{60k^{2}}{(2k-1)!}(\sqrt{2}s)^{2k-3}\leq\frac{14}{24}s+\frac{17}{180}s^{3}+2s^{5}e^{\sqrt{2}s}\,,$

as claimed. ∎

######  Lemma 14.

Let $\theta$ be given by (12). Then for all $0\leq s\leq 1/20$, we have

$\displaystyle(a(c_{4}+\theta)-1)\rho(\sqrt{a(c_{4}+\theta)})$
$\displaystyle=-12s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))\log(s)$
$\displaystyle\quad-6s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))\log\lvert
3\sin^{2}(\alpha)-\cos^{2}(\alpha)\rvert$ $\displaystyle\quad+18\log
2\,s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))+E\,,$

with

$\lvert E\rvert\leq-180s^{4}\log s+71s^{4}\,.$

###### Proof.

By Lemma 9, it holds that

$\displaystyle(x^{2}-1)\rho(x)$ $\displaystyle=-6(x^{2}-1)\log\lvert
x-1\rvert+12\log(2)(x^{2}-1)+(x^{2}-1)E_{1}$
$\displaystyle=-6(x^{2}-1)\log\lvert
x^{2}-1\rvert+18\log(2)(x^{2}-1)+(x^{2}-1)E_{1}+6(x^{2}-1)\log(1+\frac{1}{2}(x-1))\,,$
(23)

where

$\lvert E_{1}\rvert\leq-22\lvert x-1\rvert\log\lvert x-1\rvert+23\lvert
x-1\rvert\,.$

Denote also the last term in (23) by $E_{2}$. We set
$x=\sqrt{a(c_{4}+\theta)}$. Lemma 13 implies that $\lvert x-1\rvert\leq\lvert
x^{2}-1\rvert\leq 2s^{2}$. Using this and monotonicity of $r\log r$, we obtain

$\lvert(x^{2}-1)E_{1}\rvert\leq\lvert x^{2}-1\rvert(-22\lvert
x-1\rvert\log\lvert x-1\rvert+23\lvert x-1\rvert)\leq-176s^{4}\log(s)+32s^{4}$
(24)

and

$\lvert E_{2}\rvert\leq 6\lvert
x^{2}-1\rvert\lvert\log(1+\frac{1}{2}(x-1))\rvert\leq 24s^{4}\,.$ (25)

By Lemma 13, it holds that

$\displaystyle\quad-6(x^{2}-1)\log\lvert x^{2}-1\rvert$
$\displaystyle=-6s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))(1+\psi(s,\alpha))(2\log(s)+\log(3\sin^{2}(\alpha)-\cos^{2}(\alpha))+\log(1+\psi(s,\alpha)))$
$\displaystyle=-12s^{2}\log(s)(3\sin^{2}(\alpha)-\cos^{2}(\alpha))-6s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))\log\lvert
3\sin^{2}(\alpha)-\cos^{2}(\alpha)\rvert$ (26)
$\displaystyle\quad-6s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))\log(1+\psi(s,\alpha)))$
(27)
$\displaystyle\quad-6s^{2}\psi(s,\alpha)(3\sin^{2}(\alpha)-\cos^{2}(\alpha))(2\log(s)+\log\lvert
3\sin^{2}(\alpha)-\cos^{2}(\alpha)\rvert+\log(1+\psi(s,\alpha)))\,.$ (28)

The term (27) bounded by $18s^{2}\lvert\psi(s,\alpha)\rvert\leq 6s^{4}$. The
term (28) is bounded by

$-18s^{2}\log(s)\lvert\psi(s,\alpha)\rvert+4s^{2}\lvert\psi(s,\alpha)\rvert+18s^{2}\psi(s,\alpha)^{2}\leq-6s^{4}\log(s)+2s^{4}\,.$

For the second term in (23), we have

$18\log(2)(x^{2}-1)=18\log(2)s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))+18\log(2)s^{2}(3\sin^{2}(\alpha)-\cos^{2}(\alpha))\psi(s,\alpha)\,,$
(29)

with the second term bounded by $27\log(2)s^{2}\lvert\psi(s,\alpha)\rvert\leq
9\log(2)s^{4}$. Putting together the main terms (26) and (29), and the
estimates for the error terms (24), (25), (27), (28) and in (29), one obtains
the lemma. ∎

######  Lemma 15.

For all $\alpha$, it holds that

$\sum_{j=1}^{3}(3\sin^{2}(\alpha+\frac{2\pi j}{3})-\cos^{2}(\alpha+\frac{2\pi
j}{3}))\log\lvert 3\sin^{2}(\alpha+\frac{2\pi
j}{3})-\cos^{2}(\alpha+\frac{2\pi j}{3})\rvert\leq 3\log(3)\,.$

###### Proof.

Let

$a_{j}=\sin^{2}(\alpha+\frac{2\pi j}{3})-\frac{1}{3}\cos^{2}(\alpha+\frac{2\pi
j}{3})=\frac{1}{3}-\frac{2}{3}\cos(2\alpha+\frac{4\pi j}{3})\,.$

It is easy to check that

$a_{1}+a_{2}+a_{3}=1\,,\qquad a_{1}^{2}+a_{2}^{2}+a_{3}^{2}=1\,.$ (30)

Defining $b_{j}=(a_{j}+a_{j-1})/2$ (note that $a_{j+3}=a_{j}$), it follows
that

$b_{1}+b_{2}+b_{3}=1\,,\qquad b_{1}^{2}+b_{2}^{2}+b_{3}^{2}=1/2\,,$

hence $b_{1},b_{2},b_{3}\geq 0$. Using Jensen’s inequality, we deduce

$\sum_{j=1}^{3}a_{j}\log(\lvert
a_{j}\rvert)=\sum_{j=1}^{3}b_{j}\log(\frac{\lvert a_{j}\rvert\lvert
a_{j-1}\rvert}{\lvert a_{j-2}\rvert})\leq\log(\sum_{j=1}^{3}b_{j}\frac{\lvert
a_{j}\rvert\lvert a_{j-1}\rvert}{\lvert a_{j-2}\rvert})\,.$

By (30), we have that

$2a_{j}a_{j-1}=(a_{j}+a_{j-1})^{2}-(a_{j}^{2}+a_{j-1}^{2})=(1-a_{j-2})^{2}-(1-a_{j-2}^{2})=2a_{j-2}(a_{j-2}-1)\,.$

Thus, using again (30)

$\sum_{j=1}^{3}b_{j}\frac{\lvert a_{j}\rvert\lvert a_{j-1}\rvert}{\lvert
a_{j-2}\rvert}=\sum_{j=1}^{3}b_{j}(1-a_{j-2})=1\,.$

We conclude that

$\sum_{j=1}^{3}3a_{j}\log(\lvert
3a_{j}\rvert)=3\log(3)+3\sum_{j=1}^{3}a_{j}\log\lvert a_{j}\rvert\leq 3\log
3\,.$

∎

##  7\. Discussion

###  7.1. Optimal value of $\varepsilon$

An inspection of the above argument shows that $Q(f)\geq 0$ for all $f\in
V_{\varepsilon}$ as long as

$\inf_{\theta\in
H,\lvert\theta\rvert\leq\varepsilon}\frac{1}{2}\sum_{j=2}^{4}(a(\theta+c_{j})-1)\rho(\sqrt{a(\theta+c_{j})})\geq
18\pi\sup_{s\leq\varepsilon,\alpha\in[0,2\pi]}\frac{1}{2+s\frac{\psi^{\prime}(s,\alpha)}{1+\psi(s,\alpha)}}\,.$
(31)

(Non-rigorous) numerical computations suggest that this inequality holds up to
$\varepsilon=0.104$. The constant $\varepsilon^{\prime}$ in Corollary 3 could
then be increased to $0.063$.

Figure 1. The left hand side (yellow) and the right hand side (blue) of (31).

###  7.2. Fourier coefficients of $Q$

In [BTZ20], some numerical observations on the Fourier coefficients

$\hat{B}(k,l):=B(\omega_{1}^{k_{1}}\omega_{2}^{k_{2}}\omega_{3}^{k_{3}},\omega_{4}^{l_{1}}\omega_{5}^{l_{2}}\omega_{6}^{l_{3}})$

of $B$ with $k_{1}+k_{2}+k_{3}=l_{1}+l_{2}+l_{3}=0$ are discussed. Namely,
they are very large only when $k$ is very close to $l$ and when
$k_{1}^{2}+k_{2}^{2}+k_{3}^{2}\approx l_{1}^{2}+l_{2}^{2}+l_{3}^{2}$. We can
explain this using Lemma 5 as follows.

By Lemma 5 and since $\lambda_{0}=1$, for all $f\in X_{0}$

$B(f,f)=\int_{C}m(\theta)\lvert
f(\theta)\rvert^{2}\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta)+\int_{C^{2}}n(\theta)\delta(a(\theta)-a(\theta^{\prime}))f(\theta)\overline{f(\theta^{\prime})}\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta)\,\mathrm{d}\mathcal{H}^{2}_{C}(\theta^{\prime})$

for certain functions $m,n$.

The first term is a multiplier, hence it acts on the Fourier side by
convolution with a fixed bump function. This bump function decays at least
like $\lvert k-l\rvert^{-3}$, because the third derivative of $m$ is still
integrable. This explains the large coefficients when $k$ is close to $l$.

The Fourier coefficients of the second term are the Fourier coefficients of
the measure $\mu:=n(\theta)\delta(a(\theta)-a(\theta^{\prime}))$ supported on
the $3$-manifold

$M:=\\{(x,y)\in C^{2}\,:\,a(x)=a(y)\\}\subset\mathbb{R}^{6}\,.$

The measure $\mu$ has a smooth, bounded density with respect to the Hausdorff
measure on this manifold, except in the critical points of $a$. The Fourier
transform of the parts where the measure has a smooth, bounded density can be
estimated using the method of stationary phase and are of lower order than the
contribution of the critical points. To explain what happens at a critical
point (where $\det D^{2}a\neq 0$), we choose coordinates
$x_{1},x_{2},y_{1},y_{2}$ for $C^{2}$, such that the critical point of $a$ is
at $0$. After a scaling in $a$ and a linear change of variables, either

$a(x)=x_{1}^{2}+x_{2}^{2}+O(\lvert x\rvert^{3})\quad\text{or}\quad
a(x)=x_{1}^{2}-x_{2}^{2}+O(\lvert x\rvert^{3})\,.$ (32)

Thus, ignoring higher order terms,

$\delta(a(x)-a(y))\approx\delta(\lvert x\rvert^{2}-\lvert
y\rvert^{2})\quad\text{or}\quad\delta(a(x)-a(y))\approx\delta(x_{1}^{2}-x_{2}^{2}-y_{1}^{2}-y_{2}^{2})\,.$

The Fourier transforms of these measures can be explicitly computed, in fact,
they are their own Fourier transform. Now, $a$ has one local maximum and two
local minima, which together with the above discussion explain why
$\hat{B}(k,l)$ is very large on the cone $\lvert k\rvert^{2}=\lvert
l\rvert^{2}$. The contribution of all other critical points is of smaller
order, since the weight $n$ vanishes there.

This disussion can be turned into a rigorous proof that the Fourier
coefficients of $\mu$ concentrate near the cone $\lvert k\rvert^{2}=\lvert
l\rvert^{2}$. However, we can only show that they concentrate in e.g.

$\\{(k,l):\lvert\lvert k\rvert-\lvert l\rvert\rvert\leq C\lvert
k\rvert^{1/2}\\}\,,$

and not in an $O(1)$ neighbourhood of the cone, because of the higher order
terms in (32).

## References

  * [BSW+12] Jonathan M. Borwein, Armin Straub, James Wan and Wadim Zudilin  “Densities of short uniform random walks” With an appendix by Don Zagier  In _Canad. J. Math._ 64.5, 2012, pp. 961–990  DOI: 10.4153/CJM-2011-079-2
  * [BTZ20] James Barker, Christoph Thiele and Pavel Zorin-Kranich  “Band-limited maximizers for a Fourier extension inequality on the circle, II”  arXiv, 2020  DOI: 10.48550/ARXIV.2002.05118
  * [CFO+17] Emanuel Carneiro, Damiano Foschi, Diogo Oliveira e Silva and Christoph Thiele  “A sharp trilinear inequality related to Fourier restriction on the circle”  In _Rev. Mat. Iberoam._ 33.4, 2017, pp. 1463–1486  DOI: 10.4171/RMI/978
  * [CG22] Valentina Ciccone and Felipe Gonçalves  “Sharp Fourier Extension on the Circle Under Arithmetic Constraints”  arXiv, 2022  DOI: 10.48550/ARXIV.2208.09441
  * [CO14] Emanuel Carneiro and Diogo Oliveira e Silva  “Some Sharp Restriction Inequalities on the Sphere”  In _International Mathematics Research Notices_ 2015.17, 2014, pp. 8233–8267  DOI: 10.1093/imrn/rnu194
  * [CS12] Michael Christ and Shuanglin Shao  “Existence of extremals for a Fourier restriction inequality”  In _Anal. PDE_ 5.2, 2012, pp. 261–312  DOI: 10.2140/apde.2012.5.261
  * [CS12a] Michael Christ and Shuanglin Shao  “On the extremizers of an adjoint Fourier restriction inequality”  In _Adv. Math._ 230.3, 2012, pp. 957–977  DOI: 10.1016/j.aim.2012.03.020
  * [FLS16] Rupert L. Frank, Elliott H. Lieb and Julien Sabin  “Maximizers for the Stein-Tomas inequality”  In _Geom. Funct. Anal._ 26.4, 2016, pp. 1095–1134  DOI: 10.1007/s00039-016-0380-9
  * [FO17] D. Foschi and D. Oliveira e Silva  “Some recent progress on sharp Fourier restriction theory”  In _Anal. Math._ 43.2, 2017, pp. 241–265  DOI: 10.1007/s10476-017-0306-2
  * [Fos15] Damiano Foschi  “Global maximizers for the sphere adjoint Fourier restriction inequality”  In _J. Funct. Anal._ 268.3, 2015, pp. 690–702  DOI: 10.1016/j.jfa.2014.10.015
  * [FVV11] Luca Fanelli, Luis Vega and Nicola Visciglia  “On the existence of maximizers for a family of restriction theorems”  In _Bull. Lond. Math. Soc._ 43.4, 2011, pp. 811–817  DOI: 10.1112/blms/bdr014
  * [GN22] Felipe Gonçalves and Giuseppe Negro  “Local maximizers of adjoint Fourier restriction estimates for the cone, paraboloid and sphere”  In _Anal. PDE_ 15.4, 2022, pp. 1097–1130  DOI: 10.2140/apde.2022.15.1097
  * [NOT22] Giuseppe Negro, Diogo Oliveira e Silva and Christoph Thiele  “When does $e^{-\lvert\tau\rvert}$ maximize Fourier extension for a conic section?”  arXiv, 2022  DOI: 10.48550/ARXIV.2209.03916
  * [OQ21] Diogo Oliveira e Silva and René Quilodrán  “Global maximizers for adjoint Fourier restriction inequalities on low dimensional spheres”  In _J. Funct. Anal._ 280.7, 2021, pp. Paper No. 10882573  DOI: 10.1016/j.jfa.2020.108825
  * [OTZ22] Diogo Oliveira e Silva, Christoph Thiele and Pavel Zorin-Kranich  “Band-limited maximizers for a Fourier extension inequality on the circle”  In _Exp. Math._ 31.1, 2022, pp. 192–198  DOI: 10.1080/10586458.2019.1596847
  * [Pea06] Karl Pearson  “A mathematical theory of random migration”  London, Dulauco, 1906
  * [Sha16] Shuanglin Shao  “On existence of extremizers for the Tomas-Stein inequality for $S^{1}$”  In _J. Funct. Anal._ 270.10, 2016, pp. 3996–4038  DOI: 10.1016/j.jfa.2016.02.019
  * [Sha16a] Shuanglin Shao  “On smoothness of extremizers of the Tomas-Stein inequality for $S^{1}$”  arXiv, 2016  DOI: 10.48550/ARXIV.1601.07119


# Spin-wave frequency combs

T. Hula  Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics
and Materials Research, Dresden, Germany  Institut für Physik, Technische
Universität Chemnitz, 09107 Chemnitz, Germany  K. Schultheiss  Helmholtz-
Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials
Research, Dresden, Germany  F. J. T. Gonçalves  Helmholtz-Zentrum Dresden-
Rossendorf, Institute of Ion Beam Physics and Materials Research, Dresden,
Germany  L. Körber  Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion
Beam Physics and Materials Research, Dresden, Germany  Fakultät Physik,
Technische Universität Dresden, Dresden, Germany  M. Bejarano  Helmholtz-
Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials
Research, Dresden, Germany  M. Copus  Center for Magnetism and Magnetic
Nanostructures, University of Colorado, Colorado Springs, USA  L. Flacke
Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching,
Germany  Physik-Department, Technische Universität München, Munich, Germany
L. Liensberger  Walther-Meißner-Institut, Bayerische Akademie der
Wissenschaften, Garching, Germany  Physik-Department, Technische Universität
München, Munich, Germany  A. Buzdakov  Helmholtz-Zentrum Dresden-Rossendorf,
Institute of Ion Beam Physics and Materials Research, Dresden, Germany  A.
Kákay  Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and
Materials Research, Dresden, Germany  M. Weiler  Walther-Meißner-Institut,
Bayerische Akademie der Wissenschaften, Garching, Germany  Physik-Department,
Technische Universität München, Munich, Germany  Fachbereich Physik and
Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern,
Kaiserslautern, Germany  R. Camley  Center for Magnetism and Magnetic
Nanostructures, University of Colorado, Colorado Springs, USA  J. Fassbender
Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and
Materials Research, Dresden, Germany  Fakultät Physik, Technische Universität
Dresden, Dresden, Germany  H. Schultheiss<EMAIL_ADDRESS>Helmholtz-
Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials
Research, Dresden, Germany

###### Abstract

We experimentally demonstrate the generation of spin-wave frequency combs
based on the nonlinear interaction of propagating spin waves in a
microstructured waveguide. By means of time- and space-resolved Brillouin
light scattering spectroscopy, we show that the simultaneous excitation of
spin waves with different frequencies leads to a cascade of four-magnon
scattering events which ultimately results in well-defined frequency combs.
Their spectral weight can be tuned by the choice of amplitude and frequency of
the input signals. Furthermore, we introduce a model for stimulated four-
magnon scattering which describes the formation of spin-wave frequency combs
in the frequency and time domain.

Frequency combs have been attracting interest in multiple fields of research
in the last decades. For example, optical frequency combs have had great
impact on metrology as they have led to a significant enhancement in the
precision of frequency measurements Bellini and Hänsch (2000); Jones  _et al._
(2000); Fortier and Baumann (2019). In solid state magnetism, several
mechanisms are known to cause frequency spectra with equidistantly spaced
spin-wave (SW) modes, by means of nonlinear interactions known as modulation
instabilities Benner  _et al._ (2000); Wu  _et al._ (2004); Boyle  _et al._
(1997); Kalinikos  _et al._ (1991); Tsankov  _et al._ (1994); Slavin (1996);
Staudinger  _et al._ (1998). These were often discussed in the time domain as
they were found to cause soliton formation. In addition, frequency mixing of
two independently excited quasi-uniform modes has been studied in magnetic
thin films leading to the generation of additional equidistantly spaced modes
Khivintsev  _et al._ (2011); Marsh  _et al._ (2012); Marsh and Camley (2012).
Very recently, micromagnetic simulations have shown the generation of magnonic
frequency combs by nonlinear three-magnon scattering of spin waves interacting
with the intrinsic modes of magnetic skyrmions Wang  _et al._ (2021).

In microstructured magnetic waveguides, the process of four-magnon scattering
(4MS) has been reported to populate a continuum of SW states in frequency
ranges of up to several GHz around a strongly driven mode Suhl (1957);
Schultheiss  _et al._ (2012); Hula  _et al._ (2020). 4MS commonly describes
the decay of two initial magnons, the quanta of SWs, into a pair of secondary
magnons with different energies and momenta while conserving total energy and
momentum Suhl (1957); Krivosik and Patton (2010). 4MS is often considered
parasitic and undesirable Scott  _et al._ (2004); Demidov  _et al._ (2009);
Bauer  _et al._ (2015) since it leads to additional damping of a strongly
populated SW mode by coupling to the continuum of thermally populated states
within the SW band Suhl (1957); Schultheiss  _et al._ (2012); Venugopal  _et
al._ (2020).

In this letter, we take active control of 4MS in order to create tunable SW
frequency combs. Those have significant impacts for both fundamental and
applied physics as the superposition of frequencies, and associated
wavevectors, allows one to create specific high-wavevector magnons which could
not be readily created with typical geometries. In terms of applications, this
frequency comb creates a series of microwave pulses in the time domain where
the width of each pulse can be significantly adjusted by changing the
amplitude of the initial signals. Furthermore, our results experimentally
demonstrate the efficient down-conversion of spin-wave frequencies which is
important for the integration of magnonic devices into CMOS environments, an
issue currently addressed by multiple complex approaches including modulation
of optical frequency combs Moon and Girvin (2005), superconducting qubits in
cavities Fang  _et al._ (2015), and electronic circuits Yeh  _et al._ (2014).
Finally, we explore the time-scale of the nonlinear responses, a topic not
substantially addressed in earlier works.

We experimentally demonstrate the generation of such a frequency comb by
exciting SWs in a Co25Fe75 microconduit Schoen  _et al._ (2016); Korner  _et
al._ (2017); Flacke  _et al._ (2019) with a width of 2 $\mu$m, a length of 60
$\mu$m and a thickness of 30 nm, as shown in Fig. 1(a) Fab . Two 1 $\mu$m wide
microwave antennas, referred to as RF1 and RF2, are patterned on top of the
magnetic waveguide. Connecting the antennas to two microwave sources allows
for the independent excitation of SWs at potentially different power levels
$P_{1}$ and $P_{2}$. In the following, SWs generated by RF1 (RF2) will be
referred to as mode $a_{1}$ ($a_{2}$) with frequency $f_{1}$ ($f_{2}$) and
wave vector $k_{1}$ ($k_{2}$). In all measurements, an external in-plane
magnetic field of 46 mT is applied which is not sufficient to fully saturate
the waveguide perpendicular to the transport direction but still yields long
SW decay lengths, as reported for a similar geometry in Flacke  _et al._
(2019); Hula  _et al._ (2020). Note, however, that the non-collinearity at the
edges of the sample is not essential for the emergence of 4MS, which is
possible also in fully saturated specimens L’vov (1994).

Figure 1: (a) Scanning electron micrograph of the SW waveguide and the
microwave antennas RF1 and RF2. The arrows $k_{1}$ and $k_{2}$ illustrate the
wave vectors of the SWs excited by RF1 and RF2, respectively. All measurements
were performed in 1 $\mu$m distance to the left edge of RF1 as indicated by
the green dot. (b) Schematic illustration of four-magnon scattering: two
magnons with $(f_{i},k_{i})$ scatter into magnons with $(f_{+},k_{+})$ and
$(f_{-},k_{-})$, at a rate defined by $da_{k}/dt$. (c) Energy-time diagram
illustrating how spontaneous 4MS populates the SW band. (d) $\mu$BLS spectra
measured in the SW waveguide only exciting with RF1 at two different powers.
(e) Energy-time diagram for stimulated 4MS. (f) $\mu$BLS spectrum measured
when exciting with RF1 and RF2 simultaneously.

Figure 1(b) schematically shows the process of spontaneous 4MS Suhl (1957);
Schultheiss  _et al._ (2012); Hula  _et al._ (2020). Using the notation of
nonlinear SW theory Krivosik and Patton (2010); L’vov (1994), we describe the
current state (phase and oscillation magnitude) of a SW mode as a time-
dependent complex amplitude variable $a_{\nu}(t)$ (with some mode index
$\nu$), classical analoga to bosonic creation and annihilation operators
$\hat{a}_{\nu}$ in second quantization. When a mode $a_{i}$ is excited above a
certain threshold amplitude, its energy redistributes within the SW band via
scattering of two initial magnons $a_{i}$ characterized by $f_{i}$ and $k_{i}$
into a pair of secondary magnons $a_{+}$ ($f_{+}$, $k_{+}$) and $a_{-}$
($f_{-}$, $k_{-}$) under conservation of energy and momentum.

The rates $\mathrm{d}a_{k}/\mathrm{d}t$ for these scattering processes,
displayed in Fig. 1(b), describe the change in amplitudes of the participating
modes $a_{k}$ over time and can be derived using the Hamiltonian formalism for
nonlinear SW dynamics Krivosik and Patton (2010); L’vov (1994).

$\displaystyle\frac{\mathrm{d}a_{k}}{\mathrm{d}t}=$
$\displaystyle-i\omega(\mathbf{k})a_{k}-\Gamma(\mathbf{k})(a_{k}-a_{k,\mathrm{th}})$
(1) $\displaystyle+i\gamma
b_{\mathrm{RF}}e^{-i\omega_{\mathrm{RF}}t}-i\frac{\partial\cal{H}_{\mathrm{int}}}{\partial
a^{*}_{k}}$ (2)

In Eq. 2, the SW dispersion $\omega(\mathbf{k})$, the linear damping
$\Gamma(\mathbf{k})=\alpha\omega(\mathbf{k})$ with $\alpha=0.004$ typical for
Co25Fe75, a phenomenological term for thermal population
$a_{k,\mathrm{th}}=i\sqrt{N_{k,\mathrm{th}}}e^{-i\omega(k)t}$, the pump field
$b_{\mathrm{RF}}$, and the interaction Hamiltonian $\cal{H}_{\mathrm{int}}$
are included. Since in our experiments $f_{i}/2$ is located in the SW band
gap, resonant three-magnon scattering is prohibited and is therefore excluded
Krivosik and Patton (2010); Suhl (1957). Therefore, the lowest order of
interaction to be considered for our calculations is the decay of the initial
mode $a_{i}$ into the final states $a_{+}$ and $a_{-}$ and the corresponding
Hamiltonian is written as:

${\cal{H}}_{\mathrm{int}}\simeq{\cal{H}}^{(4)}=\sum_{i+-}W_{ii,+-}a_{i}a_{i}a^{\ast}_{+}a^{\ast}_{-}$
(3)

The rates given in Eq. (2) depend on the coupling coefficients $W_{ii,+-}$
between the different modes as well as the amplitudes of the three modes
$a_{i}$, $a_{+}$ and $a_{-}$ Krivosik and Patton (2010). As the coupling
coefficients are inversely proportional to the wave-vector mismatch of the
participating modes Krivosik and Patton (2010); Hula  _et al._ (2020), the
delay time for populating secondary modes via 4MS scales with the frequency
spacing $\Delta f=|f_{\pm}-f_{1}|$, as illustrated by dashed lines in Fig.
1(c). This delay time can be understood as the time needed to compensate for
the intrinsic damping of the secondary states.

In order to study stimulated 4MS experimentally, we employ Brillouin light
scattering microscopy ($\mu$BLS) Sebastian  _et al._ (2015). Figure 1(d) shows
the anti-Stokes side of $\mu$BLS spectra that we measured when exciting SWs
with antenna RF1 at a fixed frequency $f_{1}=9.25$ GHz but at two different
pumping powers. At low powers (orange line), the SW spectrum shows a single
symmetric peak at $f_{1}$. When increasing the pumping power above the
threshold for 4MS (black line), a broadening of the peak is measured as a
consequence of the energy redistribution caused by spontaneous 4MS.

If aiming at the generation of a SW frequency comb with a well-defined
frequency spectrum, we need to actively increase the scattering rate for one
specific scattering channel. Looking at Eqs. (2),(3), this can be achieved by
populating one of the secondary modes (e.g. $a_{-}$) as a seed mode using
antenna RF2 in addition to the excitation of the initial mode $a_{i}$ using
antenna RF1 (or vice versa). As energy and momentum have to be conserved under
4MS, a third mode $a_{+}$ with $f_{+}=f_{i}+\Delta f$ is generated
simultaneously, as indicated in Fig. 1(e). Thus, by actively populating the
seed mode ($a_{-}$) we control all frequency channels of the scattering
process. We refer to this process as stimulated 4MS, similar to the process of
stimulated three-magnon scattering as shown in Schultheiss  _et al._ (2019).

As depicted in Fig. 1(e), higher order scattering processes then lead to the
formation of a SW frequency comb. The secondary modes $a_{-}$ and $a_{+}$
themselves take the role of initial and seed modes in new scattering events.
Therefore, higher order processes are stimulated, e.g., with two magnons
$a_{2}=a_{-}$ scattering into modes $a_{i}$ and $a_{4}$ (with
$f_{4}=f_{-}-\Delta f$). In Fig. 1(e), this is illustrated by an additional
mode (violet circle) which appears with equidistant frequency spacing but at a
later time. The higher order processes need to compensate the intrinsic
damping of the participating states, which causes an increasing delay time, as
indicated by the slope of the dashed line.

One example of a measured SW frequency comb is shown in Fig. 1(f). The initial
mode is pumped at $f_{i}=f_{1}=10$ GHz using antenna RF1 while the seed mode
is provided by antenna RF2 at $f_{-}=f_{2}=9$ GHz. In addition to those two
frequencies, the BLS spectrum shows very distinct peaks at $f_{+}=f_{3}=11$
GHz and $f_{4}=8$ GHz.

Figure 2: (a) $\mu$BLS spectrum plotted as a function of the input frequency
of RF2 while the frequency of RF1 was fixed at 10 GHz. Both antennas were set
to a constant input power of 10 dBm. The frequency of RF2 was swept from 8 to
12 GHz. (b) Examples of the BLS spectrum measured for RF2 set to 9.5 GHz (red)
and 11 GHz (blue). The dashed lines indicate the continuum of states excited
by 4MS.

In order to demonstrate the high degree of tunability of SW frequency combs,
we measured $\mu$BLS spectra with a fixed frequency of the initial mode
($f_{i}=f_{1}=10$ GHz) but varying frequency of the seed mode $f_{2}$ between
8 GHz and 12 GHz. The microwave powers of RF1 and RF2 were kept at constant
values of $P_{1}=P_{2}=10$ dBm. The resulting BLS spectra are plotted in Fig.
2(a), with the SW intensity color coded on a logarithmic scale. The horizontal
and diagonal white dashed lines indicate the applied frequencies $f_{1}$ and
$f_{2}$. As $f_{2}$ is varied continuously to cross $f_{1}$, the frequency gap
$\Delta f=|f_{1}-f_{2}|$ decreases up to $f_{2}=f_{1}=10$ GHz and then
increases again. Depending on the value of $\Delta f$, different numbers of
indirectly excited SW modes are measured. Figure 2(a) demonstrates that the
additional frequencies, observed in our experiments, strictly follow the
spacing of the two driving frequencies $f_{1}$ and $f_{2}$ (dashed lines).

Figure 2(b) shows individual BLS spectra measured for $f_{2}=9.5$ GHz (red)
and $f_{2}=11$ GHz (blue), highlighting the different frequency spacing
between neighbouring modes of 0.5 and 1 GHz, respectively. In addition to the
SW frequency combs, we measure a continuous background from the SW band
between 8 and 12 GHz, as depicted by dashed lines. This demonstrates that both
mechanisms of spontaneous and stimulated 4MS take place at the same time. We
note that, in addition to the coupling efficiency discussed earlier, the
experimental observation of the continuum of SW states is limited towards
lower frequencies by the SW band-gap (in our case at around 7 GHz) and towards
higher frequencies by the scattering geometry of the $\mu$BLS setup
Schultheiss  _et al._ (2012); Sebastian  _et al._ (2015); Hula  _et al._
(2020).

Previous studies have shown that the population of the magnon states excited
via spontaneous 4MS increases with increasing pumping power Schultheiss  _et
al._ (2012); Hula  _et al._ (2020). In order to investigate the power
dependence of stimulated 4MS, we fix the excitation frequencies of $a_{1}$ and
$a_{2}$ at $f_{1}=10$ GHz and $f_{2}=9.5$ GHz, respectively. In addition, we
fix the pumping power of mode $a_{1}$ at $P_{1}=4.6$ dBm whereas mode $a_{2}$
is pumped with varying microwave powers $P_{2}$ ranging from 4.6 to 23 dBm.
The measured $\mu$BLS spectra as a function of $P_{2}$ are shown in Fig. 3(a)
with the BLS intensity color coded on a logarithmic scale. As expected, the
intensity of $a_{2}$ continuously grows with increasing pumping power $P_{2}$
whereas only a minor increase of the intensity of $a_{1}$ can be observed.
Besides the increase of intensity of $a_{2}$, additional modes at 8.5, 9, and
10.5 GHz appear in the spectra, especially for larger values of $P_{2}$.

Figure 3: (a) $\mu$BLS spectra measured as a function of the microwave power
$P_{2}$ pumping mode $a_{2}$ at a constant frequency $f_{2}=9.5$ GHz. Mode
$a_{1}$ is excited at a constant power $P_{1}=4.6$ dBm and constant frequency
$f_{1}=10$ GHz. (b) Power dependent intensities evaluated for the different
modes at 9 GHz (triangles), 9.5 GHz (circles), and 10 GHz (crosses). The
intensity of the continuum of spontaneous 4MS (squares) is extracted for 9.25
GHz (dashed line in a). Lines are just a guide to the eye to highlight
different slopes measured below and above 12 dBm.

The dependence of the SW spectra with respect to $P_{2}$ is discussed further
with Fig. 3(b), where the intensities extracted at 10 ($a_{1}$, crosses), 9.5
($a_{2}$, circles), 9 (triangles), and 9.25 GHz (4MS, squares) are plotted.
The signal at 9.25 GHz represents states in the continuum of the SW band that
are only excited by spontaneous 4MS.

Interestingly, the intensity of mode $a_{1}$ (crosses) is not constant despite
its fixed excitation power of $P_{1}=4.6$ dBm. Its intensity remains on the
same level up to $P_{2}=11.5$ dBm and then increases to values larger than
those initially measured for $P_{2}=4.6$ dBm. This development of the
intensity of $a_{1}$ shows that we can distinguish different scattering
regimes.

For $P_{2}<11.5$ dBm, spontaneous 4MS of both modes $a_{1}$ and $a_{2}$ into
the continuum of the SW band is the dominant scattering channel. Since $a_{2}$
is provided with more and more energy by the increasing microwave power, its
intensity rises rapidly which directly leads to a strong increase of the
intensity of the continuum of the SW band [squares in Fig. 3(b)].

For $P_{2}>11.5$ dBm, stimulated 4MS is the pronounced scattering channel and
spontaneous 4MS plays a minor role only. Since mode $a_{2}$ [circles in Fig.
3(b)] is pumped much stronger than $a_{1}$ (crosses), $a_{2}$ can be assigned
as the initial mode and $a_{1}$ the seed mode. As a result, two magnons at
$f_{i}=f_{2}=9.5$ GHz scatter in one magnon at $f_{+}=f_{1}=10$ GHz and one at
$f_{-}=9$ GHz (triangles). This not only leads to a rapid increase of the
intensity at 9 GHz above the level of 4MS but also to an increase of the
intensity of $a_{1}$. Even though the intensity of $a_{2}$ continues to
increase due to the larger pumping powers, it never exceeds the intensity of
mode $a_{1}$. This is quite remarkable since at $P_{2}=23$ dBm, $a_{2}$ is
pumped significantly stronger than $a_{1}$ ($P_{1}=4.6$ dBm) and still their
intensities remain on the same level. This demonstrates the high efficiency of
stimulated 4MS: most of the energy of mode $a_{2}$ is redistributed within the
modes of the SW frequency comb. Hence, spontaneous 4MS is less pronounced as
for $P_{2}<11.5$ dBm and the intensity of the continuum [squares in Fig. 3(b)]
increases at a much lower rate.

The mechanism of stimulated 4MS, outlined in Fig. 1(e), presumes that the SW
frequency comb is generated via a temporal cascade of 4MS events that populate
discrete modes in the SW band. To investigate this temporal evolution
experimentally, we use time-resolved BLS microscopy (TR $\mu$BLS) Sebastian
_et al._ (2015).

First, we investigated the temporal response of the system to a single
microwave pulse at $f_{1}=8.5$ GHz. The pulse width was set to 150 ns with a
rise time of 3 ns and a repetition period of 200 ns. The first 40 ns of the
measurement are shown in Fig. 4(a). The microwave pulse sets in after 17 ns,
accompanied by the characteristic frequency broadening of 4MS. Interestingly,
it takes up to 6 ns after the onset of the SW pulse for the frequency
distribution to reach a dynamic equilibrium and fill states of the continuum
with frequencies ranging from 7 to almost 10 GHz. The diagonal dashed lines
highlight that modes in the continuum with smaller frequency spacing to the
directly excited mode are populated much faster than modes with larger
frequency spacing [also see Fig. 1(c)].

Figure 4(b) shows a TR $\mu$BLS spectrum measured for a continuously excited
initial mode at $f_{1}=f_{i}=8$ GHz and a pulsed seed mode at
$f_{2}=f_{+}=8.5$ GHz starting at 15 ns with respect to the displayed time
window. Here, the additional secondary mode $a_{-}$ at 7.5 GHz is measured
simultaneously to the seed mode. Higher order modes at 7, 9, and 9.5 GHz are
measured with larger time delays, depending on their frequency spacing to the
initial mode.

Figure 4: (a) TR $\mu$BLS spectrum measured when exciting SWs with a single RF
pulse at $f_{1}=8.5$ GHz (only the first 40 ns are shown). (b) TR $\mu$BLS
spectrum measured when continuously pumping the initial mode at $f_{1}=8$ GHz
and exciting the seed mode at $f_{2}=8.5$ GHz by a microwave pulse. (c)
Temporal profiles of modes of the frequency comb obtained by the analytic
model in Eq. (2). (d) First 5 ns of the 150 ns-long time profile of $m_{z}$
and (e) FFT thereof, showing the simulated SW frequency comb.

These measurements demonstrate that in both cases a certain time is needed to
populate additional states via nonlinear interactions. The shorter delay times
for the case of stimulated 4MS in Fig. 4(b) suggest that the efficiency of
energy transfer is enhanced when only discrete modes are excited using two RF
sources.

Figure 4(c) shows the results of the calculations based on Eq. (2) for a
continuously driven initial mode at 8 GHz (black line) and a pulsed secondary
mode at 8.5 GHz setting in after 10 ns. A set of seven rate equations was
solved, referring to SW modes ranging from 6.5 to 9.5 GHz in 0.5 GHz steps.
For simplicity, only five of these modes are shown by the modulus square
$|a_{i}|^{2}$, which corresponds to the SW intensity and is proportional to
the photon counts in our time-resolved BLS experiments. We note that the delay
times obtained analytically are in qualitative agreement with those obtained
experimentally in Fig. 4(b).

In the time domain, a superposition of several modes translates into a time
modulated amplitude which enables e.g. optical frequency combs to bridge the
gap between optical and microwave frequencies Bellini and Hänsch (2000); Jones
_et al._ (2000). In order to demonstrate a similar behaviour for the observed
magnon frequency combs, we investigated the temporal evolution of the
magnetization in micromagnetic simulations using MuMax3 Khivintsev  _et al._
(2011); Vansteenkiste  _et al._ (2014) with material parameters and grid size
listed in Sim .

In the simulations, the excitation field was strictly localized to the area of
the two antennas in order to investigate the interaction of propagating waves
and avoid any far-field excitations of the antennas. The excitation
frequencies of RF1 and RF2 were set to 8 and 8.5 GHz, respectively, with field
amplitudes of 1 mT. The temporal representation of the simulated SW frequency
comb is displayed in Fig. 4(d) in form of the amplitude of the dynamic
component of the magnetization, $m_{z}(t)$. Here, we confirm a temporal
modulation of the amplitude by 80% with a 2 ns period, which corresponds to
the inverse of the 0.5 GHz frequency spacing between the two input
frequencies. Figure 4(e) shows the fast Fourier transform (FFT) of $m_{z}(t)$.
The resulting spectrum exhibits multiple SW modes in addition to the directly
excited modes at 8 and 8.5 GHz, resembling the measured SW frequency comb.

In conclusion, we observed the formation of spin-wave frequency combs
generated by the nonlinear interaction of propagating spin waves excited at
two different frequencies in a magnon conduit. A variety of spin-wave modes
can be observed, whose frequency spacing and amplitudes were manipulated
exemplarily by varying the frequency and power of one of the applied microwave
currents. The underlying mechanism can be understood in terms of stimulated
four-magnon scattering, which was found to be the dominant process in the
performed experiments. We propose that this effect can be utilized in any
magnetic material and/or geometry as long as four-magnon scattering occurs.

##  I Acknowledgement

This work was supported by the Deutsche Forschungsgemeinschaft within program
SCHU 2922/1-1. MW, LL and LF acknowledge funding by the German research
foundation (DFG) via projects WE5386/4-1 and WE5386/5-1.

## References

  * Bellini and Hänsch (2000) M. Bellini and T. W. Hänsch, Optics Letters 25, 1049 (2000).
  * Jones  _et al._ (2000) D. J. Jones, S. A. Diddams, M. S. Taubman, S. T. Cundiff, L.-S. Ma, and J. L. Hall, Optics Letters 25, 308 (2000).
  * Fortier and Baumann (2019) T. Fortier and E. Baumann, Communications Physics 2, 152 (2019).
  * Benner  _et al._ (2000) H. Benner, B. A. Kalinikos, N. G. Kovshikov, and M. P. Kostylev, JETP Letters 72, 306 (2000).
  * Wu  _et al._ (2004) M. Wu, B. A. Kalinikos, and C. E. Patton, Physical Review Letters 93, 1 (2004).
  * Boyle  _et al._ (1997) J. W. Boyle, S. A. Nikitov, A. D. Boardman, and K. Xie, Journal of Magnetism and Magnetic Materials 173, 241 (1997).
  * Kalinikos  _et al._ (1991) B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin, Journal of Applied Physics 69, 5712 (1991).
  * Tsankov  _et al._ (1994) M. A. Tsankov, M. Chen, and C. E. Patton, Journal of Applied Physics 76, 4274 (1994).
  * Slavin (1996) A. N. Slavin, Phys. Rev. Lett. 77, 4644 (1996).
  * Staudinger  _et al._ (1998) R. A. Staudinger, P. Kabos, Hua Xia, B. T. Faber, and C. E. Patton, IEEE Transactions on Magnetics 34, 2334 (1998).
  * Khivintsev  _et al._ (2011) Y. Khivintsev, J. Marsh, V. Zagorodnii, I. Harward, J. Lovejoy, P. Krivosik, R. E. Camley, and Z. Celinski, Applied Physics Letters 98, 042505 (2011).
  * Marsh  _et al._ (2012) J. Marsh, V. Zagorodnii, Z. Celinski, and R. E. Camley, Applied Physics Letters 100, 102404 (2012).
  * Marsh and Camley (2012) J. Marsh and R. E. Camley, Physical Review B - Condensed Matter and Materials Physics 86, 224405 (2012).
  * Wang  _et al._ (2021) Z. Wang, H. Y. Yuan, Y. Cao, Z.-X. Li, R. A. Duine, and P. Yan, Physical Review Letters 127 (2021), 10.1103/PhysRevLett.127.037202.
  * Suhl (1957) H. Suhl, Journal of Physics and Chemistry of Solids 1, 209 (1957).
  * Schultheiss  _et al._ (2012) H. Schultheiss, K. Vogt, and B. Hillebrands, Physical Review B 86, 054414 (2012).
  * Hula  _et al._ (2020) T. Hula, K. Schultheiss, A. Buzdakov, L. Körber, M. Bejarano, L. Flacke, L. Liensberger, M. Weiler, J. M. Shaw, H. T. Nembach, J. Fassbender, and H. Schultheiss, Applied Physics Letters 117, 042404 (2020).
  * Krivosik and Patton (2010) P. Krivosik and C. E. Patton, Physical Review B - Condensed Matter and Materials Physics 82, 184428 (2010).
  * Scott  _et al._ (2004) M. M. Scott, C. E. Patton, M. P. Kostylev, and B. A. Kalinikos, Journal of Applied Physics 95, 6294 (2004).
  * Demidov  _et al._ (2009) V. E. Demidov, J. Jersch, K. Rott, P. Krzysteczko, G. Reiss, and S. O. Demokritov, Physical Review Letters 102, 177207 (2009).
  * Bauer  _et al._ (2015) H. G. Bauer, P. Majchrak, T. Kachel, C. H. Back, and G. Woltersdorf, Nature Communications 6, 8274 (2015).
  * Venugopal  _et al._ (2020) A. Venugopal, T. Qu, and R. H. Victora, IEEE Transactions on Microwave Theory and Techniques 68, 602 (2020).
  * Moon and Girvin (2005) K. Moon and S. M. Girvin, Physical Review Letters 95, 140504 (2005).
  * Fang  _et al._ (2015) X. Fang, M. Bai, X. Ye, J. Miao, and Z. Zheng, Optics Express 23, 17111 (2015).
  * Yeh  _et al._ (2014) C. I. Yeh, W. S. Feng, and C. Y. Hsu, Electronics Letters 50, 186 (2014).
  * Schoen  _et al._ (2016) M. A. Schoen, D. Thonig, M. L. Schneider, T. J. Silva, H. T. Nembach, O. Eriksson, O. Karis, and J. M. Shaw, Nature Physics 12, 839 (2016).
  * Korner  _et al._ (2017) H. S. Korner, M. A. W. Schoen, T. Mayer, M. M. Decker, J. Stigloher, T. Weindler, T. N. G. Meier, M. Kronseder, and C. H. Back, Applied Physics Letters 111, 132406 (2017).
  * Flacke  _et al._ (2019) L. Flacke, L. Liensberger, M. Althammer, H. Huebl, S. Geprägs, K. Schultheiss, A. Buzdakov, T. Hula, H. Schultheiss, E. R. J. Edwards, H. T. Nembach, J. M. Shaw, R. Gross, and M. Weiler, Applied Physics Letters 115, 122402 (2019).
  * (29) The stripe of Co25Fe75 was processed on Si/SiO2 using electron beam lithography, magnetron sputtering and subsequent lift-off. The metallic thin film was deposited with a thickness of 30 nm. Two shorted-CPW gold antennas, processed via e-beam lithography, thermal evaporation and lift-off, were used as the microwave inputs RF1 and RF2.
  * L’vov (1994) V. S. L’vov,  _Wave Turbulence under Parametric Excitation_ (Springer-Verlag, New York, 1994).
  * Sebastian  _et al._ (2015) T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, and H. Schultheiss, Frontiers in Physics 3, 35 (2015).
  * Schultheiss  _et al._ (2019) K. Schultheiss, R. Verba, F. Wehrmann, K. Wagner, L. Körber, T. Hula, T. Hache, A. Kákay, A. A. Awad, V. Tiberkevich, A. N. Slavin, J. Fassbender, and H. Schultheiss, Physical Review Letters 122, 097202 (2019).
  * Vansteenkiste  _et al._ (2014) A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Advances 4, 107133 (2014).
  * (34) We modelled the conduit using 4096 cells along the long axis, 64 cells along the short axis and 1 cell along the film normal. For excitation we defined two rectangular regions across the stripe where dynamic out-of-plane fields could be applied. The position of these areas was similar to the fabricated sample. The following material parameters were used: $A_{ex}=26\cdot 10^{-12}$ J/m3, $M_{sat}=1700\cdot 10^{3}$ A/m and $\alpha=0.01$.