intensity there are two extra contributions in the lower left-hand and upper
right-hand corners. Using simple geometry one finds that each corner adds a
duration of size $T_{p}/4$ to the horizontal width for a combined width of
$\mathcal{T}_{p}^{\mathrm{SH}}=T_{p}+T_{c}/2$. A similar argument follows for
the joint spectral intensity leading to a combined width of
$2\pi\mathcal{B}_{c}^{\mathrm{SH}}=\Omega_{c}+\Omega_{p}/2$.

Figure 26: Schematic of the sinc-hat joint intensities showing the extra
contributions to the horizontal widths in the respective corners.

##  Appendix C Joint temporal amplitude scaling

Consider the joint temporal amplitude schematically shown in Fig. 1,
characterized by $\mathcal{T}_{p}$ and $\mathcal{T}_{c}$. To show how the
maximum value of the joint temporal amplitude scales with $\mathcal{T}_{p}$
and $\mathcal{T}_{c}$ we consider a change of variables

$\tilde{t}_{1}=\frac{1}{\sqrt{2}}\frac{t_{1}+t_{2}}{\mathcal{T}_{p}},\hskip
14.22636pt\tilde{t}_{2}=\frac{1}{\sqrt{2}}\frac{t_{1}-t_{2}}{\mathcal{T}_{c}},\hskip
14.22636ptt_{1}=\frac{1}{\sqrt{2}}(\tilde{t}_{1}\mathcal{T}_{p}+\tilde{t}_{2}\mathcal{T}_{c}),\hskip
14.22636ptt_{2}=\frac{1}{\sqrt{2}}(\tilde{t}_{1}\mathcal{T}_{p}-\tilde{t}_{2}\mathcal{T}_{c}),$
(C.1)

which are aligned with the long and short axis of the joint temporal
amplitude, see Fig. 27. The new variables $\tilde{t}_{1},\tilde{t}_{2}$ are
normalized and dimensionless; as they vary over the range
$\tilde{t}_{1},\tilde{t}_{2}\in[-1/\sqrt{2},1/\sqrt{2}]$, $t_{1},t_{2}$ vary
over the range of the joint amplitude specified by $\mathcal{T}_{p}$ and
$\mathcal{T}_{c}$. The Jacobian of this coordinate transformation is
$\text{det}(J)=\mathcal{T}_{p}\mathcal{T}_{c}$. Then setting

$\tilde{\gamma}(\tilde{t}_{1},\tilde{t}_{2})=\sqrt{\mathcal{T}_{p}\mathcal{T}_{c}}\overline{\gamma}\left(\frac{1}{\sqrt{2}}(\tilde{t}_{1}\mathcal{T}_{p}+\tilde{t}_{2}\mathcal{T}_{c}),\frac{1}{\sqrt{2}}(\tilde{t}_{1}\mathcal{T}_{p}-\tilde{t}_{2}\mathcal{T}_{c})\right)=\sqrt{\mathcal{T}_{p}\mathcal{T}_{c}}\overline{\gamma}\left(t_{1},t_{2}\right),$
(C.2)

$\tilde{\gamma}(\tilde{t}_{1},\tilde{t}_{2})$ is a normalized and
dimensionless joint amplitude that satisfies

$\int dt_{1}dt_{2}|\overline{\gamma}(t_{1},t_{2})|^{2}=\int
d\tilde{t}_{1}d\tilde{t}_{2}|\tilde{\gamma}(\tilde{t}_{1},\tilde{t}_{2})|^{2}=1.$
(C.3)

Since the rotated coordinates vary roughly over
$\tilde{t}_{1},\tilde{t}_{2}\in[-1/\sqrt{2},1/\sqrt{2}]$ and
$|\tilde{\gamma}(\tilde{t}_{1},\tilde{t}_{2})|$ is always positive, we can
infer that the approximate maximum of
$|\tilde{\gamma}(\tilde{t}_{1},\tilde{t}_{2})|$ is on the order of one. Then
rewritting Eq. (C.2), we have

$\overline{\gamma}(t_{1},t_{2})=\frac{\tilde{\gamma}(\tilde{t}_{1},\tilde{t}_{2})}{\sqrt{\mathcal{T}_{p}\mathcal{T}_{c}}},$
(C.4)

and since the location of the maximum of $|\overline{\gamma}(t_{1},t_{2})|$
must be the same as that of $|\tilde{\gamma}(\tilde{t}_{1},\tilde{t}_{2})|$,
and the latter maximum is of order unity, in general the maximum value of
$\overline{\gamma}(t_{1},t_{2})$ scales with
$1/\sqrt{\mathcal{T}_{p}\mathcal{T}_{c}}$.

Figure 27: Schematic of a general joint temporal amplitude with a pulse
duration and coherence time denoted by $\mathcal{T}_{p}$ and $\mathcal{T}_{c}$
respectively, in the original and rotated coordinate system.

##  Appendix D Relation between $K$ and $\mathcal{K}$

We begin with an alternate expression [8, 9] for the Schmidt number,

$\frac{1}{K}=\int
dt_{1}dt_{2}dt_{1}^{\prime}dt_{2}^{\prime}\overline{\gamma}(t_{1},t_{2})\overline{\gamma}^{*}(t_{1}^{\prime},t_{2})\overline{\gamma}(t_{1}^{\prime},t_{2}^{\prime})\overline{\gamma}(t_{1},t_{2}^{\prime}).$
(D.1)

which can be confirmed by using the Schmidt decomposition (Eq. (III.1)) in Eq.
(D.1) and recalling the orthogonality of the Schmidt modes. If a Whittaker-
Shannon decomposition (Eq. (V.11)) constructed with an appropriate bandwidth
limit is put in Eq. (D.1), with the use of the orthogonality relations of
$\overline{\chi}_{n}(t)$ we find

$\frac{1}{K}=\tau^{4}\sum_{a,b,c,d}\overline{\gamma}_{ab}\overline{\gamma}^{*}_{cb}\overline{\gamma}_{cd}\overline{\gamma}^{*}_{ad}=\tau^{4}\text{Tr}(\bm{\bar{\gamma}}\bm{\bar{\gamma}}^{\dagger}\bm{\bar{\gamma}}\bm{\bar{\gamma}}^{\dagger}),$
(D.2)

where we have written $\overline{\gamma}_{ab}$ as short-hand for
$\overline{\gamma}(a\tau,b\tau)$, and in the second equality we use
$\bm{\bar{\gamma}}$ to indicate the matrix with components
$\overline{\gamma}_{ab}$.

We now make use of the identity

$|\text{Tr}(\mathbf{A}\mathbf{B}^{\dagger})|^{2}\leq\text{Tr}(\mathbf{A}\mathbf{A}^{\dagger})\text{Tr}(\mathbf{B}\mathbf{B}^{\dagger}),$
(D.3)

which is the generalization of the Cauchy–Schwarz inequality to matrices under
the trace inner product [36]. For $\mathbf{A}$ an $l\times l$ matrix, we put
$\mathbf{B}=\bm{1}_{l\times l}$, the identity matrix of size $l\times l$, and
find

$|\text{Tr}(\mathbf{A})|^{2}\leq\text{Tr}(\mathbf{A}\mathbf{A}^{\dagger})l,$
(D.4)

or rather

$\text{Tr}(\mathbf{A}\mathbf{A}^{\dagger})\geq\frac{|\text{Tr}(\mathbf{A})|^{2}}{l}.$
(D.5)

Next we set $\mathbf{A}=\bm{\bar{\gamma}}\bm{\bar{\gamma}}^{\dagger}$, and
since $\text{Tr}(\bm{\bar{\gamma}}\bm{\bar{\gamma}}^{\dagger})\geq 0$ the
inequality in Eq. (D.5) leads to

$\text{Tr}(\bm{\bar{\gamma}}\bm{\bar{\gamma}}^{\dagger}\bm{\bar{\gamma}}\bm{\bar{\gamma}}^{\dagger})\geq\frac{\text{Tr}(\bm{\bar{\gamma}}\bm{\bar{\gamma}}^{\dagger})^{2}}{l}.$
(D.6)

After inputting this result into the equation (D.2) for the Schmidt number, we
arrive at

$\frac{1}{K}\geq\frac{(\tau^{2}\text{Tr}(\bm{\bar{\gamma}}\bm{\bar{\gamma}}^{\dagger}))^{2}}{l}.$
(D.7)

This can be significantly simplified by noting that the joint amplitude is
normalized to unity, and inputting the Whittaker-Shannon decomposition (Eq.
(V.11)) into

$\int dt_{1}dt_{2}|\overline{\gamma}(t_{1},t_{2})|^{2}=1,$ (D.8)

we find $\tau^{2}\text{Tr}(\bm{\bar{\gamma}}\bm{\bar{\gamma}}^{\dagger})=1$,
and then from Eq. (D.7) we have

$K\leq l.$ (D.9)

Now $l$ is the dimension of the matrix $\bm{\gamma}$, but because
$\overline{\chi}_{n}(t)$ has a width set by $\tau$ one needs on the order of
$l=\mathcal{T}_{p}/\tau$ functions to interpolate the joint amplitude; see the
discussion around Eq (V.20). If we choose $\tau=\mathcal{T}_{c}$, then
$l=\mathcal{T}_{p}/\mathcal{T}_{c}=\mathcal{K}$ and we find

$K\leq\mathcal{K},$ (D.10)

which completes the argument.

##  Appendix E Correlation functions

We begin by using the inverse relation given by Eq. (V.13) to write the
correlation functions (II.4) as

$\displaystyle\overline{G}^{(1)}(t_{1},t_{2})=\overline{\chi}_{x}^{*}(t_{1})\left\langle\Psi|B_{x}^{\dagger}B_{y}|\Psi\right\rangle\overline{\chi}_{y}(t_{2}),$
(E.1a)
$\displaystyle\overline{G}^{(2)}(t_{1},t_{2})=\overline{\chi}_{w}^{*}(t_{1})\overline{\chi}_{x}^{*}(t_{2})\left\langle\Psi|B_{w}^{\dagger}B_{x}^{\dagger}B_{y}B_{z}|\Psi\right\rangle\overline{\chi}_{y}(t_{2})\overline{\chi}_{z}(t_{1}),$
(E.1b)

we use the same summation convention as in the text. Then using the
transformation given by Eq. (VI.2) we evaluate

$\begin{split}\left\langle\Psi|B_{x}^{\dagger}B_{y}|\Psi\right\rangle&=\bra{\text{vac}}\nu_{xa}^{*}B_{a}\nu_{yb}B_{b}^{\dagger}\ket{\text{vac}}\\\
&=\nu_{xa}^{*}\nu_{ya}\\\ &=(\bm{\nu}^{*}\bm{\nu}^{T})_{xy}\\\
&=(\mathbf{U}^{*}(\text{sinh}\mathbf{P}^{*})(\text{sinh}\mathbf{P}^{T})\mathbf{U}^{T})_{xy}\\\
&=(\text{sinh}^{2}\mathbf{P})_{xy},\end{split}$ (E.2)

and

$\begin{split}&\left\langle\Psi|B_{w}^{\dagger}B_{x}^{\dagger}B_{y}B_{z}|\Psi\right\rangle\\\
&=\bra{\text{vac}}\nu^{*}_{wa}B_{a}(\mu^{*}_{xb}B_{b}^{\dagger}+\nu^{*}_{xb}B_{b})(\mu_{yc}B_{c}+\nu_{yc}B_{c}^{\dagger})\nu_{zd}B_{d}^{\dagger}\ket{\text{vac}}\\\
&=S_{wa}^{*}\mu_{xb}^{*}\mu_{yc}\nu_{zd}\bra{\text{vac}}B_{a}B_{b}^{\dagger}B_{c}B_{d}^{\dagger}\ket{\text{vac}}+\nu_{wa}^{*}\nu_{xb}^{*}\nu_{yc}\nu_{zd}\bra{\text{vac}}B_{a}B_{b}B_{c}^{\dagger}B_{d}^{\dagger}\ket{\text{vac}}\\\
&=\nu_{wa}^{*}\mu_{xa}^{*}\mu_{yc}\nu_{zc}+\nu_{wa}^{*}\nu_{xb}^{*}\nu_{yb}\nu_{za}+\nu_{wa}^{*}\nu_{xb}^{*}\nu_{ya}\nu_{zb}\\\
&=(\bm{\nu\mu}^{T})^{*}_{wx}(\bm{\nu\mu}^{T})_{zy}+(\bm{\nu}^{*}\bm{\nu}^{T})_{wz}(\bm{\nu}^{*}\bm{\nu}^{T})_{xy}+(\bm{\nu}^{*}\bm{\nu}^{T})_{wy}(\bm{\nu}^{*}\bm{\nu}^{T})_{xz}\\\
&=(\mathbf{U}(\text{sinh}\mathbf{P})(\text{cosh}\mathbf{P}))^{*}_{wx}(\mathbf{U}(\text{sinh}\mathbf{P})(\text{cosh}\mathbf{P}))_{zy}+(\text{sinh}^{2}\mathbf{P})_{wz}(\text{sinh}^{2}\mathbf{P})_{xy}+(\text{sinh}^{2}\mathbf{P})_{wy}(\text{sinh}^{2}\mathbf{P})_{xz}.\end{split}$
(E.3)

So $\overline{G}^{(1)}(t_{1},t_{2})$ is given by

$\begin{split}\overline{G}^{(1)}(t_{1},t_{2})&=\overline{\chi}_{x}^{*}(t_{1})(\text{sinh}^{2}\mathbf{P})_{xy}\overline{\chi}_{y}(t_{2})\\\
&=\bm{\overline{\chi}}^{\dagger}(t_{1})(\text{sinh}^{2}\mathbf{P})\bm{\overline{\chi}}(t_{2}),\end{split}$
(E.4)

where
$\bm{\overline{\chi}}(t)=(...,\overline{\chi}_{-1}(t),\overline{\chi}_{0}(t),\overline{\chi}_{1}(t),...)^{T}$
is the column vector formed from the set $\\{\overline{\chi}_{n}(t)\\}$ for a
given $t$, and

$\begin{split}\overline{G}^{(2)}(t_{1},t_{2})&=\overline{\chi}_{w}^{*}(t_{1})\overline{\chi}_{x}^{*}(t_{2})(\mathbf{U}(\text{sinh}\mathbf{P})(\text{cosh}\mathbf{P}))^{*}_{wx}(\mathbf{U}(\text{sinh}\mathbf{P})(\text{cosh}\mathbf{P}))_{zy}\overline{\chi}_{y}(t_{2})\overline{\chi}_{z}(t_{1})\\\
&+\overline{\chi}_{w}^{*}(t_{1})\overline{\chi}_{x}^{*}(t_{2})(\text{sinh}^{2}\mathbf{P})_{wz}(\text{sinh}^{2}\mathbf{P})_{xy}\overline{\chi}_{y}(t_{2})\overline{\chi}_{z}(t_{1})\\\
&+\overline{\chi}_{w}^{*}(t_{1})\overline{\chi}_{x}^{*}(t_{2})(\text{sinh}^{2}\mathbf{P})_{wy}(\text{sinh}^{2}\mathbf{P})_{xz}\overline{\chi}_{y}(t_{2})\overline{\chi}_{z}(t_{1})\\\
&=|\bm{\overline{\chi}}^{T}(t_{1})\mathbf{U}(\text{sinh}\mathbf{P})(\text{cosh}\mathbf{P})\bm{\overline{\chi}}(t_{2})|^{2}+[\bm{\overline{\chi}}^{\dagger}(t_{1})(\text{sinh}^{2}\mathbf{P})\bm{\overline{\chi}}(t_{1})][\bm{\overline{\chi}}^{\dagger}(t_{2})(\text{sinh}^{2}\mathbf{P})\bm{\overline{\chi}}(t_{2})]+|\bm{\overline{\chi}}^{\dagger}(t_{1})(\text{sinh}^{2}\mathbf{P})\bm{\overline{\chi}}(t_{2})|^{2}\\\
&=|\bm{\overline{\chi}}^{T}(t_{1})\mathbf{U}(\text{sinh}\mathbf{P})(\text{cosh}\mathbf{P})\bm{\overline{\chi}}(t_{2})|^{2}+\overline{G}^{(1)}(t_{1})\overline{G}^{(1)}(t_{2})+|\overline{G}^{(1)}(t_{1},t_{2})|^{2}.\end{split}$
(E.5)

##  Appendix F Mode-by-mode calculation

In this Appendix we show that the photon statistics of the sinc-hat joint
amplitude can be described on a “mode-by-mode” basis using the pseudo-Schmidt
decomposition. We then generalize this to the Whittaker-Shannon decomposition,
and show that at least approximately a “mode-by-mode” description can be
introduced.

###  F.1 Pseudo-Schmidt decomposition

We start by considering the analytical form of $\overline{G}^{(2)}(\Delta t)$
(IV.27) in the CW limit,

$\begin{split}\left.\overline{G}^{(2)}(\Delta t)\right|_{|\Delta t|\lesssim
T_{c}}&\approx\overline{G}^{(2)}(\Delta t=0)\\\
&=\frac{1}{T_{c}^{2}}(3N_{\text{mode}}^{2}+N_{\text{mode}}),\end{split}$ (F.1)

where we have approximated the “sinc” function as unity and used Eq. (IV.20);
as expected, we recover the usual standard result for $G^{(2)}(\Delta t=0)$
[18, 37], and for $\Delta t>T_{c}$ we have $\overline{G}^{(2)}(\Delta
t)\leq\overline{G}^{(2)}(\Delta t=0)$, the condition for “bunched light” [38].
Since $\overline{G}^{(2)}(\Delta t)$ varies little over a length of time
$|\Delta t|\lesssim T_{c}$, the quantity $T_{c}\overline{G}^{(2)}(\Delta t)$
is a “coincidence rate,” and

$T_{p}\left(T_{e}\left.\overline{G}^{(2)}(\Delta t)\right|_{\Delta t\lesssim
T_{c}}\right)\approx\mathcal{N}(3N_{\text{mode}}^{2}+N_{\text{mode}})$ (F.2)

is the total coincidence count over the duration of the pulse.

To gain insight into the scaling of $\overline{G}^{(2)}(\Delta t)$ with
$N_{\mathrm{mode}}$, we note that for a single pseudo-Schmidt mode labeled by
$n$, the probability of detecting $2x$ photons is [18]

$P_{n}(2x)=\frac{1}{c}\left(\frac{\sqrt{(2x)!}}{2^{x}x!}\right)^{2}\left(\frac{s}{c}\right)^{2x},$
(F.3)

where we put $s\equiv\mathrm{sinh}(r)$ and $c\equiv\mathrm{cosh}(r)$, and
$r=|\beta|/\sqrt{\mathcal{N}}$ is the squeezing parameter which is independent
of $n$ for the values of $n$ for which it is nonzero (see Eq. (IV)). Since
each pseudo-Schmidt mode is normalized we have

$1=\sum_{x=0}^{\infty}P_{n}(2x)\Rightarrow
c=\sum_{x=0}^{\infty}\left(\frac{\sqrt{(2x)!}}{2^{x}x!}\right)^{2}\left(\frac{s}{c}\right)^{2x}.$
(F.4)

Taking the derivative of both sides with respect to $r$, and simplifying using
hyperbolic trigonometry identities followed by a re-indexing of the sum, we
find

$\sum_{x=0}^{\infty}(2x)P_{n}(2x)=s^{2}=N_{\text{mode}},$ (F.5)

as expected; the average number of photons in each pseudo-Schmidt mode is
given by $N_{\text{mode}}$. Notice that we can re-write the expectation value
as

$N_{\text{mode}}=\sum_{x=0}^{\infty}(2x)P_{n}(2x)=\sum_{x=0}^{\infty}\frac{(2x)!}{(2x-1)!}P_{n}(2x)=\sum_{x=0}^{\infty}\mathcal{P}(2x,1)P_{n}(2x),$
(F.6)

where $\mathcal{P}(2x,1)$ is the number of permutations of $2x$ objects when
we select one object. Again taking the derivative of both sides with respect
to $r$, after further hyperbolic trigonometry manipulations and a re-indexing
of the sum, we find

$\sum_{x=0}^{\infty}(2x)(2x-1)P_{n}(2x)=3N_{\text{mode}}^{2}+N_{\text{mode}},$
(F.7)

where the right hand side is the familiar scaling of
$\overline{G}^{(2)}(\Delta t)$ with $N_{\mathrm{mode}}$. However, playing the
same game, we can re-write this as

$\sum_{x=0}^{\infty}(2x)(2x-1)P_{n}(2x)=\sum_{x=0}^{\infty}\frac{(2x)!}{(2x-2)!}P_{n}(2x)=\sum_{x=0}^{\infty}\mathcal{P}(2x,2)P_{n}(2x)=3N_{\text{mode}}^{2}+N_{\text{mode}},$
(F.8)

where $\mathcal{P}(2x,2)$ is the number of permutations of $2x$ objects when
we select two of them.

Thus for short time differences, on the order of $T_{c}$, the coincidence rate
is the expectation value of the total number of ways of “picking” two photons
from any pseudo-Schmidt mode, and to calculate the total coincidence count we
multiply by $\mathcal{N}$ ((F.2)), the total number of pseudo-Schmidt modes
(cf. (IV.20) and (IV.21)). Indeed, for time differences less than $T_{c}$ we
can set $n=m$ for both terms in Eq. (IV.17), so that to good approximation we
have

$G^{(2)}(t_{1},t_{2})\approx(3N_{\text{mode}}^{2}+N_{\text{mode}})\sum_{n}|\overline{\eta}_{n}(t_{1})|^{2}|\overline{\eta}_{n}(t_{2})|^{2}.$
(F.9)

For such time differences the coincidence count at a particular $t_{1},t_{2}$
is associated with only a few pseudo-Schmidt modes, unlike in the Schmidt
decomposition, where the inclusion of the contributions of a large number of
Schmidt modes is needed.

###  F.2 Whittaker-Shannon decomposition

We now consider the Whittaker-Shannon decomposition, starting with the exact
results for the correlation functions (VI.6,VI.7). The photon density is given
by

$\begin{split}\overline{G}^{(1)}(t)&=\bm{\overline{\chi}}^{\dagger}(t)(\text{sinh}^{2}\mathbf{P})\bm{\overline{\chi}}(t)\\\
&=\sum_{n,m}\overline{\chi}_{n}^{*}(t)(\text{sinh}^{2}\mathbf{P})_{nm}\overline{\chi}_{m}(t)\\\
&\approx\sum_{n}(\text{sinh}^{2}\mathbf{P})_{nn}|\overline{\chi}_{n}(t)|^{2},\end{split}$
(F.10)

where in the second line we expanded the matrix multiplication, and in the
third line approximated the double sum as just the diagonal contributions. In
reducing the double sum to a single sum we are losing significant
contributions to the photon density, because typically
$(\text{sinh}^{2}\mathbf{P})_{nm}$ will be nonzero over a small range of
$|n-m|$; further, the product of two neighbouring Whittaker-Shannon modes is
not negligible. While the final result of Eq. (F.10) provides a crude
approximate at best, it is similar to the pseudo-Schmidt result given by Eq.
(IV.16). This suggests that we should identify

$\Gamma_{n}^{2}=(\text{sinh}^{2}\mathbf{P})_{nn}$ (F.11)

as the number of photons in each mode, and indeed we find this to be the case
in Section VI.1.

We now turn our attention to $\overline{G}^{(2)}(t_{1},t_{2})$, starting with
the coherent contribution

$\begin{split}\overline{G}_{\text{coh}}^{(2)}(t_{1},t_{2})&=|\bm{\overline{\chi}}^{T}(t_{1})\mathbf{U}(\text{sinh}\mathbf{P})(\text{cosh}\mathbf{P})\bm{\overline{\chi}}(t_{2})|^{2}\\\
\
&=\sum_{n,m,p,q}\overline{\chi}^{*}_{n}(t_{1})(\mathbf{U}\text{sinh}\mathbf{P}\text{cosh}\mathbf{P})_{nm}^{*}\overline{\chi}^{*}_{m}(t_{2})\overline{\chi}_{p}(t_{1})(\mathbf{U}\text{sinh}\mathbf{P}\text{cosh}\mathbf{P})_{pq}\overline{\chi}_{q}(t_{2})\\\
&\approx\sum_{n,m}\left|(\mathbf{U}\text{sinh}\mathbf{P}\text{cosh}\mathbf{P})_{nm}\right|^{2}|\overline{\chi}_{n}(t_{1})|^{2}|\overline{\chi}_{m}(t_{2})|^{2},\end{split}$
(F.12)

where in the last line we used the same approximation that we did for the
photon density. Then if we consider times $t_{1},t_{2}$ such that
$|t_{1}-t_{2}|\lesssim\tau$, the significant contribution to the sum is when
$n=m$, and we have

$\begin{split}\overline{G}_{\text{coh}}^{(2)}(t_{1},t_{2})&\approx\sum_{n}\left|(\mathbf{U}\text{sinh}\mathbf{P}\text{cosh}\mathbf{P})_{nn}\right|^{2}|\overline{\chi}_{n}(t_{1})|^{2}|\overline{\chi}_{n}(t_{2})|^{2}.\end{split}$
(F.13)

Following the same steps for the incoherent contribution, for times
$t_{1},t_{2}$ such that $|t_{1}-t_{2}|\lesssim\tau$,
$\overline{G}^{(2)}(t_{1},t_{2})$ is given by

$\overline{G}^{(2)}(t_{1},t_{2})=\sum_{n}(\left|(\mathbf{U}\text{sinh}\mathbf{P}\text{cosh}\mathbf{P})_{nn}\right|^{2}+2\Gamma_{n}^{4})|\overline{\chi}_{n}(t_{1})|^{2}|\overline{\chi}_{n}(t_{2})|^{2}.$
(F.14)

While again a crude approximation at best, we do note the similarity between
the final result and the pseudo-Schmidt result in Eq. (F.9).

## Appendix G

We start with equation (VI.9) for $\overline{\rho}_{n}(t)$, and invert it by
multiplying by $(\text{sinh}\mathbf{P})^{-1}=\text{csch}\mathbf{P}$

$\overline{\chi}_{n}(t)=\sum_{m}(\text{csch}\mathbf{P})_{nm}\Gamma_{m}\overline{\rho}_{m}(t).$
(G.1)

Taking the term inside the squared norm for the coherent contribution (Eq.
(VI.13)) and writing it in its index representation, we insert the form of
$\overline{\chi}_{n}(t)$ in terms of $\overline{\rho}_{m}(t)$ using (G.1), so
that

$\begin{split}\bm{\overline{\chi}}^{T}(t_{1})\mathbf{U}(\text{sinh}\mathbf{P})(\text{cosh}\mathbf{P})\bm{\overline{\chi}}(t_{2})&=\overline{\chi}_{x}(t_{1})U_{xa}(\text{sinh}\mathbf{P})_{ab}(\text{cosh}\mathbf{P})_{by}\overline{\chi}_{y}(t_{2})\\\
&=(\text{csch}\mathbf{P})_{xm}\Gamma_{m}\overline{\rho}_{m}(t_{1})U_{xa}(\text{sinh}\mathbf{P})_{ab}(\text{cosh}\mathbf{P})_{by}(\text{csch}\mathbf{P})_{yn}\Gamma_{n}\overline{\rho}_{n}(t_{2})\\\
&=\Gamma_{n}\Gamma_{m}\overline{\rho}_{m}(t_{1})\overline{\rho}_{n}(t_{2})((\text{csch}\mathbf{Q})\mathbf{U}(\text{sinh}\mathbf{P})(\text{cosh}\mathbf{P})(\text{csch}\mathbf{P}))_{mn}\\\
&=\Gamma_{n}\Gamma_{m}\overline{\rho}_{m}(t_{1})\overline{\rho}_{n}(t_{2})(\mathbf{U}\text{coth}\mathbf{P})_{mn},\end{split}$
(G.2)

where in deriving this result we used the properties of
$\mathbf{P},\mathbf{Q}$ and $\mathbf{U}$ discussed in section VI, and we set
$\text{coth}\mathbf{P}=(\text{tanh}\mathbf{P})^{-1}=(\text{cosh}\mathbf{P})(\text{csch}\mathbf{P})$.
The resulting coherent contribution to the correlation function is then given
by

$\overline{G}^{(2)}_{\text{coh}}(t_{1},t_{2})=\sum_{n,m}|\Gamma_{n}\Gamma_{m}(\mathbf{U}\text{coth}\mathbf{P})_{mn}\overline{\rho}_{m}(t_{1})\overline{\rho}_{n}(t_{2})|^{2}.$
(G.3)

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&\quad+\frac{\alpha}{2}\int_{\mathbb{R}^{n}}\partial_{x_{1}}J^{s}u\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+\frac{\alpha-2}{2}}u\,dx\\\
&=\Theta_{2,1,0,2,1}(t)+\Theta_{2,1,0,2,2}(t).\end{split}$

The term $\Theta_{2,1,0,2,1}$ represents the smoothing effect after
integrating in the temporal variable.

Instead, the remainder does not contain terms that yield some smoothing, thus
we require to estimate it. In this sense, we rewrite it as

$\begin{split}\Theta_{2,1,0,2,2}(t)&=\frac{\alpha}{2}\int_{\mathbb{R}^{n}}\partial_{x_{1}}J^{s}u\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+\frac{\alpha-2}{2}}u\,dx\\\
&=-\frac{\alpha}{2}\int_{\mathbb{R}^{n}}J^{s}u\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}^{2}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+\frac{\alpha-2}{2}}u\,dx\\\
&\quad-\frac{\alpha}{2}\int_{\mathbb{R}^{n}}J^{s}u\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\right]\partial_{x_{1}}J^{s+\frac{\alpha-2}{2}}u\,dx\\\
&=\Lambda_{3}(t)+\Lambda_{4}(t).\end{split}$

We indicate how to estimate $\Lambda_{3}.$ We start by writing

$\begin{split}&\Lambda_{3}(t)=\\\
&\frac{\alpha}{2}\int_{\mathbb{R}^{n}}J^{s}\left(u\chi_{\epsilon,\tau}+u\phi_{\epsilon,\tau}+u\psi_{\epsilon}\right)\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}^{2}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+\frac{\alpha-2}{2}}\left(u\chi_{\epsilon,\tau}+u\phi_{\epsilon,\tau}+u\psi_{\epsilon}\right)dx\\\
\end{split}$

Hence by Theorem 4.3 and Lemma 5.1 we obtain

$\begin{split}|\Theta_{2,1,0,2,2}(t)|\lesssim\|J^{s}(u\chi_{\epsilon,\tau})\|_{L^{2}_{x}}^{2}+\|J^{s}(u\phi_{\epsilon,\tau})\|_{L^{2}_{x}}^{2}+\|u_{0}\|_{L^{2}_{x}}^{2}.\end{split}$

Unlike the previous step to estimate
$\|J^{s}(u\chi_{\epsilon,\tau})\|_{L^{2}}$ we only require to apply Lemma 5.5
and (3.26) we get

$\sup_{t\in(0,T)}\|J^{s}(u\chi_{\epsilon,\tau})\|_{L^{2}_{x}}^{2}<\infty.$

Instead, to estimate
$\|J^{s}\left(u\phi_{\epsilon,\tau}\right)\|_{L^{2}_{T}L^{2}_{x}}$ we combine
3.26 and the arguments described in (3.24)-(3.25) to obtain

$\|J^{s}(u\phi_{\epsilon,\tau})\|_{L^{2}_{T}L^{2}_{x}}<\infty.$

In summary,

$\int_{0}^{T}|\Lambda_{3}(t)|\,dt<\infty.$

The same arguments apply to estimate $\Lambda_{4}.$ Indeed,

$\int_{0}^{T}|\Lambda_{4}(t)|\,dt<\infty.$

Under the conditions (3.18)- (3.19) we there exist $\lambda>0$ such that

$\begin{split}&\lambda\left(\int_{\mathbb{R}^{n}}\left(J^{s+1}u\right)^{2}\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\,dx+\int_{\mathbb{R}^{n}}\left(J^{s-2}\partial_{x_{1}}u\right)^{2}\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\,dx\right)\\\
&\leq\Theta_{2,1,1,1}(t)+\Lambda_{3,1}(t)+\Theta_{2,1,0,3}(t).\end{split}$

The term that contains the kernel$\mathcal{K}_{\alpha}$ is estimated in the
same way as we did in (3.20), and in such case

$\begin{split}\int_{0}^{T}|\Theta_{2,2}(t)|\,dt<\infty.\end{split}$

The arguments of the proof are similar to the ones described in the proof of
(2.20). To handle the nonlinear part we use the decomposition

$\begin{split}\Theta_{3}(t)&=-\int_{\mathbb{R}^{n}}\chi_{\epsilon,\tau}J^{s+\frac{2-\alpha}{2}}u\,\left[J^{s+\frac{2-\alpha}{2}};\chi_{\epsilon,\tau}\right]u\partial_{x_{1}}u\,dx\\\
&\quad+\int_{\mathbb{R}^{n}}\chi_{\epsilon,\tau}J^{s+\frac{2-\alpha}{2}}u\left[J^{s+\frac{2-\alpha}{2}};u\chi_{\epsilon,\tau}\right]\partial_{x_{1}}u\\\
&\quad-\frac{1}{2}\int_{\mathbb{R}^{n}}\left(J^{s+\frac{2-\alpha}{2}}u\right)^{2}\chi_{\epsilon,\tau}^{2}\,dx-\nu_{1}\int_{\mathbb{R}^{n}}u\left(J^{s+\frac{2-\alpha}{2}}u\right)^{2}\chi_{\epsilon,\tau}\chi_{\epsilon,\tau}^{\prime}\,dx\\\
\end{split}$

the reader can notice that this decomposition is just adapted to the
regularity of the step. Although, the process required to estimate the terms
above is similar as the ones used in the $\Theta_{3}$ in the previous step, so
for the sake of brevity we will omit the details.

In summary, we find after applying Gronwall’s inequality and integrating in
time the following: for any $\epsilon>0$ and $\tau\geq 5\epsilon,$

(3.41)
$\begin{split}&\sup_{0<t<T}\int_{\mathbb{R}^{n}}\left(J^{s+\frac{2-\alpha}{2}}u(x,t)\right)^{2}\chi_{\epsilon,\tau}^{2}(\nu\cdot
x+\omega t)\,dx\\\
&\qquad+\int_{0}^{T}\int_{\mathbb{R}^{n}}(J^{s+1}u(x,t))^{2}\left(\chi_{\epsilon,\tau}\chi_{\epsilon,\tau}^{\prime}\right)(\nu\cdot
x+\omega t)\,dx\,dt\leq c.\end{split}$

As part of the inductive process we suppose that $s\in(h,h+1),h\in\mathbb{N},$
with the condition that for any $\epsilon>0$ and $\tau\geq 5\epsilon$ the
following estimate holds true:

(3.42)
$\begin{split}&\sup_{0<t<T}\int_{\mathbb{R}^{n}}\left(J^{s}u(x,t)\right)^{2}\chi_{\epsilon,\tau}^{2}(\nu\cdot
x+\omega t)\,dx\\\
&\qquad+\int_{0}^{T}\int_{\mathbb{R}^{n}}(J^{s+\frac{\alpha}{2}}u(x,t))^{2}\left(\chi_{\epsilon,\tau}\chi_{\epsilon,\tau}^{\prime}\right)(\nu\cdot
x+\omega t)\,dx\,dt\leq c.\end{split}$

As usual our starting point is the energy estimate (3.27).

As we did in the previous case we only describe the part that will provide the
smoothing effect since the remainder terms can be estimated in a standard way
with arguments described previously.

First, we rewrite $\Theta_{2}$ from (3.27) as follows

(3.43)
$\begin{split}\Theta_{2}(t)&=\frac{1}{2}\int_{\mathbb{R}^{n}}J^{s+\frac{2-\alpha}{2}}u\left[J^{\alpha}\partial_{x_{1}};\chi_{\epsilon,\tau}^{2}\right]J^{s+\frac{2-\alpha}{2}}u\,dx\\\
&\quad+\frac{1}{2}\int_{\mathbb{R}^{n}}J^{s+\frac{2-\alpha}{2}}u\left[\mathcal{K}_{\alpha}\partial_{x_{1}};\chi_{\epsilon,\tau}^{2}\right]J^{s+\frac{2-\alpha}{2}}u\,dx\\\
&=\Theta_{2,1}(t)+\Theta_{2,2}(t).\end{split}$

We shall remind that there exist pseudo-differential operators
$p_{\alpha-j}(x,D)$ for each $\,j\in\\{1,2\cdots,m\\}$ and some
$m\in\mathbb{N}$ satisfying

$\begin{split}c_{\alpha}(x,D)=p_{\alpha}(x,D)+p_{\alpha-1}(x,D)+\dots+p_{\alpha-m}(x,D)+r_{\alpha-m-1}(x,D),\end{split}$

where $p_{\alpha-j}\in\mathrm{OP}\mathbb{S}^{\alpha-j}$ and
$r_{\alpha-m-1}\in\mathrm{OP}\mathbb{S}^{\alpha-1-m}.$

We choose $m$ as being

$m=\left\lceil 2s+1-s_{n}\right\rceil.$

Thus,

$\begin{split}\Theta_{2,1}(t)&=\frac{1}{2}\sum_{j=0}^{m}\int_{\mathbb{R}^{n}}J^{s+\frac{2-\alpha}{2}}up_{\alpha-j}(x,D)J^{s+\frac{2-\alpha}{2}}u\,dx\\\
&\quad+\frac{1}{2}\int_{\mathbb{R}^{n}}J^{s+\frac{2-\alpha}{2}}ur_{\alpha-1-m}(x,D)J^{s+\frac{2-\alpha}{2}}u\,dx\\\
&=\frac{1}{2}\sum_{j=0}^{m}\int_{\mathbb{R}^{n}}J^{s+\frac{2-\alpha}{2}}up_{\alpha-j}(x,D)J^{s+\frac{2-\alpha}{2}}u\,dx\\\
&\quad+\frac{1}{2}\int_{\mathbb{R}^{n}}uJ^{s+\frac{2-\alpha}{2}}r_{\alpha-1-m}(x,D)J^{s+\frac{2-\alpha}{2}}u\,dx\\\
&=\sum_{j=0}^{m}\Theta_{2,1,j}(t)+\Theta_{2,1,m+1}(t).\end{split}$

where

$\begin{split}p_{\alpha-j}(x,D)=\sum_{|\beta|=j}c_{\beta,j}\partial_{x}^{\beta}(\chi_{\epsilon,\tau}^{2})\Psi_{\beta,j}J^{\alpha-j},\end{split}$

for each $j\in\\{1,2,\dots,m\\},$ with
$\Psi_{\beta,j}\in\mathrm{OP}\mathbb{S}^{0}.$

The key part to control the terms in the expression above relies into combine
assumption (3.46) and apply Lemma 5.5 properly to obtain that $\epsilon>0$ and
$\tau\geq 5\epsilon,$ the following estimates holds true:

(3.44)
$\sup_{0<t<T}\int_{\mathbb{R}^{n}}\left(J^{r_{1}}u(x,t)\right)^{2}\chi_{\epsilon,\tau}^{2}(\nu\cdot
x+\omega t)\,dx\leq c$

for any $r_{1}\in\left(0,s\right],$ and

(3.45)
$\begin{split}\int_{0}^{T}\int_{\mathbb{R}^{n}}(J^{r_{2}}u(x,t))^{2}\left(\chi_{\epsilon,\tau}\chi_{\epsilon,\tau}^{\prime}\right)(\nu\cdot
x+\omega t)\,dx\,dt\leq c.\end{split}$

for all $r_{2}\in\left(0,s+\frac{\alpha}{2}\right].$

The term $\Theta_{2,1,j}\,j\geq 2,$ can be estimated combining (3.44)-(3.45)
together the arguments already described in (3.35)-(3.36).

Next, the distinguished term in our decomposition is $\Theta{2,1,0}$ and it is
given by

$\begin{split}\Theta_{2,1,0}(t)&=\frac{1}{2}\int_{\mathbb{R}^{n}}J^{s+\frac{2-\alpha}{2}}uJ^{s+\frac{2+\alpha}{2}}u\partial_{x_{1}}\,\left(\chi_{\epsilon,\tau}^{2}\right)\,dx\\\
&\quad-\frac{\alpha}{2}\int_{\mathbb{R}^{n}}J^{s+\frac{2-\alpha}{2}}u\,J^{s+\frac{\alpha-2}{2}}\partial_{x_{1}}^{2}u\,\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\,dx\\\
&\quad-\frac{\alpha}{2}\sum_{|\beta|=1,\beta\neq\mathrm{e}_{1}}\int_{\mathbb{R}^{n}}J^{s+\frac{2-\alpha}{2}}u\,J^{s+\frac{\alpha-2}{2}}\partial_{x_{1}}\partial_{x}^{\beta}u\,\partial_{x}^{\beta}\left(\chi_{\epsilon,\tau}^{2}\right)\,dx\\\
&=\Theta_{2,1,0,1}(t)+\Theta_{2,1,0,2}(t)+\Theta_{2,1,0,3}(t).\end{split}$

In the first place

$\begin{split}\Theta_{2,1,0,1}(t)&=\frac{1}{2}\int_{\mathbb{R}^{n}}\left(J^{s+1}u\right)^{2}\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\,dx+\frac{1}{2}\int_{\mathbb{R}^{n}}J^{s+1}u\left[J^{\frac{\alpha}{2}};\partial_{x_{1}}(\chi_{\epsilon,\tau}^{2})\right]J^{s+\frac{2-\alpha}{2}}u\,dx\\\
&=\Theta_{2,1,0,1,1}(t)+\Theta_{2,1,0,1,2}(t).\end{split}$

The term $\Theta_{2,1,0,1,1}$ represents after integrating in time the pursuit
smoothing effect.

For $\Theta_{2,1,0,1,2}$ we rewrite it as

$\begin{split}\Theta_{2,1,0,1,2}(t)&=\frac{1}{2}\int_{\mathbb{R}^{n}}J^{s}u\left[J^{1+\frac{\alpha}{2}};\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+1-\frac{\alpha}{2}}u\,dx\\\
&\quad-\frac{1}{2}\int_{\mathbb{R}^{n}}J^{s}u\left[J;\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+1}u\,dx\\\
&=\Lambda_{1}(t)+\Lambda_{2}(t).\end{split}$

To handle $\Lambda_{1}$ we split the commutator expression as in (3.14) (after
replacing $\frac{\alpha}{2}$ by $\frac{\alpha}{2}+1$), but fixing
$m_{1}\in\mathbb{N}$ as being

$m_{1}=\left\lceil 2s+\frac{3\alpha}{2}-2-s_{n}\right\rceil.$

Notice that from such decomposition we obtain terms of lower order which are
controlled by using (3.44)-(3.45).

$\begin{split}\Theta_{2,1,0,2}(t)&=\frac{\alpha}{2}\int_{\mathbb{R}^{n}}\left(\partial_{x_{1}}J^{s}u\right)^{2}\,\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\,dx\\\
&\quad+\frac{\alpha}{2}\int_{\mathbb{R}^{n}}\partial_{x_{1}}J^{s}u\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+\frac{\alpha-2}{2}}u\,dx\\\
&=\Theta_{2,1,0,2,1}(t)+\Theta_{2,1,0,2,2}(t).\end{split}$

The term $\Theta_{2,1,0,2,1}$ corresponds to the smoothing effect of this step
after integrating in time.

Next, we rewrite

$\begin{split}\Theta_{2,1,0,2,2}(t)&=\frac{\alpha}{2}\int_{\mathbb{R}^{n}}\partial_{x_{1}}J^{s}u\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+\frac{\alpha-2}{2}}u\,dx\\\
&=-\frac{\alpha}{2}\int_{\mathbb{R}^{n}}J^{s}u\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}^{2}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+\frac{\alpha-2}{2}}u\,dx\\\
&\quad-\frac{\alpha}{2}\int_{\mathbb{R}^{n}}J^{s}u\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\right]\partial_{x_{1}}J^{s+\frac{\alpha-2}{2}}u\,dx\\\
&=\Lambda_{3}(t)+\Lambda_{4}(t).\end{split}$

As we did previously we only indicate how to estimate $\Lambda_{3}$ since for
$\Lambda_{4}$ the situation is analogous. We start by writing

$\begin{split}&\Lambda_{3}(t)=\\\
&\frac{\alpha}{2}\int_{\mathbb{R}^{n}}J^{s}\left(u\chi_{\epsilon,\tau}+u\phi_{\epsilon,\tau}+u\psi_{\epsilon}\right)\left[J^{\frac{2-\alpha}{2}};\partial_{x_{1}}^{2}\left(\chi_{\epsilon,\tau}^{2}\right)\right]J^{s+\frac{\alpha-2}{2}}\left(u\chi_{\epsilon,\tau}+u\phi_{\epsilon,\tau}+u\psi_{\epsilon}\right)dx\\\
\end{split}$

Hence by Theorem 4.3 and Lemma 5.1 we obtain

$\begin{split}|\Theta_{2,1,0,2,2}(t)|\lesssim\|J^{s}(u\chi_{\epsilon,\tau})\|_{L^{2}_{x}}^{2}+\|J^{s}(u\phi_{\epsilon,\tau})\|_{L^{2}_{x}}^{2}+\|u_{0}\|_{L^{2}_{x}}^{2}.\end{split}$

Unlike the previous step to estimate
$\|J^{s}(u\chi_{\epsilon,\tau})\|_{L^{2}}$ we only require to apply Lemma 5.5
and (3.26) we get

$\sup_{t\in(0,T)}\|J^{s}(u\chi_{\epsilon,\tau})\|_{L^{2}_{x}}^{2}<\infty.$

Instead, to estimate $\|J^{s}(u\phi_{\epsilon,\tau})\|_{L^{2}_{T}L^{2}_{x}}$
we combine 3.26 and the arguments described in (3.24)-(3.25) to obtain

$\|J^{s}(u\phi_{\epsilon,\tau})\|_{L^{2}_{T}L^{2}_{x}}<\infty.$

In summary,

$\int_{0}^{T}|\Lambda_{3}(t)|\,dt<\infty.$

The same arguments apply to estimate $\Lambda_{4}.$ Indeed,

$\int_{0}^{T}|\Lambda_{4}(t)|\,dt<\infty.$

In the case of $\Theta_{2,1,1}$ the arguments described in (3.37)-(3.39)
implies together with (3.44) and (3.45) that

$\int_{0}^{T}|\Theta_{2,1,1}(t)|\,dt<c$

for some positive constant $c.$

Hence, we focus our attention on $\Theta_{2,1,m+1},$ the error term containing
$r_{m-\alpha-1}(x,D)$ satisfies in virtue of Theorem 4.3

$\int_{0}^{T}|\Theta_{2,1,m+1}(t)|\,dt\lesssim
T\|u_{0}\|_{L^{2}_{x}}\|u\|_{L^{\infty}_{T}H^{s_{n}+}_{x}}.$

Under the conditions (3.18)- (3.19) we there exist $\lambda>0,$ such that

$\begin{split}&\lambda\left(\int_{\mathbb{R}^{n}}\left(J^{s+1}u\right)^{2}\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\,dx+\int_{\mathbb{R}^{n}}\left(J^{s-2}\partial_{x_{1}}u\right)^{2}\partial_{x_{1}}\left(\chi_{\epsilon,\tau}^{2}\right)\,dx\right)\\\
&\leq\Theta_{2,1,1,1}(t)+\Lambda_{3,1}(t)+\Theta_{2,1,0,3}(t).\end{split}$

The analysis of this term remains unchained from the one in the previous
steps.

Finally, we gather the estimates above from where we find that for any
$\epsilon>0$ and $\tau\geq 5\epsilon,$

(3.46)
$\begin{split}&\sup_{0<t<T}\int_{\mathbb{R}^{n}}\left(J^{s+\frac{2-\alpha}{2}}u(x,t)\right)^{2}\chi_{\epsilon,\tau}^{2}(\nu\cdot
x+\omega t)\,dx\\\
&\qquad+\int_{0}^{T}\int_{\mathbb{R}^{n}}(J^{s+1}u(x,t))^{2}\left(\chi_{\epsilon,\tau}\chi_{\epsilon,\tau}^{\prime}\right)(\nu\cdot
x+\omega t)\,dx\,dt\leq c.\end{split}$

This estimate finishes the inductive argument, and we conclude the proof. ∎

The attentive reader might naturally wonder if it is possible to obtain a
regularity propagation of regularity result as the one proved previously if
the dispersion is weaker when compared with that of the ZK equation. More
precisely, if we consider the equation

$\partial_{t}u-\partial_{x_{1}}\left(-\Delta\right)^{\frac{\alpha}{2}}u+u\partial_{x_{1}}u=0,\qquad
0<\alpha<1.$

This question was firstly addressed in [38] in the one dimensional case (see
[41] for a more detailed exposition). It was proved in[38] that even in the
case that the dispersion is too weak, the propagation of regularity phenomena
occurs. Although, for higher dimensions this question has not been addressed
before since it was unknown how to obtain Kato’s smoothing in these cases.

######  Remark 3.1.

The arguments described in Lemma 1.1 are strong enough and allows us to
describe

##  4\. Appendix A

The following appendix intends to provide a summary of the main results of
Pseudo differential operators we use in this work.

######  Definition 4.1.

Let $m\in\mathbb{R}.$ Let $\mathbb{S}^{m}(\mathbb{R}^{n}\times\mathbb{R}^{n})$
denote the set of functions $a\in
C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ such that for all $\alpha$ and
all $\beta$ multi-index

$\left|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)\right|\lesssim_{\alpha,\beta}(1+|\xi|)^{m-|\beta|},\quad\mbox{for
all}\quad x,\xi\in\mathbb{R}^{n}.$

An element $a\in\mathbb{S}^{m}(\mathbb{R}^{n}\times\mathbb{R}^{n})$ is called
a _symbol of order $m.$_

######  Remark 4.1.

For the sake of simplicity in the notation from here on we will suppress the
dependence of the space $\mathbb{R}^{n}$ when we make reference to a symbol in
a particular class.

######  Remark 4.2.

For $m\in\mathbb{R},$ the class of symbols $\mathbb{S}^{m}$ can be described
as

$\mathbb{S}^{m}=\left\\{a(x,\xi)\in
C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})\,|\,|a|_{\mathbb{S}^{m}}^{(j)}<\infty,\,j\in\mathbb{N}\right\\},$

where

$|a|_{\mathbb{S}^{m}}^{(j)}:=\sup_{x,\xi\in\mathbb{R}^{n}}\left\\{\left\|\langle\xi\rangle^{|\alpha|-m}\partial_{\xi}^{\alpha}\partial_{x}^{\beta}a(\cdot,\cdot)\right\|_{L^{\infty}_{x,\xi}}\Big{|}\alpha,\beta\in\mathbb{N}_{0}^{n},\,|\alpha+\beta|\leq
j\right\\}.$

######  Definition 4.2.

_A pseudo-differential operator_ is a mapping $f\mapsto\Psi f$ given by

$(\Psi
f)(x)=\int_{\mathbb{R}^{n}}e^{2\pi\mathrm{i}x\cdot\xi}a(x,\xi)\widehat{f}(\xi)\,d\xi,$

where $a(x,\xi)$ is the symbol of $\Psi.$

######  Remark 4.3.

In order to emphasize the role of the symbol $a$ we will often write
$\Psi_{a}.$ Also, we use the notation $a(x,D)$ to denote the operator
$\Psi_{a}.$

######  Definition 4.3.

If $a(x,\xi)\in\mathbb{S}^{m},$ the operator $\Psi_{a}$ is said to belong to
$\mathrm{OP\mathbb{S}^{m}.}$ More precisely, if $\nu$ is any symbol class and
$a(x,\xi)\in\nu,$ we say that $\Psi_{a}\in\mathrm{OP}\mathbb{S}^{m}.$

A quite remarkable property that pseudo-differential operators enjoy is the
existence of the adjoint operator, that is described below in terms of its
asymptotic decomposition.

######  Theorem 4.1.

Let $a\in\mathbb{S}^{m}.$ Then, there exist $a^{*}\in\mathbb{S}^{m}$ such that
$\Psi_{a}^{*}=\Psi_{a^{*}},$ and for all $N\geq 0,$

$a^{*}(x,\xi)-\sum_{|\alpha|<N}\frac{(2\pi
i)^{-|\alpha|}}{\alpha!}\partial_{\xi}^{\alpha}\partial_{x}^{\alpha}\overline{a}(x,\xi)\in\mathbb{S}^{m-N}.$

###### Proof.

See Stein [55] chapter VI or Taylor [56]. ∎

Additionally the product $\Psi_{a}\Psi_{b}$ of two operators with symbols
$a(x,\xi)$ and $b(x,\xi)$ respectively is a pseudo-differential operator
$\Psi_{c}$ with symbol $c(x,\xi).$ More precisely, the description of the
symbol $c$ is summarized in the following Theorem:

######  Theorem 4.2.

Suppose $a$ and $b$ symbols belonging to $\mathbb{S}^{m}$ and $\mathbb{S}^{r}$
respectively. Then, there is a symbol $c$ in $\mathbb{S}^{m+r}$ so that

$\Psi_{c}=\Psi_{a}\circ\Psi_{b}.$

Moreover,

$c\sim\sum_{\alpha}\frac{(2\pi
i)^{-|\alpha|}}{\alpha!}\partial_{\xi}^{\alpha}a\partial^{\alpha}_{x}b,$

in the sense that

$c-\sum_{|\alpha|<N}\frac{(2\pi
i)^{-|\alpha|}}{\alpha!}\partial_{\xi}^{\alpha}a\,\partial^{\alpha}_{x}b\in\mathbb{S}^{m+r-N},\quad\mbox{for
all integer}\quad N,\,N\geq 0.$

###### Proof.

For the proof see Stein [55] chapter VI or Taylor [56]. ∎

######  Remark 4.4.

Note that $c-ab\in\mathbb{S}^{m+r-1}.$ Moreover, each symbol of the form
$\partial_{\xi}^{\alpha}a\,\partial_{x}^{\alpha}b$ lies in the class
$\mathbb{S}^{m+r-|\alpha|}.$

A direct consequence of the decomposition above is that it allows to describe
explicitly up to an error term, operators such as commutators between pseudo-
differential operators as is described below:

######  Proposition 1.

For $a\in\mathbb{S}^{m}$ and $b\in\mathbb{S}^{r}$ we define the commutator
$\left[\Psi_{a};\Psi_{b}\right]$ by

$\left[\Psi_{a};\Psi_{b}\right]=\Psi_{a}\Psi_{b}-\Psi_{b}\Psi_{a}.$

Then, the operator
${\displaystyle\left[\Psi_{a};\Psi_{b}\right]\in\mathrm{OP}\mathbb{S}^{m+r-1},}$
has by principal symbol the Poisson bracket, i.e,

$\sum_{|\alpha|=1}^{n}\frac{1}{2\pi
i}\left(\partial_{\xi}^{\alpha}a\,\partial_{x}^{\alpha}b-\partial_{x}^{\alpha}a\,\partial_{\xi}^{\alpha}b\right)\,\,\mathrm{mod}\,\,\mathbb{S}^{m+r-2}.$

Also, certain class the pseudo-differential operators enjoy of some continuity
properties as the described below.

###### Proof.

The proof can be consulted in Stein [55] Chapter VI, Theorem 1. ∎

An interesting an useful continuity result in the Sobolev spaces is decribed
below.

######  Theorem 4.3.

Let $m\in\mathbb{R},a\in\mathbb{S}^{m},$ and $s\in\mathbb{R}$. Then, the
operator $\Psi_{a}$ extends to a bounded linear operator from
$H^{s+m}(\mathbb{R}^{n})$ to $H^{s}(\mathbb{R}^{n}).$ Moreover, there exists
$j=j(n;m,s)\in\mathbb{N}$ and $c=c(n;m;s)>0$ such that

$\left\|\Psi_{a}f\right\|_{H^{s}_{x}}\leq
c|a|_{\mathbb{S}^{m}}^{(j)}\|f\|_{H^{s+m}_{x}}$

###### Proof.

See Kumano-go [30] or Stein [55], Chapter VI. ∎

An alternative formula for the Bessel kernel.

######  Lemma 4.1.

Let $0<\delta<n+1,$ and $f$ be a tempered distribution. Then
$J^{-\delta}f=\mathcal{B}_{\delta}*f,$ where

$\mathcal{B}_{\delta}(y)=\frac{1}{(2\pi)^{\frac{n-1}{2}}2^{\frac{\delta}{2}}\Gamma\left(\frac{\delta}{2}\right)\Gamma\left(\frac{n-\delta+1}{2}\right)}e^{-|y|}\int_{0}^{\infty}e^{-|y|s}\left(s+\frac{s^{2}}{2}\right)^{\frac{n-\delta-1}{2}}\,ds$

###### Proof.

See Calderon & Zygmund [7], Lemma 4.1. ∎

######  Lemma 4.2.

The function $\Theta_{\delta}$ is non-negative and it satisfies the following
properties:

  * (a)

For $0<\delta<n,$

$\mathcal{B}_{\delta}(x)\lesssim_{\delta}e^{-|x|}\left(1+|x|^{\delta-n}\right);$

  * (b)

for $\delta=n$

$\mathcal{B}_{\delta}(x)\lesssim
e^{-|x|}\left(1+\log^{+}\left(\frac{1}{|x|}\right)\right);$

  * (c)

for $\beta$ multi-index with $|\beta|>0$ and $0<\delta<n+1$

$\left|\partial_{x}^{\beta}\mathcal{B}_{\delta}(x)\right|\leq
c_{\beta,\delta}e^{-|x|}\left(1+|x|^{-n+\delta-|\beta|}\right),\quad x\neq 0.$

###### Proof.

We refer to Calderon & Zygmund [7], Lemma 4.2. ∎

Also, the behavior of the Bessel potentials in the following cases is
necessary for our arguments.

######  Lemma 4.3.

Let $\delta>0$. The function $\mathcal{B}_{\delta}$ satisfies the following
estimates:

  * (i)

For $\delta<n$ and $|y|\rightarrow 0,$

(4.1)
$\mathcal{B}_{\delta}(y)\approx\left(\frac{\pi^{\frac{n}{2}}\Gamma\left(\frac{n-\delta}{2}\right)}{2^{\delta-n}\Gamma\left(\frac{\delta}{2}\right)}|2\pi
y|^{\tau-n}\right).$

  * (ii)

For $\delta=n$ and $|y|\rightarrow 0,$

(4.2)
$\mathcal{B}_{\delta}(y)\approx\frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}\right)}\log\left(\frac{1}{|2\pi
y|}\right).$

  * (iii)

For $\delta>n$ and $|y|\rightarrow 0,$

(4.3)
$\mathcal{B}_{\delta}(y)\approx\frac{\pi^{\frac{n}{2}}\Gamma\left(\frac{\delta-n}{2}\right)}{\Gamma\left(\frac{\delta}{2}\right)}.$

  * (iv)

For $\delta>0$ and $|y|\rightarrow\infty,$

(4.4)
$\mathcal{B}_{\delta}(y)\approx\frac{(2\pi)^{\frac{n}{2}}}{2^{\frac{\alpha-1}{2}}\pi^{-\frac{1}{2}}\Gamma\left(\frac{\delta}{2}\right)}|2\pi|^{\frac{\alpha-n-1}{2}}e^{-|2\pi
y|}.$

###### Proof.

For the proof see Aronszajn & Smith [4]. ∎

##  5\. Appendix B

In this section we present some localization tools that are quite useful to
describe the regularity phenomena we are working on.

######  Lemma 5.1.

Let $\Psi_{a}\in\mathrm{OP\mathbb{S}^{r}}.$ Let
$\alpha=\left(\alpha_{1},\alpha_{2},\dots,\alpha_{n}\right)$ be a multi-index
with $|\alpha|\geq 0.$ If $f\in L^{2}(\mathbb{R}^{n})$ and $g\in
L^{p}(\mathbb{R}^{n}),\,p\in[2,\infty]$ with

(5.1)
$\operatorname{\mathrm{dist}}\left(\operatorname{\mathrm{supp}}(f),\operatorname{\mathrm{supp}}(g)\right)\geq\delta>0,$

then,

$\left\|g\partial_{x}^{\alpha}\Psi_{a}f\right\|_{L^{2}}\lesssim\|g\|_{L^{p}}\|f\|_{L^{2}},$

where
$\partial_{x}^{\alpha}:=\partial_{x_{1}}^{\alpha_{1}}\dots\partial_{x_{n}}^{\alpha_{n}}.$

###### Proof.

See Mendez [39]. ∎

The next result can be proved by using the ideas from the proof of Lemma
above. Nevertheless, for the reader convenience we describe the main details.

######  Corollary 5.1.

Let $f,g$ be functions such that

(5.2)
$\operatorname{\mathrm{dist}}\left(\operatorname{\mathrm{supp}}(f),\operatorname{\mathrm{supp}}(g)\right)=\delta>0.$

Then, the operator

$\left(\mathcal{T}_{\psi}f\right)^{\widehat{}}(\xi):=\psi(\xi)\widehat{f}(\xi),\,f\in\mathcal{S}(\mathbb{R}^{n})$

where $\psi$ is defined as in (2.27), that is,

$\psi(\xi)=\sum_{j=1}^{\infty}{\alpha/2\choose j}\langle
2\pi\xi\rangle^{2-2j},\quad\xi\in\mathbb{R}^{n}.$

If $f\in L^{p}(\mathbb{R}^{n})$ and $g\in L^{2}(\mathbb{R}^{n}),$ then

$\left\|\left[\mathcal{T}_{\psi};f\right]g\right\|_{L^{2}}\lesssim\|f\|_{L^{p}}\|g\|_{L^{2}},$

for $p\in[2,\infty].$

###### Proof.

According to condition (5.2) we have

$\left(\left[\mathcal{T}_{\psi};f\right]g\right)(x)=\sum_{j=2}^{\infty}{\alpha/2\choose
j}f(x)\int_{\\{|x-y|>\delta\\}}g(y)\mathcal{B}_{2j-2}(x-y)\,dy.$

In virtue of Lemma 4.3 is clear that for $j>1,$

$\begin{split}&\left|f(x)\int_{\\{|x-y|>\delta\\}}g(y)\mathcal{B}_{2j-2}(x-y)\,dy\right|\\\
&\lesssim_{n}\frac{1}{2^{j}(j-2)!}\int_{\\{|x-y|>\delta\\}}|f(x)||g(y)||2\pi(x-y)|^{\frac{2j-3-n}{2}}e^{-2\pi|x-y|}\,dy.\\\
\end{split}$

Thus, by Young’s inequality

$\begin{split}\left\|\left[\mathcal{T}_{\psi};f\right]g\right\|_{L^{2}}&\lesssim_{n}\|f\|_{L^{p}}\|g\|_{L^{2}}\left(\sum_{j=2}^{\infty}\frac{\Gamma\left(j+\frac{n-3}{2}\right)}{(j-2)!j^{1+\alpha}}\right)\\\
&\lesssim_{n,\alpha}\|f\|_{L^{p}}\|g\|_{L^{2}},\end{split}$

for $p\in[2,\infty].$ ∎

The following formulas where firstly obtained by Linares, Kenig, Ponce and
Vega in the one dimensional case in their study about propagation of
regularity fo solutions of the KdV equation. Nevertheless, these results where
later extended to dimension $n,n\geq 2$ in the work of [39] for solutions of
the Zakharov-Kuznetsov equation.

######  Lemma 5.2 (Localization formulas).

Let $f\in L^{2}(\mathbb{R}^{n}).$ Let
$\nu=(\nu_{1},\nu_{2},\dots,\nu_{n})\in\mathbb{R}^{n}$ a non-null vector such
that $\nu_{j}\geq 0,\,j=1,2,\dots,n.$ Let $\epsilon>0,$ we consider the
function $\varphi_{\nu,\epsilon}\in C^{\infty}(\mathbb{R}^{n})$ to satisfy:
$0\leq\varphi_{\nu,\epsilon}\leq 1,$

$\varphi_{\nu,\epsilon}(x)=\begin{cases}0\quad\mbox{if}\quad&x\in\mathcal{H}_{\left\\{\nu,\frac{\epsilon}{2}\right\\}}^{c}\\\
1\quad\mbox{if}\quad&x\in\mathcal{H}_{\\{\nu,\epsilon\\}}\end{cases}$

and the following increasing property: for every multi-index $\alpha$ with
$|\alpha|=1$

$\partial^{\alpha}_{x}\varphi_{\nu,\epsilon}(x)\geq 0,\quad
x\in\mathbb{R}^{n}.$

  * (I)

If $m\in\mathbb{Z}^{+}$ and $\varphi_{\nu,\epsilon}J^{m}f\in
L^{2}(\mathbb{R}^{n}),$ then for all $\epsilon^{\prime}>2\epsilon$ and all
multi-index $\alpha$ with $0\leq|\alpha|\leq m,$ the derivatives of $f$
satisfy

$\varphi_{\nu,\epsilon^{\prime}}\partial^{\alpha}_{x}f\in
L^{2}(\mathbb{R}^{n}).$

  * (II)

If $m\in\mathbb{Z}^{+}$ and $\varphi_{\nu,\epsilon}\,\partial^{\alpha}_{x}f\in
L^{2}(\mathbb{R}^{n})$ for all multi-index $\alpha$ with $0\leq|\alpha|\leq
m,$ then for all $\epsilon^{\prime}>2\epsilon$

$\varphi_{\nu,\epsilon^{\prime}}J^{m}f\in L^{2}(\mathbb{R}^{n}).$

  * (III)

If $s>0,$ and $J^{s}(\varphi_{\nu,\epsilon}f)\in L^{2}(\mathbb{R}^{n}),$ then
for any $\epsilon^{\prime}>2\epsilon$

$\varphi_{\nu,\epsilon^{\prime}}\,J^{s}f\in L^{2}(\mathbb{R}^{n}).$

  * (IV)

If $s>0,$ and $\varphi_{\nu,\epsilon}J^{s}f\in L^{2}(\mathbb{R}^{n}),$ then
for any $\epsilon^{\prime}>2\epsilon$

$J^{s}\left(\varphi_{\nu,\epsilon^{\prime}}f\right)\in L^{2}(\mathbb{R}^{n}).$

###### Proof.

See [39]. ∎

A more general version that we are going to use in this work.

######  Lemma 5.3.

Let $f\in L^{2}(\mathbb{R}^{n}).$ If $\theta_{1},\theta_{2}\in
C^{\infty}(\mathbb{R}^{n})$ are functions such that:
$0\leq\theta_{1},\theta_{2}\leq 1,$ their respective supports satisfy

$\operatorname{\mathrm{dist}}\left(\operatorname{\mathrm{supp}}\left(1-\theta_{1}\right),\operatorname{\mathrm{supp}}\left(\theta_{2}\right)\right)\geq\delta,$

for some positive number $\delta,$ and for all multi-index $\beta,$ the
functions $\partial_{x}^{\beta}\theta_{1},\partial_{x}^{\beta}\theta_{2}\in
L^{\infty}(\mathbb{R}^{n}).$

Then, the following identity holds:

  * (I)

If $m\in\mathbb{Z}^{+}$ and $\theta_{1}J^{m}f\in L^{2}(\mathbb{R}^{n}),$ then
for all multi-index $\alpha$ with $0\leq|\alpha|\leq m,$ the derivatives of
$f$ satisfy

$\theta_{2}\partial^{\alpha}_{x}f\in L^{2}(\mathbb{R}^{n}).$

  * (II)

If $m\in\mathbb{Z}^{+}$ and $\theta_{1}\,\partial^{\alpha}_{x}f\in
L^{2}(\mathbb{R}^{n})$ for all multi-index $\alpha$ with $0\leq|\alpha|\leq
m,$ then

$\theta_{2}J^{m}f\in L^{2}(\mathbb{R}^{n}).$

  * (III)

If $s>0,$ and $J^{s}(\theta_{1}f)\in L^{2}(\mathbb{R}^{n}),$ then

$\theta_{2}\,J^{s}f\in L^{2}(\mathbb{R}^{n}).$

  * (IV)

If $s>0,$ and $\theta_{1}J^{s}f\in L^{2}(\mathbb{R}^{n}),$ then

$J^{s}\left(\theta_{2}f\right)\in L^{2}(\mathbb{R}^{n}).$

###### Proof.

See [39]. ∎

######  Lemma 5.4.

Let $\Psi_{a}\in\mathrm{OP\mathbb{S}^{0}}.$ Let
$\theta_{1},\theta_{2}:\mathbb{R}^{n}\longrightarrow\mathbb{R}$ be smooth
functions such that

$\operatorname{\mathrm{dist}}\left(\operatorname{\mathrm{supp}}\left(1-\theta_{1}\right),\operatorname{\mathrm{supp}}\theta_{2}\right)>\delta,$

for some $\delta>0.$

Assume that $f\in H^{s}(\mathbb{R}^{n}),\,s<0.$ If $\theta_{1}f\in
L^{2}(\mathbb{R}^{n}),$ then

$\theta_{2}\Psi_{a}f\in L^{2}(\mathbb{R}^{n}).$

###### Proof.

For the one dimensional case see [25] and the extension to the $n-$dimensional
case see [39]. ∎

######  Lemma 5.5.

Let $f\in L^{2}(\mathbb{R}^{n})$ and
$\nu=(\nu_{1},\nu_{2},\dots,\nu_{n})\in\mathbb{R}^{n}$ such that $\nu_{1}>0,$
for $j=1,2,\dots,n.$ Also assume that

$J^{s}f\in L^{2}\left(\mathcal{H}_{\\{\alpha,\nu\\}}\right),\quad s>0.$

Then, for any $\epsilon>0$ and any $r\in(0,s]$

$J^{r}f\in L^{2}\left(\mathcal{H}_{\\{\alpha+\epsilon,\nu\\}}\right).$

###### Proof.

For the one dimensional case see [25] and the extension to the $n-$dimensional
case see [39]. ∎

######  Theorem 5.1.

Let $s>0$ and $f,g\in\mathcal{S}(\mathbb{R}^{n}).$ Then,

$\left\|\left[J^{s};g\right]f\right\|_{L^{2}}\lesssim\|J^{s-1}f\|_{L^{2}}\|\nabla
g\|_{L^{\infty}}+\|J^{s}g\|_{L^{2}}\|f\|_{L^{\infty}},$

where the implicit constant does not depends on $f$ nor $g.$

###### Proof.

For the proof see the appendix in Kato and Ponce [24]. ∎

Also, the following Leibniz rule for the operator $J^{s}$ is quite useful in
our arguments

######  Theorem 5.2.

Let $s>\frac{n}{2}$ and $f,g\in\mathcal{S}(\mathbb{R}^{n}),$ then

$\|J^{s}(f\cdot
g)\|_{L^{2}}\lesssim\|J^{s}f\|_{L^{2}}\|g\|_{L^{\infty}}+\|J^{s}g\|_{L^{2}}\|f\|_{L^{\infty}},$

where the implicit constant does not depends on $f$ nor $g.$

###### Proof.

See the appendix in Kato and Ponce [24]. ∎

##  6\. Acknowledgment

I shall thank R. Freire for reading an early version of this document and its
helpful suggestions. I also would like to thank Prof. F. Linares for pointing
out several references and the suggestions in the presentation of this work. I
also thanks O. Riaño for calling my attention to this problem.

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