# Local well-posedness for the Zakharov-Kuznetsov equation in Sobolev spaces

SATOSHI OSAWA Department of Mathematics, Graduate School of Science, Kobe
University, 1-1 Rokkodai, Nada-ku, Kobe-shi, Hyogo, 657-8501, Japan.

###### Abstract

The initial value problem for two-dimensional Zakharov-Kuznetsov equation on
periodic boundary setting is shown to be locally well-posed in
$H^{s}(\mathbb{R}\times\mathbb{T})$ for $9/10<s<1$. We prove this theorem by
using bilinear estimates thinking separetely the first variable and the second
variable of space.

Keywords: Zakharov-Kuznetsov equation, Local well-posedness, Bilinear
Estimates, Bourgain’s spaces.

##  1 Introduction

We consider the initial value problem of the following Zakharov-Kuznetsov
equation on the product space $\mathbb{R}\times\mathbb{T}$:

$\displaystyle\left\\{\begin{array}[]{l}\partial_{t}u+\partial_{x}\Delta
u+u\partial_{x}u=0,\\\ u(x,y,0)=u_{0}(x,y),\end{array}\ \ \ \
(x,y)\in\mathbb{R}\times\mathbb{T},\ \ \ t\in\mathbb{R},\right.$ (ZK)

where $u=u(x,y,t)$ is a real-valued function,
$\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$, and
$\Delta=\partial^{2}_{x}+\partial^{2}_{y}$ is the Laplacian. This equation was
introduced by Zakharov and Kuznetsov in [8], as a model for the propagetion of
ionic-acoustic waves in magnetized plasma. The equation (ZK) is one of the
expanded equations of the Korteweg-de Vries (KdV) equation:

$\partial_{t}u+\partial_{x}^{3}u+u\partial_{x}u=0.$

In this paper, we prove a property of well-posedness for an initial data whose
norm is bounded in Sobolev spaces. In particular, we observe the behavior of
the solution along time development. In this paper, we try to prove a property
for the Sobolev space of lower index.

The KdV equation is completely integrable, and has infinite number of
conserved quantities. Compared to that, the Zakharov-Kuznetsov equation is not
completely integrable. However, the Zakharov-Kuznetsov equation has at least
two conservation quantities for the its flow: $L^{2}$ and $H^{1}$ versions,

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle
E[u](t):=\frac{1}{2}\int_{\mathbb{R}\times\mathbb{T}}({|\nabla
u(x,y,t)|}^{2}-\frac{1}{3}{u(x,y,t)}^{3})dxdy,\\\ \displaystyle
M[u](t):=\int_{\mathbb{R}\times\mathbb{T}}{u(x,y,t)}^{2}dxdy.\end{array}\right.$

In the non-periodic boundary setting, Grünrock and Herr [4], and Molinet and
Pilod [2] independently proved the local well-posedness in
$H^{s}(\mathbb{R}^{2})$ for $s>1/2$, by using the Fourier restriction norm
method [6] and Strichartz estimates. Subsequently, Kinoshita [10] shows the
local well-posedness for $H^{s}(\mathbb{R}^{2})$ using elaborate estimates
which are based on region decompositions. By using the $L^{2}$ and $H^{1}$
conservation laws, these conclusions specially say that the Zakharov-Kuznetsov
equation in $\mathbb{R}^{2}$ is global well-posed in $L^{2}$ and $H^{1}$,
respectively. Moreover, similar statements are researched from the perspective
of global well-posedness. Shan [13] obtained the global well-posedness in
$H^{s}(\mathbb{R}^{2})$ for $s>5/7$, and Kinoshita [10] obtained the global
well-posedness in $L^{2}(\mathbb{R}^{2})$. Shan used the I-method based on
$H^{1}$ conservation laws. Kinoshita used the classical Loomis-Whitney
inequality and decompositions according to regions and frequencies.

On the other hand, in the periodic boundary condition setting, Molinet and
Pilod [2] showed the local well-posedness in
$H^{s}(\mathbb{R}\times\mathbb{T})$ for $s\geq 1$ and the global well-
posedness in $H^{s}(\mathbb{R}\times\mathbb{T})$ for $s=1$. They used nearly
the same of strategy in $\mathbb{R}^{2}$, but generally we cannot use
elaborate region decomposition for estimates in $\mathbb{R}\times\mathbb{T}$.
Similarily, Linares, Panthee, Robert and Tzvetkov [12] obtained the well-
poseness result in $H^{s}(\mathbb{T}^{2})$. They showed that the initial value
problem for the Zakharov-Kuznetsov equation is locally well-posed in
$H^{s}(\mathbb{T}^{2})$ for $s>5/3$. Their proof is based on a variant of
Strichartz estimate. More recently, Kinoshita showed the locally well-
posedness in $H^{s}(\mathbb{T}^{2})$, for $s>1$ [11].

In this paper, we will prove the following local well-posedness of initial
value problem (ZK) in Sobolev spaces $H^{s}(\mathbb{R}\times\mathbb{T})$ of
lower index $s$.

######  Theorem 1.1.

Let $s>9/10$. For any $u_{0}\in H^{s}(\mathbb{R}\times\mathbb{T})$, there
exists $T=T(\|u_{0}\|_{H^{s}(\mathbb{R}\times\mathbb{T})})$ and a unique
solution $u$ of (ZK) such that

$u\in C([0,T]:H^{s}((\mathbb{R}\times\mathbb{T}))\cap X^{s,\frac{1}{2}+}_{T}.$

Moreover, for all $0<T^{\prime}<T$, there exists a neighborhood $\mathcal{U}$
of $u_{0}$ in $H^{s}(\mathbb{R}\times\mathbb{T})$ such that the data-to-
solution map

$S:v_{0}\in\mathcal{U}\mapsto v\in
C([0,T^{\prime}]:H^{s}((\mathbb{R}\times\mathbb{T}))\cap
X^{s,\frac{1}{2}+}_{T^{\prime}}$

is smooth, when v is a unique solution of (ZK) to the initial data $v_{0}$.

In the Section 2, we will show bilinear estimates. We will introduce some
lemmas which show relationship between measure of set and regularity of
functions, given in [1]. Moreover, by considering a polynomial called the
resonance function, we will confirm strategy of calculation. In Section 3, we
will prove Theorem 1.1 by using the propositions discussed in Section 2. We
use the frequencies decomposition to observe nonlinear interactions of two
wave functions. we calculate the norm of auxiliary space by thinking
separately the first variable and the second variable of space. At the higher
frequency part, we improved calculation because of Lemma2.9. Finally, we
applied Duhamel’s principal and evaluated the Sobolev norm of function
solution. As a result, we obtain the estimate from above for lower index $s$.

##  2 Preparation for estimates

###  2.1 Function spaces

In this subsection, we will introduce some basic function spaces. We aim to
prove the initial value problem in Lebesgue spaces or Sobolev spaces. First,
we discribe these spaces using the Fourier transform.

Let $f(x,y)$ be defined on $\mathbb{R}\times\mathbb{T}$. We denote the Fourier
transform of $f$ with respect to the variables $(x,y)$ as
$\mathcal{F}_{xy}(f)(\xi,q)$. More precisely, $y$ is a variable from
$\mathbb{T}$, we should explain $q$ as a variable connecting with the Fourier
coefficients. Therefore, we note that $\xi\in\mathbb{R}$ and $q\in\mathbb{Z}$.
Moreover, let $\mathcal{F}^{-1}_{xy}$ be the Fourier inverse transform with
respect to spacial variables. We use the absolute value of $(\xi,q)$ as
$|(\xi,q)|^{2}=3\xi^{2}+q^{2}$ in this paper. And $a\sim b$ means that there
is a constant $c$ such that $a=cb$ and $a\lesssim b$ means that there is a
constant $c$ such that $a\leq cb$. Moreover, $b+$ in statements means that
there exists $\delta>0$ such that $b+\delta$.

Then we define the Sobolev spaces for this discription: equipped with the norm

$\|f\|^{2}_{H^{s}}:=\sum_{q\in\mathbb{Z}}\int_{\mathbb{R}}({\langle|(\xi,q)|\rangle^{2}})^{s}|\mathcal{F}_{xy}(f)(\xi,q)|^{2}d\xi,$

where $\langle x\rangle=1+|x|$.

We are able to measure the size and regularity of solutions by using Sobolev
spaces. Following [2], we introduce other more complicated function spaces
$X^{s,b}$, called Bourgain’s spaces. Now let $u(x,y,t)$ be defined on
$\mathbb{R}\times\mathbb{T}\times\mathbb{R}$, and let $\hat{u}(\xi,q,\tau)$ be
Fourier tranceform of $u(x,y,t)$ in a similar manner as well. Let
$\|u\|_{L^{p}}$ be the usual Lebesgue norm, and $\|u\|_{l^{p}}$ be norm in the
usual sequence space.

######  Definition 2.1.

$\it{For}$ $s,b\in\mathbb{R},T>0,$

$\|u\|_{X^{s,b}}:=\|\langle\tau-w(\xi,q)\rangle^{b}\langle|(\xi,q)|\rangle^{s}|\hat{u}(\xi,q,\tau)|\|_{L_{\xi,\tau}^{2}l_{q}^{2}},$
$None$ $\|u\|_{X^{s,b}_{T}}:=\inf{\\{\|\bar{u}\|_{X^{s,b}};\ \
\bar{u}:\mathbb{R}\times\mathbb{T}\rightarrow\mathbb{C},\ \
\bar{u}|_{\mathbb{R}\times\mathbb{T}\times[0,T]}=u\\}},$ $None$

where $w(\xi,q)=\xi^{3}+\xi{q}^{2}$. When a norm of $u$ introduced (2.1) is
bounded, we denote $u\in X^{s,b}$.

The space $X^{s,b}_{T}$ will be used for proof of the local well-posedness
result, that is under restriction of time on $[0,T]$.

######  Remark 2.2.

The space $X^{s,b}$ will be characterized by this formula:

$\|f\|_{X^{s,b}}=\|e^{t\partial_{x}\Delta}f\|_{H^{b}_{t}H^{s}_{xy}},$ $None$

where $e^{-t\partial_{x}\Delta}$ is the operator associated linear problem of
the Zakharov-Kuznetsov equation, discribed as

$\mathcal{F}_{xy}((e^{-t\partial_{x}\Delta}f))(\xi,q)=e^{itw(\xi,q)}\mathcal{F}_{xy}(f)(\xi,q).$
$None$

There are some formulas which discribe relationship about $X^{s,b}$. We
introduce some formulas which was stated in [3]. Let $\eta\in
C^{\infty}_{0}(\mathbb{R})$ satisfy $0\leq\eta\leq 1$, $\eta_{[-1,1]}=1$, and
$\rm{supp}(\eta)\subset[-2,2]$.

######  Lemma 2.3.

Let $s\in\mathbb{R}$ and $b>1/2$. Then

$\|\eta(t)e^{-t\partial_{x}\Delta}f\|_{X^{s,b}}\lesssim\|f\|_{H^{s}},$ $None$

for all $f\in H^{s}$.

######  Lemma 2.4.

Let $s\in\mathbb{R}$. Then

$\left\|\eta(t)\int^{t}_{0}e^{-(t-t^{\prime})\partial_{x}\Delta}g(t^{\prime})dt^{\prime}\right\|_{X^{s,\frac{1}{2}+}}\lesssim\|g\|_{X^{s,-\frac{1}{2}+}},$
$None$

for all $g\in X^{s,\frac{1}{2}+}.$

######  Lemma 2.5.

For any $T>0$, $s\in\mathbb{R}$ and for all $-1/2<b^{\prime}\leq b<1/2$,

$\|u\|_{X^{s,b^{\prime}}_{T}}\lesssim T^{b-b^{\prime}}\|u\|_{X^{s,b}_{T}}.$
$None$

The polynomial $w(\xi,q)$ which appeared in (2.1) plays an important role in
characterizing of solution. Now, we define the resonance function, discribing
interaction of solutions:

$\displaystyle\mathcal{H}(\xi_{1},\xi_{2},q_{1},q_{2})$
$\displaystyle=w(\xi_{1}+\xi_{2},q_{1}+q_{2})-w(\xi_{1},q_{1})-w(\xi_{2},q_{2})$
(2.8)
$\displaystyle=3\xi_{1}\xi_{2}(\xi_{1}+\xi_{2})+\xi_{2}q^{2}_{1}+\xi_{1}q^{2}_{2}+2(\xi_{1}+\xi_{2})q_{1}q_{2}.$

###  2.2 Bilinear estimates

In this subsection, we show the bilinear estimates. First, we define cut-off
functions and the dyadic composition.

For $k\in\mathbb{N}$, we denote

$\phi(\xi)=\eta(|\xi|)-\eta(2|\xi|),\ \
\phi_{2^{k}}(\xi,q)=\phi(2^{-k}|(\xi,q)|),\ \
\psi_{2^{k}}(\xi,q,\tau)=\phi(2^{-k}(\tau-(\xi^{3}+\xi q^{2}))),$

where $(\xi,\tau)\in\mathbb{R}^{2}$ and $q\in\mathbb{Z}$. Here, we prepare
notation of interval as $I_{N}:=\rm{supp}\it{\phi_{N}}$. We define the
Littlewood-Paley decomposition as

$P_{N}(u):=\mathcal{F}^{-1}_{xy}(\phi_{N}\mathcal{F}_{xy}(u)),\ \ \
Q_{L}(u):=\mathcal{F}^{-1}(\psi_{L}\mathcal{F}(u)).$ $None$

Now we give the bilinear estimates that are crucial for proving Theorem 1.1.
We denote for $a,b\in\mathbb{R}$, $a\wedge b=\min\\{a,b\\},a\vee
b=\max\\{a,b\\}$.

######  Lemma 2.6 (Proposition 3.6 in [2]).

Suppose that $N_{1},N_{2},L_{1},L_{2}$ are dyadic numbers. Then

$\displaystyle\|(P_{N_{1}}Q_{L_{1}}u)(P_{N_{2}}Q_{L_{2}}v)\|_{L^{2}}$
$\displaystyle\lesssim(N_{1}\wedge N_{2})(L_{1}\wedge
L_{2})^{\frac{1}{2}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}.$
(2.10)

Moreover, when $4N_{2}\leq N_{1}$ or $4N_{1}\leq N_{2}$, we obtain

$\displaystyle\|(P_{N_{1}}Q_{L_{1}}u)(P_{N_{2}}Q_{L_{2}}v)\|_{L^{2}}$
$\displaystyle\lesssim\frac{(N_{1}\wedge N_{2})^{\frac{1}{2}}}{N_{1}\vee
N_{2}}(L_{1}\wedge L_{2})^{\frac{1}{2}}(L_{1}\vee
L_{2})^{\frac{1}{2}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}.$
(2.11)

Molinet and Pilod shows Lemma 2.6, using some basic lemmas introduced in [1].

When $f$ is a smooth function or a polynomial of degree 2, Lemma 2.7 and Lemma
2.8 can be written as below.

######  Lemma 2.7 (Lemma 3.8 in [2]).

Let $I$ and $J$ be two intervals on the real line and
$f:J\rightarrow\mathbb{R}$ be a smooth function. Then,

$\rm{mes}\\{\it{x}\in J:f(x)\in I\\}\lesssim\frac{|I|}{\rm{inf}_{\it{\xi\in
J}}|\it{f}^{\prime}(\xi)|},$ $None$

where $\rm{mes}\it{A}$ is the Lebesgue measure of the set A.

######  Lemma 2.8 (Lemma 3.9 in [2]).

Let $a\neq 0$, $b$, $c$ be real numbers and $I$ be an interval on the real
line. Then,

$\rm{mes}\\{\it{x}\in J:ax^{\rm{2}}+\it{bx}+c\in
I\\}\lesssim\frac{|I|^{\rm{\frac{1}{2}}}}{|a|^{\rm{\frac{1}{2}}}}.$ $None$

Now we denote a new lemma which applied at the case $N_{2}/2\leq N_{1}\leq
2N_{2}$. We denote by $R_{K}$ the Fourier multiplier operator in the variable
of $x$ as following:

$R_{K}f(x)=\int_{\mathbb{R}}\phi(\xi
K^{-1})\mathcal{F}_{x}f(\xi)e^{ix\xi}d\xi,$

where $\mathcal{F}_{x}$ is the Fourier transform of $f(x)$ with respect to $x$
only.

######  Lemma 2.9.

Let $K$ be a dyadic number. It holds that

$\displaystyle\|R_{K}\left((P_{N_{1}}Q_{L_{1}}u)(P_{N_{2}}Q_{L_{2}}v)\right)\|_{L^{2}}$
$\displaystyle\lesssim\frac{(N_{1}\wedge
N_{2})^{\frac{1}{2}}}{K^{\frac{1}{4}}}(L_{1}\wedge
L_{2})^{\frac{1}{2}}(L_{1}\vee
L_{2})^{\frac{1}{4}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}},$
(2.14)

where $K$ is a dyadic number satisfying $|\xi|\in I_{K}$.

Proof We obtain the following estimate from Cauchy-Schwarz inequality and
Plancherel’s identity:

$\displaystyle\|R_{K}((P_{N_{1}}Q_{L_{1}}u)$
$\displaystyle(P_{N_{2}}Q_{L_{2}}v))\|_{L^{2}}$
$\displaystyle=\|R_{K}((\widehat{P_{N_{1}}Q_{L_{1}}u})*(\widehat{P_{N_{2}}Q_{L_{2}}v}))\|_{L^{2}}$
$\displaystyle=\left\|R_{K}\left(\sum_{q_{1}\in\mathbb{Z}}\int_{\mathbb{R}\times\mathbb{R}}\widehat{P_{N_{1}}Q_{L_{1}}u}(\xi_{1},q_{1},\tau_{1})\cdot\widehat{P_{N_{2}}Q_{L_{2}}v}(\xi-\xi_{1},q-q_{1},\tau-\tau_{1})d\xi_{1}\tau_{1}\right)\right\|_{L^{2}}$
$\displaystyle\lesssim\sup_{(\xi,q,\tau)}|A^{K}_{\xi,q,\tau}|^{\frac{1}{2}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}$

where

$\displaystyle
A^{K}_{\xi,q,\tau}=\\{(\xi_{1},q_{1},\tau_{1})\in\mathbb{R}\times\mathbb{Z}\times\mathbb{R}:|\xi|\sim
K,|(\xi_{1},q_{1})|\in I_{N_{1}},|(\xi-\xi_{1},q-q_{1})|\in I_{N_{2}},$
$\displaystyle|\tau_{1}-w(\xi_{1},q_{1})|\in
I_{L_{1}},|\tau-\tau_{1}-w(\xi-\xi_{1},q-q_{1})|\in I_{L_{2}}\\}.$

Observe the definition of $A^{K}_{\xi,q,\tau}$, we get

$|A^{K}_{\xi,q,\tau}|\lesssim(N_{1}\wedge N_{2})^{2}(L_{1}\wedge L_{2}),$

which implies the estimate in (2.10).
On the other hand, the triangle inequality yields

$\displaystyle|\tau_{1}-w(\xi_{1},q_{1})|+|\tau-\tau_{1}-w(\xi-\xi_{1},q-q_{1})|$
$\displaystyle\leq|\tau-w(\xi_{1},q_{1})-w(\xi-\xi_{1},q-q_{1})|$
$\displaystyle=|\tau-w(\xi,q)-\mathcal{H}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})|,$

we get

$|A^{K}_{\xi,q,\tau}|\lesssim(L_{1}\wedge L_{2})|B^{K}_{\xi,q,\tau}|,$ $None$

where

$\displaystyle
B^{K}_{\xi,q,\tau}=\\{(\xi_{1},q_{1})\in\mathbb{R}\times\mathbb{Z}:|\xi|\sim
K,|(\xi_{1},q_{1})|\in I_{N_{1}},|(\xi-\xi_{1},q-q_{1})|\in I_{N_{2}},$
$\displaystyle|\tau-w(\xi,q)-\mathcal{H}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})|\lesssim
L_{1}\vee L_{2}\\}.$ (2.16)

For estimating $B^{K}_{\xi,q,\tau}$, we calculate the second order
differential of $\mathcal{H}$:

$\left|\frac{\partial^{2}\mathcal{H}}{{\partial\xi_{1}}^{2}}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})\right|=6|\xi|\sim
K,$ $None$

for $(\xi_{1},q_{1})\in B^{K}_{\xi,q,\tau}$. Let
$\widetilde{B}_{\xi,q,\tau}(q_{1})=\\{\xi_{1}\in\mathbb{R}:(\xi_{1},q_{1})\in
B^{K}_{\xi,q,\tau}\\}$ for fixed $q_{1}$. Then combining Lemma 2.8 and (2.17)
we get

$|\widetilde{B}_{\xi,q,\tau}(q_{1})|\lesssim\frac{(L_{1}\vee
L_{2})^{\frac{1}{2}}}{K^{\frac{1}{2}}},$ $None$

for all $q_{1}\in\mathbb{Z}$. Finally, it follows from that

$|B^{K}_{\xi,q,\tau}|\lesssim\frac{(N_{1}\wedge
N_{2})}{K^{\frac{1}{2}}}(L_{1}\vee L_{2}).$

Combining with (2.15), we obtain (2.14). $\Box$

##  3 Local well-posedness

In this section, we prove Theorem 1.1.

######  Proposition 3.1.

Let $9/10<s<1$. For all
$u,v:\mathbb{R}\times\mathbb{T}\times\mathbb{R}\rightarrow\mathbb{C}$ such
that $u,v\in X^{s,\frac{1}{2}+}$, we have

$\|\partial_{x}(uv)\|_{X^{s,-\frac{1}{2}+}}\lesssim\|u\|_{X^{s,\frac{1}{2}+}}\|v\|_{X^{s,\frac{1}{2}+}}.$
$None$

Proposition 3.1 instantly allows us to prove Theorem 1.1, which means local
well-posedness in Sobolev spaces.

Proof. By duality, it suffices to prove the following:

$J:=\sum_{q,q_{1}\in\mathbb{Z}}\int\Gamma^{\xi_{1},q_{1},\tau_{1}}_{\xi,q,\tau}\
\hat{u}(\xi_{1},q_{1},\tau_{1})\ \hat{v}(\xi_{2},q_{2},\tau_{2})\
\hat{w}(\xi,q,\tau)\ d\nu\lesssim\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}},$
$None$

where $\hat{u},\hat{v},\hat{w}$ are non-negative functions,

$\Gamma^{\xi_{1},q_{1},\tau_{1}}_{\xi,q,\tau}=|\xi|\langle|(\xi,q)|\rangle^{s}\langle|(\xi_{1},q_{1})|\rangle^{-s}\langle|(\xi_{2},q_{2})|\rangle^{-s}\langle\sigma\rangle^{-\frac{1}{2}+2\delta}\langle\sigma_{1}\rangle^{-\frac{1}{2}-\delta}\langle\sigma_{2}\rangle^{-\frac{1}{2}-\delta},$
$\xi=\xi_{1}+\xi_{2},\ \ \ \ \ q=q_{1}+q_{2},\ \ \ \ \
\tau=\tau_{1}+\tau_{2},\ \ \ \ \ d\nu=d\xi d\xi_{1}d\tau d\tau_{1},$
$\sigma=\tau-w(\xi,q),\ \ \ \ \ \sigma_{1}=\tau_{1}-w(\xi_{1},q_{1}),\ \ \ \ \
\sigma_{2}=\tau_{2}-w(\xi_{2},q_{2}).$

Using dyadic decomposition following (2.9), we rewrite $J$ as the following:

$J=\sum_{N,N_{1},N_{2}}J_{N,N_{1},N_{2}},$ $None$

where

$J_{N,N_{1},N_{2}}=\sum_{q,q_{1}}\int\Gamma^{\xi_{1},q_{1},\tau_{1}}_{\xi,q,\tau}\
\widehat{P_{N_{1}}u}(\xi_{1},q_{1},\tau_{1})\
\widehat{P_{N_{2}}v}(\xi_{2},q_{2},\tau_{2})\
\widehat{P_{N}w}(\xi,q,\tau)d\nu.$

We decompose $J$ to five parts:

$J_{LL\rightarrow L}:=\sum_{N_{1}\leq 4,N_{2}\leq 4,N\leq
4}J_{N,N_{1},N_{2}},$ $J_{LH\rightarrow H}:=\sum_{N_{2}\leq 4,N_{1}\leq
N_{2}/4}J_{N,N_{1},N_{2}},$ $J_{HL\rightarrow H}:=\sum_{N_{1}\leq 4,N_{2}\leq
N_{1}/4}J_{N,N_{1},N_{2}},$ $J_{HH\rightarrow L}:=\sum_{N_{1}\leq 4,N\leq
N_{1}/4\ or\ 4\leq N_{2},N\leq N_{2}/4}J_{N,N_{1},N_{2}},$ $J_{HH\rightarrow
H}:=J-(J_{LL\rightarrow L}+J_{HL\rightarrow H}+J_{LH\rightarrow
H}+J_{HH\rightarrow L}).$

We note that $J_{HH\rightarrow H}$ means the sum of $J_{N,N_{1},N_{2}}$ at the
regions of $N_{1}\geq 4,N_{2}\geq 4,N_{2}/2\leq N_{1}\leq 2N_{2},N_{1}/2\leq
N\leq 2N_{1},N_{2}/2\leq N\leq 2N_{2}$. Estimates of $J_{LH\rightarrow H}$,
$J_{HL\rightarrow H}$ and $J_{HH\rightarrow L}$ are similar to the case of
(ZK) on $\mathbb{R}^{2}$ [2]. In these cases, we use (2.11) directly.

(i) Estimate of $J_{LL\rightarrow L}$. We use Plancherel’s identity and
Hölder’s inequality to obtain

$J_{LL\rightarrow L}\lesssim\sum_{N_{1}\leq 4,N_{2}\leq 4,N\leq
4}\left\|\left(\frac{\widehat{P_{N_{1}}u}}{\langle\sigma_{1}\rangle^{\frac{1}{2}+}}\right)^{\vee}\right\|_{L^{4}}\left\|\left(\frac{\widehat{P_{N_{2}}v}}{\langle\sigma_{2}\rangle^{\frac{1}{2}+}}\right)^{\vee}\right\|_{L^{4}}\|P_{N}w\|_{L^{2}},$
$None$

where $u^{\vee}(x,y,t)$ denotes the Fourier inverse transform with respect to
$(\xi,q,\tau)$. From the Sobolev embedding $H^{1/4+}(\mathbb{R})\subset
L^{4}(\mathbb{R})$ and Plancherel’s identity, we get

$\displaystyle\left\|\left(\frac{\widehat{P_{N_{1}}u}}{\langle\sigma_{1}\rangle^{\frac{1}{2}+}}\right)^{\vee}\right\|_{L^{4}}\left\|\left(\frac{\widehat{P_{N_{2}}v}}{\langle\sigma_{2}\rangle^{\frac{1}{2}+}}\right)^{\vee}\right\|_{L^{4}}\|P_{N}w\|_{L^{2}}$
$\displaystyle\lesssim\|\langle\sigma_{1}\rangle^{-\frac{1}{4}+}\widehat{P_{N_{1}}u}\|_{L^{2}}\cdot\|\langle\sigma_{2}\rangle^{-\frac{1}{4}+}\widehat{P_{N_{2}}v}\|_{L^{2}}\cdot\|P_{N}w\|_{L^{2}}$
$\displaystyle\lesssim\|P_{N_{1}}u\|_{L^{2}}\|P_{N_{2}}v\|_{L^{2}}\|P_{N}w\|_{L^{2}},$

where $N_{1}\leq 4,N_{2}\leq 4,N\leq 4$. Thus,

$J_{LL\rightarrow L}\lesssim\sum_{N_{1}\leq 4,N_{2}\leq 4,N\leq
4}\|P_{N_{1}}u\|_{L^{2}}\|P_{N_{2}}v\|_{L^{2}}\|P_{N}w\|_{L^{2}},$

which yields

$J_{LL\rightarrow L}\lesssim\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}}.$

(ii) Estimete of $J_{LH\rightarrow H}$. We use another dyadic decomposition by
$Q_{L}$. We denote

$J_{N,N_{1},N_{2}}=\sum_{L,L_{1},L_{2}}J^{L,L_{1},L_{2}}_{N,N_{1},N_{2}},$

where

$J^{L,L_{1},L_{2}}_{N,N_{1},N_{2}}=\sum_{q,q_{1}}\int\Gamma^{\xi_{1},q_{1},\tau_{1}}_{\xi,q,\tau}\
\widehat{P_{N_{1}}Q_{L_{1}}u}(\xi_{1},q_{1},\tau_{1})\
\widehat{P_{N_{2}}Q_{L_{2}}v}(\xi_{2},q_{2},\tau_{2})\
\widehat{P_{N}Q_{L}w}(\xi,q,\tau)d\nu.$ $None$

Applying the Cauchy-Schwarz inequality, we have that the contribution of this
case to $J^{L,L_{1},L_{2}}_{N,N_{1},N_{2}}$ is bounded by

$CN_{2}N^{-s}_{1}L^{-\frac{1}{2}+2\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}\|(P_{N_{1}}Q_{L_{1}}u)(P_{N_{2}}Q_{L_{2}}w)\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}.$
$None$

By Lemma 2.6, we can show that

$\|(P_{N_{1}}Q_{L_{1}}u)(P_{N_{2}}Q_{L_{2}}w)\|_{L^{2}}\lesssim\frac{N^{\frac{1}{2}}_{1}}{N_{2}}L^{\frac{1}{2}}_{1}L^{\frac{1}{2}}_{2}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}.$
$None$

Therefore, combining (3.6) and (3.7), there is some $\delta>0$ such that for
$s>1/2$,

$\displaystyle J_{LH\rightarrow H}$ $\displaystyle\lesssim$
$\displaystyle\sum_{L,L_{1},L_{2}}L^{-\frac{1}{2}+2\delta}L^{-\delta}_{1}L^{-\delta}_{2}\sum_{4\leq
N_{2},N_{1}\leq
N_{2}/4}N^{-s+\frac{1}{2}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}$
$\displaystyle\lesssim$ $\displaystyle\|u\|_{L^{2}}\sum_{N\sim
N_{2}}N^{-s+\frac{1}{2}}\|P_{N_{2}}v\|_{L2}\|P_{N}w\|_{L^{2}}$
$\displaystyle\leq$
$\displaystyle\|u\|_{L^{2}}\left(\sum_{N_{2}}\|P_{N_{2}}v\|^{2}_{L^{2}}\right)^{\frac{1}{2}}\left(\sum_{N}N^{-2s+1}\|P_{N}w\|^{2}_{L^{2}}\right)^{\frac{1}{2}}$
$\displaystyle\lesssim$
$\displaystyle\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}}.$

(iii) Estimate of $J_{HL\rightarrow H}$. In this case, the proof following
directly from the proceeding symmetry argument. Indeed,

$J_{HL\rightarrow H}\lesssim\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}}.$

(iv) Estimate of $J_{HH\rightarrow L}$. By symmetry, we may assume $4\leq
N_{1}$ and $N\leq N_{1}/4$. Similar to the case of (ii), we have that the
contribution of this case to $J^{L,L_{1},L_{2}}_{N,N_{1},N_{2}}$ is bounded by

$C\frac{N^{s+1}}{N^{s}_{1}N^{s}_{2}}L^{-\frac{1}{2}+2\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}\|\widetilde{(P_{N_{1}}Q_{L_{1}}u)}(P_{N}Q_{L}w)\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}},$
$None$

where $\tilde{f}(\xi,q,\tau)=f(-\xi,-q,-\tau).$ We use the interpolation
inequality between (2.10) and (2.11). For any $0<\theta<1$,

$\displaystyle\|\widetilde{(P_{N_{1}}Q_{L_{1}}u)}(P_{N}Q_{L}w)\|_{L^{2}}$
$\displaystyle\lesssim\frac{(N_{1}\wedge
N)^{\frac{1}{2}(1-\theta)+\theta}}{(N_{1}\vee N)^{1-\theta}}(L_{1}\wedge
L)^{\frac{1}{2}(1-\theta)+\frac{\theta}{2}}(L_{1}\vee
L)^{\frac{1}{2}(1-\theta)}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}$
$\displaystyle\lesssim\frac{N^{\frac{1}{2}(1+\theta)}}{N^{1-\theta}_{1}}(L_{1}\vee
L)^{\frac{1}{2}(1-\theta)}(L_{1}\wedge
L)^{\frac{1}{2}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}.$ (3.9)

Then, we substitute (3.9) into (3.8). The contribution of this case to
$J^{L,L_{1},L_{2}}_{N,N_{1},N_{2}}$ is bounded by

$\displaystyle
CN^{\frac{1}{2}+\theta}N^{-s+\theta}_{1}L^{-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}L^{2\delta-\frac{\theta}{2}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}$
$\displaystyle\lesssim
N^{-s+\frac{1}{2}+2\theta}L^{-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}L^{2\delta-\frac{\theta}{2}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}.$

Thus summing up with respect to $N,N_{1}N_{2},L,L_{1},L_{2}$, we obtain

$J_{HH\rightarrow L}\lesssim\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}},$

for $0<\delta<s/4\ll 1$ and $s>1/2+2\theta$.

(v) Estimate of $J_{HH\rightarrow H}$. In this case, we divide five cases and
estimete the resonance function $\mathcal{H}$ in each situation.

(v-1) Case of $|\xi|\leq 100$. Let $k$ be an integer. We denote

$J_{N,N_{1},N_{2}}=\sum_{k\geq
0}\sum_{L,L_{1},L_{2}}J^{L,L_{1},L_{2}}_{N,N_{1},N_{2},k},$

where

$J^{L,L_{1},L_{2}}_{N,N_{1},N_{2},k}=\sum_{q,q_{1}}\int_{Y_{k}}\Gamma^{\xi_{1},q_{1},\tau_{1}}_{\xi,q,\tau}\
\widehat{P_{N_{1}}Q_{L_{1}}u}(\xi_{1},q_{1},\tau_{1})\
\widehat{P_{N_{2}}Q_{L_{2}}v}(\xi_{2},q_{2},\tau_{2})\
\widehat{P_{N}Q_{L}w}(\xi,q,\tau)d\nu,$

with

$Y_{k}=\\{(\xi,\xi_{1},\tau,\tau_{1})\in\mathbb{R}^{4}:2^{-(k+1)}100\leq|\xi|\leq
2^{-k}100\\}.$

We apply the Cauchy-Schwarz inequality, and we have that the contribution of
this case to $J^{L,L_{1},L_{2}}_{N,N_{1},N_{2},k}$ is bounded by

$C2^{-k}N^{-s}_{1}L^{-\frac{1}{2}+2\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}\|(P_{N_{1}}Q_{L_{1}}u)(P_{N_{2}}Q_{L_{2}}w)\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}.$
$None$

For $2^{-(k+1)}100\leq|\xi|\leq 2^{-k}100$, we obtain

$\left|\frac{\partial^{2}\mathcal{H}}{\partial\xi^{2}_{1}}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})\right|=6|\xi|\sim
2^{-k}.$

Using Lemma 2.9, we obtain

$\|R_{K}((P_{N_{1}}Q_{L_{1}}u)(P_{N_{2}}Q_{L_{2}}v))\|_{L^{2}}\lesssim
2^{\frac{k}{4}}N^{\frac{1}{2}}_{1}(L_{1}\vee L_{2})^{\frac{1}{4}}(L_{1}\wedge
L_{2})^{\frac{1}{2}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}.$
$None$

Therefore, combining (3.10) and (3.11), we get that (3.10) is bounded by

$C2^{-\frac{3}{4}k}N^{s-\frac{1}{2}}_{1}L^{-\frac{1}{2}+2\delta}L^{-\delta}_{1}L^{-\delta}_{2}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}.$

Summing up with respect to $N,N_{1},N_{2},L,L_{1},L_{2},k$, we have that the
contribution of this case to $J_{HH\rightarrow H}$ is bounded by

$C\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}}.$

(v-2) Case of $|\xi|\geq 100$, $|\xi_{1}|\wedge|\xi_{2}|\leq 100$. By
symmetry, we may suppose $|\xi_{2}|\leq|\xi_{1}|$. First, we calculate the
resonance function:

$\frac{\partial\mathcal{H}}{\partial\xi_{1}}(\xi_{1},\xi-\xi_{1},q,q-q_{1})=3\xi(\xi-2\xi_{1})+q(q-2q_{1}).$

We will consider the case when
$|\frac{\partial\mathcal{H}}{\partial\xi_{1}}|\geq\xi^{2}$. By using the
Cauchy-Schwarz inequality such as (v-1), we have that the contribution of this
case to $J^{L,L_{1},L_{2}}_{N,N_{1},N_{2}}$ is bounded by

$\sum_{K}CKN^{-s}_{1}L^{-\frac{1}{2}+2\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}\|R_{K}((P_{N_{1}}Q_{L_{1}}u)(P_{N_{2}}Q_{L_{2}}w))\|_{L^{2}}\|R_{K}P_{N}Q_{L}w\|_{L^{2}}.$

Recall the proof of Lemma 2.9. Indeed of (2.17), we use
$\frac{\partial\mathcal{H}}{\partial\xi_{1}}\gtrsim K$ along which Lemma 2.7
to obtain

$|\widetilde{B}_{\xi,q,\tau}(q_{1})|\lesssim\frac{(L_{1}\vee
L_{2})^{\frac{1}{2}}}{K^{2}},$

for (2.18). Then,

$\|R_{K}((P_{N_{1}}Q_{L_{1}}u)(P_{N_{2}}Q_{L_{2}}v))\|_{L^{2}}\lesssim
K^{-1}N^{\frac{1}{2}}_{1}(L_{1}\vee L_{2})^{\frac{1}{2}}(L_{1}\wedge
L_{2})^{\frac{1}{2}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}.$

Combining the Cauchy-Schwarz inequality and bilinear estimate, we obtain that
the contribution of this case to $J^{L,L_{1},L_{2}}_{N,N_{1},N_{2}}$ is
bounded by

$\displaystyle\sum_{K\lesssim
N}KN^{-s}_{1}L^{-\frac{1}{2}+2\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}K^{-1}N^{\frac{1}{2}}_{1}(L_{1}\vee
L_{2})^{\frac{1}{2}}(L_{1}\wedge L_{2})^{\frac{1}{2}}$
$\displaystyle\times\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}\|R_{K}P_{N}Q_{L}w\|_{L^{2}}$
$\displaystyle\lesssim\sum_{K\lesssim
N}N^{-s+\frac{1}{2}}L^{-\frac{1}{2}+2\delta}L^{-\delta}_{1}L^{-\delta}_{2}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}\|R_{K}P_{N}Q_{L}w\|_{L^{2}}.$

So summing up with respect to $K,N,N_{1},N_{2},L,L_{1},L_{2}$, the right-hand
side is bounded by $C\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}}$ for $s>1/2$.

Next, we consider the case when
$|\frac{\partial\mathcal{H}}{\partial\xi_{1}}|\ll\xi^{2}$. In this case when
$|\xi-\xi_{1}|\gtrsim 1$, the proof is similar to the case (i) and (ii).
Without loss of generality, we may suppose that
$|\xi_{2}|=|\xi-\xi_{1}|=o(1)$, where $o$ is the Landau symbol. In this case,
we have

$\displaystyle\frac{\partial\mathcal{H}}{\partial\xi_{1}}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})$
$\displaystyle=$ $\displaystyle
3\xi(\xi-\xi_{1})-3\xi\xi_{1}+q(q-q_{1})-qq_{1}$ $\displaystyle=$
$\displaystyle 6\xi(\xi-\xi_{1})-3\xi^{2}+q(q-q_{1})-qq_{1}$ $\displaystyle=$
$\displaystyle o(\xi)+o(\xi^{2})+q(q-q_{1})-qq_{1}.$

To satisfy the hypothesis of resonance function
$|\frac{\partial\mathcal{H}}{\partial\xi_{1}}|\ll\xi^{2}$, $N\sim N_{1}\sim
N_{2}$ and $|\xi-\xi_{1}|=o(1)$, we need
$\min\\{|q|,|q_{1}|,|q_{2}|\\}\gtrsim|\xi|$ to set
$q(q-q_{1})-qq_{1}=q(q_{2}-q_{1})\sim\xi^{2}$. Next we calculate the resonance
function $\mathcal{H}$,

$\displaystyle\mathcal{H}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})=3\xi\xi_{1}(\xi-\xi_{1})+(\xi-\xi_{1})(q^{2}-(q-q_{1})^{2})+(q-q_{1})\xi_{1}(q+q_{1}).$

We easily confirm that

$|q|\sim|q_{1}|\sim|q_{2}|\sim|\xi|\sim N,\ 3\xi\xi_{1}(\xi-\xi_{1})\sim
o(\xi^{2}),\ (\xi-\xi_{1})(q^{2}-(q-q_{1})^{2})\sim o(\xi^{2}).$

In the case of $|\mathcal{H}|\gtrsim\xi^{2}$, we note that
$\max\\{|\sigma|,|\sigma_{1}|,|\sigma_{2}|\\}\gtrsim\xi^{2}.$ We suppose
$\max\\{|\sigma|,|\sigma_{1}|,|\sigma_{2}|\\}=|\sigma|$, without loss of
generality. Then $\langle\sigma\rangle^{-\frac{1}{2}}|\xi|\sim 1$. The
bilinear estimate (2.11) along with the argument similar to the case (ii)
implies that the contribution of this case to $J_{HH\rightarrow H}$ is bounded
by $C\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}}$.

On the other hand, in the case of $|\mathcal{H}|\ll\xi^{2}$, it holds that
$|(q-q_{1})\xi_{1}(q+q_{1})|\lesssim o(\xi^{2})$. We obtain $q+q_{1}=0$. Write
$q_{1}=-q,q_{2}=q-q_{1}=2q$. We rewrite resonance function as
$\mathcal{H}=3(\xi-\xi_{1})(\xi\xi_{1}-q^{2})$ in this case. We regard this
formula as a quadratic equation of $\xi_{1}$ as the following:

$\xi\xi^{2}_{1}-(\xi^{2}+q^{2})\xi_{1}-\xi q^{2}+\frac{\mathcal{H}}{3}=0,$

which is solved by factoring

$\xi_{1}=\frac{(\xi^{2}-q^{2})\pm\sqrt{(\xi^{2}+q^{2})^{2}-4\xi^{2}q^{2}+\frac{4}{3}\xi\mathcal{H}}}{2\xi}.$
$None$

Now, the discussion in [7] tells us that it is suffice that the contribution
of this case to $J_{HH\rightarrow H}$ is bounded by
$C\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}}$ when

$I:=\sup_{(\xi,q,\tau)}\frac{|\xi|^{2}}{\langle\sigma\rangle^{2(1-\frac{1}{2}-\delta)}N^{2s}}\sum_{q_{1}}\int_{|\xi-\xi_{1}|\lesssim
1}\frac{d\xi_{1}}{\langle\sigma-\mathcal{H}\rangle^{2(\frac{1}{2}+\delta)}}<\infty.$
$None$

It suffice to say $I<\infty$. We use the change of the variable (3.12):

$d\xi_{1}=\frac{d\mathcal{H}}{\pm
3\sqrt{(\xi^{2}-q^{2})^{2}-\frac{3}{4}\xi\mathcal{H}}}.$ $None$

For simplicity, we suppose $\xi>0$. The estimate in Lemma 7.15 of [9] shows

$\displaystyle\left|\int_{|\xi-\xi_{1}|\lesssim
1}\frac{d\xi_{1}}{\langle\sigma-\mathcal{H}\rangle^{2(\frac{1}{2}+\delta)}}\right|$
$\displaystyle=\left|\int_{|\xi-\xi_{1}|\lesssim
1}\frac{d\mathcal{H}}{\langle\sigma-\mathcal{H}\rangle^{1+2\delta}\times\left(\pm
3\sqrt{(\xi^{2}-q^{2})^{2}-\frac{3}{4}\xi\mathcal{H}}\right)}\right|$
$\displaystyle\lesssim\int^{\infty}_{-\infty}\frac{d\mathcal{H}}{\langle\mathcal{H}\rangle^{1+2\delta}\frac{\sqrt{3\xi}}{2}\times
3\sqrt{\frac{3}{4}\frac{(\xi^{2}-q^{2})^{2}}{\xi}-\mathcal{H}}}$
$\displaystyle\lesssim\frac{1}{|\xi|^{\frac{1}{2}}+|\xi^{2}-q^{2}|}.$ (3.15)

Then

$\displaystyle\frac{|\xi|^{2}}{\langle\sigma\rangle^{2(1-\frac{1}{2}-\delta)}N^{2s}}\sum_{q_{1}=-q}\int_{|\xi-\xi_{1}|\lesssim
1}\frac{d\xi_{1}}{\langle\sigma-\mathcal{H}\rangle^{2(\frac{1}{2}+\delta)}}$
$\displaystyle\lesssim\frac{|\xi|^{2}}{\langle\sigma\rangle^{1-2\delta}N^{2s}}\frac{1}{|\xi|^{\frac{1}{2}}+|\xi^{2}-q^{2}_{1}|}$
$\displaystyle\lesssim\langle\xi\rangle^{2-\frac{1}{2}-2s}=\langle\xi\rangle^{\frac{3}{2}-2s},$

which is bounded for $s>3/4$.

Next, we consider case where $|\xi|,|\xi_{1}|,|\xi_{2}|\geq 100$. In this
case, following proof in [2] based on the calculation

$\left|\frac{\partial\mathcal{H}}{\partial\xi_{1}}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})\right|=\left||(\xi,q)|^{2}-|(\xi_{1},q_{1})|^{2}\right|,$

we devide the region into two subregions such that

$\displaystyle\rm{(v-3)}\ \ \ \it{i}\in\\{\rm{0},1,2\\},\
\it{j}\in\\{\rm{0},1,2\\},\ \ \exists(\it{i},j)\ \ \
||(\xi_{\it{i}},q_{i})|^{\rm{2}}-|(\xi_{\it{j}},q_{j})|^{\rm{2}}|\geq\it{N}^{\rm{6}/5}L^{\rm{6}\delta},$
(3.16) $\displaystyle\rm{(v-4)}\ \ \ \forall\it{i},j\in\\{\rm{0},1,2\\},\ \ \
||(\xi_{\it{i}},\it{q}_{i})|^{\rm{2}}-|(\xi_{\it{j}},\it{q}_{j})|^{\rm{2}}|\leq\it{N}^{\rm{6}/5}\it{L}^{\rm{6}\delta},$
(3.17)

where we rewrite $\xi=\xi_{0}$ and $q=q_{0}$.

Case of (v-3) Without loss of generality, we suppose that (3.16) holds for
$i=0$, and $j=1$. In a similar way to the case (v-2), the contribution of the
case to $J^{L,L_{1},L_{2}}_{N,N_{1},N_{2}}$ is bounded by

$CNN^{-s}_{1}L^{-\frac{1}{2}+2\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}\frac{(N_{1}\wedge
N_{2})^{\frac{1}{2}}}{N^{\frac{3}{5}}L^{3\delta}}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}.$
$None$

Because of $|\xi|\lesssim N_{1}$, we deduce

$\lesssim
N^{\frac{9}{10}-s}_{1}L^{-\frac{1}{2}-\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}.$
$None$

We sum up the right hand-side with respect to $N,N_{1},N_{2},L,L_{1},L_{2}$
that the contribution of the case to $J_{HH\rightarrow H}$ is bounded by

$C\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}},$

for $s>9/10$.

To continue this formula in another region, we divide into two cases again.
We can rewrite resonance function $\mathcal{H}$ as

$\displaystyle\mathcal{H}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})$
$\displaystyle=3\xi\xi_{1}\xi_{2}+\xi_{1}q^{2}-\xi
q^{2}_{1}-2\xi_{1}qq_{1}+2\xi qq_{1}$
$\displaystyle=-3\xi\xi_{1}\xi_{2}+(6\xi\xi_{1}\xi_{2}+\xi_{1}q^{2}-\xi
q^{2}_{1}-2\xi_{1}qq_{1}+2\xi qq_{1})$
$\displaystyle=:-3\xi\xi_{1}\xi_{2}+P(\xi,\xi_{1},q,q_{1}).$ (3.20)

The straightforward calculation gives us the estimation of
$P(\xi,\xi_{1},q,q_{1})$:

$\displaystyle|P(\xi,\xi_{1},q,q_{1})|\leq$
$\displaystyle|\xi_{i_{1}}|(||(\xi_{i_{1}},q_{i_{1}})|^{2}-|(\xi_{i_{0}},q_{i_{0}})|^{2}|+||(\xi_{i_{1}},q_{i_{1}})|^{2}-|(\xi_{i_{2}},q_{i_{2}})|^{2}|)$
$\displaystyle+|\xi_{i_{2}}|(||(\xi_{i_{2}},q_{i_{2}})|^{2}-|(\xi_{i_{0}},q_{i_{0}})|^{2}|+||(\xi_{i_{2}},q_{i_{2}})|^{2}-|(\xi_{i_{1}},q_{i_{1}})|^{2}|)$
$\displaystyle\leq 4|\xi_{i_{1}}|N^{\frac{6}{5}}L^{6\delta}$ (3.21)

where $\\{i_{0},i_{1},i_{2}\\}=\\{0,1,2\\}$ such that
$|\xi_{i_{0}}|\leq|\xi_{i_{1}}|\leq|\xi_{i_{2}}|$. So we get

$|\mathcal{H}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})|\geq|\xi_{i_{1}}|(3|\xi_{i_{0}}||\xi_{i_{2}}|-4N^{\frac{6}{5}}L^{6\delta}).$
$None$

We rewrite $\xi_{i_{0}}=\xi_{min}$, $\xi_{i_{1}}=\xi_{med}$, and
$\xi_{i_{2}}=\xi_{max}$. We consider two cases according to (3.22) :

$\displaystyle\rm{(v-4)}\ \ \
|\xi_{\it{max}}||\xi_{min}|\gg\it{N}^{\rm{6}/5}\it{L}^{\rm{6}\delta},$
$\displaystyle\rm{(v-5)}\ \ \
|\xi_{\it{max}}||\xi_{min}|\lesssim\it{N}^{\rm{6}/5}\it{L}^{\rm{6}\delta}.$

Case of (v-4) From the hypothesis, we note that

$\displaystyle\max\\{|\sigma|,|\sigma_{1}|,|\sigma_{2}|\\}$
$\displaystyle\gtrsim|\mathcal{H}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})|$ (3.23)
$\displaystyle\gtrsim\min\\{|\xi|,|\xi_{1}|,|\xi_{2}|\\}\cdot(\max\\{|\xi|,|\xi_{1}|,|\xi_{2}|\\})^{2}.$

We consider the case when $|\sigma|$ attains maximal of the left-hand side of
the above inequality. Using (3.23), we confirm this inequality:

$|\xi|N^{\frac{1}{2}-s}_{1}\langle\sigma\rangle^{-\frac{1}{2}+3\delta}\
\lesssim\
|\xi_{max}|^{-6\delta}N^{\frac{1}{2}-s}_{1}|\xi_{min}|^{-\frac{1}{2}+3\delta}\
\lesssim\ 1,$

for $s>1/2$. Moreover, we use calculation of resonance function:

$\left|\frac{\partial^{2}\mathcal{H}}{\partial\xi^{2}_{1}}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})\right|=6|\xi|\gtrsim
1.$ $None$

In a similar way to the case of (v-2), the contribution of this case to
$J_{HH\rightarrow H}$ is bounded by

$C\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}},$

for $s>1/2$.

Case of (v-5) Finally, we treat the case of (v-5). Applying the Fourier
multiplier operator $R_{K},R_{K_{1}},R_{K_{2}}$ to $w,u$ and $v$,
respectively, we get the contribution this case to
$J^{L,L_{1},L_{2}}_{N,N_{1},N_{2}}$ is bounded by

$C\sum_{K,K_{1},K_{2}}\frac{K}{N^{s}_{1}}L^{-\frac{1}{2}+2\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}\|R_{K}((P_{N_{1}}Q_{L_{1}}R_{K_{1}}u)(P_{N_{2}}Q_{L_{2}}R_{K_{2}}v))\|_{L^{2}}\|P_{N}Q_{L}R_{K}w\|_{L^{2}}.$
$None$

Using the hypothesis and, we consider dyadic numbers of (3.25):

$\frac{K}{N^{s}_{1}}L^{-\frac{1}{2}+2\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2}\lesssim
KN^{-s}(K^{-\frac{3}{4}}_{min}K^{-\frac{3}{4}}_{max}(N^{\frac{5}{6}}L^{6\delta})^{\frac{3}{4}})L^{-\frac{1}{2}+2\delta}L^{-\frac{1}{2}-\delta}_{1}L^{-\frac{1}{2}-\delta}_{2},$
$None$

where $K_{max}\sim|\xi_{max}|$ and $K_{min}\sim|\xi_{min}|$. Notice that
$(K^{-3/4}_{min}K^{-3/4}_{max}(N^{6/5}L^{6\delta})^{3/4})\gtrsim 1$ and

$\left|\frac{\partial^{2}\mathcal{H}}{\partial\xi^{2}_{1}}(\xi_{1},\xi-\xi_{1},q_{1},q-q_{1})\right|=6|\xi|\sim
K.$

So we use Lemma 2.9 to obtain

$\displaystyle\|R_{K}((P_{N_{1}}Q_{L_{1}}R_{K_{1}}u)(P_{N_{2}}Q_{L_{2}}R_{K_{2}}v))\|_{L^{2}}$
(3.27) $\displaystyle\lesssim(K_{1}\wedge K_{2})^{\frac{1}{2}}(L_{1}\wedge
L_{2})^{\frac{1}{2}}K^{-\frac{1}{4}}(L_{1}\vee
L_{2})^{\frac{1}{4}}\|P_{N_{1}}Q_{L_{1}}R_{K_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}R_{K_{2}}v\|_{L^{2}}.$

Since $K(K_{1}\wedge
K_{2})^{\frac{1}{2}}K^{-\frac{1}{4}}K^{-\frac{3}{4}}_{min}K^{-\frac{3}{4}}_{max}\lesssim
K^{-\frac{1}{4}}_{min}$, we deduce that (3.25) is bounded by

$\displaystyle
C\sum_{K,K_{1},K_{2}}K^{-\frac{1}{4}}_{min}L^{-\frac{1}{2}+\frac{11}{2}\delta}L^{-\delta}_{1}L^{-\delta}_{2}(N^{-\frac{5}{6}})^{\frac{3}{4}}N^{-s}\
\times$
$\displaystyle\|P_{N_{1}}Q_{L_{1}}R_{K_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}R_{K_{2}}v\|_{L^{2}}\|P_{N}Q_{L}R_{K}w\|_{L^{2}}.$
(3.28)

Notice that $N^{-s}_{1}(N^{6/5})^{3/4}\sim N^{9/10-s}$. Therefore, when
$s>9/10$, we can confirm that (3.28) is bounded by

$CL^{-\frac{1}{2}+\frac{11}{2}\delta}L^{-\delta}_{1}L^{-\delta}_{2}N^{\frac{9}{10}-s}\|P_{N_{1}}Q_{L_{1}}u\|_{L^{2}}\|P_{N_{2}}Q_{L_{2}}v\|_{L^{2}}\|P_{N}Q_{L}w\|_{L^{2}}.$

Summing over $N_{1},N_{2},N,L_{1},L_{2},L$, we have that the contribution of
the case is bounded by $C\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}}$. So we
obtain

$J_{HH\rightarrow H}\lesssim\|u\|_{L^{2}}\|v\|_{L^{2}}\|w\|_{L^{2}},$

for $s>9/10$, which complete the proof of Proposition 3.1. $\Box$

Proof of Theorem 1.1. Let $s>9/10$, $a>0$, $0<\delta\ll 1$ and $0<T<1$. We
define

$Z_{a,T}:=\\{u\in
X^{s,\frac{1}{2}+\delta}_{T}:\|u\|_{X^{s,\frac{1}{2}+\delta}_{T}}<a\\},$

and the operator $\Phi_{u_{0}}(u)$ as follows:

$\Phi_{u_{0}}(u):=\eta(t)e^{-t\partial_{x}\Delta}u_{0}+\eta(t)\int^{t}_{0}e^{-(t-t^{\prime})\partial_{x}\Delta}\partial_{x}(u^{2}(t^{\prime}))dt^{\prime}.$

We shall show that $\Phi_{u_{0}}$ is a contraction on $Z_{a,T}$, where each
$T<T_{0}$ for some constance $0<T_{0}<1$. Using Lemma 2.3, Lemma 2.4, Lemma
2.5 and Proposition 3.1, it holds

$\displaystyle\|\Phi_{u_{0}}(u)\|_{X^{s,\frac{1}{2}+\delta}_{T}}$
$\displaystyle\leq
c\|u_{0}\|_{H^{s}}+c\|\eta(t)\partial_{x}u^{2}(\cdot,\cdot,t)\|_{X^{s,-\frac{1}{2}+\frac{\delta}{2}}_{T}}$
$\displaystyle\leq
c\|u_{0}\|_{H^{s}}+cT^{\delta}\|\partial_{x}u^{2}(\cdot,\cdot,t)\|_{X^{s,-\frac{1}{2}+\delta}_{T}}$
$\displaystyle\leq
c\|u_{0}\|_{H^{s}}+cT^{\delta}\|u\|^{2}_{X^{s,\frac{1}{2}+\delta}_{T}}$
$\displaystyle\leq c\|u_{0}\|_{H^{s}}+cT^{\delta}a^{2}.$ (3.29)

Setting $c\|u_{0}\|_{H^{s}}=a/2$ and

$cT^{\delta}a<\frac{1}{3},$

we have $\|\Phi_{u_{0}}(u)\|_{X^{s,\frac{1}{2}}_{T}}<a$, which concludes that
$\Phi_{u_{0}}:Z_{a,T}\rightarrow Z_{a,T}$.
Now we consider any $u,u^{\prime}\in Z_{a,T}$. Using Lemma 2.3, Lemma 2.4,
Lemma 2.5, Proposition 3.1 and (3.29),

$\displaystyle\|\Phi_{u_{0}}(u)-\Phi_{u_{0}}(u^{\prime})\|_{X^{s,\frac{1}{2}+\delta}}$
$\displaystyle=\|\eta(t)\int^{t}_{0}e^{-(t-t^{\prime})\partial_{x}\Delta}(u^{2}-u^{\prime
2})(t^{\prime})dt^{\prime}\|_{X^{s,\frac{1}{2}+\delta}}$ $\displaystyle\leq
cT^{\delta}a\|u+u^{\prime}\|_{X^{s,\frac{1}{2}+\delta}_{T}}\|u-u^{\prime}\|_{X^{s,\frac{1}{2}+\delta}_{T}}$
$\displaystyle\leq
2cT^{\delta}a\|u-u^{\prime}\|_{X^{s,\frac{1}{2}+\delta}_{T}}<\frac{2}{3}\|u-u^{\prime}\|_{X^{s,\frac{1}{2}+\delta}},$
(3.30)

which implies that $\Phi_{u_{0}}$ is a contraction map from $Z_{a,T}$ into
itself. So we obtain a unique solution $u=\Phi_{u_{0}}(u)$ in $Z_{a,T}$, i.e.,

$u(t)=\eta(t)e^{-t\partial_{x}\Delta}+\eta(t)\int^{t}_{0}e^{-(t-t^{\prime})\partial_{x}\Delta}\partial_{x}(u^{2}(t^{\prime}))dt^{\prime}.$
$None$

Finally, we will prove $u\in C([0,T]:H^{s}(\mathbb{R}\times\mathbb{T}))$. Note
that

$\|u\|_{X^{s,b}}=\|e^{-t\partial_{x}\Delta}u\|_{H^{b}_{t}H^{s}_{xy}.}$

By using the Sobolev embedding theorem $H^{1/2+}(\mathbb{R})\hookrightarrow
C(\mathbb{R})$ and the fact that $e^{-t\partial_{x}\Delta}$ is lead to
continuous in $L^{\infty}([0,T]:H^{s})$, we get

$u\in C([0,T]:H^{s}(\mathbb{R}\times\mathbb{T})).$

Here we complete the proof of Theorem 1.1. $\Box$

## Acknowledgement

The author would like to express the deepest appreciation to Professor Hideo
Takaoka for his kind support and reassuring guidance. Without his help, this
paper would not have been possible. In addition, the author want to thank
Naoya Takahashi and Rika Takiguchi for their contribution to the discussion.

## References

  * [1] L. Molinet, J. C. Saut, N. Tzvetkov,  Global well-posedness for the KP-II equation on the background of a non-localized solution,  Ann. Inst. H. Poincaré, Anal. Non Linéaire, 28 (2011), 653–676.
  * [2] L. Molinet, D. Pilod,  Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications,  Ann. Inst. H. Poincaré, Anal. Non Lineairé, 32 (2015), 347–371.
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  * [7] C. E. Kenig, G. Ponce, L. Vega,  A bilinear estimate with applications to the KdV equation,  J. Amer. Math. Soc, 9 (1996), 573–603.
  * [8] E. A. Zalharov, V.E.Kuznetsov,  On three dimensional solitons,  Sov. Phys. JEPT, 39 (1974), 285–286.
  * [9] F. Linares, G. Ponce,  Introduction to nonlinear dispersive equations,  Springer, (2009).
  * [10] S. Kinoshita,  Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D,  arXiv:1905.01490v1 [math.AP].
  * [11] S. Kinoshita, R. Schippa,  Loomis-Whitney-type inequalities and low regularity well-posedness of the periodic Zakharov-Kuznetsov equation,  arXiv:2003.02694v1 [math.AP].
  * [12] F. Linares, M. Panthee, T. Robert, and N. Tzvetkov On the periodic Zakharov-Kuznetsov equation,  arXiv:1809.02027v1 [math.AP].
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E-mail adress, Satoshi Osawa<EMAIL_ADDRESS>


# Frequency limits of sequential readout for sensing AC magnetic fields using
nitrogen-vacancy centers in diamond

Santosh Ghimire  Korea Research Institute of Standards and Science, Daejeon
34113, Republic of Korea  Seong-Joo Lee  Korea Research Institute of Standards
and Science, Daejeon 34113, Republic of Korea  Sangwon Oh  Korea Research
Institute of Standards and Science, Daejeon 34113, Republic of Korea  Jeong
Hyun Shim<EMAIL_ADDRESS>Korea Research Institute of Standards and
Science, Daejeon 34113, Republic of Korea  Department of Applied Measurement
Science, University of Science and Technology, Daejeon 34113, Republic of
Korea

###### Abstract

The nitrogen-vacancy (NV) centers in diamond have ability to sense
alternating-current (AC) magnetic fields with high spatial resolution.
However, the frequency range of AC sensing protocols based on dynamical
decoupling (DD) sequences has not been thoroughly explored experimentally. In
this work, we aimed to determine the sensitivity of ac magnetic field as a
function of frequency using sequential readout method. The upper limit at high
frequency is clearly determined by Rabi frequency, in line with the expected
effect of finite DD-pulse width. In contrast, the lower frequency limit is
primarily governed by the duration of optical repolarization rather than the
decoherence time (T2) of NV spins. This becomes particularly crucial when the
repetition (dwell) time of the sequential readout is fixed to maintain the
acquisition bandwidth. The equation we provide successfully describes the
tendency in the frequency dependence. In addition, at the near-optimal
frequency of 1 MHz, we reached a maximum sensitivity of 229
pT/$\sqrt{\mathrm{Hz}}$ by employing the XY4-(4) DD sequence.

##  I Introduction

Sensing alternating-current (AC) magnetic fields in the range of kilohertz to
megahertz has been used in various applications, including nuclear magnetic
resonance (NMR),[1] magnetic induction tomography,[2] and magnetic
communications.[3] Solid-state quantum spins, such as negatively charged
nitrogen-vacancy (NV) centers in diamond[4, 5, 6], are among the promising
sensors for ac magnetic fields due to their high spatial resolution, which
ranges from micro to nanometer.[4, 5, 7, 8, 9, 10, 11, 12, 6, 13, 14, 15] The
controllable density and depth of the quantum spins near the surface of host
materials provide a reduced standoff distance from sources, resulting in such
high resolution. In addition, the protocols for sensing ac magnetic fields
allow control over both sensible frequency and bandwidth.[16] Moreover, the
bandwidth that NV spins provide is often significantly wider than that of
radio-frequency optically pumped magnetometers.[17] The boardband sensing
using solid-state spin sensors enables the high-fidelity reception of rapidly
varying or modulated ac signals.

AC-field sensing protocols called dynamical decoupling (DD) exploit a multiple
number of phase-refocusing $\pi$ pulses.[18, 16, 19] The stroboscopic pulse
train periodically flips the quantum states and hence filters out a specific
frequency, reducing the frequency window and detuning the quantum spins from
surrounding magnetic noises. This increases the coherence time T2 of quantum
spins [20, 21, 22] and thus enhances sensitivity to AC magnetic fields.[5, 23,
24] However, when detecting an oscillating signal with a time-varying or
modulated envelope, such as nuclear spin precessions in NMR, the signal often
persists longer than the T2 duration. To address this issue, sequential
readout (SR) scheme or quantum heterodyne (Qdyne) has been developed, based on
the repetition of a block consisting of a DD sequence and projective readout.
The SR method works because the phase of the AC magnetic field continues even
though the quantum states collapse during the projective readout.[25, 26, 27]
In principle, the spectral resolution achievable with SR is limited only by
the stability of external clocks synchronizing instrumentations, regardless of
T2.[26] This feature enabled sensing of an AC magnetic field with sub-
millihertz resolution[27] and acquiring of high-resolution NMR spectra from
micron-scale liquid samples.[1, 28, 29]

The range of frequency to which the SR method is sensitive is certainly
crucial for its applications in the field. As the sensing frequency increases,
the time interval $\tau$ in Fig. 1 (a) decreases, and eventually the
interrogation window closes. Considering the finite width of the $\pi$ pulses,
one can expect Rabi-oscillation frequency to be the high-frequency limit. The
low-frequency limit may be determined by decoherence process, as the maximum
$\tau$ will be imposed by the value of T2. These expectations, however, have
not been experimentally investigated yet. In this study, we investigate the
frequency dependence of AC-field sensitivity using an ensemble of NV centers
in diamond. By accounting for the finite widths of $\pi$ pulses and the
duration of optical readout/repolarization, we successfully explain the
obtained AC-field sensitivity.

Figure 1: (a) The sequence of XY4-(k) in the presence of an AC magnetic field
commensurate with the $\pi$ pulses. (b) The normalized weighing function
$\overline{W}$ (see the main text) is plotted as a function of $\alpha$
(=$\tau_{\pi}/\tau).$ (c) The bandwidth curve of the AC sensing employing
XY4-(4) is shown at the center frequency ($f_{ac}^{(0)}$) of 1 MHz. The
theoretical curve fitted with only a single parameter to match the dip height
is shown in orange.

##  II Results and discussion

###  II.1 The effect of finite-width $\pi$ pulses

In the following two subsections, we provide readers with a concise reminder
of how the finite width of $\pi$ pulses influences the conditions of AC
sensing protocols. Furthermore, we highlight the key features of the SR
method, which will be exploited to interpret the frequency dependance of the
sensitivity. For an oscillating field $B(t)=B_{ac}\cos(2\pi
f_{ac}t+\phi_{ac})$ to be commensurate with the $\pi$ pulses, as illustrated
in Fig. 1 (a), the frequency $f_{ac}$ should satisfy the condition below[16,
30]

$f_{ac}=\frac{1}{2\tau(1+\alpha)},$ (1)

where $\alpha=\tau_{\pi}/\tau$ given $\tau$ and $\tau_{\pi}$ as illustrated in
Fig. 1(a). For Carl-Purcell type sequences, the accumulated phase $\Phi$ can
be written as $\Phi$ = $\gamma B_{ac}N\tau(1+\alpha)W$, in which $N$ is the
total number of the $\pi$ pulses. According to Ref. [30], the weighting
function $W$ is given as

$W(\alpha,\beta,\phi_{ac})=\frac{\sin(N\beta)}{N\beta}\left[1-\frac{\cos(\beta\frac{\alpha}{1+\alpha})}{\cos(\beta)}\right]\cos(N\beta+\phi_{ac}),$
(2)

where $\beta$ is defined as $\pi f_{ac}\tau(1+\alpha)$. If the frequency
$f_{ac}$ satisfies the condition of Eq. (1), $\beta=\pi/2$. Then, $\alpha$ and
the phase offset $\phi_{ac}$ crucially affect the accumulated phase $\Phi$. In
Fig. 1(b), the normalized weighting function $\overline{W}(\alpha)$ (=
$\frac{\pi}{2}W(\alpha,\phi_{ac}=0)$) is depicted as a function of $\alpha$.
If $\alpha<0.2$, the reduction of $\overline{W}(\alpha)$ is less than 4 $\%$.
As $\alpha$ further increases, $\overline{W}(\alpha)$ shows a significant
decrease, which implies that the DD sequence becomes highly insensitive to AC
magnetic fields. In Eq. (2), $\frac{\alpha}{1+\alpha}$ can be substituted by
$\frac{f_{ac}}{f_{\mathrm{Rabi}}}$, in which Rabi frequency
$f_{\mathrm{Rabi}}$ is defined as $\frac{1}{2\tau_{\pi}}$. As $f_{ac}$
approaches to $f_{\mathrm{Rabi}}$, $\alpha$ becomes infinitely high. This
explains that the AC-field sensing protocol is not able to detect the
frequencies higher than Rabi frequency ($f_{\mathrm{Rabi}}$).

Figure 2: The principle of sequential readout is illustrated. The top exhibits
the unit block and its duration is $\mathrm{T_{SR}}$. The unit block includes
DD (XY4-(4)) sequence and optical readout/repolarization. If $\mathrm{T_{SR}}$
is commensurate with the period of $B_{ac}(t)$, a constant signal S0 (t) will
be observed. If not, an under-sampled signal Ssens (t) is acquired and its
frequency is given by Eq. (3).

###  II.2 Sequential readout method

The principle of the SR method is illustrated in Fig.2. The unit block of SR
consists of a DD sequence for sensing AC fields and laser pulse for optical
readout and repolarization. The total duration $\mathrm{T_{SR}}$ splits into
the interrogation time, $\tau_{sens}=N\tau(1+\alpha)$, and the rest $\delta$.
The duration of $\delta$ includes the laser pulse, $\mathrm{T_{Laser}}$ and a
time margin that is practically inevitable. In general $\delta$ may not be
commensurate with $B_{ac}(t)$ as the case in Fig.2. According to Eq. (2), the
weighting function depends on the phase $\phi_{ac}$ of the ac field
$B_{ac}(t)$. At the $i$th sampling time ti (=$i\mathrm{T_{SR}}$), the starting
phase can be expressed as $2\pi f_{ac}i\delta$, which is illustrated by the
increasing areas in light orange color in Fig. 2. Because the SR method
entails an undersampling of $B_{ac}(t)$, the obtained signal S${}_{sens}(t)$
has a down-converted frequency $f_{sens}$ given as

$f_{sens}=f_{ac}-\overline{n}f_{\mathrm{SR}},$ (3)

in which the integer
$\overline{n}=\mathrm{argmin}_{n}\left(|f_{ac}-nf_{\mathrm{SR}}|\right)$, and
the sampling frequency $f_{\mathrm{SR}}=1/\mathrm{T_{SR}}$ (see the
supplementary material for derivation). Eq. (3) implies that the measured
frequency $f_{sens}$ is determined by the sampling frequency
$\mathrm{f_{SR}}$, irrespective of DD sequence. But, the frequency $f_{ac}$
should be within the sensible bandwidth ($f_{\mathrm{BW}}$ in Fig. 1 (c)).
Then, the amplitude of the measured signal, $A_{sens}$, can be expressed as

$A_{sens}=S\cdot C\cdot\sin\left(\Phi_{(\phi_{ac}=0)}\right).$ (4)

$S$ is the intensity of the optical signal acquired during
$\mathrm{T_{Laser}}$, and $C$ is the contrast induced by spin-manipulating
pulses within the DD sequence. The term of $\sin(\Phi)$ appears when the
readout pulse of the DD sequence is $\frac{\pi}{2}(Y)$ as shown in Fig.2. For
a weak AC magnetic field, $\sin(\Phi)$ can be approximated as $\Phi$.

Figure 3: (a) Fourier transform of the time-domain signal Ssens (t) obtained
by the SR method. The rms noise floor is calibrated from the intensity of the
applied AC field, 0.3211 $\mu$T (rms), and the sensitivity is 229
pT/$\sqrt{\mathrm{Hz}}$. (b) The time-domain data of the oscillating signal of
NV fluorescence measured for 1 s, and (c) its zoomed view for 5 ms.

###  II.3 Bandwidth

We experimentally determined the key features of NV AC-magnetometry using the
XY4-(4) sequence. We choose this sequence because it exhibited the highest
slope in the variation of the contrast as a function of AC-field amplitude
(see the section 4 of the supplementary material). Since the readout pulse is
$\frac{\pi}{2}(X)$, the output signal is proportional to $\cos(\Phi)$, and the
formation of a dip happens when the frequency of external AC field satisfies
Eq. (1). In the presence of 1 MHz AC field with unknown intensities, we
scanned $\tau$ and found the dips at theoretically expected values (see the
section 3 of the supplementary material). For the main peak, $\tau$ should be
420 ns because $\tau_{\pi}$ = 80 ns in our experiment. With such $\tau$ and
$\tau_{\pi}$ ($\alpha=0.19$), the intensity of the external 1MHz AC field was
calibrated. As the amplitude $B_{ac}$ of the AC field increases, the contrast
of NV magnetometry follows a periodic oscillation as $\cos(\Phi)$ curve. The
first minimum point corresponds to the phase accumulation of $\pi$. From the
equation of $\Phi=\pi$, the amplitude $B_{ac}(\pi)$ of AC magnetic signal for
the $\pi$ phase accumulation can be obtained as below

$B_{ac}(\pi)=\frac{\pi^{2}f_{ac}}{\gamma N\overline{W}(\alpha)}.$ (5)

The numerically calculated value of $\overline{W}(0.19)$ (= 0.968) was used
for our calibration.

In order to measure the bandwidth of the XY4-(4) sequence, we swept the
frequency of AC field across 1 MHz. The obtained data was compared with the
theoretical prediction $\cos(\Phi(f_{ac}))$, where $\Phi(f_{ac})=\frac{\gamma
NB_{ac}}{\pi f_{ac}^{(0)}}\overline{W}(f_{ac})$. In Eq. (2),
$\beta=\frac{\pi}{2}f_{ac}/f_{ac}^{(0)}$, $f_{ac}^{(0)}$ = 1 MHz, $N$ = 16,
and $\phi_{ac}$ = 0. The intensity of an ac field we applied is 1.9 $\mu$T.
The orange line in Fig.1 (c) is the result of the fitting, where we only used
a single parameter to match the height of the dip. The obtained data shows a
good agreement with the theoretical prediction (orange), and the bandwidth
$f_{BW}$ is found out to be approximately 50 kHz.

###  II.4 Sensitivity

Figure 4: The frequency dependence of AC sensitivity based on the SR method is
investigated. (a) Theoretical prediction according to Eq. (6) is plotted in
orange color. The red dashed line on the left side represents Rabi frequency
limit (6.3 MHz). The lower limit is shown in orange dashed line (160 kHz). The
blue dashed line (82 kHz) indicates T2 limit, which is obtained by omitting
$S(\mathrm{T_{Laser}})$ from Eq. (6). (b) Experimentally obtained AC
sensitivity as a function of frequency is in accordance with theoretical
prediction.

To measure the sensitivity of AC magnetic fields, we used the SR method, of
which main principle is described above. For a period of 1 s, we acquired the
oscillating signal, $S_{sens}(t)$. In the presence of a reference AC-field
having a calibrated rms intensity, the sensitivity can be estimated from the
signal to noise ratio (SNR) after Fourier transformation. In the experiment,
we used the XY4-(4) sequence to measure the sensitivity. $\mathrm{T_{SR}}$ is
set to 50 $\mu$s and repeated 20,000 times. The $\tau$ value is adjusted for 1
MHz. But, the actual frequency of the reference AC-field was slightly detuned
by 4 kHz ($f_{ac}$ = 996 kHz) to avoid being commensurate with the sampling
rate $f_{\mathrm{SR}}$. The detuning causes insignificant deterioration of the
obtained signal, because 4 kHz is far less than the bandwidth fBW ( = 50 KHz).
The noise floor, which is indicated by the red dashed line below in Fig. 3(a),
can be converted to detectable magnetic field $\delta B$ by multiplying the
ratio between the rms intensity of the applied magnetic field (0.3211 $\mu$T)
and the peak intensity at 4 kHz in the Fourier transformed spectrum (0.3211
$\mu$T corresponds to 34 mVPP shown in Fig. S4 in the supplementary material).
Because the total measurement time corresponds to 1 s, the sensitivity $\eta$
can be obtained as $\eta$ = $\delta B$/$\sqrt{\mathrm{Hz}}$. As shown in Fig.
3 (a), we obtained the sensitivity of 229 pT/$\sqrt{\mathrm{Hz}}$. The main
peak appears at 4 kHz, which is consistent with Eq. (3). The 2nd harmonic peak
is observed due to the non-linearity in the response to the amplitude of AC
field (refer to Fig. S4 in the supplementary material). Fig.3 (b) shows the
time series data of oscillating signal of NV fluorescence measure for 1 s, and
Fig. 3(c) is its zoomed view for 5 ms.

###  II.5 Frequency dependence of sensitivity

We measured the frequency dependence of the sensitivity obtained by the SR
method. If the dwell time $\mathrm{T_{SR}}$ is fixed, the sensitivity in Fig.
3 will be proportional to the amplitude $A_{sens}$ of the time-domain signal
in Eq. (4). In varying the sensing frequency, $\tau$ should be adjusted to
meet the condition of Eq. (1) for each $f_{ac}$ and thereby $\beta=\pi/2$ in
Eq. (2). For producing AC magnetic fields in experiment, we used a single turn
coil, of inductance is not negligible. Knowing the AC field intensity at 1 MHz
attained from the calibration procedure in the sec. 2.3, we obtained the
actual AC field intensities at other frequencies by comparing the current
values (see the current variation in the supplementary material). For
theoretical prediction, the parameters $S$, $C$, and $\Phi$ in Eq. (4) need to
be expressed in terms of $f_{ac}$. $\overline{W}$ is relatively simple because
$\frac{\alpha}{1+\alpha}=\frac{f_{ac}}{f_{\mathrm{Rabi}}}$ in Eq. (2) and
$\phi_{ac}$ can be set $0$. The contrast $C$ in Eq. (4) decays as a function
of $\tau$ due to decoherence process like
$C(\tau/T_{2})=C_{0}\exp(-(N\tau/T_{2})^{p})$, in which $p$ and $T_{2}$ are
estimated from decoherence curve fitting (see the supplementary material for
details). $\tau$ varies with $f_{ac}$ following Eq. (1) as $\tau$ =
$1/2(1/f_{ac}-1/f_{\mathrm{Rabi}})$. Fig. 5(a) shows that the signal intensity
$S$ increases with $\mathrm{T_{Laser}}$, because the degree of NV spin
polarization to $m_{S}=0$ state becomes higher. This repolarization process
can be expressed as
$S(\mathrm{T_{Laser}}/T_{p})=S_{0}(1-\exp(-\mathrm{T_{Laser}}/T_{p}))$ with
the polarization time constant $T_{p}$. The numerical fitting found $T_{p}$ to
be 15 $\mu s$. Finally, $\mathrm{T_{Laser}}$ is related with $f_{ac}$ through
$\mathrm{T_{SR}}$. The relationship is described by $\mathrm{T_{Laser}}$ =
$\mathrm{T_{SR}}$ \- $N\tau(1+\alpha)$ = $\mathrm{T_{SR}}$ \- $N/2f_{ac}$.
Including them altogether, we can describe how the sensitivity varies as a
function of $f_{ac}$,

$\eta(f_{ac})\propto\frac{1}{S(\mathrm{T_{Laser}}/T_{p})C(\tau/T_{2})\Phi(f_{ac})}\sqrt{\frac{\mathrm{T_{SR}}}{N\tau(1+\alpha)}}.$
(6)

The first terms originate from the inverse of the amplitude in Eq. (4), and
the last term represent the effect of the overhead time.[22, 24] The
theoretical prediction is shown as orange curve in Fig. 4 (a). The
experimental data in Fig.4 (b) shows a good agreement with it. The high-
frequency limit is clearly determined by the Rabi frequency, being
approximately 6.3 MHz in the present study, which is indicated by the red
dashed line. The low-frequency limit is close to 200 kHz (yellow dashed line).
We found that the effect of decoherence causing the decay of the contrast
$C(\tau/T_{2})$ alone cannot explain it. The blue curve in Fig. 4 (a) is
obtained by omitting $S(\mathrm{T_{Laser}}/T_{p})$ from Eq. (6) and its low-
frequency limit is below 100 kHz, indicated by blue dashed line. This
discrepancy originates from the reduction of optical signal intensity $S$ when
the duration of TLaser decreases (Fig. 5 (a)). Given TSR, lowering $f_{ac}$
accompanies decreasing TLaser as seen by the fitting equation
$\mathrm{T_{Laser}}$ = $\mathrm{T_{SR}}$ \- $2N/f_{ac}$ in Fig. 5 (b). The
reduced signal intensity results in the deterioration of the sensitivity. In
addition, similar to previous works[22, 24], the theoretical prediction
described by Eq. (6) has the optimal frequency, which is found to be within
the range from 800 kHz to 1 MHz. We achieved a sensitivity of 229
pT/$\sqrt{\mathrm{Hz}}$ at 1 MHz (Fig. 3(a)).

Figure 5: (a) NV fluorescence intensity is measured as a function of optical
readout duration. The repolarization time $t_{p}$ is estimated to be 15 $\mu$s
from exponential fitting. (b) The duration TLaser of optical readout and
repolarization varies according to the sensing frequency because the dwell
time TSR is fixed. The values of TLaser used in a series of measurement nearly
follow the relation, $\mathrm{T_{Laser}}$ = $\mathrm{T_{SR}}$ \- $N/2f_{ac}$.

In Eq. (6), the term $S(\mathrm{T_{Laser}}/T_{p})$ sets it apart from earlier
theoretical predictions[22, 24]. Governed by the ratio
$\mathrm{T_{Laser}}/T_{p}$, mitigating the impact of
$S(\mathrm{T_{Laser}}/T_{p})$ and thus reaching the lower frequency limit
determined by $\mathrm{T_{2}}$ can be accomplished through two approaches. The
first method is associated with the optical power density irradiated to NV
centers. The power density for optical readout/repolarization should be high
enough to sufficiently polarize NV spins within $\mathrm{T_{Laser}}$. With a
laser spot diameter of 0.2 mm and laser power of 1.4 W in this work, the power
density utilized is approximately 45 W/mm2. However, this value still falls
significantly short of the saturation density of NV spins ($>$ 5 kW/mm2) as
estimated from single NV experiments.[14] Given the considerably high
saturation power density for NV spins, practical challenges might arise in
achieving complete optical repolarization during the $\mathrm{T_{Laser}}$ time
with an ensemble of NV centers. The second method involves increasing
$\mathrm{T_{Laser}}$. This leads to the increase in the dwell time
$\mathrm{T_{SR}}$ of the SR method. Essentially, the SR method emulates the
acquisition of a highly under-sampled signal as shown in Fig. (2). If a
sinusoidal signal in the time domain is subjected to a certain amount of
Gaussian noise, the noise floor of the under-sampled signal decreases at a
higher sampling rate. This effect stems from the fact that higher sampling
rates distribute the noise across a wider frequency. Hence, during the
measurements for Fig. 4 (b), we maintained a fixed value for $\mathrm{T_{SR}}$
to prevent variations in the noise floor due to changes in bandwidth. When
employing sequential readout, the selection of $\mathrm{T_{SR}}$ is imperative
in light of the desired bandwidth. What Eq. (6) implies is that the
consideration of $\mathrm{T_{Laser}}$ and $T_{p}$ plays an essential role in
anticipating the lower frequency limit.

##  III Conclusion

In this work, we present a combination of experimental and theoretical
investigations into the frequency dependent behavior of AC magnetic field
sensitivity through the use of sequential readout. Our findings unveil that
the upper limit is determined by the finite-width of $\pi$ pulses, while the
lower limit can be governed by the duration of optical repolarization for NV
ensembles when the optical power density is often below the saturation limit.
With the dwell time of the SR method held constant, our theoretical model,
which encompasses these two factors, demonstrates a reasonable concordance
with the experiment results. Furthermore, through the employment of the
XY4-(4) DD sequence, we obtained a maximum sensitivity of 229
pT/$\sqrt{\mathrm{Hz}}$ at 1 MHz. Considering the sensitivity reported in the
previous study[1], ranging from 30 to 70 pT/$\sqrt{\mathrm{Hz}}$, we hold the
conviction that the achieved sensitivity empowers us to conduct NV-NMR in
micron-scale, where (thermal) Boltzmann polarization remains predominant.[31]
This work not only offers valuable insight but also furnishes practical
guidelines for the applications of AC magnetometry using NV spins in the range
of kilohertz to megahertz. These applications may include other areas, such as
magnetic induction tomography, in which NV spins are expected to facilitate
high spatial resolution imaging.

## acknowledgments

This research was supported by a grant (GP2022-0010) from Korea Research
Institute of Standards and Science, and Institute of Information and
communications Technology Planning $\&$ Evaluation (IITP) grant funded by the
Korea government (MSIT) (No.2019-000296).

##  Appendix A Experimental methods

###  A.1 optical and microwave system

A static magnetic field of 5.4 mT is aligned along one of the four NV
orientations in diamond. A continuous-wave laser (532 nm) is used to excite
the NV spins. The laser passes through an acousto-optic modulator (AOM). An
arbitrary waveform generator is used to store the dynamical decoupling (DD)
sequences, and a signal generator to produce the microwave frequency. After
mixing these two signals by frequency mixer, 100 watt amplifier amplifies the
signal and delivered to the diamond sample through 2 mm microwave loop below
the sample. Timing signals are generated from a programmable timing generator
and control the outputs of AWG, AOM and microwave through the RF switches. The
fluorescence from the diamond NV centers is collected with a parabolic
concentrator and delivered to the photodiode. A long pass filter is placed
between the concentrator and photodiode, allowing only NV fluorescence to pass
through. The collected signal from the photodiode is amplified by a current
amplifier and then recorded by digital-to-analog converter. Further details of
experimental method are described in the supplementary material.

###  A.2 Diamond sensor

A thin nitrogen-doped ([14N] $\simeq$ 10 ppm) diamond layer (12C $>$
99.99$\%$, 40 $\mu$m thick) is grown by chemical vapor deposition (CVD) on top
of an electronic grade diamond plate by Applied Diamond. The dimensions of the
diamond plate are approximately 2 × 2 × 0.54 mm3 . The diamond is electron
irradiated (1 MeV, 1 × 1019/cm2 ), and annealed in a vacuum at 800 ∘C for 4
hours and 1000 ∘C for 2 hours, sequentially.

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