\section{Introduction}

\subsection{Setup} In this note we continue the study initiated in \cite{COSS17, Qu15} on sharp Fourier restriction theory on hyperboloids.
Let us start by recalling the basic terminology and the main definitions.

\smallskip

Throughout this work we adopt the following normalization for the Fourier transform in $\mathbb{R}^{d+1}$:
\begin{equation}\label{NormalizeFT}
\widehat{g}(\zeta)=\int_{\mathbb{R}^{d+1}} e^{-iz\cdot\zeta}\, g(z)\,\d z.
\end{equation}
If $\xi \in \mathbb{R}^d$, we define $\langle \xi \rangle := (1 + |\xi|^2)^{\frac12}$. The hyperboloid $\mathbb{H}^d \subset \mathbb{R}^{d+1}$ is the surface defined by\footnote{A simple rescaling argument transfers all the results of this paper to the hyperboloids $\mathbb{H}^d_s=\big\{(\xi,\tau)\in\mathbb{R}^d\times\mathbb{R}: \tau=(s^2+|\xi|^2)^{\frac12}\big\}$, $s > 0$.}
\begin{equation*}
\mathbb{H}^d=\big\{(\xi,\tau)\in\mathbb{R}^d\times\mathbb{R}: \tau=\langle \xi \rangle\big\},
\end{equation*}
and comes equipped with the Lorentz-invariant measure
\begin{equation}\label{defsigma}
\d\sigma(\xi,\tau)=\ddirac{\tau-\jp{\xi}}\frac{\d \xi\,\d \tau}{\langle \xi \rangle},
\end{equation}
which is defined by duality on an appropriate dense class of functions via the identity
$$\int_{\mathbb{H}^d}\varphi(\xi,\tau)\,\d\sigma(\xi,\tau)=\int_{\mathbb{R}^d} \varphi(\xi,\langle \xi \rangle)\frac{\d \xi}{\langle \xi \rangle}.$$
The {\it Fourier extension operator} on $\mathbb{H}^d$ (or adjoint Fourier restriction operator) is given by
\begin{equation}\label{eq:DefExtensionOp}
T(f)(x,t):=\int_{\mathbb{R}^d} e^{ix\cdot \xi} e^{it\langle \xi \rangle} f(\xi)\frac{\d \xi}{\langle \xi \rangle},
\end{equation}
where $(x,t)\in\mathbb{R}^d\times\mathbb{R}$ and $f$ belongs to the Schwartz class in $\mathbb{R}^d$.
Throughout this note we identify a function $f:\mathbb{H}^d\to\mathbb{C}$ with a complex-valued function defined on $\mathbb{R}^d$. The norm in $L^p(\mathbb{H}^d) = L^p(\mathbb{H}^d, \sigma)$ is then given by
$$\|f\|_{L^p(\mathbb{H}^d)}=\left(\int_{\mathbb{R}^d} |f(\xi)|^p \frac{\d \xi}{\langle \xi \rangle}\right)^{\frac1p}.$$
With the Fourier transform normalized as in \eqref{NormalizeFT}, note that
\begin{equation}\label{TintermsofHat}
T(f)(x,t)=\widehat{f\sigma}(-x,-t).
\end{equation}
The seminal work of Strichartz \cite[Theorem 1, Cases III (b)(c)]{St77} establishes the estimate
\begin{equation}\label{ExtensionInequality}
\|T(f)\|_{L^p(\mathbb{R}^{d+1})}\leq {\bf H}_{d,p} \, \|f\|_{L^2(\mathbb{H}^d)}\,,
\end{equation}
with a finite constant ${\bf H}_{d,p}$ (independent of $f$), provided that
  \begin{equation}\label{AdmissibleRange}
   \begin{cases}
    6 \leq p< \infty, \text{ if } d=1;\\
  \frac{2(d+2)}{d} \leq p\leq  \frac{2(d+1)}{d-1}, \text{ if } d\geq 2.
      \end{cases}
\end{equation}
We reserve the symbol ${\bf H}_{d,p}$ for the optimal constant
\begin{equation}\label{Def_Sharp_Const}
{\bf H}_{d,p}:=\sup_{0\neq f\in L^2(\mathbb{H}^d)} \frac{\|T(f)\|_{L^p(\mathbb{R}^{d+1})}}{\|f\|_{L^2(\mathbb{H}^d)}},
\end{equation}
and say that a nonzero function $f\in L^2(\mathbb{H}^d)$ is an {\it extremizer} of \eqref{ExtensionInequality} if it realizes the supremum in \eqref{Def_Sharp_Const}, and we call a nonzero sequence  $\{f_n\} \subset L^2(\mathbb{H}^d)$ an {\it extremizing sequence} of \eqref{ExtensionInequality} if the ratio ${\|T(f_n)\|_{L^p(\mathbb{R}^{d+1})}}/{\|f_n\|_{L^2(\mathbb{H}^d)}}$ converges to ${\bf H}_{d,p}$ {as $n\to\infty$}.

\subsection{Main theorem} The first result to address the sharp form of \eqref{ExtensionInequality} is due to Quilodr\'{a}n \cite{Qu15}, in which he computes the exact values of ${\bf H}_{d,p}$ in the endpoint cases $(d,p) = (2,4),(2,6)$ and $(3,4)$, and establishes the non-existence of extremizers in these cases.\footnote{By contrast, the recent work \cite{Qu18} establishes existence of extremizers for the endpoint $L^2$ to $L^4$ adjoint Fourier restriction inequality on the one-sheeted hyperboloid in dimension 4.} A crucial element of his proof is the fact that the Lebesgue exponents $p$ under consideration are even integers, a fact that allows one to use the convolution structure of the problem via an application of Plancherel's theorem. In \cite{Qu15}, Quilodr\'{a}n also raises two interesting questions: What is the value of the sharp constant at the endpoint $(d,p) = (1,6)$ (the remaining case with $p$ even); and do extremizers exist in the non-endpoint cases.

\smallskip

The precursor \cite{COSS17} of the present work contains two main results. The first result \cite[Theorem 1]{COSS17} is the explicit computation of the optimal constant ${\bf H}_{d,p}$ in the case $(d,p) = (1,6)$ and the proof that extremizers do not exist in this case. The second result \cite[Theorem 2]{COSS17} establishes the existence of extremizers in all non-endpoint cases of \eqref{ExtensionInequality} in dimensions $d \in \{1,2\}$. The proof of the latter result is obtained  by establishing {that} extremizing sequences converge modulo certain symmetries of the problem. {In the present case}, by a {\it symmetry} we mean an operator $S:L^2(\mathbb{H}^d)\rightarrow L^2(\mathbb{H}^d)$ such that
$$\|Sf\|_{L^2(\mathbb{H}^{d})}=\|f\|_{L^2(\mathbb{H}^{d})}
\text{ and }
\|T(Sf)\|_{L^p(\mathbb{R}^{d+1})}=\|T(f)\|_{L^p(\mathbb{R}^{d+1})}.$$\noindent {Such an operator} can shift the mass of sequences and destroy strong convergence while still mantaining its extremizing properties, hence the study of these symmetries is fundamental. In the {case of the hyperboloid}, one has to account for the action of the Lorentz group and space-time modulations (and their compositions), which we introduce in more detail in the next section. In \cite{COSS17}, the convergence is obtained via a direct and self-contained approach that explores the convolution structure of the problem at the lower endpoint (which is an even integer in these low dimensions). The drawback of this particularly simple proof is that it does not work in the higher dimensional cases $d \geq 3$.

\smallskip

In this note we return to this problem and extend the result of \cite[Theorem 2]{COSS17} to dimensions $d\geq 3$. Our main result is the following.

\begin{theorem}\label{Thm1}
Let $d\geq 3$.
Extremizers for inequality \eqref{ExtensionInequality} exist if $\frac{2(d+2)}{d} < p < \frac{2(d+1)}{d-1}$. In fact, given any extremizing sequence $\{f_n\}$, there {exist} symmetries $S_n$ such that $\{S_nf_n\}$ converges in $L^2(\mathbb{H}^d)$ to an extremizer $f$, {after passing to a subsequence}.
\end{theorem}

The main new ingredient of the proof, when compared to that of \cite[Theorem 2]{COSS17}, is the use of machinery from bilinear restriction theory to obtain a refined version of inequality \eqref{ExtensionInequality}.
As in \cite{Ca17, COSS17}, we exploit the fact that the hyperboloid is well approximated by the paraboloid and the cone.
The geometric construction underlying the bilinear restriction machinery accounts for this fact:  in some sense, it interpolates between the two endpoint cases, which we will refer to as the {\it elliptic} and the {\it conic} regimes, respectively.

\smallskip

Estimates for Fourier extension operators are related to estimates for dispersive partial differential equations. In our case, the extension operator $T$ defined in \eqref{eq:DefExtensionOp} is related to the Klein--Gordon equation $\partial_t^2u = \Delta_x u -u$ for $(x,t)\in\mathbb{R}^d\times\mathbb{R}$. Defining the (half) {\it Klein--Gordon propagator} as
\begin{equation*}
    e^{it\sqrt{1-\Delta}}g(x):= \frac{1}{(2\pi)^d}\int_{\mathbb{R}^d} e^{i x\cdot\xi}\,e^{it\langle\xi\rangle}\,\widehat{g}(\xi)\,\d\xi,
\end{equation*}
one readily sees that
\begin{equation}\label{July19_12:43am}
T(f)(x,t)=(2\pi)^{d} \,e^{it\sqrt{1-\Delta}}\, g(x),
\end{equation}
with $\widehat{g}(\xi)=\langle \xi \rangle^{-1}f(\xi)$.
Therefore, inequality \eqref{ExtensionInequality} can be restated as
\begin{equation*}
   \|e^{it\sqrt{1-\Delta}}g\|_{ L_{x,t}^p(\mathbb{R}^d\times\mathbb{R})}\leq (2\pi)^{-d} \,{\bf H}_{d,p}\,\|g\|_{H^{\frac{1}{2}}(\mathbb{R}^d)},
\end{equation*}
where for $s \geq 0$ we denote by $H^s(\mathbb{R}^d)$ the nonhomogeneous Sobolev space, defined as
$$H^{s}(\mathbb{R}^d)=\Big\{g\in L^2(\mathbb{R}^d):~\|g\|_{H^s(\mathbb{R}^d)}^2:=\int_{\mathbb{R}^d}|\widehat{g}(\xi)|^2\jp{\xi}^{2s}\d\xi<\infty\Big\}.$$
The reader should keep in mind this equivalent formulation, since some of the results we quote from \cite[Section 6]{COSS17} are stated in terms of the Klein--Gordon propagator.

\smallskip

Extremal problems related to Fourier restriction theory have garnished a lot of attention in recent years, and a large body of {work} has emerged. Several authors have investigated the interface between bilinear restriction theory and these extremal questions, {both from the restriction side and the partial differential equations point of view}. Here we mention the works \cite{BV07,Ca17,FLS16,FS17,JSS,KSV12,Ra12}, all of which deal with these connections. Many other authors have contributed to the development of the area, and we refer the reader to \cite{COSS17} for an exposition of related literature on sharp Fourier restriction theory.

\subsection{Outline} We discuss the Lorentz symmetry of the problem in Section \ref{sec:Prelims}, where we also establish an annular decoupling inequality which implies a modest  gain of control over extremizing sequences.
The actual proof starts in Section \ref{sec:ang_rest}, with a simple but useful argument that allows us to restrict the angular support of the functions under consideration.
In Section \ref{sec:Caps}, we describe a geometric decomposition of space into {\it caps} and {\it sectors}, and the corresponding bilinear restriction estimates that will play a key role in the analysis.
As in \cite{COSS17}, the crux of the matter is the construction of a {\it distinguished region}, i.e. the lift of a cap or a sector to the hyperboloid that contains a positive universal proportion of the total mass in an extremizing sequence.
We establish this fact via a refined Strichartz inequality, formulated as Theorem \ref{Sharpened} and proved in Section \ref{sec:RefinedStrichartz}.
Once the existence of a special region has been established in dimensions $d \geq3$, the proof of Theorem \ref{Thm1} is finished by invoking the concentration-compactness material of \cite[Section 6]{COSS17}, which was already tailor-made to receive the input in any dimension. The details are outlined in Section \ref{sec:CC}.

\subsection{Notation} {\it Universal} quantities will be allowed to depend only on the dimension $d$ and the Lebesgue exponent $p$.
In a similar spirit, given $A,B\geq 0$, we write $A\simeq B$ (resp. $A \lesssim B$) and say that $A,B$ are {\it comparable} if there exists a finite constant $C=C(d,p)>0$, such that $\frac1C B\leq A\leq CB$ (resp. $A \leq CB$).
A number $N$ is said to be {\it dyadic} if it is an integral power of 2, i.e. $N\in2^{\mathbb{Z}}$.

\section{Preliminaries}\label{sec:Prelims}
\subsection{Lorentz boosts}
The Lorentz group, denoted $\mathcal{L}$, is defined as the group of invertible linear transformations in $\mathbb{R}^{d+1}$ that preserve the bilinear form
$(x,y)\in\mathbb{R}^{d+1}\times\mathbb{R}^{d+1}\mapsto x\cdot Jy,$ where $J=\text{diag}(-1,\ldots,-1,1)$.
In particular, if $L\in\mathcal{L}$, then $|\det L|=1$.
Denote the subgroup of $\mathcal{L}$ that preserves $\mathbb{H}^d$ by $\mathcal{L}^+$.
A one-parameter subgroup of $\mathcal{L}^+$ is $\{L^t\}_{t\in(-1,1)}$, where the linear map $L^t:\mathbb{R}^{d+1}\to\mathbb{R}^{d+1}$ is defined via
$$L^t(\xi_1,\ldots,\xi_d,\tau)=\left(\frac{\xi_1+t\tau}{\sqrt{1-t^2}},\xi_2,\ldots,\xi_d,\frac{\tau+t\xi_1}{\sqrt{1-t^2}}\right).$$
Given an orthogonal matrix $A\in\text{O}(d)$, the map $(\xi,\tau)\mapsto(A\xi,\tau)$ belongs to $\mathcal{L}^+$.
A way to parametrize more general
Lorentz boosts is as follows.
Given a frequency parameter $\nu\in\mathbb{R}^d$, we define the Lorentz boost in the direction $\nu$ as
\begin{equation}\label{eq:Lorentz}
L_\nu(\xi,\tau):=
(\xi^\perp+\jp{\nu}\xi^\parallel-\nu \tau,\jp{\nu}\tau-\nu\cdot \xi).
\end{equation}
Here $\xi^\perp$ and $\xi^\parallel$  denote the components of $\xi$ which are orthogonal and parallel to $\nu$, respectively.
The boost $L_\nu$ preserves space-time volume since its determinant is one,
and acts on $\mathbb{R}^d$ via
\begin{equation}\label{eq:RdLorentz}
L_\nu^\flat(\xi):=\xi^\perp+\jp{\nu}\xi^\parallel-\nu \jp{\xi}.
\end{equation}
Note that $L_{\nu}^{-1}=L_{-\nu}$, and likewise $(L_{\nu}^\flat)^{-1}=L_{-\nu}^\flat$.
We also have that $L_\nu(\nu,\jp{\nu})=(0,1)$, and correspondingly $L_\nu^\flat(\nu)=0$.
For $p\in[1,\infty]$, $L\in\mathcal{L}^+$, and $f\in L^p(\mathbb{H}^d)$, define the composition $L^*f=f\circ L$.
Then one easily checks that
$$\|L^*f\|_{L^p(\mathbb{H}^{d})}=\|f\|_{L^p(\mathbb{H}^{d})}
\text{ and }
\|T(L^*f)\|_{L^p(\mathbb{R}^{d+1})}=\|T(f)\|_{L^p(\mathbb{R}^{d+1})}.$$

\subsection{Annular decoupling}\label{sec:AnnularDec}
The extension operator $T$ defined in \eqref{eq:DefExtensionOp} satisfies more general mixed-norm estimates of which \eqref{ExtensionInequality} is a particular case.
As pointed out in \cite{KO11} and the references therein, the inequality
\begin{equation}\label{MixedNormKGStrichartz}
\|T(f)\|_{L^q_t L^r_x(\mathbb{R}^{d+1})}\lesssim \|\jp{\xi}^{\frac1q-\frac1r} f\|_{L^2_\xi(\mathbb{H}^d)}
\end{equation}
holds, provided
$q\in[2,\infty]$,
$r\in[2,2d/(d-2)]$
($r\in[2,\infty]$ if $d\in\{1,2\}$), and
$$\frac2q+\frac{d-1+\theta}r=\frac{d-1+\theta}2,\;(q,r)\neq(2,\infty),$$
for some $\theta\in[0,1]$.
A pair $(q,r)$ of Lebesgue exponents satisfying these conditions will be referred to as an {\it admissible pair}.
Certain instances of inequality \eqref{MixedNormKGStrichartz} together with a variant of the Littlewood--Paley decomposition yield an annular decoupling inequality which we now prove.

\smallskip

We will use a dyadic frequency decomposition.
To implement it,
let $N\geq 1$ be a dyadic number.
Given $f\in L^2(\mathbb{H}^d)$, we denote by $f_N$ the smoothed out restriction of $f$ to frequencies $|\xi|\simeq N$. More precisely, fix a smooth radial bump function $\psi:\mathbb{R}^d\to[0,1]$ supported in the ball $\{\xi\in\mathbb{R}^d:|\xi|\leq\frac{11}{10}\}$ and equal to 1 on the unit ball $\{\xi\in\mathbb{R}^d:|\xi|\leq1\}$, and define
\begin{displaymath}
f_N(\xi) := \left\{ \begin{array}{ll}
\psi(\xi){f}(\xi), & \textrm{if }N=1,\\
\big(\psi(\frac{\xi}N)-\psi(\frac{2\xi}N)\big){f}(\xi), & \textrm{if $N>1$.}
\end{array} \right.
\end{displaymath}
Note that ${\rm supp}(f_1) \subseteq \{\xi\in\mathbb{R}^d:|\xi| \leq 2\}$ and ${\rm supp}(f_N) \subseteq \{\xi\in\mathbb{R}^d:\tfrac N2 \leq |\xi|\leq 2N\}$, for $N >1$.
The following annular decoupling is in the spirit of \cite{JSS, KSV12}.
\begin{proposition}\label{annulardec}
Let $d\geq 3$ and $\frac{2(d+2)}{d} \leq p \leq  \frac{2(d+1)}{d-1}$.
 Then
\begin{equation}\label{Annular_decoupling}
\|T(f)\|^p_{L^p(\mathbb{R}^{d+1})}\lesssim
\sup_{N\in 2^{\mathbb{Z}_{\geq 0}}} \|T(f_N)\|_{L^p(\mathbb{R}^{d+1})}^{p-2}
\|f\|_{L^{2}(\mathbb{H}^d)}^2,
\end{equation}
for every $f\in L^2(\mathbb{H}^d)$.
\end{proposition}

\begin{proof}
By the Littlewood--Paley square function estimate, we have that
\begin{equation}\label{eq:LPConsequence}
\|T(f)\|_{L_{x,t}^p}^p
\simeq\Big\|\Big(\sum_N |T (f_N)|^2\Big)^{\frac12}\Big\|_{L^p_{x,t}}^p.
\end{equation}
Indeed,  $\mathcal{F}_x[T(f)](\xi,t)=e^{it\langle\xi\rangle}\langle\xi\rangle^{-1}f(\xi)$, where $\mathcal{F}_x$ denotes the Fourier transform  in the variable $x\in\mathbb{R}^d$.
Standard Littlewood--Paley theory yields
$$\|T(f)(\cdot, t)\|_{L^p_x}^p\simeq\Big\|\Big(\sum_N |T(f_N)(\cdot,t)|^2\Big)^{\frac12}\Big\|_{L^p_x}^p,$$
for each fixed $t\in\mathbb{R}$.
Estimate \eqref{eq:LPConsequence} then follows from integration in the time variable $t$.

Since $d\geq 3$, we have that $\frac p2\leq 2$, and thus the sequence space embedding $\ell^{\frac p2}\hookrightarrow\ell^2$ implies
$$\Big(\sum_N |T (f_N)|^2\Big)^{\frac 12}\leq \Big(\sum_N |T (f_N)|^{\frac p2}\Big)^{\frac 2p}.$$
We can estimate
\begin{equation}\label{triangleineq}
\|T(f)\|_{L^p_{x,t}}^p
\lesssim\int_{\mathbb{R}^{d+1}} \Big(\sum_N |T(f_N)|^{\frac p2}\Big)^2
=\int_{\mathbb{R}^{d+1}} \sum_{N,M} |T (f_M) T (f_N)|^{\frac p2}
\lesssim \sum_{M} \sum_{N\leq M} \|T (f_M)T (f_N)\|_{L^{\frac p2}_{x,t}}^{\frac p2},
\end{equation}
where the last inequality follows from Fubini's theorem and symmetry.
We control each of the summands of the right-hand side of \eqref{triangleineq} using the mixed-norm estimates \eqref{MixedNormKGStrichartz}.
With this purpose in mind, fix admissible pairs $(q_0,r_0)$ and $(q_1,r_1)$ with $q_1<p<q_0$ and $r_0<p<r_1$, which additionally satisfy
\begin{equation}\label{eq:MixedNormExponents}
\frac2p=\frac1{q_0}+\frac1{q_1}=\frac1{r_0}+\frac1{r_1}.
\end{equation}
Then, invoking H\"older's inequality twice, we have that
\begin{align*}
\|T (f_M)T (f_N)\|_{L^{\frac p2}}^{\frac p2}
&\leq\|T (f_M)\|_{L^p}^{\frac p2-1}\|T (f_N)\|_{L^p}^{\frac p2-1}\|T (f_M)T (f_N)\|_{L^{\frac p2}}\\
&\leq\|T (f_M)\|_{L^p}^{\frac p2-1}\|T (f_N)\|_{L^p}^{\frac p2-1}\|T (f_M)\|_{L_t^{q_0}L_x^{r_0}}\|T (f_N)\|_{L_t^{q_1}L_x^{r_1}}\\
&\lesssim
\|T (f_M)\|_{L^p}^{\frac p2-1}\|T (f_N)\|_{L^p}^{\frac p2-1}
\|\langle\xi\rangle^{\frac1{q_0}-\frac1{r_0}} f_M\|_{L^2(\mathbb{H}^d)}
\|\langle\xi\rangle^{\frac1{q_1}-\frac1{r_1}} f_N\|_{L^2(\mathbb{H}^d)}.
\end{align*}
where the last line is a consequence of \eqref{MixedNormKGStrichartz}.
Since $\langle\xi\rangle\simeq M$ inside the support of $f_M$, and similarly for $f_N$, from this and  \eqref{eq:MixedNormExponents} it follows that
$$\|T (f_M)T (f_N)\|_{L^{\frac p2}}^{\frac p2}
\lesssim
\Big(\frac NM\Big)^{\frac1{q_1}-\frac1{r_1}}
\|T (f_M)\|_{L^p}^{\frac p2-1}\|T (f_N)\|_{L^p}^{\frac p2-1}
\|f_M\|_{L^2(\mathbb{H}^d)}
\|f_N\|_{L^2(\mathbb{H}^d)}.
$$
Going back to \eqref{triangleineq} and noting that $\frac1{q_1}-\frac1{r_1}>0$,
we use H\"older's inequality and the elementary estimate $2ab\leq a^2+b^2$ with $a=\|f_N\|_{L^2(\mathbb{H}^d)}$ and $b=\|f_M\|_{L^2(\mathbb{H}^d)}$,
and sum a geometric series to  finally conclude that
\begin{align*}
\sum_{M} \sum_{N\leq M} \|T (f_M)T (f_N)\|_{L^{\frac p2}}^{\frac p2}
&\leq\sup_N \|T (f_N)\|_{L^p}^{p-2}
\sum_{M} \sum_{N\leq M}
\Big(\frac NM\Big)^{\frac1{q_1}-\frac1{r_1}}
\|f_M\|_{L^2(\mathbb{H}^d)}
\|f_N\|_{L^2(\mathbb{H}^d)}\\
&\lesssim\sup_N \|T (f_N)\|_{L^p}^{p-2}
\|f\|^2_{L^2(\mathbb{H}^d)}.
\end{align*}
This finishes the proof of the proposition.
\end{proof}

\section{Beginning of the proof: angular restriction} \label{sec:ang_rest}

Let $\{f_n\}_{n\in\mathbb{N}} \subset L^2(\mathbb{H}^d)$ be an extremizing sequence for \eqref{ExtensionInequality}. We may assume that $\|f_n\|_{L^2(\mathbb{H}^d)} = 1$ and that $\|T(f_n)\|_{L^p(\mathbb{R}^{d+1})} \to {\bf H}_{d,p}$ as $n \to \infty$. Recall from the Introduction that each $f_n$ is regarded as a function on $\mathbb{R}^d$.
Given $K\in\mathbb{N}$,  consider a finite partition of the unit sphere $\mathbb{S}^{d-1} =\{\xi\in \mathbb{R}^d: |\xi|=1\}$ into $K$ disjoint regions,
$$\mathbb{S}^{d-1} = \bigcup_{k=1}^{K} C_k^*.$$
Given a function $f:\mathbb{R}^d \to \mathbb{C}$, let $f^{(k)} := f \ \mathbbm 1_{R_k}$, where $R_k = \{\xi \in \mathbb{R}^d : \xi/|\xi| \in C_k^*\}$. In this way we split $\mathbb{R}^d$ into $K$ angular sectors. The triangle inequality implies
$$\|T(f_n)\|_{L^p(\mathbb{R}^{d+1})} \leq \sum_{k=1}^{K} \|T(f_n^{(k)})\|_{L^p(\mathbb{R}^{d+1})}.$$
Observe that, possibly after extraction of a subsequence, there exists $k_0 \in \{1,2,\ldots, K\}$ such that $\{f_n^{(k_0)}\}_{n\in\mathbb{N}}$ is a {\it quasi-extremizing sequence} for \eqref{ExtensionInequality}.
By this we mean that $\|f_n^{(k_0)}\|_{L^2(\mathbb{H}^d)} \leq 1$, and
\begin{equation}\label{quasi-ext}
\|T(f_n^{(k_0)})\|_{L^p(\mathbb{R}^{d+1})}\geq \delta_1,
\end{equation}
for every $n\in\mathbb{N}$ and some universal $\delta_1 >0$ (we may take for instance $\delta_1 = \frac{{\bf H}_{d,p}}{2K}$).

\smallskip

Under these circumstances, we will establish the existence of
a universal ball $B \subset \mathbb{R}^d$ centered at the origin,
a universal $\delta_2 >0$,
and a sequence of Lorentz transformations $\{L_n\}_{n\in\mathbb{N}}$ such that
$$\|L_n ^* f_n^{(k_0)}\|_{L^2(B)} \geq \delta_2,$$
for every $n\in\mathbb{N}$.
This naturally implies
$$\|L_n^* f_n\|_{L^2(B)} \geq \delta_2,$$
for every $n\in\mathbb{N}$.
The latter inequality is of the sort which is required in order to invoke the machinery from \cite[Section 6]{COSS17} and conclude the proof of Theorem \ref{Thm1}.

\smallskip

Throughout the upcoming Sections \ref{sec:Caps} and \ref{sec:RefinedStrichartz} we will thus assume that our functions are supported in a small angular region $R_1$
(the corresponding $C_1^* \subset \mathbb{S}^{d-1}$ is described at the beginning of Section \ref{sec:Caps}).
Henceforth, such functions will be referred to as {\it admissible}.

\section{Caps, sectors and bilinear estimates}\label{sec:Caps}
As mentioned in the Introduction, one of the key ingredients in the proof of Theorem \ref{Thm1} is the use of tools from bilinear restriction theory. Classical works on the topic include \cite{Ta03, TVV98, Wo01}.
\smallskip

In this section, we define the appropriate geometric regions and the notion of separation between them, and establish the bilinear restriction estimates that will be of relevance in the sequel.

\subsection{Definition of dyadic regions}
Let $d \geq 3$ be a fixed dimension.
Consider the $(d-1)$-dimensional cube
$$C_1 = \{\eta = (\eta_1, \eta_2, \ldots, \eta_{d-1}) \in \mathbb{R}^{d-1} : |\eta_i| \leq \ell, \; i=1,2,\ldots,d-1\}$$
of sidelength $2\ell$ centered at the origin. The quantity $\ell < \tfrac14$ is a small fixed number which depends only on the dimension $d$, and shall be appropriately chosen in due course. Given a dyadic number $M \in 2^{\mathbb{Z}_{\leq0}}$, let $\Gamma_M$ denote the usual dyadic decomposition of the cube $C_1$ into cubes of sidelength $2\ell M$ on $\mathbb{R}^{d-1}$. In particular, $\Gamma_1 = \{C_1\}$, and $\Gamma_M$ consists of $M^{-(d-1)}$ {\it essentially disjoint} cubes
(i.e. the intersection of any two distinct cubes is a Lebesgue null-set).
 Let $* \ : C_1 \to \mathbb{S}^{d-1}$ be the lift of a point in $C_1$ to a point in the unit sphere $\mathbb{S}^{d-1} \subset \mathbb{R}^d$, defined via
$$\eta^* = (\eta, (1 - |\eta|^2)^{\frac12}).$$
For each cube $Q \in \Gamma_M$, let
$$Q^*  = \{\eta^* : \eta \in Q\}$$
denote the lift of the cube $Q$, and let $\Gamma^*_M$ denote the collection of the lifted cubes of $\Gamma_M$.

\smallskip

For the purposes of the present construction, we may think of distances in $C^*_1 \subset \mathbb{S}^{d-1} \subset \mathbb{R}^d$ as being almost the same as Euclidean distances in $C_1 \subset \mathbb{R}^{d-1}$. More precisely, given any constant $\varepsilon_1 >0$, we may choose $\ell = \ell(d,\varepsilon_1)>0$ sufficiently small, such that
\begin{equation}\label{Sphere correction}
|\eta - \zeta|\leq\text{dist}(\eta^*, \zeta^*) \leq (1 + \varepsilon_1) \,|\eta - \zeta|
\end{equation}
for all $\eta, \zeta \in C_1 \subset \mathbb{R}^{d-1}$. Here $| \cdot | $ denotes Euclidean distance in $\mathbb{R}^{d-1}$, and dist$( \cdot, \cdot)$ denotes the geodesic distance on $\mathbb{S}^{d-1} \subset \mathbb{R}^d$. We may take for instance $\varepsilon_1 = \frac1{100}$.

\smallskip

Given $N\in2^{\mathbb{Z}_{>0}}$, define the restricted dyadic annulus
\begin{gather}\label{eq:DefAnnulus}
\mathcal{A}_N:=\big\{\xi\in\mathbb{R}^d: \tfrac12 N\leq|\xi|\leq 2N \ {\rm and} \ \xi/|\xi| \in C_1^*\big\},
\end{gather}
and set $\mathcal{A}_1:=\{\xi\in\mathbb{R}^d: |\xi|\leq 2  \ {\rm and} \ \xi/|\xi| \in C_1^*\}$.

\smallskip

Given $N\in2^{\mathbb{Z}_{\geq0}}$, let $r\in 2^\mathbb{Z}$ be such that $0<r\leq N$.
If $0<r\leq1$, then we further decompose the restricted annulus $\mathcal{A}_N$ into an essentially disjoint union of regions
\begin{equation}\label{height1}
\mathcal{A}_N^{(j)}:=\big\{\xi\in\mathcal{A}_N: \tfrac12 N(1+3j r)\leq|\xi|\leq \tfrac12 N(1+3(j+1)r)\big\},
\end{equation}
for $j\in J:=\{0,1,\ldots,r^{-1}-1\}$.
If $1< r\leq N$, then we unify the notation below by letting $J = \{0\}$ and $\mathcal{A}_N^{(0)}:=\mathcal{A}_N$.
In both cases we then have that $\# J=\max\{1,r^{-1}\}$.

\smallskip

Given $N\in2^{\mathbb{Z}_{\geq0}}$, and $r\in 2^\mathbb{Z}$ such that $0<r\leq N$, let $M = r/N$ and consider
$$\mathcal{D}_{N,r}
:=\big\{\kappa_{N,r}^{j,k}: (j,k)\in J\times \{1,2,\ldots, M^{-(d-1)}\}\big\},$$
where the regions $\kappa=\kappa_{N,r}^{j,k}$ are defined as
\begin{equation}\label{eq:DefTau}
\kappa_{N,r}^{j,k}:=\big\{\xi\in \mathcal{A}_N^{(j)}: \xi/|\xi| \in Q_k^*\big\},
\end{equation}
and $Q_k^*$ is a cube in the collection $\Gamma_M^*$.
The center of a region $\kappa=\kappa_{N,r}^{j,k}$ as in \eqref{eq:DefTau} is \mbox{defined to be}
\begin{equation}\label{eq:DefCenter}
c(\kappa):=\tfrac12 N\big(1+3\min\{1,r\} (j+\tfrac12)\big)\,\omega_k^*,
\end{equation}
where $\omega_k^*\in\mathbb{S}^{d-1}$ is the lift of the center $ \omega_k$ of the cube $Q_k \in \Gamma_M$.

\smallskip

If $0<r\leq 1$, then an element of $\mathcal{D}_{N,r}$ is called an {\it $r$-cap at scale $N$}.
If $1<r\leq N$, then an element of $\mathcal{D}_{N,r}$ is called an {\it $r$-sector at scale $N$}.
The Lebesgue measure of an $r$-cap  at scale $N$ is comparable to $Nr^d$, and
the Lebesgue measure of an $r$-sector at scale $N$ is comparable to $Nr^{d-1}$.

\smallskip

By a {\it region} we will continue to mean a set which is either a cap or a sector.
For fixed $N,r$, the regions in $\mathcal{D}_{N,r}$ are essentially disjoint.
If $r < N$, then each $\kappa \in \mathcal{D}_{N,r}$ is contained in a unique $\kappa^\circ \in \mathcal{D}_{N,2r}$, and we refer to $\kappa^\circ$ as the {\it parent} of $\kappa$.
In a similar spirit, each $\kappa \in \mathcal{D}_{N,r}$ has either $2^{d-1}$ or $2^{d}$ {\it children}, according to the change of regime when $r=1$.

\smallskip

The construction outlined above can be regarded as a hybrid between a dyadic decomposition on $\mathbb{R}^{d}$ (caps) and on $\mathbb{R}^{d-1}$ (sectors), and is convenient to treat the elliptic and conic regimes in a unified way.

\subsection{Separated regions}\label{sec:separated}

We call two regions {\it adjacent} if their closures intersect, possibly at boundary points.
We say that two regions $\kappa,\kappa'\in \mathcal{D}_{N,r}$ are {\it separated}, and write $\kappa\sim\kappa'\in \mathcal{D}_{N,r}$, if $\kappa,\kappa'$ are not adjacent, their parents are not adjacent, their $2$-parents (i.e. grandparents) are not adjacent, \ldots, their $(d-1)$-parents are not adjacent, and their $d$-parents are adjacent. Naturally, this assumes that $r \leq N/2^d$, so that $\kappa,\kappa'$ indeed have ancestors up to the $d$-th generation.
The main reason why we climb up $d$ degrees in the genealogical tree when defining separation is to ensure that certain  naturally arising geometric regions which contain $\kappa, \kappa'$  are also ``separated''. In fact, as will become clear from the proof below, around $k$ generations up in the tree with $k \simeq \log_2 d$ would morally suffice.

If $\kappa,\kappa'\in \mathcal{D}_{N,r}$ are separated regions, then either: (i) the angular distance between $c(\kappa)$ and $c(\kappa')$ (which is $\simeq N |\omega_k^* - \omega_{k'}^*|$) is comparable to $r$; or (ii) the radial distance between $c(\kappa)$ and $c(\kappa')$ is comparable to $N r$. Note that option (ii) is only available if $0 < r < 2^{-d}$.

\smallskip

Defining the regions and the separation between them in this way, we ensure that the union in the forthcoming expression \eqref{eq:Whitney} is essentially disjoint, an important step in the proof of the refined Strichartz estimate.

\subsection{Bilinear estimates} If $\kappa\in\mathcal{D}_{N,r}$ is a dyadic region as defined in the previous subsection, then  we set $f_{\kappa} := f \mathbbm 1_{\kappa}$. The main result of this section is the following.
 \begin{proposition}
 Let $d\geq 3$ and $\frac{2(d+2)}{d} \leq p\leq  \frac{2(d+1)}{d-1}$. Then there exists an exponent $1\leq s<2$, which can be taken arbitrarily close to $2$, for which the following bilinear extension estimates hold, uniformly in $N,r,f,g$. Let $f,g\in L^2(\mathbb{R}^d)$ be admissible functions, and let $N\geq 1$ be a dyadic number.

\begin{enumerate}
\item[(i)] If $0<r\leq 1$ is a dyadic number, and $\kappa\sim\kappa'\in\mathcal{D}_{N,r}$, then
\begin{equation}\label{Bilinear1}
\|T(f_\kappa) T(g_{\kappa'})\|_{L^{\frac p2}(\mathbb{R}^{d+1})}\lesssim_s N^{-\frac2s} r^{\frac{2d}{s'}-\frac{2(d+2)}{p}} \|f_\kappa\|_{L^s(\mathbb{R}^d)}\|g_{\kappa'}\|_{L^s(\mathbb{R}^d)}.
\end{equation}

\item[(ii)] If $1<r\leq N$ is a dyadic number, and $\kappa\sim\kappa'\in\mathcal{D}_{N,r}$, then
 \begin{equation}\label{Bilinear2}
\|T(f_\kappa)  T(g_{\kappa'})\|_{L^{\frac p2}(\mathbb{R}^{d+1})}\lesssim_s N^{-\frac2s} r^{\frac{2(d-1)}{s'}-\frac{2(d+1)}{p}} \|f_\kappa\|_{L^s(\mathbb{R}^d)}\|g_{\kappa'}\|_{L^s(\mathbb{R}^d)}.
\end{equation}
\end{enumerate}
 \end{proposition}

\begin{proof}
We first establish the estimate in the elliptic regime $0<r\leq1$.
The proof consists of a rescaling of the bilinear extension result of Tao \cite{Ta03}.
We start by constructing affine transformations that map separated caps $\kappa\sim\kappa'\in\mathcal{D}_{N,r}$ into unit separated regions.

\smallskip

{\it Boosted caps.}
Let $N\geq 1$ and $0<r\leq 1$ be dyadic numbers, and let $\kappa\sim\kappa'\in\mathcal{D}_{N,r}$.
Let $\widetilde{\kappa}, \widetilde{\kappa}'$ denote the {\it lifts} of the caps $\kappa,\kappa'$ into the hyperboloid $\mathbb{H}^d$, defined as
\begin{equation}\label{lift_def}
\widetilde{\kappa}=\{(\xi,\jp{\xi}):\xi\in\kappa\},\qquad \widetilde{\kappa}'=\{(\xi,\jp{\xi}):\xi\in\kappa'\}.
\end{equation}
Let $\xi_0=c(\kappa)$ denote the center of the cap $\kappa$ as in \eqref{eq:DefCenter},
and let $L_{\xi_0}$ be the Lorentz transformation defined in \eqref{eq:Lorentz} with $\nu=\xi_0$.
Then $L_{\xi_0}$ maps $\widetilde{\kappa}, \widetilde{\kappa}'$ into the lifts $\widetilde{\lambda}, \widetilde{\lambda}'$ of
sets  $\lambda:=L_{\xi_0}^\flat (\kappa)$ and $\lambda':=L_{\xi_0}^\flat (\kappa')$ which are contained in $r$-separated cubes of sidelength comparable to $r$.
Moreover, we can take the center of the cube containing $\lambda$ to be $L^\flat_{\xi_0}(\xi_0)=0$.
Recall that the Lorentz boost $L_{\xi_0}$ is volume preserving, $\det (L_{\xi_0})=1$.
Moreover, on $\kappa\cup\kappa'$, the map $L_{\xi_0}^\flat$
has Jacobian determinant $\det (DL_{\xi_0}^\flat)\simeq N^{-1}$.

\smallskip

{\it Parabolic rescaling.}
The region $\{(\xi,\jp{\xi})\in\mathbb{R}^d\times\mathbb{R}: |\xi|\lesssim 1\}$ is of elliptic type, in the terminology of \cite[Section 9]{Ta03}.
The parabolic rescaling
$$P_r(\xi,\tau):=\big(\tfrac\xi r,\tfrac{\tau-1}{r^2}\big),$$
maps the lifts $\widetilde{\lambda}, \widetilde{\lambda}'$ defined above to the lifts $\widetilde{\rho},\widetilde{\rho}'$ into the compact hypersurface
\begin{equation}\label{eq:Sigmar}
\Sigma_r:=\big\{\big(\xi,\tfrac{\jp{r\xi}-1}{r^2}\big): |\xi|\lesssim1\big\}
\end{equation}
of $O(1)$-separated sets $\rho,\rho'$ of diameter comparable to 1.
Let $P_r^\flat:\mathbb{R}^d\to\mathbb{R}^d$ denote the map $\xi\mapsto r^{-1}{\xi}$,
whose Jacobian determinant satisfies $\det (D P_r^\flat)=r^{-d}$.
Note that $P^\flat_{r^{-1}}\circ P^\flat_r=\text{Id}$, and that $P_r$ is an affine map whose linear part has determinant equal to $r^{-(d+2)}$.

\smallskip

{\it Bilinear extension of caps.}
With $\rho,\rho'$ as defined above, set $f_\rho:=f\mathbbm 1_\rho$ and $g_{\rho'}:=g\mathbbm 1_{\rho'}$.
Let $\mathcal{E}_r$ denote the Fourier extension operator associated to the hypersurface $\Sigma_r$ defined in \eqref{eq:Sigmar},
$$\mathcal{E}_r (f)(x,t):=\int_{\mathbb{R}^d} e^{ix\cdot\xi} e^{it\Phi_r(\xi)} f(\xi)\,\d\xi,$$
with phase function given by $\Phi_r(\xi):=\frac{\jp{r\xi}-1}{r^2}$.
The hypersurfaces $\{\Sigma_r\}_{0<r\leq 1}$ are uniformly elliptic in the sense of \cite{TVV98}.
As a consequence of Tao's bilinear extension theorem for general elliptic hypersurfaces \cite[Section 9]{Ta03},
the  estimate
\begin{equation}\label{eq:TaoBilinear}
\|\mathcal{E}_r (f_\rho)\mathcal{E}_r (g_{\rho'})\|_{L^q}
\lesssim \|f_\rho\|_{L^2}\|g_{\rho'}\|_{L^2},\qquad q>\tfrac{d+3}{d+1},
\end{equation}
holds, uniformly in $0<r\leq1$.
Using the Riesz--Th\"orin convexity theorem to interpolate the latter inequality with the trivial estimate
\begin{equation}\label{eq:trivial}
\|\mathcal{E}_r (f_\rho)\mathcal{E}_r (g_{\rho'})\|_{L^\infty}
\leq \|f_\rho\|_{L^1}\|g_{\rho'}\|_{L^1},
\end{equation}
we conclude the existence of $s_0<2$, such that
\begin{equation}\label{eq:PreChange}
\|\mathcal{E}_r (f_\rho)\mathcal{E}_r (g_{\rho'})\|_{L^{\frac p2}}
\lesssim \|f_\rho\|_{L^s}\|g_{\rho'}\|_{L^s},
\end{equation}
for every $s\in(s_0,2)$.
We claim that \eqref{Bilinear1} follows from \eqref{eq:PreChange} by a standard change of variables, which we now present in detail.
Start by noting that
$f_\rho=f_\kappa\circ L_{-\xi_0}^\flat\circ P_{r^{-1}}^\flat.$
It follows that
\begin{equation}\label{eq:Change1}
\|f_\rho\|_{L^s}^s
=\int_{\rho} |f_\kappa(L_{-\xi_0}^\flat\circ P_{r^{-1}}^\flat(\xi))|^s \d\xi
\simeq (Nr^d)^{-1}\|f_\kappa\|_{L^s}^s,
\end{equation}
since on $\kappa$ the change of variables $\xi=(P_r^\flat\circ L_{\xi_0}^\flat)(\zeta)$ has Jacobian determinant comparable to
$$\det (DP_r^\flat)\det (DL_{\xi_0}^\flat)\simeq r^{-d}N^{-1}.$$
On the other hand, a straightforward computation shows that
$$\mathcal{E}_r (f_\rho)(x,t)=r^{-d}e^{-i\frac{t}{r^2}} \int_{\mathbb{R}^{d+1}} e^{i(\frac xr,\frac t{r^2})\cdot(\xi,\tau)} f_\kappa(L_{-\xi_0}(\xi,\tau)) \ddirac{\tau^2-\jp{\xi}^2}\jp{\xi} \,\d \xi\,\d \tau,$$
and so another change of variables $L_{-\xi_0}(\xi,\tau)=(\xi',\tau')$ yields
$$\mathcal{E}_r (f_\rho)(x,t)=r^{-d}e^{-i\frac{t}{r^2}} \int_{\mathbb{R}^{d}} e^{iL_{\xi_0}^T(\frac xr,\frac t{r^2})\cdot(\xi,\jp{\xi})} f_\kappa(\xi)\jp{L_{\xi_0}^\flat(\xi)} \frac{\d \xi}{\jp{\xi}}.$$
This in turn can be rewritten as
$$\mathcal{E}_r (f_\rho)(x,t)=r^{-d}e^{-i\frac{t}{r^2}}T(f_\kappa{\jp{L_{\xi_0}^\flat(\cdot)}})(L_{\xi_0}^T(\tfrac xr,\tfrac t{r^2})),$$
and so, in particular,
$$|\mathcal{E}_r (f_\rho)(x,t)|=r^{-d}|T(f_\kappa{\jp{L_{\xi_0}^\flat(\cdot)}})((L_{\xi_0}^T\circ D_r)(x,t))|,$$
where $D_r$ denotes the parabolic dilation $D_r(x,t):=(\frac xr, \frac t{r^2})$.
It follows that
\begin{equation}\label{eq:Change2}
\big\|\mathcal{E}_r (f_\rho) \mathcal{E}_r (g_{\rho'})\big\|_{L^{\frac p2}}^{\frac p2}=r^{-dp} r^{d+2} \big\|T(f_\kappa{\jp{L_{\xi_0}^\flat(\cdot)}}) T(g_{\kappa'}{\jp{L_{\xi_0}^\flat(\cdot)}})\big\|_{L^{\frac p2}}^{\frac p2}.
\end{equation}
Since $\jp{L_{\xi_0}^\flat(\xi)}\simeq 1$ if $\xi \in\kappa\cup\kappa'$, inequality \eqref{Bilinear1} is now easily seen to follow from \eqref{eq:PreChange}, \eqref{eq:Change1} and \eqref{eq:Change2}. This concludes the verification of the elliptic case.

\smallskip

For the conic case $1<r\leq N$, we can follow a similar path, invoking either Wolff's bilinear estimates for the cone \cite{Wo01} or a variant on Tao's estimates for the paraboloid noted in \cite{LeeVargas kflat}.
We choose to take a shortcut, noting that Candy's recent work \cite{Ca17} on bilinear restriction estimates for general phases already implies the adequate rescaled substitute of \eqref{eq:TaoBilinear} in the conic regime.
More precisely, \cite[Theorem 1.10]{Ca17} specializes to the inequality
\begin{equation}\label{eq:FromTim}
\|T(f_\kappa)T(g_{\kappa'}) \|_{L^q}\lesssim N^{-1}r^{d-1-\frac{d+1}q}\|f_\kappa\|_{L^2}\|g_{\kappa'}\|_{L^2},\;\;\;q>\tfrac{d+3}{d+1}.
\end{equation}
As before, this can be interpolated with the trivial
$$\|T(f_\kappa)T(g_{\kappa'}) \|_{L^\infty}\lesssim N^{-2}\|f_\kappa\|_{L^1}\|g_{\kappa'}\|_{L^1}$$
to yield \eqref{Bilinear2}.
The proof is now complete.
\end{proof}

\section{A refined Strichartz estimate} \label{sec:RefinedStrichartz}
There exists a well-established program, using tools from Littlewood--Paley theory, Whitney-type decompositions and quasi-orthogonality, to derive refined inequalities of Strichartz type from bilinear restriction estimates, see for instance the works \cite{BV07, KV13, KSV12, Ra12}.

The goal of this section is to establish the following refinement of inequality \eqref{ExtensionInequality} which holds for admissible functions in each dyadic annulus.
\begin{theorem}\label{Sharpened}
Let $d\geq 3$ and $\frac{2(d+2)}{d} \leq p\leq  \frac{2(d+1)}{d-1}$.
Then there exists $\gamma\in (0,1-\frac2p)$ such that the following inequality holds
\begin{align}
\|T (f_N)\|_{L^p(\mathbb{R}^{d+1})}^p
\lesssim&
\Bigg[
\sup_{0<r\leq 1}(r^d)^{(\frac p2-\frac{d+2}{d})(1-\gamma)} \Big(\sup_{\kappa\in\mathcal{D}_{N,r}}
 \|T (f_\kappa)\|_{L^p(\mathbb{R}^{d+1})}^{p\gamma}\Big)\notag\\
&+
\sup_{1<r\leq N} (r^{d-1})^{(\frac p2-\frac{d+1}{d-1})(1-\gamma)}\Big(\sup_{\kappa\in\mathcal{D}_{N,r}}
 \|T (f_\kappa)\|_{L^p(\mathbb{R}^{d+1})}^{p\gamma}\Big)
\Bigg]
\|f_N\|_{L^2(\mathbb{H}^d)}^{p(1-\gamma)},\label{sharpenedextensionineq}
\end{align}
for every dyadic number $N\geq 1$ and admissible function $f\in L^2(\mathbb{H}^d)$.
\end{theorem}

\noindent {\it Remark.}
Both exponents in $r$ appearing on the right-hand side of inequality \eqref{sharpenedextensionineq} are favorable:
$\frac p2-\frac{d+2}{d}\geq 0$ (in case $r\leq1$) and
$\frac p2-\frac{d+1}{d-1}\leq 0$ (in case $r>1$), with strict inequality except for the case of endpoint exponents.\\

We start with two technical lemmata which bound certain quantities that will naturally appear in the course of the proof of Theorem \ref{Sharpened}.

\begin{lemma}\label{lem:almostorthogonal}
Let $d\geq 3$ and $\frac{2(d+2)}{d} \leq p\leq  \frac{2(d+1)}{d-1}$.
Then the following inequality holds
\begin{equation}\label{pstarineq}
\Big\|\sum_{0<r\leq N}\sum_{\kappa\sim\kappa'\in\mathcal{D}_{N,r}}
T(f_\kappa) T(f_{\kappa'})\Big\|_{L^{\frac p2}(\mathbb{R}^{d+1})}^{\frac p2}
\lesssim
\sum_{0<r\leq N}\sum_{\kappa\sim\kappa'\in\mathcal{D}_{N,r}} \big\|T (f_\kappa) T (f_{\kappa'})\big\|_{L^{\frac p2}(\mathbb{R}^{d+1})}^{\frac p2},
\end{equation}
for every dyadic number $N\geq 1$ and  admissible function $f\in L^2(\mathbb{H}^d)$.
\end{lemma}

\begin{proof}

Let $\kappa\in\mathcal{D}_{N,r}$ be given, and let $\xi_0=c(\kappa)$ denote its center as in \eqref{eq:DefCenter}.
For every $\xi\in\kappa$, one easily checks that
\begin{gather}
\big| |\xi|- |\xi_0| \big|\lesssim \min\{1,r\} N,\label{eq:Property1}\\
\big(|\xi||\xi_0|-\xi\cdot\xi_0\big)^{\frac12}\lesssim r.\label{eq:Property2}
\end{gather}
Indeed, inequality \eqref{eq:Property1} follows from the fact that the length along the radial direction
of $r$-caps and $r$-sectors at scale $N$ is comparable to $rN$ and to $N$, respectively,
and  inequality \eqref{eq:Property2} amounts to the fact that the angle between the vectors $\xi$ and $\xi_0$ is $O(\frac rN)$.
Now, given $\kappa\sim\kappa'\in\mathcal{D}_{N,r}$, with corresponding centers $\xi_0=c(\kappa)$ and $\xi_0'=c(\kappa')$, the following estimate follows from the definition of the separation relation $\sim$:
\begin{equation}\label{eq:Property3}
\frac{\big||\xi_0|-|\xi_0'|\big|}{N^2}+\frac{\big(|\xi_0||\xi_0'|-\xi_0\cdot \xi_0'\big)^{\frac12}}{N}\simeq\frac rN.
\end{equation}
Let $\widetilde{\kappa}$ and $\widetilde{\kappa}'$ be the lifts of the regions $\kappa$ and $\kappa'$ into the hyperboloid $\mathbb{H}^d$ as defined in \eqref{lift_def}.

\smallskip

We aim to use \cite[Lemma 2.2]{Ra12} (which is a slightly more general version of \cite[Lemma A.9]{KV13} and \cite[Lemma 6.1]{TVV98}) to obtain the quasi-orthogonality proposed in \eqref{pstarineq}. Our first task is to understand the geometry of the sumset
$$\widetilde{\kappa} + \widetilde{\kappa}' = \big\{ (\xi+\xi',\jp{\xi}+\jp{\xi'}): (\xi,\xi')\in\kappa\times\kappa'\big\}\subset\mathbb{R}^{d+1}.$$
Using \eqref{eq:Property1}, \eqref{eq:Property2} and \eqref{eq:Property3}, one may reason as in \cite[Proof of Prop. 15]{COSS17} to further check that\footnote{Here we use the notation $\jp{x}_s:=(s^2+|x|^2)^{\frac12}$. Estimates \eqref{eq:vertical}--\eqref{eq:angular} also appear in \cite[Proof of Theorem 2.6]{Ca17}.}
\begin{gather}
\jp{\xi}+\jp{\xi'} - \jp{\xi+\xi'}_2 \simeq\tfrac{r^2}N,\label{eq:vertical}\\
\big||\xi+\xi'|-|\xi_0+\xi_0'|\big|\lesssim\min\{1,r\}N,\label{eq:radial}\\
\big(|\xi+\xi'| |\xi_0+\xi_0'| - (\xi+\xi')\cdot (\xi_0+\xi_0')\big)^{\frac12}\lesssim r.\label{eq:angular}
\end{gather}

\smallskip

\noindent {\it Step 1.} Observe that \eqref{eq:vertical}, \eqref{eq:radial} and \eqref{eq:angular} imply that the sumsets $\widetilde{\kappa} + \widetilde{\kappa}'$ are {\it almost disjoint}, in the following sense: There exists a universal constant such that, for any pair $(\kappa,\kappa')$ with $\kappa\sim\kappa'\in\mathcal{D}_{N,r}$, the number of pairs $(\rho,\rho')$ with $\rho\sim\rho'\in\mathcal{D}_{N,s}$ and
\begin{equation}\label{non-empty-intersection}
\big(\widetilde{\kappa} + \widetilde{\kappa}') \cap \big(\widetilde{\rho} + \widetilde{\rho}') \neq \emptyset
\end{equation}
is bounded by this constant. In fact, if \eqref{non-empty-intersection} occurs, then estimate \eqref{eq:vertical} implies the existence of universal constants $a, b \in \mathbb{Z}$ such that $2^a r \leq s \leq 2^b r$. Let $\eta_0 = c(\rho)$ denote the center of $\rho$.
Once $s$ is trapped, then \eqref{eq:Property1}, \eqref{eq:Property3} and \eqref{eq:radial}  imply that the lengths of $|\eta_0|$ and $|\xi_0|$ are not far from each other,  in the sense that
\begin{equation}\label{modulus_not_so_far}
\big| |\eta_0|- |\xi_0| \big|\lesssim\min\{1,r\}N.
\end{equation}
In a similar way, \eqref{eq:Property2}, \eqref{eq:Property3} and \eqref{eq:angular} together imply that the angle between $\eta_0$ and $\xi_0$ is controlled, that is
\begin{equation}\label{angle_not_so_far}
\big(|\eta_0||\xi_0|-\eta_0\cdot \xi_0\big)^{\frac12} \lesssim r.
\end{equation}

Expressions \eqref{modulus_not_so_far} and \eqref{angle_not_so_far} imply that, given $\xi_0$, the number of possible choices for $\eta_0$ in the dyadic decomposition is finite and universally bounded. For each possible $\eta_0 = c(\rho)$, the number of regions $\rho'$ separated from $\rho$ is also finite and universally bounded.

\smallskip

\noindent {\it Step 2.} Observe that \begin{equation}\label{eq:suppFT}
\text{supp } \mathcal{F}_{t,x}[T(f_\kappa) T(f_{\kappa'})]\subset \widetilde{\kappa}+\widetilde{\kappa}',
\end{equation}
where $\mathcal{F}_{t,x}$ denotes the space-time Fourier transform. In order to use \cite[Lemma 2.2]{Ra12}, it is convenient to place the sumsets $\widetilde{\kappa}+\widetilde{\kappa}'$ inside regions which are geometrically simpler but still almost disjoint. Expression \eqref{eq:vertical} already implies that
\begin{equation}\label{eq:defGammakk}
\widetilde{\kappa}+\widetilde{\kappa}' \subset \big\{(\xi,\tau) \in \mathbb{R}^{d}\times\mathbb{R} : \jp{\xi}_2 + c_1 \tfrac{r^2}{N}\leq \tau \leq \jp{\xi}_2 + c_2 \tfrac{r^2}{N}  \ \ {\rm and} \ \ \xi \in \kappa+\kappa'\big\} =: \Gamma_{\kappa,\kappa'},
\end{equation}
for some universal constants $c_1, c_2$.  Note that equations \eqref{eq:radial} and \eqref{eq:angular} imply that the set $\kappa + \kappa'$ lies inside a rectangle centered at $\gamma_0:=\xi_0 + \xi'_0$, of height comparable to $\min\{1,r\}N$ (the major axis being aligned with the vector $\gamma_0$) and of  sidelength comparable to $r$. Denote this rectangle by $R_{\kappa,\kappa'}$.
Consider a centered dilation\footnote{More generally, given a parallelepiped $P$ and  $\lambda>0$, we denote by $\lambda\cdot P$ the {\it centered dilate} of $P$.
In other words, if $c_P$ denotes the center of $P$, then $\lambda\cdot P:=\lambda(P-c_P)+c_P$.}
$R_{\kappa,\kappa'}^*:=(1+\alpha)\cdot R_{\kappa,\kappa'}$
of $R_{\kappa,\kappa'}$, with $\alpha>0$ sufficiently small and independent of $(\kappa,\kappa')$,  such that the sets
\begin{equation*}
\Sigma_{\kappa,\kappa'}:=\big\{(\xi,\tau) \in \mathbb{R}^{d}\times\mathbb{R} : \jp{\xi}_2 + \tfrac{c_1}2 \tfrac{r^2}{N}\leq \tau \leq \jp{\xi}_2 + 2c_2 \tfrac{r^2}{N}  \ \ {\rm and} \ \ \xi \in R_{\kappa,\kappa'}^*\big\}
\end{equation*}
still have bounded overlap. We may now decompose the collection $\{(\kappa,\kappa'):\kappa\sim\kappa'\}$ as a union of a finite (universal) number of subsets whose corresponding $\{\Sigma_{\kappa,\kappa'}\}$ are pairwise disjoint.
By the triangle inequality, it suffices to bound the sum over just one of these subsets, which we henceforth denote by $\mathcal{T}$.

\smallskip
\noindent {\it Step 3.} We claim the existence of a universal number $K$ with the following property:
For every $(\kappa,\kappa')\in\mathcal{T}$,
there exist  parallelepipeds $\{P_\ell=P_\ell(\kappa,\kappa')\}_{\ell=1}^K$ with disjoint interiors, satisfying
\begin{equation*}
    \Gamma_{\kappa,\kappa'}\subset \bigcup_{\ell=1}^K P_\ell,
\end{equation*}
and such that $(1+\beta)\cdot P_\ell\subset \Sigma_{\kappa,\kappa'}$, for some universal $\beta>0$.

\smallskip
Indeed, given a point $\gamma\in \mathbb{R}^d$, define $T(\gamma)$ to be the tangent plane to the hyperboloid $\mathbb{H}^d_2$ at the point $(\gamma, \langle \gamma \rangle_2)$, i.e.
$$T(\gamma):=\{(\gamma,\langle\gamma\rangle_2)+v:\, v \in \mathbb{R}^{d+1},\, v\bot (\gamma/\langle\gamma\rangle_2, -1)\}.$$
Let $e_1, e_2, \ldots, e_{d+1}$ denote the canonical basis vectors in $\mathbb{R}^{d+1}$. Without loss of generality, assume  $\gamma_0$ to be parallel to $e_d$. At a vector $te_d$, the slope of the tangent to the hyperbola $\{(te_d, \jp{t}_2): t\in\mathbb{R}\}$ equals $t/\jp{t}_2$. We may then consider a point $\gamma = te_d$ sufficiently close to $\gamma_0$, and  the corresponding hyperplane
$$T(\gamma)=\{(\gamma, \jp{\gamma}_2) + (x_1, x_2, \ldots, x_{d-1}, x_d, x_d \,t/\jp{t}_2) \  \textrm{with each} \ x_i \in \mathbb{R} \}.$$ Lifting the rectangle $R_{\kappa,\kappa'}$ to the hyperplane $T(\gamma)$  amounts to choosing $|(x_1, x_2, \ldots ,x_{d-1})| \lesssim r$ and $x_d \simeq \min\{1,r\}N$. Set $y=(x_1, x_2, \ldots ,x_{d-1})$, and assume
\begin{equation}\label{Cond_1_Rectangles}
|y| \leq c_3 r
\text{ and  }
|x_d| \leq c_4 \min\{1,r\}N,
\end{equation}
for some constants $c_3, c_4$ which are yet to be  chosen. Under these assumptions, we may estimate the largest displacement in the vertical direction $e_{d+1}$ between the hyperplane $T(\gamma)$ and the hyperboloid $\mathbb{H}_2^d$ as follows.
Recalling that $t \simeq N$, this displacement is given by
\begin{equation*}
\sqrt{4 + (t+x_d)^2 + |y|^2} - \left( \sqrt{4 + t^2} + \frac{t x_d}{\sqrt{4 + t^2}}\right) \simeq \frac{4x_d^2 + |y|^2(4 +t^2)}{N^3}.
\end{equation*}
By choosing the constants $c_3, c_4$ sufficiently small (but universal), we can bound this displacement from above by $\delta\frac{c_1}{10}\frac{r^2}{N}$, where $\delta<1$ is chosen so $c_2+\frac{\delta c_1}{10}<\frac{3c_2}{2}$, i.e. $\delta < \frac{5c_2}{c_1}$, where $c_1, c_2$  are the universal constants appearing in the definition \eqref{eq:defGammakk} of $\Gamma_{\kappa,\kappa'}$. This implies the existence of a  constant $K$, such that the original rectangle $R_{\kappa,\kappa'}$ can be decomposed into a union of $K$ smaller rectangles $\{R_\ell=R_\ell({\kappa,\kappa'})\}_{\ell=1}^K$ of the same size having disjoint interiors and verifying conditions \eqref{Cond_1_Rectangles}.
We again emphasize that, once $c_3$ and $c_4$ are chosen, the number $K$ is universal.

\smallskip
For each $\ell$, let $\alpha_\ell$  be the center of the rectangle $R_\ell$, and let $T(R_\ell)$ denote the lift of $R_\ell$ into the hyperplane $T(\alpha_\ell)$. Define the region $P_\ell = P_\ell(\kappa,\kappa')  \subset \mathbb{R}^{d+1}$ as the sumset
$$P_\ell := T(R_\ell) + \{s e_{d+1} : c_1 \tfrac{r^2}{N} \leq s \leq (c_2 + \tfrac{\delta c_1}{10}) \tfrac{r^2}{N}\}.$$
 Note that each $P_\ell$ is a parallelepiped lying above the hyperboloid $\mathbb{H}^d_2$ of height comparable to $ r^2/N$.
 Moreover, distinct elements of the family $\{P_\ell\}_{\ell=1}^K$ have disjoint interiors. Further observe that
\begin{equation}\label{eq:support_gamma}
    \widetilde{\kappa}+\widetilde{\kappa}' \subset \Gamma_{\kappa,\kappa'} \subset \bigcup_{\ell=1}^{K} P_\ell.
\end{equation}
It follows from the construction of $R_{\kappa,\kappa'}^*$ and $\{R_\ell\}$ that there exists $\beta>0$, such that $(1+\beta)\cdot R_\ell \subset R_{\kappa,\kappa'}^*$, for every $\ell\in\{1,2,\ldots,K\}$. From the aforementioned displacement considerations and the choice of $\delta$ (by possibly choosing a smaller  $\beta$, depending only on $c_1,c_2$), we may guarantee that the parallelepipeds $\{P_\ell\}$  further satisfy
\begin{equation}\label{eq:dilation inclusion}
    (1+\beta)\cdot P_\ell\subset \Sigma_{\kappa,\kappa'}.
\end{equation}
This concludes the the verification of claim.
\smallskip

\noindent {\it Step 4.}  Define $\psi_\ell:=\mathbbm{1}_{P_\ell}$.
The estimate
\begin{equation}\label{eq:parallelepid_boundedness}
\|f\ast \widehat{\psi}_\ell\|_{L^q(\mathbb{R}^{d+1})}    \leq C\|f\|_{L^q(\mathbb{R}^{d+1})},
\end{equation}
which holds for any exponent $q>1$, follows from a simple application of the boundedness of the Hilbert transform,
yielding a constant $C=C_{q,d}<\infty$ that does not depend
 on $\ell$ nor on $(\kappa,\kappa')$.
 By the support considerations from \eqref{eq:suppFT} and \eqref{eq:support_gamma}, we have that
$$T(f_\kappa)T(f_{\kappa'})=\sum_{\ell=1}^K(T(f_\kappa)T(f_{\kappa'}))\ast \widehat{\psi}_\ell.$$
\smallskip
By the triangle inequality, it suffices to establish the estimate
\begin{equation}\label{eq:almost_there}
\Big\|\sum_{({\kappa,\kappa'})\in\mathcal{T}}
(T(f_\kappa)T(f_{\kappa'}))\ast \widehat{\psi}_\ell\Big\|_{L^{\frac p2}(\mathbb{R}^{d+1})}^{\frac p2}
\lesssim
\sum_{({\kappa,\kappa'})\in\mathcal{T}}  \big\|T (f_\kappa) T (f_{\kappa'})\big\|_{L^{\frac p2}(\mathbb{R}^{d+1})}^{\frac p2},
\end{equation}
for each $\ell\in\{1,2,\ldots,K\}$.
From  \cite[Lemma 2.2]{Ra12}, we have that
\begin{equation}\label{eq:almost_there2}
\Big\|\sum_{({\kappa,\kappa'})\in\mathcal{T}}
(T(f_\kappa)T(f_{\kappa'}))\ast \widehat{\psi}_\ell\Big\|_{L^{\frac p2}(\mathbb{R}^{d+1})}^{\frac p2}
\lesssim
\sum_{({\kappa,\kappa'})\in\mathcal{T}}  \big\|(T(f_\kappa)T(f_{\kappa'}))\ast \widehat{\psi}_\ell\big\|_{L^{\frac p2}(\mathbb{R}^{d+1})}^{\frac p2}.
\end{equation}
In fact, the Fourier transform of each function $(T(f_\kappa)T(f_{\kappa'}))\ast \widehat{\psi}_\ell$ is supported in $P_\ell$, and as we have seen there exists $\beta>0$ such that the elements in the family $\{(1+\beta)\cdot P_\ell\}_{(\kappa,\kappa')\in\mathcal{T}}$ are pairwise disjoint. Moreover, for each $(\kappa,\kappa')\in\mathcal{T}$, one easily constructs a function $\varphi=\varphi(\kappa,\kappa')$ satisfying
\begin{align*}
\begin{split}
    {\rm supp}{(\varphi)}&\subset (1+\beta)\cdot P_\ell, \\
    \varphi(x)&\equiv 1,\, \mathrm{if} \,\,x\in P_\ell, \\
    \|\widehat{\varphi}\|_{L^1(\mathbb{R}^{d+1})}&\leq C,
\end{split}
\end{align*}
where the constant $C$ is uniform in $(\kappa,\kappa')$.
One just has to observe that each parallelepiped $P_\ell$ is an affine image of the unit cube.
Therefore \eqref{eq:almost_there2} follows from a direct application of \cite[Lemma 2.2]{Ra12}.
To finish, invoke \eqref{eq:parallelepid_boundedness} with $q=p/2>1$ in order to obtain \eqref{eq:almost_there} from \eqref{eq:almost_there2}.
The proof is now complete.
\end{proof}

\begin{lemma}\label{BegoutVargasLemma}
Let $d\geq 3$ and $\frac{2(d+2)}{d} \leq p\leq  \frac{2(d+1)}{d-1}$.
Let $1\leq s<2$ and $0<\gamma<1-\frac2p$.
Then the following inequality holds
\begin{equation}\label{BegoutVargas}
\sum_{0<r\leq N}\sum_{\kappa\in\mathcal{D}_{N,r}} \big(|\kappa|^{1-\frac2s} \|f_\kappa\|_{L^s(\mathbb{R}^d)}^2\big)^{\frac p2(1-\gamma)}\lesssim \|f_N\|_{L^2(\mathbb{R}^d)}^{p(1-\gamma)},
\end{equation}
for every dyadic number $N\geq 1$ and admissible function $f\in L^2(\mathbb{H}^d)$.
\end{lemma}

\begin{proof}
We may assume that $\|f_N\|_{L^2} = 1$.  The strategy, suggested by the proof of \cite[Theorem 1.3]{BV07}, amounts to decomposing the function $f_N$ into low and high frequencies, depending on the size of the region $\kappa$.
More precisely, write
$$f_N
=\mathbbm 1_{\{|{f}|\leq |\kappa|^{-\frac12}\}}{f_N}+\mathbbm 1_{\{|{f}|> |\kappa|^{-\frac12}\}}{f_N}
=:{f_N^{\leq}}+{f_N^{>}}.$$
Set $\alpha:=\frac p2(1-\gamma)$.
To estimate the low frequencies, use H\"older's inequality to bound
$$ \|f_N^{\leq}\|_{L^s(\kappa)}
\leq
|\kappa|^{\frac1s-\frac1{2\alpha}}\|f_N^{\leq}\|_{L^{2\alpha}(\kappa)},$$
which holds provided $2\alpha\geq s$, or equivalently $\gamma\leq1-\frac sp$.
In this case,
\begin{equation} \label{E:sum fN leq}
\sum_{0<r\leq N}\sum_{\kappa\in\mathcal{D}_{N,r}}
\big(|\kappa|^{1-\frac2s} \|f_N^{\leq}\|_{L^s(\kappa)}^2\big)^{\alpha}\lesssim
\sum_{0<r\leq N}\sum_{\kappa\in\mathcal{D}_{N,r}}
|\kappa|^{\alpha-1}\|f_N^{\leq}\|^{2\alpha}_{L^{2\alpha}(\kappa)}.
\end{equation}
Let $V_{N,r}$ denote the volume of a region $\kappa \in \mathcal D_{N,r}$.  Recall that $V_{N,r} \simeq Nr^{d}$ when $0 < r \leq 1$, and that $V_{N,r} \simeq Nr^{d-1}$ when $1 < r < N$.
The right-hand side of \eqref{E:sum fN leq} can be estimated as follows:
\begin{equation}\label{geom_series}
\sum_{0<r\leq N}\sum_{\kappa\in\mathcal{D}_{N,r}}
|\kappa|^{\alpha-1}\|f_N^{\leq}\|^{2\alpha}_{L^{2\alpha}(\kappa)}
\lesssim
\int_{|\xi|\simeq N}
\Bigg(\sum_{\substack{0<r\leq N: \\ |f(\xi)|\leq (V_{N,r})^{-\frac12}}}
(V_{N,r})^{\alpha-1}\Bigg)
 |f(\xi)|^{2\alpha} \d\xi.
\end{equation}
 Thus the sum on the right-hand side of \eqref{geom_series} amounts to  two geometric series, both of which can be estimated by their largest terms:
$$\int_{|\xi|\simeq N}
\Bigg(\sum_{\substack{0<r\leq N: \\ |f(\xi)|\leq (V_{N,r})^{-\frac12}}}
(V_{N,r})^{\alpha-1}\Bigg)
 |f(\xi)|^{2\alpha} \d\xi
 \lesssim
 \int_{|\xi|\simeq N} |f(\xi)|^{-2(\alpha-1)}|f(\xi)|^{2\alpha}\d\xi=\|f_N\|_{L^2}^2=1.$$
Note that the latter inequality only holds provided $\alpha-1>0$, or equivalently $\gamma<1-\frac2p$, which is a valid constraint since $p>2$.

\smallskip

To estimate the high frequencies, use Minkowski's inequality to bound
$$\sum_{0<r\leq N}\sum_{\kappa\in\mathcal{D}_{N,r}} \left(|\kappa|^{1-\frac2s} \|f_N^>\|_{L^s(\kappa)}^2\right)^{\alpha}
\leq
\left(\sum_{0<r\leq N}\sum_{\kappa\in\mathcal{D}_{N,r}} |\kappa|^{\frac s2-1} \|f_N^>\|_{L^s(\kappa)}^s\right)^{\frac{2\alpha}s},
$$
which  as before holds provided $2\alpha\geq s$.
The right-hand side of this expression can be estimated as before:
\begin{align*}
\sum_{0<r\leq N}\sum_{\kappa\in\mathcal{D}_{N,r}} |\kappa|^{\frac s2-1} \|f_N^>\|_{L^s(\kappa)}^s
&\lesssim
\int_{|\xi|\simeq N}\Bigg(\sum_{\substack{0<r\leq N: \\ |f(\xi)|> (V_{N,r})^{-\frac12}}} (V_{N,r})^{\frac s2-1} \Bigg) |f(\xi)|^s \d\xi\\
&\lesssim\int_{|\xi|\simeq N} |{f}(\xi)|^{-2(\frac s2-1)} |{f}(\xi)|^s \d\xi= \|f_N\|_{L^2}^2= 1.
\end{align*}
This concludes the verification of \eqref{BegoutVargas}.
\end{proof}

We are now ready for the proof of the refined Strichartz inequality.
\begin{proof}[Proof of Theorem \ref{Sharpened}]
We recall the following simple geometric observation:
Given dyadic numbers $N\geq 1$ and $0<r\leq N$,
and a region $\kappa\in\mathcal{D}_{N,r}$, the number of regions $\kappa' \in\mathcal{D}_{N,r}$ which are separated from $\kappa$ is universally bounded.
In other words,
\begin{equation}\label{DoubletoSingle}
\#\{\kappa': \kappa\sim\kappa'\in\mathcal{D}_{N,r}\}\lesssim_d 1.
\end{equation}
Via a standard decomposition argument, see \cite{BV07, TVV98}, we have that
\begin{equation}\label{Step3First}
\|T(f_N)\|_{L^p}^p
=\Bigg\|\sum_{0<r\leq N}\sum_{\kappa\sim\kappa'\in\mathcal{D}_{N,r}}
T (f_\kappa) T (f_{\kappa'})\Bigg\|_{L^{\frac p2}}^{\frac p2}.
\end{equation}
To verify this, recall the definition \eqref{eq:DefAnnulus} of the restricted annulus $\mathcal{A}_N$, and consider the diagonal
$$\Gamma:=\{(\xi,\eta)\in\mathcal{A}_N\times\mathcal{A}_N: \xi=\eta\}.$$
Then the following Whitney-type decomposition is a consequence of the construction performed in Section \ref{sec:Caps}:
\begin{equation}\label{eq:Whitney}
(\mathcal{A}_N\times\mathcal{A}_N)\setminus\Gamma
=\bigcup_{0<r\leq N}\bigcup_{\kappa\sim\kappa'\in\mathcal{D}_{N,r}} \kappa\times\kappa'.
\end{equation}
Identity \eqref{Step3First} follows from this by writing $\|T (f_N)\|_{L^p}^p=\|T (f_N)^2\|_{L^{\frac p2}}^{\frac p2}$.
By Lemma \ref{lem:almostorthogonal}, we then have
\begin{equation}\label{eq:applyL5}
\|T (f_N)\|_{L^p}^p\lesssim
\sum_{0<r\leq N}\sum_{\kappa\sim\kappa'\in\mathcal{D}_{N,r}} \big\|T (f_\kappa) T(f_{\kappa'})\big\|_{L^{\frac p2}}^{\frac p2}.
\end{equation}
On the one hand, each of these summands can be bounded by H\"older's inequality as follows:
\begin{equation}\label{Holder}
\|T (f_\kappa) T (f_{\kappa'})\|_{L^{\frac p2}}
\leq
\|T (f_\kappa)\|^\gamma_{L^p}\|T (f_{\kappa'})\|^\gamma_{L^{p}}
\|T (f_\kappa) T (f_{\kappa'})\|^{1-\gamma}_{L^{\frac p2}},
\end{equation}
where $\gamma \in (0,1)$ is a parameter to be chosen below.
On the other hand, we can split the sum on the right-hand side of \eqref{eq:applyL5} into two pieces, depending on whether $0<r\leq 1$ or $1<r\leq N$.
Let us focus on the first sum, that over caps.
We claim that
\begin{multline}\label{AllTogether}
\sum_{0<r\leq 1}\sum_{\kappa\sim\kappa'\in\mathcal{D}_{N,r}} \big\|T (f_\kappa) T (f_{\kappa'})\big\|_{L^{\frac p2}}^{\frac p2}
\lesssim
N^{-\frac p2(1-\gamma)}
\left(\sup_{0<r\leq 1} \sup_{\kappa\in\mathcal{D}_{N,r}}
(r^{d})^{(\frac p2-\frac{d+2}{d})(1-\gamma)} \|T (f_\kappa)\|_{L^p}^{p\gamma}\right)\\
\times\sum_{0<r\leq 1}\sum_{\kappa\in\mathcal{D}_{N,r}} \big(|\kappa|^{1-\frac2s} \|f_\kappa\|_{L^s}^2\big)^{\frac p2(1-\gamma)}.
\end{multline}
\noindent This follows from  inequality \eqref{Holder},
the case $f=g$ of the bilinear extension estimate \eqref{Bilinear1},
and the observation \eqref{DoubletoSingle} that allows to bound the double sum $\sum_{\kappa\sim\kappa'}$ by a single sum $\sum_\kappa$.
One just has to recall that the Lebesgue measure of an $r$-cap at scale $N$ is comparable to $Nr^{d}$.

Lemma \ref{BegoutVargasLemma} then implies that the last factor on the right-hand side of inequality \eqref{AllTogether} is  $O(\|f_N\|_{L^2}^{p(1-\gamma)})$, provided $\gamma<1-\frac2p$.
As a consequence, the following inequality for $r$-caps at scale $N$ holds:
\begin{multline}\label{sectors}
\sum_{0<r\leq 1}\sum_{\kappa\sim\kappa'\in\mathcal{D}_{N,r}} \big\|T (f_\kappa) T (f_{\kappa'})\big\|_{L^{\frac p2}}^{\frac p2}\lesssim
 \left(\sup_{0<r\leq 1} \sup_{\kappa\in\mathcal{D}_{N,r}}
(r^d)^{(\frac p2-\frac{d+2}{d})(1-\gamma)} \|T (f_\kappa)\|_{L^p}^{p\gamma}\right)
\|f_N\|_{L^2(\mathbb{H}^d)}^{p(1-\gamma)}.
\end{multline}
In a similar way,  recalling that the Lebesgue measure of an $r$-sector at scale $N$ is comparable to $Nr^{d-1}$,
and using \eqref{Bilinear2} instead of \eqref{Bilinear1},
one can show the corresponding inequality for $r$-sectors at scale $N$,
\begin{multline}\label{caps}
\sum_{1<r\leq N}\sum_{\kappa\sim\kappa'\in\mathcal{D}_{N,r}} \big\|T (f_\kappa) T (f_{\kappa'})\big\|_{L^{\frac p2}}^{\frac p2}
\lesssim
\left(\sup_{1<r\leq N} \sup_{\kappa\in\mathcal{D}_{N,r}}
(r^{d-1})^{(\frac p2-\frac{d+1}{d-1})(1-\gamma)} \|T (f_\kappa)\|_{L^p}^{p\gamma}\right)
\|f_N\|_{L^2(\mathbb{H}^d)}^{p(1-\gamma)},
\end{multline}
under the same assumption  $\gamma<1-\frac2p$.
Inequality \eqref{sharpenedextensionineq} follows from \eqref{eq:applyL5}, \eqref{sectors} and \eqref{caps}.
The proof is now complete.
 \end{proof}

\section{End of the proof: concentration-compactness}\label{sec:CC}

As we left off in Section \ref{sec:ang_rest}, let $\{f_n\}_{n\in\mathbb{N}} \subset L^2(\mathbb{H}^d)$ be an extremizing sequence for \eqref{ExtensionInequality}, with $\|f_n\|_{L^2(\mathbb{H}^d)} = 1$ for all $n \in \mathbb{N}$, and let $\{f_n^{(k_0)}\}_{n\in\mathbb{N}}$ be a quasi-extremizing sequence in the sense of \eqref{quasi-ext}. Assuming without loss of generality that $k_0 =1$, the sequence $\{f_n^{(1)}\}_{n\in\mathbb{N}}$ belongs to our class of admissible functions considered in Sections \ref{sec:Caps} and \ref{sec:RefinedStrichartz}.

\smallskip

From Proposition \ref{annulardec}, for each $n\in\mathbb{N}$, there exists $N = N_n\in2^{\mathbb{Z}_{\geq0}}$ such that
$$\big\|T\big((f_{n}^{(1)})_N)\big)\big\|_{L^p(\mathbb{R}^{d+1})}\geq \delta_3,$$
where $\delta_3>0$ is a universal constant.

\smallskip

If $\frac{2(d+2)}{d} \leq p <  \frac{2(d+1)}{d-1}$, then Theorem \ref{Sharpened} ensures for each $n\in\mathbb{N}$ the existence of a dyadic number $r=r_n$ satisfying $r \leq  2^\alpha$ for a universal constant $\alpha$, and of a region $\kappa=\kappa_n \in \mathcal{D}_{N,r}$, such that
$$\big\|T\big((f_{n}^{(1)})_\kappa\big)\big\|_{L^p(\mathbb{R}^{d+1})}\geq \delta_4,$$
where $\delta_4>0$ is a universal constant. This implies at once that
$$\|(f_{n}^{(1)})_\kappa\|_{L^2(\mathbb{H}^{d})}\geq \delta_5,$$
where $\delta_5>0$ is a universal constant.
Set $L_n := L_{c(\kappa)}$, where $c(\kappa)$ denotes as usual the center of the region $\kappa$.
Since $r \leq 2^\alpha$, a standard computation shows that the image $L_n^\flat(\kappa)$ is contained in a universal ball $B\subset\mathbb{R}^d$ centered at the origin.
Therefore
$$\|L_n^*f_{n}^{(1)}\|_{L^2(B)} \geq \delta_6,$$
where $\delta_6>0$ is a universal constant. As already observed in Section \ref{sec:ang_rest}, this plainly implies that
$$\|L_n^*f_{n}\|_{L^2(B)} \geq \delta_6.$$
This establishes the existence of distinguished region when $\frac{2(d+2)}{d} \leq p <  \frac{2(d+1)}{d-1}$.

\smallskip

We can now invoke the machinery of \cite[Section 6]{COSS17}, which only works when $\frac{2(d+2)}{d} < p \leq  \frac{2(d+1)}{d-1}$, to arrive at the existence of extremizers stated in  Theorem \ref{Thm1} in the non-endpoint range $\frac{2(d+2)}{d} < p <  \frac{2(d+1)}{d-1}$.
We provide the details below.

\smallskip

By \cite[Proposition 18]{COSS17}, there exists $(x_n,t_n)\in\mathbb{R}^d\times\mathbb{R}$ such that the sequence $\{h_n\}_{n\in\mathbb{N}}$ defined by
$$h_n(\xi):=e^{ix_n\cdot \xi}e^{i t_n\langle \xi\rangle}f_n(\xi)$$
admits a subsequence that converges weakly to a nonzero limit, say $h\neq 0$, in $L^2(\mathbb{H}^d)$.
For this subsequence, possibly after extracting a further subsequence, \cite[Proposition 19]{COSS17}  implies
$$T(h_n)(x,t)\to T(h)(x,t), \text{ as }n\to\infty,$$
for almost every $(x,t)\in\mathbb{R}^d\times\mathbb{R}$.
The existence of extremizers then follows from a straightforward application of  \cite[Proposition 1.1]{FVV11}.
This completes the proof of Theorem \ref{Thm1}.

\section*{Acknowledgements}
The authors are thankful to Rupert Frank {and Ren\'e Quilodr\'an} for helpful comments and suggestions.
E.C.~acknowledges support from CNPq - Brazil, FAPERJ - Brazil and the Simons Associate Scheme from the International Centre for Theoretical Physics (ICTP) - Italy.
D.O.S.~was partially supported by the Hausdorff Center for Mathematics, the Deutsche Forschungsgemeinschaft through the Collaborative Research Center 1060, and the College Early Career Travel Fund of the University of Birmingham.
M.S. ~acknowledges support from the grant PICT 2014-1480 (ANPCyT).
B.S.~was supported by a National Science Foundation grant (DMS-1600458).


\section{Introduction}
\label{sec:intro}

A hallmark feature of 3D turbulence is the forward energy cascade, transporting kinetic energy from large scales to ever smaller scales as described by the celebrated theory of \citet{Kolmogorov1941}. In many geophysical and astrophysical flows, however, velocity fluctuations are largely suppressed in one direction as a consequence of, for example, confinement \citep{Benavides2017,Musacchio2017,Musacchio2019}, strong magnetic fields \citep{Alexakis2011,Seshasayanan2014} or fast rotation \citep{Smith1996,Seshasayanan2018,Pestana2019,VanKan2020}, rendering the flow quasi-2D. This leads to the development of an inverse energy flux, akin to fully 2D turbulence \citep{Kraichnan1967,Batchelor1969}, transporting energy from smaller to larger scales. Ultimately, this can lead to accumulation of kinetic energy at the largest available scales, followed by condensation into a vertically coherent large-scale vortex (LSV) structure at the domain size \citep[see][for a recent review]{Alexakis2018}. These LSVs are believed to play a crucial role in for example the formation of the Earth's magnetic field \citep{Aurnou2015,Guervilly2015,Roberts2013}.

Following the framework that is brought forward in \citet{Alexakis2018}, we aim to classify the transition from a 3D forward cascading state to the condensate state by considering the behaviour of an order parameter that measures the strength of the LSV as a function of a control parameter of the flow throughout this transition. Then one can observe either a smooth transition, a continuous transition with discontinuous derivative, or a discontinuous transition. This categorisation of the transition into the condensate state has shown to be an insightful approach in various other quasi-2D flow systems \citep{Alexakis2015,Seshasayanan2018,VanKan2019,Yokoyama2017}.

These earlier works, however, have focused on more artificial, idealised flow models where the turbulent forcing occurs at a single well-defined length scale. Here, we characterise the LSV transition in a natural, broadband-forced system of rotating convection, which is ubiquitous in geophysical and astrophysical flows. In this system, \citet{Favier2019} have shown a bistability of an LSV with a non-LSV state at the same parameter values, depending on the initial conditions. The natural buoyant forcing over a broad range of scales obfuscates the strict separation of the injection, dissipation and condensation scale. Although one may expect that in such natural and vigorously fluctuating turbulent systems, any transitions between different states are washed out and become gradual, we find that the transition towards the condensate state is in fact sharp and discontinuous.

Such abrupt transitions between turbulent states in a more general sense are a remarkable feature of fluid turbulence and have received much recent interest, being observed in various different flow settings, e.g. in torque measurements of Taylor-Couette and Von Kármán flows \citep{Huisman2014,Ravelet2004,Saint-Michel2013}, in states of stochastically forced 2D and 3D turbulence \citep{Bouchet2009,Bouchet2019,Iyer2017}, and in reversals of large scale dynamics in thin layers \citep{Sommeria1986,Michel2016,Dallas2020}. These types of abrupt transitions are surmised to play an important role in, for example, climate research \citep{Weeks1997,Herbert2020,Jackson2018} and in understanding the geomagnetic reversal \citep{Petrelis2009,Berhanu2007}.

\begin{figure}
    \centering
    \includegraphics[width=0.7\linewidth]{lsv_example.pdf}
    \caption{Snapshot of horizontal kinetic energy (in units of $U^2$) of the LSV-forming case $Ra=1.7\times10^7$, truncated at three-quarter height to reveal a cross section of the LSV.}
    \label{fig:lsv_example}
\end{figure}

\section{Numerical approach}\label{sec:numerical_approach}
We consider the canonical system of rotating Rayleigh-B\'{e}nard convection, in which the flow is driven by buoyancy through a temperature difference $\Delta T$ between the bottom and top of the domain, whilst being simultaneously affected by strong background rotation $\Omega$ along the vertical axis. The input space to this problem is described by three dimensionless numbers: the Rayleigh number $Ra=g\alpha\Delta T H^3/(\nu\kappa)$, quantifying the strength of the thermal forcing, the Ekman number ${E\mkern -1.5mu k}=\nu/(2\Omega H^2)$, representing the (inverse) strength of rotation and the Prandtl number $\Pran=\nu/\kappa$, containing the diffusive properties of the fluid. Here, $g$ denotes gravitational acceleration, $H$ is the domain height and $\alpha$, $\nu$ and $\kappa$ represent the thermal expansion coefficient, kinematic viscosity and thermal diffusivity of the fluid, respectively. The system is non-dimensionalized into convective units using $H$, $\Delta T$ and the free-fall velocity $U=\sqrt{g\alpha\Delta T H}$.

We solve the full governing set of Boussinesq Navier-Stokes equations through direct numerical simulation (DNS), employing the finite-difference code described in \citet{Verzicco1996} and \citet{Ostilla-Monico2015} on a Cartesian grid with periodic sidewalls and stress-free boundary conditions at the top and bottom. For the width $D$ of the domain, we choose $D/H=10\mathcal{L}_c$ with $\mathcal{L}_c=4.8{E\mkern -1.5mu k}^{1/3}$ the most unstable wavelength at onset of convection \citep{Chandrasekhar1961}. The complete set of input parameters as well as the employed resolutions are provided in table~\ref{tab:input}. Note that we include the convective Rossby number $Ro=U/(2\Omega H)={E\mkern -1.5mu k}\sqrt{Ra/\Pran}$ for reference. A validation of the grid resolution is provided in appendix~\ref{appA}.

\begin{table}
    \renewcommand\arraystretch{1.1}
    \centering
    \caption{The different series of input parameters used in this work.}\label{tab:input}
    \makebox[\textwidth][c]{
    \begin{tabular}[t]{lccccccc}
        \toprule
        &\#runs&$Ra$&${E\mkern -1.5mu k}$&$\Pran$&$Ro$&$D/H$&Resolution\\
        \midrule
        Low-$Ra$ transition&14&$[2\times10^6 : 1\times10^7]$&$10^{-4}$&1&$[0.14:0.32]$&2.24&$256\times256\times128$\\
        Intermediate $Ra$&2&$[1.3\times10^7:1.7\times10^7]$&$10^{-4}$&1&$[0.36:0.41]$&2.24&$256\times256\times136$\\
        High-$Ra$ transition&30&$[2\times10^7 : 5\times10^7]$&$10^{-4}$&1&$[0.45:0.71]$&2.24&$256\times256\times144$\\
        Ensemble&100&$6\times10^6$&$10^{-4}$&1&0.24&2.24&$128\times128\times72$\\
        \bottomrule
    \end{tabular}
    }
\end{table}

For numerical convenience, we use $\Pran=1$ and ${E\mkern -1.5mu k}=10^{-4}$, for which stable LSVs have been observed in earlier DNSs over a limited range of $Ra$ \citep{Favier2014,Guervilly2014,Favier2019}. Upon increasing $Ra$ from the onset of convection, the two boundaries of the region of existence of the LSV are crossed. At the low-$Ra$ transition, the LSV develops as sufficient turbulent forcing is obtained to set-up the upscale transport into the condensate, whereas at the high-$Ra$ transition, the LSV breaks down as too strong thermal forcing renders the flow insufficiently rotationally constrained, breaking the quasi-2D conditions for upscale energy flux \citep{Favier2014,Guervilly2014}. We carry out a total of 46 runs at varying $Ra$, concentrated around both LSV transitions.

In order to analyse the LSV, we decompose the flow field $\bm{u}=u\bm{e}_x+v\bm{e}_y+w\bm{e}_z$ into a 2D (vertically averaged) barotropic flow and a 3D (depth-dependent) baroclinic flow \citep[following][]{Guzman2020,Favier2019,Favier2014,Julien2012,Maffei2021,Rubio2014}, i.e. $\bm{u}=\bm{u}^\textrm{2D}+\bm{u}^\textrm{3D}$, where
\begin{equation}
    \bm{u}^\textrm{2D}=\overbar{u}\bm{e}_x+\overbar{v}\bm{e}_y,\quad \bm{u}^\textrm{3D}=\underbrace{(u-\overbar{u})}_{u'}\bm{e}_x+\underbrace{(v-\overbar{v})}_{v'}\bm{e}_y+w\bm{e}_z,
\end{equation}
where the overbar $\overbar{\vphantom{a}\textrm{...}}$ denotes vertical averaging. Since the LSV is a largely vertically coherent structure (see figure~\ref{fig:lsv_example}), it resides primarily in the 2D field, whereas the turbulent baroclinic fluctuations are captured by the 3D field. Accordingly, we decompose the total kinetic energy $E_\textrm{tot}=\tfrac{1}{2}\langle|\bm{u}|^2\rangle$ into 2D and (horizontal and vertical) 3D contributions $E_\textrm{tot}=E^\textrm{2D}+E_H^\textrm{3D}+E_V^\textrm{3D}$ as
\begin{equation}
    E^\textrm{2D}=\tfrac{1}{2}\langle\overbar{u}^2+\overbar{v}^2\rangle,\quad E_H^\textrm{3D}=\tfrac{1}{2}\langle u'^2+v'^2\rangle,\quad E_V^\textrm{3D}=\tfrac{1}{2}\langle w^2\rangle,
\end{equation}
where angular brackets $\langle ... \rangle$ represent an average over the full spatial domain. We also consider the energy spectrum of the 2D flow from its Fourier transform $\hat{\bm{u}}^\textrm{2D}_{k_x k_y}$ as
\begin{equation}
    \hat{E}^\textrm{2D}(K)=\sum_{K\leq\sqrt{k_x^2+k_y^2}<K+1} \tfrac{1}{2}|\hat{\bm{u}}^\textrm{2D}_{k_x k_y}|^2,
\end{equation}
where we normalise the wavenumber $K$ by the box-size mode $2\pi/D$, such that the LSV occupies the $K=1$ mode of the spectrum.

\section {Results and discussion}\label{sec:results}
In figure~\ref{fig:kins}, the different components of kinetic energy are provided over the range of considered $Ra$, crossing both LSV transitions. From the (largest mode of) 2D energy that captures the LSV, we find a substantial discontinuity at both boundaries of the LSV state. At the high-$Ra$ transition, we find that this transition is hysteretic: the morphology of the flow depends on its history c.q. its initial conditions. These findings are in line with \citet{Favier2019}, showing this bistability of an LSV and a non-LSV state for one case in this parameter range. To study this hysteresis loop, we initialise simulations using flow snapshots from a preceding $Ra$. For decreasing $Ra$ from a non-LSV state, the lower branch of the hysteresis loop is followed (open diamonds), whilst for increasing $Ra$ from an LSV state, the flow adheres to the upper hysteretic branch (filled squares), see figure~\ref{fig:kins}. Note the remarkably large discontinuity in the lower hysteretic branch, where the flow transitions directly from a non-LSV state into nearly the strongest LSV forming state. Hysteresis in the LSV transition has also been observed in a rotating flow system with a \emph{sharp bandwidth} Taylor-Green forcing \citep{Yokoyama2017}.

\begin{figure}
    \centering
    \includegraphics[width=0.55\linewidth]{kins.pdf}
    \caption{Averaged kinetic energy components as a function of $Ra$. Filled squares: upper hysteretic branch of the high-Ra transition; open diamonds: lower hysteretic branch. The cyan dashed-dotted line denotes the low-$Ra$ LSV transition, whilst the magenta and red lines denote the LSV transition of the lower and upper branch of the high-$Ra$ transition, respectively.}
    \label{fig:kins}
\end{figure}

At the low-$Ra$ transition, on the other hand, no hysteresis is observed (using increments in $Ra$ of $\sim$2\%). Considering the cases directly above the LSV transition, however, we find that the growth of the LSV from a non-LSV state is non-monotonic: the flow shows an evident plateau during which the LSV has not yet developed, before finally growing relatively suddenly into the stable LSV state, see figure~\ref{fig:barrier}a. The flow in this plateau state shows morphological similarities with the jet state observed in rectangular domains \citep{Guervilly2017,Julien2018}, but alternates between being predominantly in the $x$- and $y$-direction. We hypothesise that the peculiar growth behaviour found here signifies a memoryless abrupt growth process, much akin to the nucleation \& growth type of dynamics that is observed in plentiful different systems throughout physics \citep{Matsumoto2002,Watanabe2010,Garmann2019,Metaxas2019}. To substantiate this conjecture, we simulate an ensemble of 100 additional runs at $Ra=6\times10^6$ with statistically perturbed initial conditions, using a reduced resolution of $128\times128\times72$ for computational feasibility. The hypothesised abrupt memoryless growth would then predict an exponential distribution of the waiting time spent in the metastable plateau state (stage B in figure~\ref{fig:barrier}a).

\begin{figure}
    \includegraphics[width=\textwidth]{barrier.pdf}
    \caption{Panel (a): time evolution of horizontal kinetic energy of seven of the runs in the ensemble of the LSV-forming case $Ra=6\times 10^6$ close to the transition. We distinguish four stages: an initialisation phase (A), a metastable underdeveloped plateau state showing alternating jets (B), (quick) growth of the LSV (C) and the stably developed LSV state (D). The horizontal dashed line denotes the threshold to be crossed and sustained for 2000 convective time units ($U^{-1}H$) before the LSV is said to be completely developed, which defines the time point $t_\textrm{LSV}$, depicted by the blue cross. Panel (b): cumulative distribution of $t_\textrm{LSV}$ of the ensemble (blue solid line) with 95\% confidence bounds (blue dotted lines). It is fitted by an exponential distribution (red dashed line) as provided by (\ref{eq:fit_dist}) and in the figure. Inset (I) shows the histogram corresponding to the same distribution.}
    \label{fig:barrier}
\end{figure}

To investigate the distribution of these waiting times, we define a time point $t_\textrm{LSV}$ at which the LSV is said to have stably developed once a threshold of horizontal kinetic energy is surpassed and sustained for 2000 convective time units, see figure~\ref{fig:barrier}a. The obtained empirical cumulative distribution is then fitted with an exponential distribution
\begin{equation}\label{eq:fit_dist}
    CDF(t_\textrm{LSV})=1-\exp\bigg(-\frac{t_\textrm{LSV}-t_0}{\underbrace{\mu(t_\textrm{LSV})-t_0}_{\tau_W}}\bigg),
\end{equation}
where the fit parameter $t_0$ can be interpreted as the (fixed) contributions of the initialisation, growth and stable LSV stages (A, C and D in figure~\ref{fig:barrier}a). Here, $\mu(t_\textrm{LSV})$ denotes the mean of $t_\textrm{LSV}$, providing the maximum-likelihood estimate for the typical waiting time $\tau_W$ in this distribution. Figure~\ref{fig:barrier}b shows that there is excellent agreement between the hypothesised and the obtained distribution: the exponential distribution remains everywhere in between the 95\% confidence bounds of the empirical distribution. Signs of exponentiality of waiting times have been observed in the LSV-forming system of \emph{sharp bandwidth} forced thin-layer turbulence \citep{VanKan2019a}. Our findings indicate that, indeed, turbulent fluctuations randomly trigger the growth of the LSV, giving rise to this memoryless abrupt growth dynamics. Similar sudden growth behaviour is also found near the ends of the hysteresis loop in the high-$Ra$ transition, although an extended analysis of how the mean transition time evolves for changing $Ra$, repeating the ensemble average approach for all points in the considered parameter space, is currently out of computational reach for rotating convection.

Considering the observed hysteretic behaviour as well as the exponentially distributed waiting times, an analogy of this transition with first-order phase transitions in equilibrium statistical mechanics seems befitting \citep{Binder1987}. However, although it is conjectured that in particular the relatively more weakly dissipative large scales of the flow may be in resemblance with thermal equilibrium states \citep{Alexakis2018,Bouchet2012}, ultimately, the chaotic and dissipative nature of turbulence makes the analogy with equilibrium statistical mechanics indirect. A more immediate interpretation of the transition in this fluctuating dynamical system would be in terms of non-linear bifurcations. Then, the condensate transition as observed here can be interpreted as a subcritical non-linear bifurcation, giving rise to two distinct attractors (indeed, the LSV state and the non-LSV state) which remain separated in phase space. Such noise induced transitions between attractors are also known to be memoryless, yielding exponentially distributed waiting times \citep{Kraut1999}.

To understand how the LSV is energetically sustained, we compute the mode-to-mode kinetic energy transfer \citep[see][]{Dar2001,Alexakis2005,Mininni2005,Mininni2009,Verma2017,Verma2019}, distinguishing the 3D to 2D (baroclinic to barotropic) transport \citep[following][]{Rubio2014,Guzman2020}
\begin{subequations}
\begin{equation}
    T_\textrm{3D}(K,Q)=-\Big\langle\bm{u}^\textrm{2D}_K\cdot(\overbar{\bm{u}^\textrm{3D} \cdot \boldsymbol{\nabla} \bm{u}^\textrm{3D}_Q})\Big\rangle,
\end{equation}
and the 2D to 2D (barotropic to barotropic) transport
\begin{equation}
    T_\textrm{2D}(K,Q)=-\Big\langle\bm{u}^\textrm{2D}_K\cdot(\bm{u}^\textrm{2D}\cdot\boldsymbol{\nabla}\bm{u}^\textrm{2D}_Q)\Big\rangle,
\end{equation}
\end{subequations}
describing the energetic transport into the Fourier-filtered 2D flow field $\bm{u}^\textrm{2D}_K$ of mode $K$ from 3D and 2D modes $Q$ through triadic interactions arising from the advective term of Navier-Stokes. If $T_\textrm{3D},T_\textrm{2D} > 0$, there is a net transfer of kinetic energy from mode $Q$ to mode $K$ and vice-versa. We also consider the transport into 3D mode $K$ from the full (unfiltered) flow components $\mathcal{T}_\textrm{3D}(K)=\sum_Q T_\textrm{3D}(K,Q)$ and $\mathcal{T}_\textrm{2D}(K)=\sum_Q T_\textrm{2D}(K,Q)$, by summing over the donating modes $Q$.

The results for the shell-to-shell energy transfer throughout the LSV transitions are provided in figure~\ref{fig:transfer_maps}. The two main energy fluxes are apparent in both $T_\textrm{3D}$ and $T_\textrm{2D}$. Near the diagonal, one can observe the direct forward cascade, transporting energy from $Q$ to slightly higher modes $K$. Note here, that while the $T_\textrm{2D}$ self-interaction must be symmetric by definition $T_\textrm{2D}(K,Q)=-T_\textrm{2D}(Q,K)$, this does not apply to $T_\textrm{3D}$ as it describes the energetic interactions between scales of the 2D component and 3D component of the flow. In the bottom row $K=1$, on the other hand, the upscale energy flux into the LSV can be appreciated. This energy flux is non-local: energy is transported directly from virtually all scales in the system into the box scale of the LSV, without participation of intermediate scales.

\begin{figure}
    \centering
    \includegraphics[width=0.7\linewidth]{transfer_maps.pdf}
    \caption{Time-averaged kinetic energy transport from 3D (left panels) or 2D (right panels) modes $Q$ to 2D modes $K$, i.e. $T_\textrm{3D}(K,Q)$ and $T_\textrm{2D}(K,Q)$, respectively, in units of $U^3H^{-1}$. Blue lines denote $\mathcal{T}_\textrm{3D}(K)$ (left) and $\mathcal{T}_\textrm{2D}(K)$ (right). The low-$Ra$ transition is crossed from (a,b) [$Ra=5.6\times10^6$] to (c,d) [$Ra=5.7\times10^6$], whilst the high-$Ra$ transition of the upper hysteretic branch is crossed from (e,f) [$Ra=3.10\times10^7$] to (g,h) [$Ra=3.13\times10^7$], as also depicted in figure~\ref{fig:trans}.}
    \label{fig:transfer_maps}
\end{figure}

Figure~\ref{fig:trans} shows the energetic transport into the box-size mode as a function of $Ra$. Note that this considers the transport from the full, unfiltered 3D and 2D flow components into the LSV, that is, a sum over the donating scales in the bottom row $K=1$ of the transfer maps in figure~\ref{fig:transfer_maps}. It makes clear that also the upscale transport into the LSV exhibits an evident discontinuous transition, both in $\mathcal{T}_\textrm{3D}(K=1)$ as well as, albeit to a lesser degree, in $\mathcal{T}_\textrm{2D}(K=1)$. Importantly, the figure indicates that it is the 3D transport that is the dominant component in the forcing of the LSV.

We argue that this upscale transport provides a clue to understand the physical mechanism behind the observed sudden growth and hysteretic behaviour. As also detailed in \citet{Rubio2014} and \citet{Favier2019}, the upscale transport contains a positive feedback loop, where the presence of the LSV itself enhances the upscale transport into the box-size mode. This agrees with our observation that the energetic transport into the LSV increases discontinuously as the LSV is created. The exact nature of how the LSV interacts energetically with its 3D turbulent background is an interesting non-trivial question to explore in future work, which goes beyond considerations purely in Fourier space as well, by looking at individual vortex interactions, for example. The existence of such a positive feedback loop, however, seems intuitive: the predominantly cyclonic LSV locally increases the total vorticity (background rotation + flow vorticity), thereby strengthening the quasi-2D conditions and hence the upscale transport. This mechanism allows the LSV to develop once its growth is triggered by rare turbulent fluctuations and lets the LSV remain stably self-sustained over the hysteresis loop.

\begin{figure}
    \centering
    \includegraphics[width=0.55\linewidth]{trans.pdf}
    \caption{Total 3D $\mathcal{T}_\textrm{3D}(K=1)$ (blue) and 2D $\mathcal{T}_\textrm{2D}(K=1)$ (red) transport of kinetic energy into the 2D $K=1$ mode averaged over time as a function of $Ra$. Symbols and vertical lines are as in figure~\ref{fig:kins}. Labels (a-h) depict the corresponding transfer maps in figure~\ref{fig:transfer_maps}.}
    \label{fig:trans}
\end{figure}

Since we consider specifically a cross section of the full parameter space for varying $Ra$, the influence of other parameters on the transitional dynamics, such as ${E\mkern -1.5mu k}$, $\Pran$ and also the aspect ratio $D/H$ (since the domain width is the principal dynamical scale of the LSV), remains an open question. The morphology of the LSV in the asymptotically reduced model for ${E\mkern -1.5mu k}\to0$ is studied in its total parameter space in more detail in \citet{Maffei2021}. For the transition specifically, however, one can argue that as both attractors are expected to shift in continuous fashion through the phase space, only quantitative changes to the observed transitional dynamics are expected as the other control parameters are varied in vicinity to the computationally tractable values considered here. Nonetheless, the possibility that the discontinuity in the transition vanishes in a certain limit if both attractors would shift to coincide cannot be ruled out from the current simulations; the asymptotically reduced model seems appropriate to investigate this premise for the limit ${E\mkern -1.5mu k}\to0$.

\section{Conclusions}\label{sec:conclusions}
We have described the fluid turbulence transition into a quasi-2D condensate state in a natural broadband-forced system of rotating Rayleigh-B\'{e}nard convection, where the transition is sharply discontinuous, in spite of the lack of a clear separation of scales. We provide evidence of memoryless abrupt growth dynamics and hysteresis in these transitions, raising the picture of a double attractor phase space with a subcritical noise-induced transition between them. Furthermore, the correspondence of our findings with certain aspects of the LSV transition in other, artificially forced, flow systems, as remarked in the text, ultimately shows that this peculiar type of transition is a relevant and robust phenomenon that is expected to survive even in the geo- and astrophysically relevant flow of rotating convection, being one of the principal sources of fluid motion in nature.

\section*{Acknowledgments}
We thank W.G. Ellenbroek for careful reading of the manuscript. A.J.A.G. and R.P.J.K. received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 678634). We are grateful for the support of the Netherlands Organisation for Scientific Research (NWO) for the use of supercomputer facilities (Cartesius) under Grants No. 2019.005 and No. 2020.009.

\section*{Declaration of Interests}
The authors report no conflict of interest.