#  A note on Strichartz estimates for the wave equation with orthonormal
initial data

Neal Bez  Department of Mathematics, Graduate School of Science and
Engineering, Saitama University, Saitama 338-8570, Japan
<EMAIL_ADDRESS>,  Shinya Kinoshita  Department of Mathematics,
Tokyo Institute of Technology, Meguro-ku, Tokyo, 152-8551, Japan
<EMAIL_ADDRESS>and  Shobu Shiraki  Departamento de Matemática,
Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal
<EMAIL_ADDRESS>
###### Abstract.

This note is concerned with Strichartz estimates for the wave equation and
orthonormal families of initial data. We provide a survey of the known results
and present what seems to be a reasonable conjecture regarding the cases which
have been left open. We also provide some new results in the maximal-in-space
boundary cases.

###### Key words and phrases:

###### 2010 Mathematics Subject Classification:

This work was supported by JSPS Kakenhi grant numbers 19H00644, 19H01796,
22H00098 and 23H01080 (Bez), 21J00514, 22KJ0446 (Kinoshita), and 19H01796,
22H00098 (Shiraki). The third author is also supported by Centro de Análise
Matemática, Geometria e Sistemas Dinâmicos (CAMGSD)

##  1\. Introduction

The (one-sided) wave propagator $e^{it\sqrt{-\Delta}}$ is given by

$e^{it\sqrt{-\Delta}}f(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{i(x\cdot\xi+t|\xi|)}\widehat{f}(\xi)\,\mathrm{d}\xi$

for sufficiently nice initial data $f$ and we often write
$Uf(t,x)=e^{it\sqrt{-\Delta}}f(x)$. Our interest here are Strichartz estimates
of the form

(1.1)
$\bigg{\|}\sum_{j}\lambda_{j}|Uf_{j}|^{2}\bigg{\|}_{L^{\frac{q}{2}}_{t}L^{\frac{r}{2}}_{x}}\lesssim\|\lambda\|_{\ell^{\beta}}$

where $(f_{j})_{j}$ is a (possibly infinite) family of orthonormal functions
in the homogeneous Sobolev space $\dot{H}^{s}(\mathbb{R}^{d})$,
$s=\frac{d}{2}-\frac{d}{r}-\frac{1}{q}$ (we refer the reader forward to the
end of this section for the meaning of $\lesssim$). Such an estimate obviously
implies the classical Strichartz estimate

(1.2) $\|Uf\|_{L^{q}_{t}L^{r}_{x}}\lesssim\|f\|_{\dot{H}^{s}}$

by considering a family of data for which all but one member is zero, and
rescaling. Therefore, in what follows, we restrict attention to exponents $q$
and $r$ which admit a classical Strichartz estimate. Conversely, it is also
clear that (1.2) implies (1.1) with $\beta=1$ by simply using the triangle
inequality (and, in particular, not making use of the orthogonality) and so
the challenge is to establish (1.1) for a range of $\beta$ (depending on $q$
and $r$) which is as large as possible.

Before going forward to describe the current situation regarding (1.1), we
pause to briefly clarify which exponents $q$ and $r$ admit a classical
Strichartz estimate (1.2). For $d\geq 2$, let us say that
$(q,r)\in[2,\infty]\times[2,\infty]$ is _admissible_ if

$\frac{1}{q}\leq\frac{d-1}{2}\bigg{(}\frac{1}{2}-\frac{1}{r}\bigg{)}.$

Then it is widely known that if $(q,r)$ is admissible and $r<\infty$ then
(1.2) holds (see, for example, [15] and the references therein). The boundary
case

(1.3)
$\|Uf\|_{L^{q}_{t}L^{\infty}_{x}}\lesssim\|f\|_{\dot{H}^{\frac{d}{2}-\frac{1}{q}}}$

presents difficulties. Indeed, it is known that (1.3) fails when $(q,d)=(4,2)$
and when $q=2$ for all $d\geq 3$ (see [8, Theorem 3] for the former claim and
[13, Theorem 1.1] for the latter). Also (1.3) fails when $q=\infty$ thanks to
the failure of the associated Sobolev embedding. For the remaining $q$ (that
is, $q\in(4,\infty)$ when $d=2$, or $q\in(2,\infty)$ when $d\geq 3$) estimate
(1.3) holds; see, for example [8, Theorem 3].

Returning to the wider framework in (1.1), Frank and Sabin [10] provided the
first progress by establishing this estimate in the sharp admissible case

(1.4) $\frac{1}{q}=\frac{d-1}{2}\bigg{(}\frac{1}{2}-\frac{1}{r}\bigg{)}$

with $q=r=\frac{2(d+1)}{d-1}$ and $\beta\leq\frac{2r}{r+2}$. Formally
interpolating with the trivial estimate with $\beta=1$ at $(q,r)=(\infty,2)$,
this yields (1.1) in the sharp admissible case (1.4) for
$r\in[2,\frac{2(d+1)}{d-1}]$ and $\beta\leq\frac{2r}{r+2}$. As far as we are
aware, this interpolation argument seems to require the use of an analytic
family of operators (since the regularity exponent $s$ varies); we refer the
reader forward to Section 2 for arguments of a similar nature.

To discuss results which have been established since [10], it is very
convenient to introduce the notation $\beta_{\sigma}(q,r)$, where $\sigma>0$,
for the exponent satisfying

$\frac{\sigma}{\beta_{\sigma}(q,r)}=\frac{1}{q}+\frac{2\sigma}{r}.$

Observe that $\beta_{\sigma}(q,r)=\frac{2r}{r+2}$ in the case
$\frac{1}{q}=\sigma(\frac{1}{2}-\frac{1}{r})$, and so the result in [10] may
be viewed as the statement that (1.1) holds for
$\beta\leq\beta_{\frac{d-1}{2}}(q,r)$ whenever (1.4) and
$r\in[2,\frac{2(d+1)}{d-1}]$. It is also trivial to see that
$\beta_{\sigma}(q,r)$ is increasing as a function of $\sigma$ for each fixed
$(q,r)$.

In terms of upper bounds on the optimal value of the exponent $\beta$, it was
established in [5] that

(1.5) $\beta\leq\min\bigg{\\{}\beta_{\frac{d}{2}}(q,r),\frac{q}{2}\bigg{\\}}$

is necessary in order for (1.1) to be true. For completeness, in the Appendix
of this note, we give a sketch of the proof. One interesting consequence is
that there is no hope of raising $\beta$ above 1 in the case where $q=2$ and
so we omit this case from the subsequent discussion. We also note that
$\beta_{\frac{d}{2}}(q,r)\leq\frac{q}{2}$ if and only if
$\frac{d}{r}\leq\frac{d-1}{q}$.

The result in [10] was extended in [5] and as a result the following is now
known to be true.

######  Theorem 1.1 (see [5, 10]).

Let $d\geq 2$, $q\in(2,\infty),r\in[2,\infty)$, and
$\frac{1}{q}\leq\frac{d-1}{2}(\frac{1}{2}-\frac{1}{r})$. Then the estimate
(1.1) holds in each of the following cases.

_(i)_ $\beta\leq\beta_{\frac{d-1}{2}}(q,r)$ when
$\frac{d-1}{r}>\frac{d-2}{q}$.

_(ii)_ $\beta<\frac{q}{2}$ when $\frac{d-1}{r}\leq\frac{d-2}{q}$.

It is perhaps beneficial to view this result with Figures 1 and 2 in mind. The
points $A,B,C,D,E$ are given as follows.

$\displaystyle A=\bigg{(}\frac{d-2}{2d},\frac{d-1}{2d}\bigg{)},\quad
B=\bigg{(}\frac{d-1}{2(d+1)},\frac{d-1}{2(d+1)}\bigg{)},$ $\displaystyle
C=\bigg{(}\frac{1}{2},0\bigg{)},\quad D=\bigg{(}0,\frac{1}{2}\bigg{)},\quad
E=\bigg{(}\frac{d-3}{2(d-1)},\frac{1}{2}\bigg{)}.$

When $(\frac{1}{r},\frac{1}{q})$ belongs to the interior of the region $OAC$
or the line segment $(A,C]$ (i.e. not including the endpoint at $A$), then
Theorem 1.1 guarantees that (1.1) holds for
$\beta\leq\beta_{\frac{d-1}{2}}(q,r)$. For $d\geq 3$ and in the interior of
region $ODEA$, or the line segments $(O,A]$, $[A,E)$, we have that (1.1) holds
for $\beta<\frac{q}{2}$. Thanks to (1.5), this means that Theorem 1.1(ii) is
sharp up to the critical case $\beta=\frac{q}{2}$.

###  1.1. A conjecture

The most pressing issue appears to be closing the gap between the necessary
condition (1.5) and the sufficient condition
$\beta\leq\beta_{\frac{d-1}{2}}(q,r)$ given in Theorem 1.1(i). For this it
seems reasonable to us to believe that Theorem 1.1(i) can be improved up to
the necessary condition in (1.5). This amounts to the following conjecture.

######  Conjecture 1.2.

Let $d\geq 2$, $q\in(2,\infty),r\in[2,\infty)$, and
$\frac{1}{q}\leq\frac{d-1}{2}(\frac{1}{2}-\frac{1}{r})$. Then the estimate
(1.1) holds in each of the following cases.

_(i)_ $\beta\leq\beta_{\frac{d}{2}}(q,r)$ when $\frac{d}{r}>\frac{d-1}{q}$.

_(ii)_ $\beta<\frac{q}{2}$ when $\frac{d}{r}\leq\frac{d-1}{q}$.

In Figures 1 and 2, the following points also arise:

$A^{\prime}=\bigg{(}\frac{d-1}{2(d+1)},\frac{d}{2(d+1)}\bigg{)},\quad
E^{\prime}=\bigg{(}\frac{d-2}{2d},\frac{1}{2}\bigg{)},\quad
F=\bigg{(}\frac{(d-1)^{2}}{2(d^{2}+1)},\frac{d(d-1)}{2(d^{2}+1)}\bigg{)}.$

Conjecture 1.2(i) says that (1.1) holds for
$\beta\leq\beta_{\frac{d}{2}}(q,r)$ in the interior of $OFC$ and the line
segment $[C,F)$, and Conjecture 1.2(ii) says that (1.1) holds for
$\beta<\frac{q}{2}$ in the interior of $ODEF$, or $(O,F)$, or $[F,E)$.

Observe that the line segment $[C,E^{\prime}]$ corresponds to the sharp
admissible line

$\frac{1}{q}=\frac{d}{2}\bigg{(}\frac{1}{2}-\frac{1}{r}\bigg{)}$

for the Strichartz estimates associated with the Schrödinger propagator
$e^{it\Delta}$. The analogue of (1.1) for Schrödinger propagator takes the
form

(1.6)
$\bigg{\|}\sum_{j}\lambda_{j}|e^{it\Delta}f_{j}|^{2}\bigg{\|}_{L^{\frac{q}{2}}_{t}L^{\frac{r}{2}}_{x}}\lesssim\|\lambda\|_{\ell^{\beta}}$

where $(f_{j})_{j}$ is a family of orthonormal functions in the homogeneous
Sobolev space $\dot{H}^{s}(\mathbb{R}^{d})$,
$s=\frac{d}{2}-\frac{d}{r}-\frac{2}{q}$, and interestingly this family of
estimates is much better understood than (1.1). In particular, it is known
(see [9, 10, 11, 4]) that for $q\in(2,\infty),r\in[2,\infty)$, and
$\frac{1}{q}\leq\frac{d}{2}(\frac{1}{2}-\frac{1}{r})$, the estimate (1.6)
holds in each of the following cases.

_(i)_ $\beta\leq\beta_{\frac{d}{2}}(q,r)$ when $\frac{d}{r}>\frac{d-1}{q}$
(i.e. the interior of $OA^{\prime}C$ and $[C,A^{\prime})$).

_(ii)_ $\beta<\frac{q}{2}$ when $\frac{d}{r}\leq\frac{d-1}{q}$ (i.e. the
interior of $ODE^{\prime}A^{\prime}$ and the line segments $(O,A^{\prime}]$,
$[A^{\prime},E^{\prime})$).

Furthermore, the restriction in (1.5) is also known to be necessary for (1.6)
too. Thus, apart from certain critical/boundary cases, we have an almost
complete understanding of when the estimates (1.6) are valid. We refer the
reader to Frank _et. al_ [9] for the origin of this line of study for the
Schrödinger equation, and to [10, 11, 4] for further developments on (1.6).

$\tfrac{1}{q}$$\tfrac{1}{r}$$E$$E^{\prime}$$D$$A$$A^{\prime}$$C$$B$$F$$\frac{d-3}{2(d-1)}$$\frac{d-2}{2d}$$\frac{1}{2}$$\frac{1}{4}$$O$$\frac{1}{2}$

Figure 1. The case $d\geq 3$ (note that $D=E$ when $d=3$). On the lines
$[C,E]$ and $[O,A]$ we have
$\frac{1}{q}=\frac{d-1}{2}(\frac{1}{2}-\frac{1}{r})$ and
$\frac{d-2}{q}=\frac{d-1}{r}$, respectively.

$\tfrac{1}{q}$$\tfrac{1}{r}$$A$$A^{\prime}$$C$$F$$E^{\prime}$$B$$\frac{1}{2}$$\frac{1}{4}$$O$$\frac{1}{2}$

Figure 2. The case $d=2$. On the line $[C,A]$ we have
$\frac{1}{q}=\frac{1}{2}(\frac{1}{2}-\frac{1}{r})$.

$\tfrac{1}{\beta}$$\tfrac{1}{r}$$1$$0$$\frac{1}{2}$$\tfrac{d-2}{2d}$$\frac{d-1}{d}$$\frac{1}{r_{d}}$$\frac{d(d-1)}{d^{2}+1}$$\frac{d-3}{2(d-1)}$$?$

Figure 3. For $d\geq 3$, the gap between the necessary condition (1.5)
(represented by the dark gray region) and Theorem 1.1(i) (represented by the
light gray region) in the sharp admissible case
$\frac{1}{q}=\frac{d-1}{2}(\frac{1}{2}-\frac{1}{r})$. The exponent $r_{d}$ is
given by $r_{d}=\frac{2(d^{2}+1)}{(d-1)^{2}}$.

$\tfrac{1}{\beta}$$\tfrac{1}{r}$$1$$\frac{1}{2}$$0$$\frac{1}{2}$$?$$\frac{2}{5}$$\frac{1}{10}$

Figure 4. For $d=2$, the gap between the necessary condition (1.5)
(represented by the dark gray region) and Theorem 1.1(i) (represented by the
light gray region) in the sharp admissible case
$\frac{1}{q}=\frac{1}{2}(\frac{1}{2}-\frac{1}{r})$.

###  1.2. Boundary cases

The boundary cases $q=\infty$ and $r=\infty$ have been excluded from the
statements of Theorem 1.1 and Conjecture 1.2, and for the remainder of this
note we mainly focus on the case $r=\infty$. When $q=\infty$, we may simply
use the Sobolev inequality for orthonormal functions proved by Lieb [17] to
quickly obtain (1.1) with $q=\infty$, $r\in(2,\infty)$ and
$\beta<\frac{r}{2}=\beta_{\frac{d}{2}}(\infty,r)$; in light of the necessary
condition (1.5), we see that this result is sharp up to the critical case
$\beta=\frac{r}{2}$.

When $r=\infty$, matters appear to be less simple and for the remainder of
this note we shall be concerned with the estimate

(1.7)
$\bigg{\|}\sum_{j}\lambda_{j}|Uf_{j}|^{2}\bigg{\|}_{L^{\frac{q}{2}}_{t}L^{\infty}_{x}}\lesssim\|\lambda\|_{\ell^{\beta}},$

where $(f_{j})_{j}$ is a family of orthonormal functions in
$\dot{H}^{\frac{d}{2}-\frac{1}{q}}(\mathbb{R}^{d})$. Also we recall from our
discussion of the classical Strichartz estimates (1.2) that we are interested
in the range $q\in(4,\infty)$ when $d=2$, and $q\in(2,\infty)$ when $d\geq 3$.
As such, we introduce the notation

$q_{d}:=\begin{cases}4\ &(d=2),\\\ 2&(d\geq 3).\end{cases}$

Our first new result is the following.

######  Theorem 1.3.

For any $d\geq 2$, the estimate (1.7) holds with $q\in(q_{d},\infty)$ and
$\beta<\frac{q}{2}$.

Although it is unclear to use whether one can extend Theorem 1.3 to the
critical case $\beta=\frac{q}{2}$, we are at least able to show the following
restricted weak-type estimates.

######  Theorem 1.4.

For any $d\geq 3$ and $q\in(q_{d},\infty)$, the estimate

(1.8)
$\Bigl{\|}\sum_{j}\lambda_{j}|Uf_{j}|^{2}\Bigr{\|}_{L_{t}^{\frac{q}{2},\infty}L_{x}^{\infty}}\lesssim\|\lambda\|_{\ell^{\frac{q}{2},1}}$

holds whenever $(f_{j})_{j}$ is a family of orthonormal functions in
$\dot{H}^{\frac{d}{2}-\frac{1}{q}}(\mathbb{R}^{d})$.

The rest of the paper is devoted to proving Theorems 1.3 and 1.4, and this is
taken up in Sections 2 and 3, respectively. Prior to that, we introduce a
small amount of notation.

### Notation

For non-negative quantities $A$, $B$, the notation $A\lesssim B$ (or $B\gtrsim
A$) denotes an estimate $A\leq CB$ for some constant $C>0$, and we write
$A\sim B$ when both $A\lesssim B$ and $B\lesssim A$.

Let $\varphi\in C_{0}^{\infty}$ be supported in the annulus
$\\{\xi\in\mathbb{R}^{n}:2^{-1}<|\xi|<2\\}$ and satisfy

$\sum_{k\in{\mathbb{Z}}}\varphi(2^{-k}\xi)=1,\qquad\xi\in{\mathbb{R}}^{d}\setminus\\{0\\}.$

Define the family of the Littlewood–Paley operators
$(P_{k})_{k\in{\mathbb{Z}}}$ by
$\widehat{P_{k}f}(\xi)=\varphi_{k}(\xi)\widehat{f}(\xi)$, where
$\varphi_{k}(\xi)=\varphi(2^{-k}\xi)$ for each $k\in\mathbb{Z}$. Also, let
$\chi\in C_{0}^{\infty}$ be supported in the ball
$\\{\xi\in\mathbb{R}^{n}:|\xi|<1\\}$ with $\chi(0)=1$.

For $1\leq p<\infty$, $1\leq q\leq\infty$, we write $L^{p,q}(\mathbb{R}^{d})$
for the associated Lorentz function space and $\ell^{p,q}$ for the associated
Lorentz sequence space. We refer the reader to [12, 21] for the definition and
wider discussion on these space. In particular, it is well known that

$\displaystyle L^{p,p}({\mathbb{R}}^{d})=L^{p}({\mathbb{R}}^{d}),\quad
L^{p,q_{1}}({\mathbb{R}}^{d})\subseteq
L^{p,q_{2}}({\mathbb{R}}^{d})\quad\text{when }q_{1}\leq q_{2}.$

##  2\. A sketch proof of Theorem 1.3

In [3], the analogue of (1.7) for the fractional Schrödinger propagators has
been obtained.

######  Theorem 2.1 (see [3]).

Let $a\in(0,\infty)\setminus\\{1\\}$ and $d\geq 1$. For $q\in(q_{d+1},\infty)$
and $\beta<\frac{q}{2}$, the estimate

(2.1)
$\bigg{\|}\sum_{j}\lambda_{j}|e^{it(-\Delta)^{\frac{a}{2}}}f_{j}|^{2}\bigg{\|}_{L^{\frac{q}{2}}_{t}L^{\infty}_{x}}\lesssim\|\lambda\|_{\ell^{\beta}}$

holds whenever $(f_{j})_{j}$ is a family of orthonormal functions in
$\dot{H}^{\frac{d}{2}-\frac{a}{q}}(\mathbb{R}^{d})$.

One may follow the argument in [3] to prove Theorem 1.3 with very minor
modifications, and so we only present a very brief sketch here and refer the
reader to [3] for further details.

Key to the argument is the dispersive estimate of the integral kernel
associated with the wave propagator:

(2.2) $\Bigl{|}\int
e^{i(x\cdot\xi+t|\xi|)}|\xi|^{-d+\frac{2}{q}+i\kappa}\chi(\xi)\,\mathrm{d}\xi\Bigr{|}\lesssim
C(\kappa)|t|^{-\frac{2}{q}},$

where $d\geq 2$, $\frac{4}{d-1}<q<\infty$, $\kappa\in\mathbb{R}$, and
$C(\kappa)>0$ satisfies $C(\kappa)\lesssim e^{\varepsilon|\kappa|}$ for an
arbitrary small $\varepsilon>0$. The corresponding estimate for the fractional
Schrödinger propagator

(2.3) $\Bigl{|}\int
e^{i(x\cdot\xi+t|\xi|^{a})}|\xi|^{-d+\frac{2a}{q}+i\kappa}\chi(\xi)\,\mathrm{d}\xi\Bigr{|}\lesssim
C(\kappa)|t|^{-\frac{2}{q}}$

holds for $d\geq 1$, $a\in(0,\infty)\backslash\\{1\\}$, $\frac{4}{d}\leq
q<\infty$, $\kappa\in\mathbb{R}$, and $C(\kappa)>0$ as above (for a proof of
(2.2) and (2.3), see for example [14] and [13]). Notice that the condition on
$q$ in (2.2) corresponds to the one for (2.3), but $d$ is replaced by $d-1$
and the endpoint $q=\frac{4}{d-1}$ is excluded; the lack of an endpoint
estimate here turns out to be harmless since our goal is $\beta<\frac{q}{2}$.

The argument proceeds using a duality principle observed by Frank–Sabin [10,
Lemma 3] in an abstract setting. In our case, we see that (1.7) with $U$
replaced by $\chi(|D|)U$ is equivalent to

(2.4)
$\|WU\chi^{2}(|D|)|D|^{-(d-1+\frac{2}{\widetilde{q}})}U^{*}\overline{W}\|_{\mathcal{C}^{\beta^{\prime}}}\lesssim\|W\|_{L_{t}^{\widetilde{q}}L_{x}^{2}}^{2},$

where $\beta^{\prime}$ denotes the (standard) conjugate of $\beta$, and
$\widetilde{q}$ denotes the (half-)conjugate of $q$ given by

$\frac{1}{\beta}+\frac{1}{\beta^{\prime}}=1,\qquad\frac{1}{q}+\frac{1}{\widetilde{q}}=\frac{1}{2}.$

Here, $\mathcal{C}^{\beta^{\prime}}$ with $\beta^{\prime}\geq 1$ denotes the
Schatten space on $L^{2}(\mathbb{R}^{d+1})$ defined as

$\mathcal{C}^{\beta^{\prime}}=\\{T\in\mathrm{Com}(L^{2}({\mathbb{R}}^{d+1})):\|T\|_{\mathcal{C}^{\beta^{\prime}}}:=\|(\mu_{j}(T))_{j\in{\mathbb{N}}}\|_{\ell^{\beta^{\prime}}}<\infty\\},$

where $\mathrm{Com}(L^{2}({\mathbb{R}}^{d+1}))$ denotes the set of compact
operators on $L^{2}({\mathbb{R}}^{d+1})$ and $(\mu_{j}(T))_{j}$ denotes the
family of the singular values of $T$ (i.e. non-zero eigenvalues of
$\sqrt{T^{*}T}$). Note that establishing (1.7) with $\chi(|D|)U$ suffices
since one can rescale to deduce the same estimate with $\chi(\varepsilon|D|)U$
for any $\varepsilon>0$, and then take a limit $\varepsilon\to 0$.

The space $\mathcal{C}^{1}$ consists of trace-class operators and
$\mathcal{C}^{\infty}$ consists of compact operators (endowed with the
standard operator norm). Hence $\mathcal{C}^{\beta^{\prime}}$ with
$1<\beta^{\prime}<\infty$ is regarded as an intermediate space. Of particular
importance to the analysis in this note is that the class $\mathcal{C}^{2}$
consists of Hilbert–Schmidt operators and that there is a very useful
expression of the $\mathcal{C}^{2}$ norm; if $T$ is given

$Tf(x)=\int_{{\mathbb{R}}^{d+1}}K(x,y)f(y)\,\mathrm{d}y,\qquad f\in
L^{2}({\mathbb{R}}^{d+1}),$

then
$\|T\|_{\mathcal{C}^{2}}=\|K\|_{L^{2}(\mathbb{R}^{d+1}\times\mathbb{R}^{d+1})}$.
For more details about Schatten spaces, see Simon’s textbook [20].

Our goal is to prove (2.4) for $\widetilde{q}\in(2,\widetilde{q}_{d})$ and
$\beta^{\prime}>\frac{\widetilde{q}}{2}$. We write $\mathcal{U}=\chi(|D|)U$
and divide into two cases depending on $\widetilde{q}$.

Case 1: $d\geq 3$ and $4\leq\widetilde{q}<\infty$: We aim to get the following
$\mathcal{C}^{2}$ and $\mathcal{C}^{\infty}$ bilinear estimates: if
$2<\widetilde{q_{0}}<4$, $2<\widetilde{q_{1}}<\infty$,

(2.5)
$\displaystyle\|W_{1}\mathcal{U}|D|^{-(d-1+\frac{2}{\widetilde{q_{0}}})+i\kappa}\mathcal{U}^{*}\overline{W}_{2}\|_{\mathcal{C}^{2}}\lesssim
C(\kappa)\|W_{1}\|_{L_{t}^{\widetilde{q_{0}},4}L_{x}^{2}}\|W_{2}\|_{L_{t}^{\widetilde{q_{0}},4}L_{x}^{2}},$
(2.6)
$\displaystyle\|W_{1}\mathcal{U}|D|^{-(d-1+\frac{2}{\widetilde{q_{1}}})+i\kappa}\mathcal{U}^{*}\overline{W}_{2}\|_{\mathcal{C}^{\infty}}\lesssim
C(\kappa)\|W_{1}\|_{L_{t}^{\widetilde{q_{1}},\infty}L_{x}^{2}}\|W_{2}\|_{L_{t}^{\widetilde{q_{1}},\infty}L_{x}^{2}},$

where $C(\kappa)\lesssim e^{\varepsilon|\kappa|}$ for an arbitrary small
$\varepsilon>0$. Then, by applying a bilinear version of Stein’s analytic
interpolation to the above estimates, we may obtain (2.4) for the case
$4\leq\widetilde{q}<\infty$. For this purpose, the complex power $i\kappa$ is
essential in (2.5) and (2.6). One may also notice that, thanks to the Lorentz
improvements on the right-hand sides of (2.5) and (2.6), the outcome is also
slightly stronger than (2.4), namely, in Case 1 and for
$\beta^{\prime}>\frac{\widetilde{q}}{2}$ one obtains

$\|W\mathcal{U}|D|^{-(d-1+\frac{2}{\widetilde{q}})}\mathcal{U}^{*}\overline{W}\|_{\mathcal{C}^{\beta^{\prime}}}\lesssim\|W\|_{L_{t}^{\widetilde{q},2\beta^{\prime}}L_{x}^{2}}^{2}.$

The proofs of (2.5) and (2.6) are based on the dispersive estimate for the
wave propagator (2.2) and the Lorentz refinement of the
Hardy–Littlewood–Sobolev inequality due to O’Neil [19]. The interested reader
is encouraged to visit [3] where further details regarding the bilinear
analytic interpolation of (2.5) and (2.6) can also be found.

Case 2: $d\geq 2$ and $2<\widetilde{q}<4$: We approach this case by first
establishing the somewhat weaker estimates

(2.7)
$\|W_{1}\mathcal{U}|D|^{-(d-1+\frac{2}{\widetilde{q}})+i\kappa}\mathcal{U}^{*}\overline{W}_{2}\|_{\mathcal{C}^{\beta^{\prime}}}\lesssim
C(\kappa)\|W_{1}\|_{L_{t}^{\widetilde{q},2}L_{x}^{2}}\|W_{2}\|_{L_{t}^{\widetilde{q},2}L_{x}^{2}},$

where $\beta^{\prime}>\frac{\widetilde{q}}{2}$ and $C(\kappa)\lesssim
e^{\varepsilon|\kappa|}$ for an arbitrary small $\varepsilon>0$. In fact, we
can offset the loss in (2.7) (in the sense of
$L_{t}^{\widetilde{q},2}\subsetneq L_{t}^{\widetilde{q}}$ when
$\widetilde{q}>2$) by capitalizing on the gain in (2.5) (in the sense of
$L_{t}^{\widetilde{q_{0}}}\subsetneq L_{t}^{\widetilde{q_{0}},4}$ if
$\widetilde{q_{0}}<4$), and use bilinear analytic interpolation once again to
obtain the desired estimate (2.4) in Case 2 (see [3] for further details of
this step).

To prove (2.7), we use a frequency-localization argument and use a different
bilinear interpolation argument in this spirit of Keel–Tao [15] based on the
estimates

(2.8)
$\displaystyle\|W_{1}\mathcal{U}P_{k}|D|^{-2s+i\kappa}P_{k}\mathcal{U}^{*}\overline{W}_{2}\|_{\mathcal{C}^{1}}$
$\displaystyle\lesssim(2^{k})^{d-2s}\|W_{1}\|_{L_{t}^{2}L_{x}^{2}}\|W_{2}\|_{L_{t}^{2}L_{x}^{2}},$
(2.9)
$\displaystyle\|W_{1}\mathcal{U}P_{k}|D|^{-2s+i\kappa}P_{k}\mathcal{U}^{*}\overline{W}_{2}\|_{\mathcal{C}^{2}}$
$\displaystyle\lesssim
C(\kappa)(2^{k})^{d-2s-1+\frac{1}{\widetilde{q_{1}}}+\frac{1}{\widetilde{q_{2}}}}\|W_{1}\|_{L_{t}^{\widetilde{q_{1}}}L_{x}^{2}}\|W_{2}\|_{L_{t}^{\widetilde{q_{2}}}L_{x}^{2}},$

where $\widetilde{q_{1}},\widetilde{q_{2}}\in[2,\infty)$ are such that
$\frac{1}{\widetilde{q_{1}}}+\frac{1}{\widetilde{q_{2}}}>\frac{1}{2}$, and
$C(\kappa)\lesssim e^{\varepsilon|\kappa|}$ for an arbitrary small
$\varepsilon>0$. As in Case 1, (2.9) readily follows from a stronger form of
the dispersive estimate (2.2) with decay $(1+|t|)^{-d/2}$. The estimate (2.8)
is a manifestation of the frequency localization111Note that (2.8) without the
frequency localization is not permitted due to the failure of the associated
Sobolev embedding. and Bessel’s inequality for orthonormal sequences, and for
its proof, it is convenient to make use of the representation

$\|T\|_{\mathcal{C}^{\beta^{\prime}}}=\sup_{(\phi,\psi)\in\mathcal{B}}\|\langle
T\phi_{j},\psi_{j}\rangle_{L^{2}({\mathbb{R}}^{d+1})}\|_{\ell^{\beta^{\prime}}}$

for Schatten norms (see [20, Proposition 2.6]). Here, $\mathcal{B}$ denotes
the set of pairs of orthonormal sequences in $L^{2}({\mathbb{R}}^{d+1})$.

##  3\. Proof of Theorem 1.4

Key to our proof of (1.8) is a clever summation idea going back to Bourgain
[6] (which has been used in several papers; see also [7, Section 6.2] and [16,
Lemma 2.3] for example) which allows us to pass from frequency-local to
frequency-global estimates.

######  Lemma 3.1 (see [4]).

Let $q_{0},q_{1},r\in[2,\infty]$, $\beta_{0},\beta_{1}\in[2,\infty]$ and
$(g_{j})_{j}$ be a uniformly bounded sequence in $L_{t}^{q_{0}}L_{x}^{r}\cap
L_{t}^{q_{1}}L_{x}^{r}$. Suppose there exist $\varepsilon_{0}$,
$\varepsilon_{1}>0$ such that

$\Big{\|}\sum_{j}\lambda_{j}|P_{k}g_{j}|^{2}\Big{\|}_{L_{t}^{\frac{q_{0}}{2},\infty}L_{x}^{\frac{r}{2}}}\lesssim
2^{-\varepsilon_{0}k}\|\lambda\|_{\ell^{\beta_{0}}}$

and

$\Big{\|}\sum_{j}\lambda_{j}|P_{k}g_{j}|^{2}\Big{\|}_{L_{t}^{\frac{q_{1}}{2},\infty}L_{x}^{\frac{r}{2}}}\lesssim
2^{\varepsilon_{1}k}\|\lambda\|_{\ell^{\beta_{1}}}$

for all $k\in\mathbb{Z}$, then

$\Big{\|}\sum_{j}\lambda_{j}|g_{j}|^{2}\Big{\|}_{L_{t}^{\frac{q}{2},\infty}L_{x}^{\frac{r}{2}}}\lesssim\|\lambda\|_{\ell^{\beta,1}},$

where $\frac{1}{q}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}}$,
$\frac{1}{\beta}=\frac{1-\theta}{\beta_{0}}+\frac{\theta}{\beta_{1}}$ and
$\theta=\frac{\varepsilon_{0}}{\varepsilon_{0}+\varepsilon_{1}}$.

The above lemma was proved in [4, Proposition 2.2]. Let us give a sketch of
the argument when $q_{1}=\infty$ (since this case does not immediately follow
from the argument in [4] as it stands).

###### Proof of Lemma 3.1 when $q_{1}=\infty$.

It is enough to show the desired inequality for $\lambda=\textbf{1}_{E}$
(here, $\textbf{1}_{E}(j)=0$ if $j\in E$, and zero otherwise), so that we aim
for

(3.1)
$|J_{\nu}|\lesssim\left(\frac{(\\#E)^{\frac{1}{\beta}}}{\nu}\right)^{\frac{q}{2}},$

where $\nu>0$ and

$J_{\nu}:=\Big{\\{}t\in\mathbb{R}:\Big{\|}\sum_{j\in
E}|g_{j}(t,\cdot)|^{2}\Big{\|}_{L_{x}^{\frac{r}{2}}}>\nu\Big{\\}}.$

For a fixed $M\in\mathbb{Z}$ (chosen later), set

$J_{\nu,M}^{0}:=\Big{\\{}t\in\mathbb{R}:\Big{\|}\sum_{j\in E}|\sum_{k\geq
M+1}P_{k}g_{j}(t,\cdot)|^{2}\Big{\|}_{L_{x}^{\frac{r}{2}}}>\frac{\nu}{2}\Big{\\}}$

and

$J_{\nu,M}^{1}:=\Big{\\{}t\in\mathbb{R}:\Big{\|}\sum_{j\in E}|\sum_{k\leq
M}P_{k}g_{j}(t,\cdot)|^{2}\Big{\|}_{L_{x}^{\frac{r}{2}}}>\frac{\nu}{2}\Big{\\}}.$

Regarding $J_{\nu,M}^{0}$, let us observe that

$\Big{\|}\sum_{j\in E}|\sum_{k\geq
M+1}P_{k}g_{j}(t,\cdot)|^{2}\Big{\|}_{L_{t}^{\frac{q_{0}}{2},\infty}L_{x}^{\frac{r}{2}}}\lesssim
2^{-\varepsilon_{0}M}(\\#E)^{\frac{1}{\beta_{0}}}.$

In fact, applying Minkowski’s inequality twice gives

$\Big{\|}\sum_{j\in E}|\sum_{k\geq
M+1}P_{k}g_{j}(t,\cdot)|^{2}\Big{\|}_{L_{t}^{\frac{q_{0}}{2},\infty}L_{x}^{\frac{r}{2}}}\lesssim\Big{(}\sum_{k\geq
M+1}\Big{\|}\sum_{j}|P_{k}g_{j}|^{2}\Big{\|}_{L_{t}^{\frac{q_{0}}{2},\infty}L_{x}^{\frac{r}{2}}}^{\frac{1}{2}}\Big{)}^{2},$

which is further bounded by (up to some constant)
$2^{-\varepsilon_{0}M}(\\#E)^{\frac{1}{\beta_{0}}}.$ Hence, one readily sees
that

$|J_{\nu,M}^{0}|\lesssim\nu^{-\frac{q_{0}}{2}}\Big{\|}\sum_{j\in
E}|\sum_{k\geq
M+1}P_{k}g_{j}(t,\cdot)|^{2}\Big{\|}_{L_{t}^{\frac{q_{0}}{2},\infty}L_{x}^{\frac{r}{2}}}^{\frac{q_{0}}{2}}\lesssim
2^{-\frac{\varepsilon_{0}Mq_{0}}{2}}\nu^{-\frac{q_{0}}{2}}(\\#E)^{\frac{q_{0}}{2\beta_{0}}}.$

A similar calculation reveals that there exists $C>0$ such that

$\Big{\|}\sum_{j\in E}|\sum_{k\leq
M}P_{k}g_{j}(t,\cdot)|^{2}\Big{\|}_{L_{t}^{\infty}L_{x}^{\frac{r}{2}}}\leq
C2^{\varepsilon_{1}M}(\\#E)^{\frac{1}{\beta_{1}}},$

and now we select $M$ to be the largest integer such that

$C2^{\varepsilon_{1}M}(\\#E)^{\frac{1}{\beta_{1}}}<\frac{\nu}{2}.$

With this choice, we have $J_{\nu,M}^{1}=\emptyset$ and
$2^{M}\sim\nu^{\frac{1}{\varepsilon_{1}}}(\\#E)^{-\frac{1}{\beta_{1}\varepsilon_{1}}}.$
Since $|J_{\nu}|\leq|J_{\nu,M}^{0}|$, we obtain (3.1) after a straightforward
computation. ∎

###### Proof of Theorem 1.4.

Let $(f_{j})_{j\in{\mathbb{N}}}$ be a family of orthonormal functions in
$L^{2}({\mathbb{R}}^{d})$. First note that

(3.2)
$\Big{\|}\sum_{j}\lambda_{j}|U|D|^{-s}P_{k}f_{j}|^{2}\Bigr{\|}_{L_{t}^{\infty}L_{x}^{\infty}}\lesssim(2^{k})^{d-2s}\|\lambda\|_{\ell^{\infty}}.$

This is the estimate dual to (2.8) (which can easily be verified directly by
an application of Bessel’s inequality as in [4]).

First we consider the case $d\geq 4$ in which case it is known (see for
example [8, Theorem 1]) that the classical Strichartz estimate (1.3) holds
with $f$ replaced by $P_{0}f$. Thus, by recaling,

(3.3)
$\|U|D|^{-s}P_{k}f_{j}\|_{L_{t}^{2}L_{x}^{\infty}}\lesssim(2^{k})^{\frac{d-1}{2}-s}$

holds for each $f_{j}$, and therefore the triangle inequality implies

(3.4)
$\Bigl{\|}\sum_{j}\lambda_{j}|U|D|^{-s}P_{k}f_{j}|^{2}\Bigr{\|}_{L_{t}^{1}L_{x}^{\infty}}\lesssim(2^{k})^{d-1-2s}\|\lambda\|_{\ell^{1}}.$

Although (3.2) and (3.3) are valid for all $s\in\mathbb{R}$, if we fix
$q\in(2,\infty)$ and apply these estimates with
$s_{q}=\frac{d}{2}-\frac{1}{q}$, then Lemma 3.1 immediately gives the desired
estimate

$\Bigl{\|}\sum_{j}\lambda_{j}|U|D|^{-s_{q}}f_{j}|^{2}\Bigr{\|}_{L_{t}^{\frac{q}{2},\infty}L_{x}^{\infty}}\lesssim\|\lambda\|_{\ell^{\frac{q}{2},1}}.$

The case when $d=3$ requires a bit more work to show (1.8) for $2<q<\infty$
since (3.3) with $d=3$ is known to be false (see [18]). Thus, instead of
(3.4), we aim for

(3.5)
$\Bigl{\|}\sum_{j}\lambda_{j}|U|D|^{-s}P_{k}f_{j}|^{2}\Bigr{\|}_{L_{t}^{\frac{q}{2}}L_{x}^{\infty}}\lesssim(2^{k})^{3-\frac{2}{q}-2s}\|\lambda\|_{\ell^{\frac{q}{2}}}$

with $q$ larger than but abritrarily close to $2$ (here the implicit constant
blows up as $q\to 2$). It is clear that Lemma 3.1 combined with (3.2) and
(3.5) yields (1.8).

Let us check (3.5). First, by rescaling, we may assume $k=0$. Using duality
[10, Lemma 3], it suffices to show

(3.6)
$\|WS\overline{W}\|_{\mathcal{C}^{2^{m}}}\lesssim\|W\|_{L_{t}^{2^{m+1}}L_{x}^{2}}^{2},$

where $S:=UP_{0}P_{0}^{*}U^{*}$ and $m$ a sufficiently large natural number.
Recall that the $\mathcal{C}^{2}$-norm can be expressed by the $L^{2}$-norm of
its integral kernel, and in addition, from the definition of singular values,
it holds that
$\|T\|_{\mathcal{C}^{2^{m+1}}}^{2}=\|T^{*}T\|_{\mathcal{C}^{2^{m}}}$.
Therefore, since $S$ is self-adjoint, we have

$\|WS\overline{W}\|_{\mathcal{C}^{2^{m}}}^{2^{m-1}}=\|(WS\overline{W})^{2^{m-1}}\|_{\mathcal{C}^{2}}.$

Hence our task is to get an $L^{2}$ bound for the integral kernel
$\mathcal{K}_{2^{m-1}}$ of $(WS\overline{W})^{2^{m-1}}$. The notation for the
variables in the forthcoming argument requires a little care and our choices
will become clear as we proceed.

Let us write $\zeta_{j}=(t_{j},x_{j})\in\mathbb{R}^{1+3}$. Then

$(WS\overline{W})F(\zeta_{1})=\int_{\mathbb{R}^{4}}F(\zeta_{2})\mathcal{K}_{1}(\zeta_{1},\zeta_{2})\,\mathrm{d}\zeta_{2}$

with
$\mathcal{K}_{1}(\zeta_{1},\zeta_{2})=W(\zeta_{1})\overline{W}(\zeta_{2})K(\zeta_{1},\zeta_{2})$
and

$K(\zeta_{1},\zeta_{2})=\int_{\mathbb{R}^{3}}|\varphi(\xi)|^{2}e^{i((x_{1}-x_{2})\cdot\xi+(t_{1}-t_{2})|\xi|)}\,\mathrm{d}\xi.$

To handle this, we shall invoke the following frequency-localized dispersive
estimate (see e.g. [14]):

(3.7) $\sup_{x_{1},x_{2}}|K(\zeta_{1},\zeta_{2})|\lesssim\langle
t_{1}-t_{2}\rangle^{-1},$

where $\langle t\rangle:=1+|t|$.

For $m=2$ we have

$\displaystyle(WS\overline{W})^{2}G(\zeta_{1})=\int_{\mathbb{R}^{4}}G(\zeta_{3})\Big{(}\int_{\mathbb{R}^{4}}W(\zeta_{1})|W(\zeta_{2})|^{2}\overline{W}(\zeta_{3})K(\zeta_{1},\zeta_{2})K(\zeta_{2},\zeta_{3})\,\mathrm{d}\zeta_{2}\Big{)}\mathrm{d}\zeta_{3}$

and from this we have an expression for
$\mathcal{K}_{2}(\zeta_{1},\zeta_{3})$. Inductively,

$\displaystyle\mathcal{K}_{2^{m}}(\zeta_{1},\zeta_{2^{m}+1})$
$\displaystyle=\int_{\mathbb{R}^{4}}\mathcal{K}_{2^{m-1}}(\zeta_{1},\zeta_{2^{m-1}+1})\mathcal{K}_{2^{m-1}}(\zeta_{2^{m-1}+1},\zeta_{2^{m}+1})\,\mathrm{d}\zeta_{2^{m-1}+1}$
$\displaystyle=\int_{(\mathbb{R}^{4})^{2^{m}-1}}W(\zeta_{1})\prod_{j=2}^{2^{m}}|W(\zeta_{j})|^{2}\overline{W}(\zeta_{2^{m}+1})\prod_{k=1}^{2^{m}}K(\zeta_{k},\zeta_{k+1})\,\prod_{\ell=2}^{2^{m}}\mathrm{d}\zeta_{\ell}$

for all $m\geq 1$, and consequently

$\displaystyle|\mathcal{K}_{2^{m}}(\zeta_{1},\zeta_{2^{m}+1})|\lesssim|W(\zeta_{1})|\bigg{(}\int_{\mathbb{R}^{2^{m}-1}}\prod_{j=2}^{2^{m}}h(t_{j})\prod_{k=1}^{2^{m}}\langle
t_{k}-t_{k+1}\rangle^{-1}\,\prod_{\ell=2}^{2^{m}}\mathrm{d}t_{\ell}\bigg{)}|W(\zeta_{2^{m}+1})|,$

where $h(t):=\|W(t,\cdot)\|_{L^{2}_{x}}^{2}$.

Now we are ready to estimate the relevant Schatten norm of $WS\overline{W}$:

$\displaystyle\|WS\overline{W}\|_{\mathcal{C}^{2^{m}}}^{2^{m}}$
$\displaystyle=\int_{(\mathbb{R}^{4})^{2}}|\mathcal{K}_{2^{m-1}}(\zeta_{1},\zeta_{2^{m-1}+1})|^{2}\,\mathrm{d}\zeta_{1}\mathrm{d}\zeta_{2^{m-1}+1}$
$\displaystyle\lesssim\int_{\mathbb{R}^{2}}h(t_{1})\bigg{(}\int_{\mathbb{R}^{2^{m-1}-1}}\prod_{j=2}^{2^{m-1}}h(t_{j})\prod_{k=1}^{2^{m-1}}\langle
t_{k}-t_{k+1}\rangle^{-1}\,\prod_{\ell=2}^{2^{m-1}}\mathrm{d}t_{\ell}\bigg{)}^{2}h(t_{2^{m-1}+1})\,\mathrm{d}t_{1}\mathrm{d}t_{2^{m-1}+1}$

By expanding the square and an appropriate choice of labelling222If we use
$t_{2},\ldots,t_{2^{m-1}},t^{*}_{2},\ldots,t^{*}_{2^{m-1}}$ for the variables
arising from expanding the square, then the claimed formula follows by
relabelling $t_{j}=t^{*}_{2^{m}+2-j}$ for $j=2^{m-1}+2,\ldots,2^{m}$., we see
that

(3.8)
$\displaystyle\|WS\overline{W}\|_{\mathcal{C}^{2^{m}}}^{2^{m}}\lesssim\int_{\mathbb{R}^{2^{m}}}\prod_{j=1}^{2^{m+1}}f_{j}(L_{j}\textbf{t})\,\mathrm{d}\textbf{t},$

where $L_{j}:\mathbb{R}^{2^{m}}\ni\textbf{t}\mapsto
v_{j}\cdot\textbf{t}\in\mathbb{R}$ with

$v_{j}=\begin{cases}e_{j}\quad&\text{if $j=1,\dots,2^{m}$},\\\
e_{j-2^{m}}-e_{j+1-2^{m}}\quad&\text{if
$j=2^{m}+1,\dots,2^{m+1}$},\end{cases}$

under the relation $e_{2^{m}+1}=e_{1}$, and

$f_{j}(t)=\begin{cases}h(t)\quad&\text{if $j=1,\dots,2^{m}$,}\\\ \langle
t\rangle^{-1}\quad&\text{if $j=2^{m}+1,\dots,2^{m+1}$}.\end{cases}$

One may notice that the term on the right-hand side of (3.8) has the structure
of the left-hand side of the Brascamp–Lieb inequality and, in fact, we shall
invoke the following characterization of finiteness of the Brascamp–Lieb
constant (in the rank-one case) due to Barthe [1]. Let $n$, $M\in{\mathbb{N}}$
and for $I=\\{1,\ldots,M\\}$ we denote by $\textbf{1}_{I}$ the vector of
$\mathbb{R}^{M}$ such that $j$-th element of $\textbf{1}_{I}$ is $1$ if $j\in
I$ and $0$ elsewhere.

######  Lemma 3.2 (see Proposition 3 in [1]).

There exists $C<\infty$ such that

$\int_{\mathbb{R}^{n}}\prod_{j=1}^{M}f_{j}(x\cdot v_{j})\,\mathrm{d}x\leq
C\prod_{j=1}^{M}\|f_{j}\|_{L^{p_{j}}(\mathbb{R})}$

holds for all non-negative $f_{j}$ in $L^{p_{j}}(\mathbb{R})$ if and only if
the following holds. The vector
$\frac{1}{p}=(\frac{1}{p_{1}}\,\dots,\frac{1}{p_{M}})$ belongs to the convex
hull of $\textbf{1}_{I}$ for subsets such that $(v_{i})_{i\in I}$ forms a
basis of $\mathbb{R}^{n}$.

We will use this lemma with $n=2^{m}$ and $M=2^{m+1}$. Let
$I_{0}=\\{2^{m}+1,2^{m}+2,\dots,2^{m+1}\\}$ so that
$\textbf{1}_{I_{0}}=(0,\dots,0;1,\dots,1)$; i.e. the first half of elements
are all $0$ and the rests are all $1$ (the separator “;” is positioned so that
the components have been divided into exactly two equal-sized parts). For
integers $i$, $j$ with $1\leq i\leq 2^{m}<j\leq 2^{m+1}$ the transposition
operator swapping $i$-th element and $j$-th element, denoted by $[i,j]$,
provides

$[i,j]\textbf{1}_{I_{0}}=(0,\dots,0,1,0,\dots,0;1,\dots,1,0,1,\dots,1).$

From the definition of $(v_{j})_{j}$, if $1\leq i\leq 2^{m}$, the vectors

$(v_{j})_{j\in(\\{i\\}\cup(I_{0}\setminus\\{2^{m}+i\\}))}$

form a basis of ${\mathbb{R}}^{2^{m}}$. In $\mathbb{R}^{2^{m+1}}$, considering
$2^{m}$ points
$[1,2^{m}+1]\textbf{1}_{I},\dots,[2^{m},2^{m+1}]\textbf{1}_{I}$, one may
discover that the middle point of the plane spanned by those vectors, namely,

$(2^{-{m}},\dots,2^{-{m}};1-2^{-{m}},\dots,1-2^{-{m}})$

lies on the boundary of the convex hull of characteristic vectors. Hence, if
we set

$\displaystyle(p_{1}^{-1},\dots,p_{2^{m}}^{-1};p_{2^{m}+1}^{-1},\dots,p_{2^{m+1}}^{-1})=(2^{-{m}},\dots,2^{-{m}};1-2^{-{m}},\dots,1-2^{-{m}}),$

then Lemma 3.2 implies that

$\displaystyle\int_{\mathbb{R}^{2^{m}}}\prod_{j=1}^{2^{m+1}}f_{j}(L_{j}\textbf{t})\,\mathrm{d}\textbf{t}\lesssim\prod_{j=1}^{2^{m+1}}\|f_{j}\|_{L_{t}^{p_{j}}}\lesssim\|W\|_{L_{t}^{{}^{2^{m+1}}}L_{x}^{2}}^{2^{m+1}},$

which ends the proof. ∎

## Appendix: Necessary conditions

We provide a justification of the necessary condition (1.5) for (1.1). The
proof here is not new and may be found in [5, Proposition 9].

### The necessity of $\beta\leq\frac{q}{2}$

The argument we give here is the same as in the proof of [5, Proposition 9];
since $r<\infty$ was a blanket assumption throughout [5], here we consider the
only case $r=\infty$.

Consider the family $(f_{j})_{j\geq 1}$ given by

$f_{j}(x)=Ug(-j,x)$

where the function $g$ satisfies

$\widehat{g}(\xi)=c\frac{\mathrm{1}_{[\pi,3\pi]}(|\xi|)}{|\xi|^{s+\frac{d-1}{2}}}$

and the constant $c>0$ will be chosen shortly. Parseval’s identity and
changing to polar coordinates reveals

$\langle
f_{j},f_{k}\rangle_{\dot{H}^{s}}=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{i(j-k)|\xi|}|\widehat{g}(\xi)|^{2}|\xi|^{2s}\,\mathrm{d}\xi=\frac{c^{2}|\mathbb{S}^{d-1}|}{(2\pi)^{d}}\int_{\pi}^{3\pi}e^{i(k-j)\rho}\,\mathrm{d}\rho$

and so $(f_{j})_{j\geq 1}$ is an orthonormal family in
$\dot{H}^{s}(\mathbb{R}^{d})$ upon an appropriate choice of $c$.

Also, assuming $\lambda_{j}>0$ for each $j$ we have

$\displaystyle\bigg{\|}\sum_{j}\lambda_{j}|Uf_{j}|^{2}\bigg{\|}_{L^{\frac{q}{2}}_{t}L^{\infty}_{x}}^{\frac{q}{2}}$
$\displaystyle\geq\int_{\mathbb{R}}\bigg{(}\sum_{j}\lambda_{j}|Ug(t-j,0)|^{2}\bigg{)}^{\frac{q}{2}}\mathrm{d}t$
$\displaystyle\geq\sum_{n\geq
1}\lambda_{n}^{\frac{q}{2}}\int_{n}^{n+\varepsilon_{0}}|Ug(t-n,0)|^{q}\,\mathrm{d}t$
$\displaystyle=C\sum_{n\geq 1}\lambda_{n}^{\frac{q}{2}}$

where $\varepsilon_{0}>0$ is a sufficiently small constant and

$C=\int_{0}^{\varepsilon_{0}}|Ug(t,0)|^{q}\,\mathrm{d}t=c^{q}\int_{0}^{\varepsilon_{0}}\bigg{|}\int_{|\xi|\in[\pi,3\pi]}e^{it|\xi|}\frac{\mathrm{d}\xi}{|\xi|^{s+\frac{d-1}{2}}}\bigg{|}^{q}\mathrm{d}t.$

The constant $C$ is clearly finite, and a sufficiently small choice of
$\varepsilon_{0}$ guarantees $|t|\xi||\leq\frac{1}{10}$ for $\xi\in[\pi,3\pi]$
and thus

$\displaystyle C$ $\displaystyle\geq
c^{q}\int_{0}^{\varepsilon_{0}}\bigg{|}\int_{|\xi|\in[\pi,3\pi]}\cos(t|\xi|)\frac{\mathrm{d}\xi}{|\xi|^{s+\frac{d-1}{2}}}\bigg{|}^{q}\mathrm{d}t$
$\displaystyle\geq\bigg{(}\frac{c}{2}\bigg{)}^{q}\int_{0}^{\varepsilon_{0}}\bigg{(}\int_{|\xi|\in[\pi,3\pi]}\frac{\mathrm{d}\xi}{|\xi|^{s+\frac{d-1}{2}}}\bigg{)}^{q}\mathrm{d}t>0.$

Hence, if (1.1) holds we deduce
$\|\lambda\|_{\frac{q}{2}}\lesssim\|\lambda\|_{\beta}$ and this implies
$\beta\leq\frac{q}{2}$.

### The necessity of $\beta\leq\beta_{\frac{d}{2}}(q,r)$

Let $B(x,r)$ be the ball in $\mathbb{R}^{d}$ centred at $x$ with radius $r$.
Consider the family $(f_{j})_{j=1}^{N}$ given by

$\widehat{f_{j}}(\xi)=\mathrm{1}_{B(0,1)}(R(\xi-v_{j})).$

Here, $v_{1},\ldots,v_{N}\in\mathbb{R}^{d}$ are fixed vectors satisfying
$|v_{j}|\in[1,2]$ and chosen so that the balls
$\\{B(v_{j},\frac{1}{R})\\}_{j=1}^{N}$ are disjoint. In particular, this means
that the functions $(\widehat{f_{j}})_{j=1}^{N}$ have disjoint support and
thus $(f_{j})_{j=1}^{N}$ is an orthogonal family. Moreover, we choose a
maximal collection of vectors $v_{1},\ldots,v_{N}$ under the above
constraints, which means $N\sim R^{d}$.

Next we claim that $|Uf_{j}|\gtrsim R^{-d}1_{B(0,c^{\prime}R)}$ for an
appropriately small choice of the constant $c^{\prime}$. To see this, first
assume that $v_{j}=\rho e_{d}$ for some $\rho\in[1,2]$, and write

$\displaystyle x\cdot\xi+t|\xi|$
$\displaystyle=\bar{x}\cdot\bar{\xi}+(x_{d}+t)(\xi_{d}-\rho)+t(|\xi|-\xi_{d})+\rho(x_{d}+t)$
$\displaystyle=\bar{x}\cdot\bar{\xi}+(x_{d}+t)(\xi_{d}-\rho)+t\frac{|\bar{\xi}|^{2}}{|\xi|+\xi_{d}}+\rho(x_{d}+t)=:\Phi_{t,x}(\xi)+\rho(x_{d}+t).$

Here, we use the notation $x=(\bar{x},x_{d})$. If $c^{\prime}$ is chosen
sufficiently small, then for $(t,x)\in B(0,c^{\prime}R)$ and $\xi$ in the
support of $\widehat{f_{j}}$ (in particular,
$|\bar{\xi}|\leq\frac{1}{R},|\xi_{d}-\rho|\leq\frac{1}{R}$) we get

$|\Phi_{t,x}(\xi)|\leq\frac{1}{10}$

and thus

$|Uf_{j}(t,x)|=\bigg{|}\int_{B(\rho
e_{d},\frac{1}{R})}e^{i\Phi_{t,x}(\xi)}\,\mathrm{d}\xi\bigg{|}\gtrsim\frac{1}{R^{d}}$

as claimed. Taking $\lambda_{j}=\|f_{j}\|_{\dot{H}^{s}}^{2}$ and
$g_{j}=\|f_{j}\|^{-1}_{\dot{H}^{s}}f_{j}$ for each $j=1,\ldots,N$, and
recalling that $N\sim R^{d}$, we easily deduce that

$\displaystyle\bigg{\|}\sum_{j=1}^{N}\lambda_{j}|Ug_{j}|^{2}\bigg{\|}_{L^{\frac{q}{2}}_{t}L^{\frac{r}{2}}_{x}}=\bigg{\|}\sum_{j=1}^{N}|Uf_{j}|^{2}\bigg{\|}_{L^{\frac{q}{2}}_{t}L^{\frac{r}{2}}_{x}}\gtrsim
R^{\frac{2}{q}+\frac{2d}{r}-d}.$

If we assume (1.1) then

$R^{\frac{2}{q}+\frac{2d}{r}-d}\lesssim\bigg{(}\sum_{j=1}^{N}\|f_{j}\|_{\dot{H}^{s}}^{2\beta}\bigg{)}^{\frac{1}{\beta}}\lesssim
R^{\frac{d}{\beta}-d}$

since $\|f_{j}\|_{\dot{H}^{s}}^{2}\sim R^{-d}$. Therefore
$\frac{2}{q}+\frac{2d}{r}\leq\frac{d}{\beta}$ and this yields the necessary
condition $\beta\leq\beta_{\frac{d}{2}}(q,r)$.

######  Acknowledgements.

The first author would like to express his appreciation to Sanghyuk Lee and
Shohei Nakamura for a number of extremely helpful discussions related to this
work. The second author shows his deep gratitude to Yutaka Terasawa and
Satoshi Masaki for the opportunity at the RIMS conference “Harmonic Analysis
and Partial Differential Equations” in 2022. The third author also thanks
Mitsuru Sugimoto for his continuous encouragement.

## References

  * [1] Barthe, F., On a reverse form of the Brascamp-Lieb inequality, Invent. Math., 134 (1998) 355–361.
  * [2] Bergh, J., Löfström, J., Interpolation Spaces, Springer, Berlin, 1976.
  * [3] Bez, N., Kinoshita, S., Shiraki, S., Boundary Strichartz estimates for orthonormal systems, preprint.
  * [4] Bez, N., Hong, Y., Lee, S., Nakamura, S., Sawano, Y., On the Strichartz estimates for orthonormal systems of initial data with regularity, Adv. Math., 354 (2019), 106736, 37 pp.
  * [5] Bez, N., Lee, S., Nakamura, S., Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates, Forum of Math. Sigma, 9 (2021) e1.
  * [6] Bourgain, J., Estimations de certaines fonctions maximales, C. R. Acad. Sci. Paris Sér. I Math., 310 (1985) 499–502.
  * [7] Carbery, A., Seeger, A., Wainger, S., Wright, J., Classes of singular integral operators along variable lines, J. Geom. Anal., 9 (1999) 583–605.
  * [8] Fang, D., Wang, C., Some remarks on homogeneous estimates for wave equation, Nonlinear Anal. 65 (2006) 697–706.
  * [9] Frank, R., Lewin, M., Lieb, E. H., Seiringer, R., Strichartz inequality for orthonormal functions, J. Eur. Math. Soc., 16 (2014) 1507–1526.
  * [10] Frank, R., Sabin, J., Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates, Amer. J. Math., 139 (2017) 1649–1691.
  * [11] Frank, R., Sabin, J., The Stein–Tomas inequality in trace ideals, Séminaire Laurent Schwartz – EPD et applications (2015-2016), Exp. No. XV, 12 pp., 2016.
  * [12] Grafakos, L., Classical Fourier analysis, third edition, Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014.
  * [13] Guo, Z., Li, J., Nakanishi, K., Yan, L., On the boundary Strichartz estimates for wave and Schrödinger equations, J. Differential Equations, 265 (2018) 5656–5675.
  * [14] Guo, Z., Peng, L., Wang, B., Decay estimates for a class of wave equations, J. Funct. Anal., 254 (2008) 1642–1660.
  * [15] Keel, M., Tao, T., Endpoint Strichartz estimates, Amer. J. Math., 120 (1998) 955–980.
  * [16] Lee, S., Some sharp bounds for the cone multiplier of negative order in ${\mathbb{R}}^{3}$, Bull. London Math. Soc., 35 (2003) 373–390.
  * [17] Lieb, E. H., An $L^{p}$ bound for the Riesz and Bessel potentials of orthonormal functions, J. Funct. Anal., 51 (1983) 159–165.
  * [18] Montgomery-Smith, S. J., Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations, Duke Math. J., 91 (1998), 393–408.
  * [19] O’Neil, R., Convolution operators and $L(p,q)$ spaces, Duke Math. J., 30 (1963) 129–142.
  * [20] Simon, B., Trace ideals and their applications, Vol. 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979.
  * [21] Stein, E. M., Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, (1971).


# Theoretical Particle Limiting Velocity From The Bicubic Equation: Neutrino
Example

Josip Šoln
JZS Phys-Tech, Vienna, Virginia 22182
E mail<EMAIL_ADDRESS>
( January 2014)

###### Abstract

There has been a lot of interest in measuring the velocities of massive
elementary particles, particularly the neutrinos. Some neutrino experiments at
first observed superluminal neutrinos, thus violating the velocity of light
$c$ as a limiting velocity. But, after eliminating some mistakes, such as, for
the OPERA experiments plugging the cable correctly and calibrating the clock
correctly, the measured neutrino velocity complied with $c$. Pursuing the
theoretical side of particle limiting velocities, here directly from the
special relativistic kinematics, in which all physical quantities are in the
overall mathematical consistency with each other, one treats formally the
velocity of light $c$ as yet to be deduced particle limiting velocity, and
derives the bicubic equation for the particle limiting velocity in the
arbitrary reference frame. The Lorentz invariance (LI) of the energy-momentum
dispersion relation assumes the velocity of light $c$ to be universal limiting
velocity of any particle. This expects the physical solutions of the bicubic
equation to be constrained in a sense that any physical limiting velocity
solution should equal numerically to $c$, preferably exactly, or at least, in
a extremely good approximation. A rather large numerical departure from $c$
means solution which, if it is physical, would indicate the significant degree
of Lorentz violation (LV). However, this LV could be false if experimentally
particle parameters were read wrongly yielding different from $c$ solution for
the physical limiting velocity. Still, one may allow possible anticipation of
finding some LV in neutrino physics. The three solutions for the squares of
limiting velocities, denoted as $c_{1}^{2},c_{2}^{2}$ and $c_{3}^{2}$, depend
on the particle $m$, $E$ and $v$ (mass, energy and ordinary velocity) through
inverse sinusoidal functions. As $c_{1}^{2}$, $c_{3}^{2}>0$ and $c_{2}^{2}$
$<0,$ only $c_{1}^{2}$ and $c_{3}^{2}$ have chances to be physical while
$c_{2}^{2}$ is unphysical. Furthermore, with the inverse sinusoidal functions
principal values dependences, $c_{1}^{2}$ and $c_{3}^{2}$ being positive are
complementary limiting velocity squares, at least one of them physical and
presumed luminal, while $c_{2}^{2}$ being negative is definitely unphysical.
However, $c_{2}^{2}$ can become $c_{1}^{2}$ and $c_{3}^{2}$ when transformed
from the principal values region into the multiple values region of the
inverse sinusoidal functions. With these solutions one can treat physical
limiting velocities, for any particle, electron, neutrino, photon, etc. The
OPERA $17GeV$ muon neutrino velocity experiments are discussed through the
limiting velocity $c_{3}$ because the calculated neutrino $m_{\nu}c^{2}$ of
$0.076$ $eV$ , being negligible, makes $c_{1}$ unphysical. Furthermore,
because in OPERA experiments, $m_{\nu}c^{2}<<E_{\nu}$, one finds out that
$c_{3}=c(1+\Delta)\simeq c$ because $\Delta$ is negligible (it varies from
$O(-10^{-6})$ to $O(10^{-6})$ ). This implies basically the LI of the neutrino
energy-momentum dispersion relation.

1\. Introduction

There have been the whole series of neutrino velocity experiments such as the
OPERA collaborations with the detector in the CNGS beam [1] (versions 1, 2 and
3), the OPERA detector the CNGS beam using the 2012 dedicated data [2]
(versions 1 and 2) as well as the ICARUS detector in the CNGS beam [3]
(versions 1 and 2).

These neutrino velocity experiments are not simple to carry out and a lot of
them had a variety mistakes . For instance in [1] (version 1) a cable was
incorrectly plugged and there was a miscallibrated atomic clock. Both mistakes
were found and the OPERA collaborators made proper corrections and after a
precise measurment of the neutrino velocity in agreement with $c$, the
velocity of light, published the result in [1] (version 3). OPERA
collaborators also published the results from a measurement of a special
bunched neutrino beam [2] (version 2) giving the precision measurement of the
muon neutrino and muon anti-neutrino velocities, in good agreement with $c$,
the velocity of light. Other, so called Gran Sasso laboratory repeated the
measurements and obtained $c$, the velocity of light, for the neutrino
velocity [3] (version 2). Now, the masses of flavor neutrinos, whose
velocities one can measure, are not yet known precisely but are calculated as
exactly as possible from the provided masses of the mass state neutrinos.
Nevertheless, the accepted notion from the special relativity, also in cases
like these, expects the neutrino velocity not to exceed the velocity of light
$c$, considered in the special relativity as the universal limiting velocity.

In Section 2,from relativistic kinematics one formulates the sixth order
bicubic equation for the square of the limiting velocity and, as exposed in
[4], can be solved as a cubic equation for $c^{2}$. The bicubic equation
yields three solutions $c_{i}^{2},$ $i=1,2,3$, depending on $m,v,$ and $E$
(particle mass, velocity and energy). One expects that at least one solution
is physical and luminal and as such supports the LI; that is when evaluated to
be numerically, either exactly or practically exactly equal to $c$, so that
its substitution in place of $c$, will not change at all the energy-momentum
dispersion relation or it will change it insignificantly. The three limiting
velocity solutions depend on the inverse sinusoidal function principal values
in such a way that the complementary $c_{1}$ and $c_{3}$ are real and, at
least one of them, physical while $c_{2}$ is imaginary and as such unphysical.
However, for the specific multiple values of the inverse sinusoidal function,
$c_{2}$ can become $c_{1}$ and $c_{3}$. The important thing is the fact that
$c_{1}$ and $c_{3}$ are complementary limiting velocities since they together
can cover all the allowed particle parameters, $m,v,$ and $E$ , while each of
them is limited to particular values. Besides the exact solutions, Section 2
contains also the perturbative solutions for $c_{i}^{2},$ $i=1,2,3$, basically
in terms of $(mv^{2}/E)$ . These perturbative solutions are often very
convenient for determining as to which of the limiting velocities is physical,
$c_{1}$ or $c_{3}$ , either luminal $(=,\simeq c)$ or not $(\neq c)$ that is,
applicable for the particle in question.

In Section 3, after deducing the three flavor neutrino masses, one finds out
that the physical parameter structure of the OPERA [2] muon neutrino velocity
experiment is such that the Taylor series expansion, in terms of $(mv^{2}/E)$,
strongly suggests $c_{3}$ as the luminal solution. Along the same lines, one
notices that $c_{1}$ is unphysical in the OPERA [2] experiments.The same is
true for other neutrino velocity experiments [1] and [3]. Furthermore, because
the muon neutrino mass being negligible, one finds out that in fact
$c_{3}\simeq c$. To verify this perturbative result, one performs the
calculation also with exact non-perturbative expression for $c_{3}$. The
result is the same as from the pertubative calculation.

Conclusion and final remarks are given in Section 4. Here also the comparisons
with other approaches from the literature for discussing the Lorentz
invariance and Lorentz violation, either through changes in the Dirac equation
or by explicitly changing the relativistic kinematics, are given.

2\. Particle limiting velocities from the velocity bicubic equation

The velocity of light, $c$, in the special relativity particle kinematics is
considered the universal relativistically invariant limiting velocity. Here,
with the desire of having $c$ on an equal basis with other physical
parameters, one treats it as yet to be analythically formulated limiting
velocity and starts with the same kinematics.

$\overrightarrow{p}=\frac{E\text{
}\overrightarrow{v}}{c^{2}},E^{2}=\frac{m^{2}c^{4}}{1-\frac{v^{2}}{c^{2}}}$
(1,2)

which defines the particle momentum $\overrightarrow{p}$, and energy $E,$
through its mass $m$ and velocity $\overrightarrow{v}$. Momentum and energy
from (1) and (2) are related through the mass shell condition,

$\overrightarrow{p}^{2}c^{2}-E^{2}=-m^{2}c^{4}$ (3.1)

whose change, if any, caused by replacing $c$ with limiting velocity solutions
from the bicubic equation (to be discussed), could indicate either LI or LV,
providing that particle parameters, $m,v,$ and $E$ are known.

As in the neutrino velocity experiments [1,2,3], one has the known energy and
the velocity of fixed direction, it is convenient to continue with just
relation (2) by transforming it into the bicubic equation for the particle
limiting velocity $c:$

$m^{2}c^{6}=E^{2}c^{2}-E^{2}v^{2}$ (3.2)

Next, one rewrites it in the mathematically more familiar forms with solutions
characterized by the discriminant satisfying $D<0$ ,

$\displaystyle\left[\left(\frac{c}{v}\right)^{2}\right]^{3}-\left(\frac{E}{mv^{2}}\right)^{2}\left(\frac{c}{v}\right)^{2}+\left(\frac{E}{mv^{2}}\right)^{2}$
$\displaystyle=$ $\displaystyle 0,\text{ \
}q=-p=\left(\frac{E}{mv^{2}}\right)^{2},\text{ }$ $\displaystyle D$
$\displaystyle=$
$\displaystyle\left(\frac{q}{2}\right)^{2}+\left(\frac{p}{3}\right)^{3}$ (1)
$\displaystyle=$
$\displaystyle\frac{1}{4}\left(\frac{E}{mv^{2}}\right)^{4}\left[1-\frac{4}{27}\left(\frac{E}{mv^{2}}\right)^{2}\right]<0,$
$\displaystyle\TCItag{3.3}$

$z=\frac{3\sqrt{3}mv^{2}}{2E};\text{ }D<0:\text{ }-1<z<1$ (4)

According to [4], the solutions for (3.2,3) plus (4) can be written as

$\displaystyle c_{1}^{2}$ $\displaystyle=$ $\displaystyle
2v^{2}\sqrt{\frac{\left|p\right|}{3}}\cos\left(\frac{\theta}{3}+\frac{\pi}{6}\right),\text{
}$ $\displaystyle c_{2}^{2}$ $\displaystyle=$
$\displaystyle-2v^{2}\sqrt{\frac{\left|p\right|}{3}}\cos\left(\frac{\theta}{3}-\frac{\pi}{6}\right),\text{
}$ $\displaystyle c_{3}^{2}$ $\displaystyle=$
$\displaystyle-2v^{2}\sqrt{\frac{\left|p\right|}{3}}\cos\left(\frac{\theta}{3}+\frac{\pi}{2}\right)$
$\displaystyle\cos\left(\theta+\frac{\pi}{2}\right)$ $\displaystyle=$
$\displaystyle-\frac{q}{2\left(\frac{\left|p\right|}{3}\right)^{\frac{3}{2}}}=\frac{-3\sqrt{3}mv^{2}}{2E},$
$\displaystyle\text{ }\theta$ $\displaystyle=$
$\displaystyle-\frac{\pi}{2}+\cos^{-1}\left(\frac{-3\sqrt{3}mv^{2}}{2E}\right)=\sin^{-1}\left(\frac{3\sqrt{3}mv^{2}}{2E}\right)\TCItag{5}$
(2)

However, in order to see more of the physics, the exact solutions from (5),
with the help from relations (3.3) and (4), are rewritten in such a way as to
exhibit more explicitly the $m$, $v$, and $E$ parameters in them:

$\displaystyle c_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{2E}{\sqrt{3}m}\sin\left[\frac{\pi}{3}-\frac{1}{3}\sin^{-1}\left(\frac{3\sqrt{3}mv^{2}}{2E}\right)\right]>0,\TCItag{6.1}$
(3) $\displaystyle c_{2}^{2}$ $\displaystyle=$
$\displaystyle-\frac{2E}{\sqrt{3}m}\cos\left[\frac{1}{3}\sin^{-1}\left(\frac{3\sqrt{3}mv^{2}}{2E}\right)-\frac{\pi}{6}\right]<0,\TCItag{6.2}$
(4) $\displaystyle c_{3}^{2}$ $\displaystyle=$
$\displaystyle\frac{2E}{\sqrt{3}m}\sin\left[\frac{1}{3}\sin^{-1}\left(\frac{3\sqrt{3}mv^{2}}{2E}\right)\right]\text{
}>0\TCItag{6.3}$ (5)

Noting that with the variable $z$ from relation (4) the inverse sinus
function, $\sin^{-1}\left(z\right)$, in (6.1,2,3) refer to the principal
values that lie in the $\left(-\pi/2\ \ to\ \ \pi/2\right)$ range where either
of the positive $c_{1}^{2}$ and $c_{3}^{2}$ can be physical while the negative
$c_{2}^{2}$ is definitely unphysical. However, $c_{2}^{2}$ can become physical
in the multiple values ranges. Denote $c_{i}^{2},i=1,2,3$ dependence on $z$ as
$c_{i}^{2}\left[\sin^{-1}\left(z\right)\right],i=1,2,3$ . Then assume that
alternately in $c_{2}^{2}$ the range of $\sin^{-1}\left(z\right)$ is changed
from $\left(-\pi/2\ to\ \ \pi/2\right)$ to $(3\pi/2$ $to$ $5\pi/2)$ and to
$(\left(-5\pi/2\right)to$ $\left(-3\pi/2\right)).$This is simply achieved by
replacing $c_{2}^{2}\left[\sin^{-1}\left(z\right)\right]$ in (6.2) alternately
with $c_{2}^{2}\left[\sin^{-1}\left(z\right)+2\pi\right]$ and
$c_{2}^{2}\left[\sin^{-1}\left(z\right)-2\pi\right]$. Then the simple
evaluations, with the help from (6.1,2,3), shows that

$c_{2}^{2}\left[\sin^{-1}\left(z\right)+2\pi\right]=c_{3}^{2}\left[\sin^{-1}\left(z\right)\right];\text{
}c_{2}^{2}\left[\sin^{-1}\left(z\right)-2\pi\right]=c_{1}^{2}\left[\sin^{-1}\left(z\right)\right]$
(6.4,5))

In (6.4,5) the respective multivalue ranges in $c_{2}^{2}$ are $\left(3\pi/2\
\ to\ \ 5\pi/2\right)$ and $\left(-5\pi/2\ \ to\ -\ 3\pi/2\right)$ while in
$c_{3}^{2}$and $c_{1}^{2}$ the range is $\left(-\pi/2\ \ to\ \ \pi/2\right)$.
As one sees , here the complementary limiting velocities $c_{1}$ and $c_{3}$
are the only ones of the physical significances. Importance of (6.4,5) is in
the fact that sometimes one has to change the ”coordinates” in order to find
out the same or perhaps even the new physics. These different ranges,
principle values and multiple values do not change the fact that according to
(4) $|z|<1$. The imaginary $c_{2}$ moves the ”imaginary” physics to the ”real”
physics with $\sin^{-1}\left(z\right)$, being changed to
$\sin^{-1}\left(z\right)\pm 2\pi$ now with the multiple value ranges.

Next, it is illustrative to perform the Taylor series expansions of
$c_{i}^{2},i=1,2,3$ (6.1,2,3) . Except for the few first terms, they are done
basically in terms of $(mv^{2}/E)$ with inequalities between $v,E$ and $m$ as
indicated in each of the series. Extra $v^{2}$ factor in each term makes the
whole expression to have dimension of $v^{2}$ .

$\displaystyle c_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{E}{m}-\frac{v^{2}}{2}-\frac{3mv^{4}}{8E}-\frac{m^{2}v^{6}}{2E^{2}}+O\left[v^{2}\left(\frac{mv^{2}}{E}\right)^{3}\right]>0,v^{2}\text{
}<\frac{2E}{3\sqrt{3}m},\TCItag{7.1}$ (6) $\displaystyle c_{2}^{2}$
$\displaystyle=$
$\displaystyle-\frac{E}{m}-\frac{v^{2}}{2}+\frac{3mv^{4}}{8E}-\frac{m^{2}v^{6}}{2E^{2}}+O\left[v^{2}\left(\frac{mv^{2}}{E}\right)^{3}\right]<0,v^{2}\text{
}<\frac{2E}{3\sqrt{3}m},\TCItag{7.2}$ (7) $\displaystyle c_{3}^{2}$
$\displaystyle=$ $\displaystyle
v^{2}+\frac{m^{2}v^{6}}{E^{2}}+\frac{69}{32}\frac{m^{4}v^{10}}{E^{4}}+O\left[v^{2}\left(\frac{mv^{2}}{E}\right)^{6}\right]>0,m<\frac{2E}{3\sqrt{3}v^{2}}\TCItag{7.3}$
(8)

Already from exact solutions (6.1,2,3) as well as now from the Taylor series,
one sees that all three limiting velocities are different from each other in
the principle values region. For instance, $c_{1}^{2}$ and $c_{2}^{2}$ diverge
for $m=0$ but are finite for $v=0$. So $c_{1}^{2}$ and the unphysical
$c_{2}^{2}$ need to have $m\neq 0$. The different behaviors of $c_{1}^{2}$ and
$c_{3}^{2}$ for either $m\rightarrow 0$ or $v\rightarrow 0$ emphasizes their
complementarity. This small excursion, suggests defining the physical
$c_{1}^{2}$ and $c_{3}^{2}$ in the principle values region satisfying

$Phyhsical:c_{1}^{2},\text{ }c_{3}^{2}\neq 0,\infty$ (8.1)

Here, consistent with (8.1), is the summary of important situations that can
occur for $c_{1}^{2}$ and $c_{3}^{2}$

$\displaystyle v$ $\displaystyle=$ $\displaystyle 0,m\neq
0:\frac{E_{0}}{m}=c_{1}^{2}=c^{2};\text{ }c_{3}^{2}=0\text{
}(unphysical),\TCItag{8.2}$ (9) $\displaystyle m$ $\displaystyle=$
$\displaystyle 0,E\text{ }finite;c_{3}=v=c\text{ }(photon);\text{
}c_{1}=\infty\text{ }(unphysical),\TCItag{8.3}$ (10) $\displaystyle m$
$\displaystyle\neq$ $\displaystyle 0,E\text{
}\rightarrow\infty;c_{3}\rightarrow v;\text{ }c_{1}\rightarrow\infty\text{
}(unphysical).\TCItag{8.4}$ (11)

What examples (8.2,3,4) show clearly is that in these particular situations
the physically acceptable limiting velocity is either given by $c_{1}$ or
$c_{3}$ which further indicates to their complementarity. The relation (8.2)
is to be understood as a definition of $E(v)$ at $v=0$ where
$c_{1}^{2}=c^{2}$. As $c_{1}$ and $c_{3}$ are the limiting velocity solutions
of the bicubic equation (3.3), it is appropriate to see what effect will be
caused if one sets either $c_{1}$ or $c_{3}$ in place of $c$ in the energy
momentum relation (3.1). These substitutions leave the energy momentum
relation (3.1) either LI or, to a degree, LV under the Lorentz transformations
with the following respective general possible values for $c_{1}$ or $c_{3}$:

$LI:c_{1}=c\text{ }or\text{ }c_{3}=c;\text{ }LV:c_{1}\neq c\text{ }or\text{
}c_{3}\neq c$ (8.5)

Here it is assumed that either $c_{1}$ or $c_{3}$ is LI but not both of them
at the same time. Also, the degree of the LV would depend on how strongly
$c_{1}\neq c$ $or$ $c_{3}\neq c$.

Despite their complementarity, It is necessary to investigate whether it can
happen that for a given particle one can have $c_{1}=c_{3}$ ? Imposing this
equality, from (6.1) and (6.2), with $z$ as defined in (4), one arrives at the
following sequence of equations,

$\displaystyle c_{1}^{2}$ $\displaystyle=$ $\displaystyle
c_{3}^{2}:\sin\left[\frac{\pi}{3}-\frac{1}{3}\sin^{-1}\left(z\right)\right]=\sin\left[\frac{1}{3}\sin^{-1}\left(z\right)\right],\TCItag{9.1}$
(12) $\displaystyle\sin^{-1}\left(z\right)$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}:\text{ }z=1\TCItag{9.2}$ (13)

Since relation (9.2) is in contradiction to the relation (4), which excludes
$z=1$, one concludes that $c_{1\text{ }}$and $c_{3}$, while complementary,
cannot be equal to each other for the same particle,$c_{1}\neq c_{3}$ . Hence,
if for instance $c_{3\text{ }}=c$ then $c_{1}\neq c$ and so on.

3\. Limiting velocity of the neutrino

Here one is specifically interested in applying the formalism of obtaining the
limiting velocity for the muon neutrino, $\nu_{\mu}$, with the physical
parameters from the OPERA experiment [2]. From the perturbative expressions
(7) , the indication is that $c_{1}$ and $c_{3}$ are respectively, the
unphysical and physical limiting velocities, in this case. To see whether the
physical $c_{3\text{ }}$is also luminal, that is, leading to $c$ and LI, one
first expresses $c_{3}$ perturbatively from (7.3) and then, for verification
purposes, also exactly from (6.3).

The formalism in relations (6) and (7) demand, in addition to $E$ and $v$ also
the value of the mass, here denoted for the muon neutrino as $m_{\nu}$. As in
the reference [2] the value of $m_{\nu}$ is not given, one has to first find
which value is presently favored, although in OPERA experiment [2] with the
neutrino energy of $\ E_{\nu}\left(\mu\right)=17GeV$, the calculated neutrino
mass even if exact, will be very likely negligible as compared to the energy.
Now,there are three flavor neutrinos, denoted as $\nu_{e}$, $\nu_{\mu}$ and
$\nu_{\tau}$, the electron, muon and tau neutrino.Their masses $m_{\nu}(e)$,
$m_{\nu}(\mu)$ and $m_{\nu}(\tau)$ are derived from the masses of the
independent mass-state three neutrinos with masses $m_{1}$,$m_{2\text{, }}$and
$m_{3}$. In the discussion of the $\mu-\tau$ symmetry, these masses have been
given by S. Gupta et al. [5] as,

$m_{1}c^{2}=0.067\text{ }eV,\text{ }m_{2}c^{2}=0.068\text{ }eV,\text{
}m_{3}c^{2}=0.084\text{ }eV\text{,}$ (10.1)

The flavor neutrino masses are defined in [5] with the help of the Harrison et
al. neutrino mixing matrix [6], $U_{\alpha,i},$ $\alpha=e,\mu.\tau$
$;i=1,2,3$, connecting the flavor neutrino states to the mass-state neutrino
states (see,also [7] ]). Hence, using $U_{\alpha,i}$ as in references [6] and
[7], according to Gupta et al. [5], the flavor neutrino masses are defined as

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle e,\upsilon,\tau;\text{
}i=1,2,3:\text{
}m_{\nu}\left(\alpha\right)=\left[\mathop{\textstyle\sum}\nolimits_{i}\left|U_{\alpha,i}\right|^{2}m_{i}^{2}\right]^{\left(\frac{1}{2}\right)};$
$\displaystyle\text{ }\left(U_{\alpha,i}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{lll}$\sqrt{\frac{2}{3}}$&$\sqrt{\frac{1}{3}}$&$0$\\\
$-\sqrt{\frac{1}{6}}$&$\sqrt{\frac{1}{3}}$&$-\sqrt{\frac{1}{2}}$\\\
$-\sqrt{\frac{1}{6}}$&$\sqrt{\frac{1}{3}}$&$\sqrt{\frac{1}{2}}$\end{tabular}\right)\TCItag{10.2}$
(17)

yielding

$m_{\nu}\left(e\right)c^{2}=0.067\text{ }eV,\text{
}m_{\nu}\left(\mu\right)c^{2}=0.076\text{ }eV,\text{
}m_{\nu}\left(\tau\right)c^{2}=0.076\text{ }eV$ (11)

With these values and from [2] the collection of data for the OPERA muon
neutrino velocity experiment is

$\displaystyle E_{\nu}\left(\mu\right)$ $\displaystyle=$ $\displaystyle
17GeV,\text{ }m_{\nu}\left(\mu\right)c^{2}=0.076\text{ }eV,\text{ }$
$\displaystyle v_{\nu}\left(\mu\right)$ $\displaystyle=$ $\displaystyle
c(1+\Delta)$ $\displaystyle-1.8\times 10^{-6}$ $\displaystyle\leq$
$\displaystyle\Delta\leq 2.3\times 10^{-6}\TCItag{12}$ (18)

Because one has that $\
m_{\nu}\left(\mu\right)<\left(2E_{\nu}\left(\mu\right)/3\sqrt{3}v_{\nu}^{2}\left(\mu\right)\right)$
for any $v_{\nu}\left(\mu\right)$ from (12), one easily deduces, according to
(7.1), that approximate numerical value of $c_{1}$ is $\ c_{1}$ $\simeq
4.73\times 10^{5}c$ .Such a large value makes $c_{1}$ unphysical. Expecting
that $c_{3}$ is physical, one calculates it with more precision first
perturbatively from (7.3),

$c_{3}^{2}=v_{\nu}^{2}\left(\mu\right)\left[1+\left(\frac{m_{\nu}\left(\mu\right)c^{2}}{E_{\nu}\left(\mu\right)}\right)^{2}\left(\frac{v_{\nu}\left(\mu\right)}{c}\right)^{4}+O\left(\left(\frac{m_{\nu}\left(\mu\right)c^{2}}{E_{\nu}\left(\mu\right)}\right)^{4}\left(\frac{v_{\nu}\left(\mu\right)}{c}\right)^{8}\right)\right]$
(13)

Furthermore with
$\left(m_{\nu}\left(\mu\right)c^{2}/E_{\nu}\left(\mu\right)\right)\simeq
4.5\times 10^{-12}$ and $|\Delta|<<1$, one obtains for $c_{3}$ the
perturbative solution in the form,

$\frac{c_{3}}{c}\simeq\left(1+\Delta\right)\left(1+\frac{1}{2}\left(\frac{m_{\nu}\left(\mu\right)c^{2}}{E_{\nu}\left(\mu\right)}\right)^{2}(1+\Delta)^{4}\right)=\left(1+\Delta\right)$
(14.1)

Exact expression for $c_{3}$ from (6.3) with the OPERA physical parameters is
as follows

$\displaystyle\left(\frac{c_{3}}{c}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{2E_{\nu}\left(\mu\right)}{\sqrt{3}m_{\nu}\left(\mu\right)c^{2}}\sin\left[\frac{1}{3}\sin^{-1}\left(\frac{3\sqrt{3}m_{\nu}\left(\mu\right)c^{2}\left(1+\Delta\right)^{2}}{2E_{\nu}\left(\mu\right)}\right)\right]$
(19) $\displaystyle=$ $\displaystyle(1+\Delta)^{2}\TCItag{14.2}$

In deriving (14.2), one simply takes into account that
$\left(m_{\nu}\left(\mu\right)c^{2}/E_{\nu}\left(\mu\right)\right)<<1$ and
that in $(1+\Delta)$ , $|\Delta|<<1$ so that only first terms in Taylor
expansions of sin and sin-1 functions need to be retained. Hence , both
solutions, being equal and basically luminal, yield LI with $c$ as the
solution for $c_{3}$:

$\ \ \ \ \ \ \ (1-1.8\times 10^{-6})c\leq\ c_{3}\leq(1+2.3\times 10^{-6})c\
:\text{ }c_{3}\simeq c$ (14.3)

The result in (14.3) is what Einstein envisioned long time ago.

What one notices here is the fact that for OPERA experiments [2], through a
particular collection of neutrino physical parameters, such as mass, ordinary
velocity and energy, the bicubic equation yields the luminal limiting velocity
solution, that is, with the velocity of light $c$ and with the LI.

Now, on one example one can show how the luminal limiting velocity solution
with the LI, can become the superluminal solution with the LV. Simply, in
(14.1) and (14.2) replace the negligible $\Delta$ with a small but finite and
positive $\Delta$. In doing so, one basically obtains he LV from reference
[8],implied by the change in the particle special relativistic velocity,
written as $c/\left(p^{2}+m^{2}c^{2}\right)^{\frac{1}{2}}+\Delta c.$It is
easily seen that in the situation where the mass is negligible, as is in the
OPERA experiments, this $\Delta$ should be the same as $\Delta$ in relations
(14.1,2), and if negligible, as in relations (14), should allow the LI rather
than the LV under the Lorentz transformations. Although so far no verifiable
LV showed up in neutrino physics, on should, nevertheless, keep an open mind
also for such a possibility with subluminal or superluminal anticipations. The
LV formulations with superluminal particles through the Dirac equation have
been done, for example, in [9] and [10].

4\. Conclusion and final remarks

Identifying the velocity of light $c$ in the relativistic kinematics as a
limiting velocity yet to be determined, one is lead naturally to the bicubic
equation for the limiting velocity. Of the three resulting solutions one,
$c_{2}$, is imaginary while two other solutions, $c_{1}$ and $c_{3}$, are real
and complementary with different emphasis on particle parameter dependences.
Of course, if the particle parameters choose, say $c_{1}$ $=c$ ,then as argued
in (8.1) to (8.4) $c_{3\text{ }}$will be unphysical. The remarkable point in
determining the limiting velocity of any particle from the bicubic equation is
that the particle physical parameters will yield for it most likely $c$, no
matter where one measures its mass, energy and the ordinary velocity.

It appears that what one needs are the velocity experiments done with rather a
masssive particle which allow full participation of the particle mass in
determining of its limiting velocity . A natural candidate for such a limiting
velocity determination is the electron whose mass is very well known and the
energy can be chosen so as not to render the mass negligible.

Acknowledgment

The author is grateful to Dr. M. Dracos for informing him that the neutrino
beam has 17 GeV average energy in the OPERA experiments. Descriptions and
explanations of neutrino velocity experiments by an anonymous physicist
knowledgeable with OPERA and other experiments is gratefully acknowledged.

References

[1] T. Adam et al., ”Measurement of neutrino velocity with OPERA detector in
the CNGS beam.”; arXiv:1109.4897.

[2] T. Adam et al., ”Measurement of neutrino velocity with the OPERA detector
in the CNGS beam using the 2012 dedicated data”; arXiV:1212.1276

(17 Dec. 2012), published in JHEP 1210, 093 (2012).

[3] M. Antonello et al., ”Precision measurements of the neutrino velocity with
the ICARUS detector in the CNGS beam.”; arXiv:1208.2629 (26 Sept., 2012).
Phys. Lett. B713, 17 (2012). .

[4] R. S. Burrington, Handbook of Mathematical Tables and Formulas, McGraw-
Hill, 1973; H. J. Bartsch, Taschenbuch Matematischen Formeln (Handbook of
Mathematical Formulas),

Verlag Harri Deutsch, 1975.

[5] S. Gupta, A. S. Joshipura and K. M. Patel, ”How Good is mu-tau Symmetry
After Results on Non-Zero Theta(13)?”; arXiv: 1301.7130 (30 Jan., 2013).

[6] P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B 530, 167
(2002).

[7] Josip Šoln, Phys. Scr. 80, 025101 (2009); arXiv: 0908.1763.

[8] G. Amelino-Camelia, ”Phenomenology of Philosophy of Science”: OPERA data ;
arXiv: 1206.3554 (2012).

[9] S. R. Coleman and S. L. Glashow, Phys. Rev. D59, 116008 (1999).

[10] L. Zhou, B. Q. Ma, Astropart. Phys. 41, 24-27 (2013); arXiv: 1109-6097
(2013).