\label{thm:global_existence_cauchy}
Let $N=2$ or $N=3$. Assume that $\gamma$ satisfies
\begin{equation}
    \begin{cases}
        \gamma \in [1, \infty) & \text{if } N=2, \\
        \gamma \in [1, 8/3) & \text{if } N=3.
    \end{cases}
\end{equation}
Let $\bar{\rho} > 0$ be the far field behavior of density. Assume that the initial data $(\rho_0, u_0)$ satisfies
\begin{equation}
    0 < \underline{\varrho} \le \rho_0 \le \bar{\varrho}, \quad (\rho_0 - \bar{\rho}) \in H^3(\mathbb{R}^N), \quad u_0 \in H^2(\mathbb{R}^N),
\end{equation}
where $\underline{\varrho}$ and $\bar{\varrho}$ are positive constants. Then the Cauchy problem  \eqref{NSK_simplified}-\eqref{far field} admits a unique global strong solution $(\rho, u)$ satisfying for any $0 < T < \infty$ and $(x,t) \in \mathbb{R}^N \times [0, T]$,
\begin{equation}
    (C(T))^{-1} \le \rho(x,t) \le C(T),
\end{equation}
and
\begin{equation}
\begin{aligned}
    &(\rho - \bar{\rho}) \in C([0, T]; H^3) \cap L^2(0, T; H^4), \quad \rho_t \in C([0, T]; H^1) \cap L^2(0, T; H^2), \\
    &u \in C([0, T]; H^2) \cap L^2(0, T; H^3), \quad u_t \in L^\infty(0, T; L^2) \cap L^2(0, T; H^1),
\end{aligned}
\end{equation}
where the constant $C(T) > 0$ depends on the initial data and $T$.

In this paper, we are concerned with the global well-posedness of the compressible Navier-Stokes-Korteweg equations in the whole space $\mathbb{R}^N$. This system describes the motion of a viscous compressible fluid endowed with internal capillarity and reads as follows
\begin{equation}\label{NSK} \begin{cases} \partial_t \rho + \nabla \cdot(\rho u) = 0,
\\ \partial_t (\rho u) + \nabla \cdot(\rho u \otimes u) + \nabla P(\rho) =\nabla \cdot \mathbb{S} + \nabla \cdot \mathbb{K}. \end{cases} \end{equation}
The viscous stress tensor $\mathbb{S}$ is given by Newton's rheological law
\begin{equation}
    \mathbb{S} = 2\mu(\rho) \mathcal{D}(u) + \lambda(\rho) \nabla \cdot u \mathbb{I},
\end{equation}
where $\mathcal{D}(u) = \frac{1}{2}(\nabla u + \nabla u^T)$ denotes the symmetric deformation tensor, and $\mathbb{I}$ is the identity matrix in $\mathbb{R}^N$. The scalar functions $\mu(\rho)$ and $\lambda(\rho)$ represent the shear and bulk viscosity coefficients, respectively, which satisfy the physical condition $\mu(\rho) > 0$ and $2\mu(\rho) + N\lambda(\rho) \ge 0$.
In order to ensure the stability of the system and utilize the BD entropy structure, we assume the viscosity coefficients satisfy the relation $$\lambda(\rho) = 2(\rho \mu'(\rho) - \mu(\rho)).$$
In the context of the Korteweg theory, the capillarity tensor $\nabla \cdot\mathbb{K}$ is typically defined as
\begin{equation}
    \nabla \cdot \mathbb{K} = \nabla \left( \rho \kappa(\rho) \Delta \rho + \frac{\kappa(\rho) + \rho \kappa'(\rho)}{2} |\nabla \rho|^2 \right) - \nabla \cdot \left( \kappa(\rho) \nabla \rho \otimes \nabla \rho \right).
\end{equation}
where $\kappa(\rho) > 0$ is the coefficient of capillarity.\par
The study of capillary fluids originates from the pioneering work of Van der Waals and Korteweg \cite{Korteweg,J.F. Van derWaals}. In their theory of capillarity, the fluid energy is assumed to depend not only on the standard thermodynamic variables but also on the gradient of the density. This concept was later formalized in the modern language of continuum mechanics by Dunn and Serrin in the 1980s, leading to the so-called Korteweg-type models. \par
  When $\kappa(\rho) = 0$, $\mu=\rho$ and $\lambda=0$, system \eqref{NSK} reduces to the well-known viscous shallow water equations.
Significant progress has been made for the viscous shallow water equations in recent years. Regarding general multi-dimensional initial data, significant breakthroughs concerning the global well-posedness of weak solutions were achieved by Li-Xin \cite{Li-Xin} and Vasseur-Yu \cite{Vasseur-Yu}. Independently, these authors established the global existence of weak solutions for the compressible system with $\mu(\rho) = \rho$ and $\lambda(\rho) = 0$, admitting arbitrarily large data and vacuum states. More specifically, they obtained global weak solutions for $\gamma \in (1, \infty)$ when $N=2$, and for $\gamma \in (1, 3)$ when $N=3$. It is worth noting that Li-Xin \cite{Li-Xin} extended their analysis to a wider class of viscosity coefficients satisfying the BD entropy relation. Concerning the global smooth solutions for multi-dimensional shallow water equations with arbitrarily large initial data, Huang-Meng-Zhang \cite{Huang-Meng-Zhang-V} pioneered the proof of global classical solutions for the two-dimensional initial-boundary value problem under the assumption of radial symmetry with $\gamma\ge\frac{3}{2}$. Subsequently, Gu-Huang \cite{Gu-Huang} extended the range of the exponent to $\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\par
 When $\kappa(\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\mu(\rho) = \rho$ and capillarity $\kappa(\rho) = \rho^{-1}$, Charve-Haspot \cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\mu(\varrho) = \varrho^\alpha$ and capillarity $\kappa(\varrho) = \varrho^\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity.  For the multi-dimensional case, building upon their earlier local theory \cite{Hattori-Li}, Hattori-Li \cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \par
However, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.
\par
Throughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients
$$\mu(\rho) = \rho, \quad \lambda(\rho) = 0,$$
and we assume that $\kappa(\rho)$ satisfies
$$\kappa(\rho)=\frac{1}{\rho}.$$
With the specific choice of the capillarity coefficient $\kappa(\rho)$, the divergence of the Korteweg tensor takes the following form
\begin{equation}
\nabla \cdot \mathbb{K} = \nabla \cdot (\rho \nabla \nabla \log \rho).
\end{equation}
Consequently, under the assumption of shallow-water viscosity ($\mu(\rho)=\rho, \lambda(\rho)=0$) and the specific capillarity coefficient $\kappa(\rho)=1/\rho$, equation \eqref{NSK} can be rewritten as
\begin{equation}\label{NSK_simplified} \begin{cases} \partial_t \rho + \nabla \cdot(\rho u) = 0, \\ \partial_t (\rho u) + \nabla \cdot(\rho u \otimes u) + \nabla P(\rho) = \nabla \cdot \left(2 \rho \mathcal{D}(u) \right) + \nabla \cdot (\rho \nabla \nabla \log \rho). \end{cases} \end{equation}
Now we investigate the global existence of strong solutions to system \eqref{NSK_simplified} in $\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\rho_0, u_0)$ satisfying
\begin{equation}
\rho(x, 0) = \rho_0(x), \quad u(x, 0) = u_0(x), \quad \text{for } x \in \mathbb{R}^N,
\end{equation}
with far field behavior
\begin{equation}\label{far field}
    \rho \left( x,t \right) \rightarrow \bar{\rho}>0,\quad u\left( x,t \right) \rightarrow 0,\quad \mathrm{as}\, |x|\rightarrow \infty .
\end{equation}
We now state the main result on the global existence of strong solutions to the Cauchy problem for system \eqref{NSK_simplified}-\eqref{far field} with arbitrarily large initial data.

e assumption of radial symmetry with $\gamma\ge\frac{3}{2}$. Subsequently, Gu-Huang \cite{Gu-Huang} extended the range of the exponent to $\gamma > 1$ and generalized the result to the three-dimensional case with $1 < \gamma < 3$. Concurrently, the global existence of large solutions for the associated Cauchy problem was proved independently by Chen-Zhang-Zhu \cite{Chen-Zhang-Zhu}. Moreover, Huang-Meng-Zhang \cite{Huang-Meng-Zhang-V} successfully proved the global classical well-posedness for general isentropic compressible Navier-Stokes equations satisfying the BD entropy condition in 2D and 3D, treating the shallow water model as a specific instance.\par
 When $\kappa(\rho) >0$, we review some related works regarding the well-posedness of the Navier-Stokes-Korteweg system with general viscosity coefficients. For the one-dimensional case, considering the system with specific density-dependent viscosity $\mu(\rho) = \rho$ and capillarity $\kappa(\rho) = \rho^{-1}$, Charve-Haspot \cite{Charve-Haspot} established the existence of global strong solutions in the whole space, allowing for large non-vacuum initial data. Furthermore, they demonstrated that these solutions converge to the entropic weak solutions of the compressible Euler equations. Moreover, Germain-LeFloch \cite{Germain-LeFloch} proved the global existence of finite energy weak solutions for the Cauchy problem with general density-dependent coefficients and demonstrated their convergence to the entropy solutions of the Euler system. For the case with power-law viscosity $\mu(\varrho) = \varrho^\alpha$ and capillarity $\kappa(\varrho) = \varrho^\beta$ satisfying specific conditions, Antonelli-Bresch-Spirito \cite{Antonelli-Bresch-Spirito} established the existence of global weak solutions for the periodic problem with large data. Furthermore, investigating the Cauchy problem in Lagrangian coordinates, Chen et al. \cite{Chen-Chai} obtained global classical solutions for large initial data away from vacuum, considering density-dependent viscosity and capillarity. For the multi-dimensional case, building upon their earlier local theory \cite{Hattori-Li}, Hattori-Li \cite{Hattori-Li-2} established the global existence of solutions for the constant-coefficient Cauchy problem with small, non-vacuum initial data. For the case where both viscosity and capillarity coefficients depend on the density, Danchin-Desjardins \cite{Danchin_Desjardins} obtained global smooth solutions. Their result holds for small perturbations of a non-vacuum state in functional spaces that are critical with respect to the physical energy. Bresch and Desjardins \cite{B-D} investigated the two-dimensional viscous shallow water equations extended by a capillary term. They established the global existence of weak solutions in the presence of vacuum and demonstrated their convergence towards the strong solution of the viscous quasi-geostrophic system with a free surface. \par
However, the global existence of strong solutions with arbitrarily large initial data for the multi-dimensional Navier-Stokes-Korteweg system has long remained an open problem. It was not until recently that the first author Huang \cite{Huang-Meng-Zhang} in his newly preprint resolved this by establishing global strong solutions on the two- and three-dimensional periodic torus, provided that the initial density is bounded from above and below. However, extending their method to the whole space presents new difficulties. In this paper, we overcome these obstacles to establish the global existence of strong solutions for the multi-dimensional Navier-Stokes-Korteweg system with large initial data in the whole space. By employing the novel critical inequality established by the first author in his newly preprint \cite{Huang-Meng-Zhang}, we successfully establish the lower bound of the density by employing a novel De Giorgi iteration method. It is worth noting that our work does not require any radial symmetry assumption. To the best of our knowledge, this can be regarded as the first result concerning global large solutions for the corresponding Cauchy problem.
\par
Throughout the rest of this paper, we focus on the case of shallow-water viscosity coefficients
$$\mu(\rho) = \rho, \quad \lambda(\rho) = 0,$$
and we assume that $\kappa(\rho)$ satisfies
$$\kappa(\rho)=\frac{1}{\rho}.$$
With the specific choice of the capillarity coefficient $\kappa(\rho)$, the divergence of the Korteweg tensor takes the following form
\begin{equation}
\nabla \cdot \mathbb{K} = \nabla \cdot (\rho \nabla \nabla \log \rho).
\end{equation}
Consequently, under the assumption of shallow-water viscosity ($\mu(\rho)=\rho, \lambda(\rho)=0$) and the specific capillarity coefficient $\kappa(\rho)=1/\rho$, equation \eqref{NSK} can be rewritten as
\begin{equation}\label{NSK_simplified} \begin{cases} \partial_t \rho + \nabla \cdot(\rho u) = 0, \\ \partial_t (\rho u) + \nabla \cdot(\rho u \otimes u) + \nabla P(\rho) = \nabla \cdot \left(2 \rho \mathcal{D}(u) \right) + \nabla \cdot (\rho \nabla \nabla \log \rho). \end{cases} \end{equation}
Now we investigate the global existence of strong solutions to system \eqref{NSK_simplified} in $\mathbb{R}^N$, where $N=2, 3$. The system is supplemented with the prescribed initial data $(\rho_0, u_0)$ satisfying
\begin{equation}
\rho(x, 0) = \rho_0(x), \quad u(x, 0) = u_0(x), \quad \text{for } x \in \mathbb{R}^N,
\end{equation}
with far field behavior
\begin{equation}\label{far field}
 \rho \left( x,t \right) \rightarrow \bar{\rho}>0,\quad u\left( x,t \right) \rightarrow 0,\quad \mathrm{as}\, |x|\rightarrow \infty .
\end{equation}
We now state the main result on the global existence of strong solutions to the Cauchy problem for system \eqref{NSK_simplified}-\eqref{far field} with arbitrarily large initial data.

\begin{equation}\label{far field}
    \rho \left( x,t \right) \rightarrow \bar{\rho}>0,\quad u\left( x,t \right) \rightarrow 0,\quad \mathrm{as}\, |x|\rightarrow \infty .
\end{equation}

\begin{document}
\maketitle
\begin{abstract}
In this paper, we establish global strong solutions for arbitrarily large initial data to the multi-dimensional compressible Navier-Stokes–Korteweg system, also referred to as the quantum Navier–Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95–133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $\gamma \ge 1$, and $N=3$ with $1 \le \gamma < 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood–Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier–Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data.
\end{abstract}
\bigskip
\noindent \textbf{Keywords:} compressible Navier-Stokes-Korteweg system; quantum Navier-Stokes system; global large strong solutions; Nash-Moser iteration; De Giorgi iteration; Cauchy problem.

In the first phase, we work within the framework of Besov spaces, utilizing maximal regularity estimates for parabolic equations to secure the upper bound. The second and most crucial step is establishing the density lower bound. Although Haspot \cite{Haspot 1} investigated the isothermal NSK system, his approach failed to yield a lower bound for the density. Our strategy draws inspiration from a key inequality introduced by the first author in \cite{Huang-Meng-Zhang}. However, the technique in \cite{Huang-Meng-Zhang} cannot be directly applied to the whole space problem with non-vacuum far-field conditions. To overcome this obstacle, we have developed a novel truncated De Giorgi iteration method.

\begin{proof}
By H\"{o}lder's inequality, Sobolev embedding and \eqref{eq:19}, we have
\begin{equation}\label{the initial estimate for Psi}
\begin{aligned}
 \Psi(r(p+2)) &= \int^{T}_{0}\int_{\mathbb{R}^{3}}\rho|v|^{\frac{5}{3}(p+2)}\,dx\,dt \\
 &\leq \int^{T}_{0}\left( \int \rho^{\frac{3}{2}}|v|^{p+2}\,dx \right)^{\frac{2}{3}} \left( \int|v|^{3(p+2)}\,dx \right)^{\frac{1}{3}}\,dt \\
 &\leq \|\rho\|_{L_{T}^{\infty}L^{\infty}}^{\frac{1}{3}} \sup_{0 \le t \le T}\left( \int \rho|v|^{p+2}\,dx \right)^{\frac{2}{3}} \int^{T}_{0} \left\| |v|^{\frac{p+2}{2}} \right\|_{L^{6}}^{2}\,dt \\
 &\leq C \sup_{0 \le t \le T}\left( \int \rho|v|^{p+2}\,dx\right)^{\frac{2}{3}} \|\rho^{-1}\|_{L_{T}^{\infty}L^{\infty}} \\
 &\quad \times \left( \int^{T}_{0}\int \rho \left| \nabla|v|^{\frac{p+2}{2}} \right|^{2}\,dx\,dt + \int^{T}_{0}\int \rho|v|^{p+2}\,dx\,dt \right) \\
 &\le CV_{T} \left( \sup_{0 \le t \le T}\int \rho|v|^{p+2}\,dx + \int^{T}_{0}\int \rho \left| \nabla|v|^{\frac{p+2}{2}} \right|^{2}\,dx\,dt \right)^{r}.
\end{aligned}
\end{equation}
To estimate the right hand side, multiplying $\eqref{parabolic_system}_2$ by $|v|^{p}v$ and integrating by parts yields
\begin{equation}
 \frac{1}{p+2} \frac{d}{dt}\int \rho|v|^{p+2}dx+p\int \rho|v|^{p}\left| \nabla \left|v \right| \right| ^{2}dx+\int \rho|v|^{p}|\nabla v|^{2}dx=\int P(\rho)\nabla\cdot\left(\left| v \right| ^{p}v\right)dx.
\end{equation}
By Young's inequality and \eqref{eq:19}, we have
\begin{equation}
 \begin{aligned}
 \int P(\rho)\nabla \cdot\left(\left| v \right| ^{p}v\right)dx
=& \int \rho^{\gamma}(\left| v \right| ^{p}\nabla \cdot v+ v\cdot { \nabla \left( \left| v \right| ^{p} \right) } )dx\\
\leq& \int\rho^{\gamma}\left| v \right| ^{p} \left| \nabla v \right|dx + p\int \rho^{\gamma}|v|^{p}\left| \nabla \left| v \right| \right|dx \\
\leq & \frac{1}{2} \int \rho|v|^{p}|\nabla v|^{2}dx+ \frac{p}{2}\int \rho|v|^{p}|\nabla \left| v \right| |^{2} dx+ C(p+1)\int \rho^{2\gamma-1}|v|^{p}dx\\
 \leq& \frac{1}{2}\int \rho |v|^{p}\left| \nabla v \right| ^{2}dx+ \frac{p}{2}\int \rho|v|^{p}|\nabla \left| v \right| |^{2}dx + C(p+1)\left\lVert \rho \right\rVert _{L^{\infty}}^{2(\gamma-1)}\int \rho|v|^{p}dx\\
 \leq& \frac{1}{2}\int \rho |v|^{p}\left| \nabla v \right| ^{2}dx+ \frac{p}{2}\int \rho|v|^{p}|\nabla \left| v \right| |^{2}dx\\
 &+ C(p+1)\sup_{0 \le t \le T}\left( \int \rho|v|^{2}dx \right)^{\frac{2}{p}}\left( \int \rho|v|^{p+2} dx\right)^{1- \frac{2}{p}}.
\end{aligned}
\end{equation}
Combining \eqref{v energy}, we have
\begin{equation}
 \frac{1}{p+2}\frac{d}{dt}\int \rho |v|^{p+2}dx + \frac{1}{2}\int \rho |v|^p |\nabla v|^2dx + \frac{p}{2}\int \rho |v|^p |\nabla |v||^2dx \leq C(p+1)\left( \int \rho |v|^{p+2} dx \right)^{1-\frac{2}{p}}.
\end{equation}
Integrating over $t\in[0,T]$, we obtain
\begin{equation}
 \begin{aligned}
&\sup_{0 \le t \le T}\frac{1}{p+2}\int{\rho}|v|^{p+2}dx+ \frac{1}{2}\int_{0}^{T}\int \rho |v|^p |\nabla v|^2dxdt + \frac{p}{2}\int_{0}^{T}\int \rho |v|^p |\nabla |v||^2dxdt \\
\leq &C(p+1)\int^{T}_{0}\left( \int \rho |v|^{p+2} dx \right)^{1-\frac{2}{p}}dt+\frac{2}{p+2}\int \rho_{0}|v_{0}|^{p+2}\\
\leq& C(p+1){T^{*}}^{\frac{2}{p}}\left( \int^{T}_{0}\int \rho|v|^{p+2}dxdt \right)^{1- \frac{2}{p}}+ \frac{2}{p+2}\int \rho_{0}|v_{0}|^{p+2}.
\end{aligned}
\end{equation}
Substituting this into the RHS of \eqref{the initial estimate for Psi}, we get
\begin{equation}
 \begin{aligned}
\Psi(r(p+2))&\leq C{V}_{T} \left( C(p+2)^2 \left( \int^{T}_{0}\int \rho |v|^{p+2} dxdt \right)^{\frac{p-2}{p}} + C \int \rho_0 |v_0|^{p+2} dx \right) ^{r }\\
&\leq C{V}_{T} (p+2)^{2r } \left( \int^{T}_{0}\int \rho |v|^{p+2} dxdt \right)^{r } + C{V}_{T} (p+2)^{2r } + C{V}_{T} \left( \int \rho_0 |v_0|^{p+2} dx \right)^{r }.
\end{aligned}
\end{equation}
Denoting the constant in the RHS as $C_{3}\geq1$, and noting that
\begin{equation}
 \left\| \rho_0^{\frac{1}{p+2}} v_0 \right\|_{L^{p+2}} \leq \left\| \rho_0^{\frac{1}{2}} v_0 \right\|_{L^2}^{\frac{p}{p+2}} \|v_0\|_{L^\infty}^{\frac{2}{p+2}} \leq \left\| \rho_0^{\frac{1}{2}} v_0 \right\|_{L^2} + \|v_0\|_{L^\infty},
\end{equation}
the proof is complete.
\end{proof}
The following proposition plays a pivotal role in deriving the positive lower bound for the density, as it establishes a crucial link between the density and the effective velocity.
\begin{proposition}
There exists a constant $c_{v}\geq 1$ depending on $T^{*}$, $q$, $\gamma$, $E_{0}$, $\left\lVert \rho_{0}^{\frac{1}{q+2}}v_{0} \right\rVert_{L^{q+2}}$, and $\left\lVert \rho_{0} \right\rVert_{L^{\infty}}$ such that
\begin{equation}\label{eq:v_L_infty_est}
 \left\lVert v \right\rVert _{L_{T}^{\infty}\left(L^{\infty}\left(\mathbb{R}^3\right)\right)}\leq c_{v}(\log V_{T} )^{\frac{1}{2}}.
\end{equation}
\end{proposition}

\begin{remark}
    For the sake of brevity, we restrict our proof to the three-dimensional case where $\gamma \in (1, \frac{8}{3})$. For the critical 3D case $(\gamma=1)$ and the 2D case $(\gamma \ge 1)$, although the density upper bound estimates for these cases were addressed in Haspot \cite{Haspot 1} and Yu-Wu \cite{Yu-Wu}, neither work derived the density lower bound. Consequently, they were unable to establish the global existence of solutions. We emphasize that the approach developed in this paper can be successfully applied to establish the density lower bound for the critical 3D case $(\gamma=1)$ and the 2D case $(\gamma \ge 1)$ as well. Therefore, we are able to obtain global solutions for both of these cases. It is worth noting that the technique used by Haspot \cite{Haspot 1} relies on the assumption that the estimate for $\|\rho^{\frac{1}{q+2}} v\|_{L^\infty_T L^{q+2}}$ is independent of $q$ to derive the density lower bound. Since such $q$-independence does not hold in the present context, Haspot's method is not applicable for securing a strictly positive lower bound.
\end{remark}
\begin{remark}
    Huang-Meng-Zhang \cite{Huang-Meng-Zhang} established this result for the periodic domain. In this work, we extend their findings to the more challenging whole space setting by introducing a new De Giorgi iteration method to effectively handle the far field behavior.
\end{remark}
Provided that the density remains strictly positive, we define the effective velocity $v$ as
\begin{equation}
v = u + \nabla \log \rho.
\end{equation}
This transformation converts the original system \eqref{NSK}  into the following parabolic system
\begin{equation}\label{parabolic_system}
\begin{cases}
\partial_t \rho + \nabla \cdot(\rho v) - \Delta \rho = 0, \\
\rho \partial_t v + \rho u \cdot \nabla v + \nabla P(\rho) = \nabla \cdot(\rho \nabla v).
\end{cases}
\end{equation}
Consequently, the initial effective velocity $v_0$ is defined by
\begin{equation}\label{initial data}
v_0 = u_0 + \nabla \log \rho_0.
\end{equation}
Let us outline the main strategy of the proof. Our proof is organized into three main parts: establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.

In the first phase, we work within the framework of Besov spaces, utilizing maximal regularity estimates for parabolic equations to secure the upper bound. The second and most crucial step is establishing the density lower bound. Although Haspot \cite{Haspot 1} investigated the isothermal NSK system, his approach failed to yield a lower bound for the density. Our strategy draws inspiration from a key inequality introduced by the first author in \cite{Huang-Meng-Zhang}. However, the technique in \cite{Huang-Meng-Zhang} cannot be directly applied to the whole space problem with non-vacuum far-field conditions. To overcome this obstacle, we have developed a novel truncated De Giorgi iteration method.

We now proceed to outline the derivation of these density bounds.\\
\textbf{Upper bound of $\rho$.} \par
To derive the upper bound of the density, we fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation. Specifically, closing the density estimate requires bounding the norm $\|\rho^{1/(q+2)}v\|_{L^{q+2}}$ for some $q > 1$. In Proposition \ref{PRO3.4}, by means of a precise domain decomposition analysis, we established that for any $\gamma \in (1, \frac{8}{3})$, there exists a $q$ such that the norm $\|\rho^{\frac{1}{q+2}} v\|_{L^{q+2}}$ remains bounded.\\
\textbf{Lower bound of $\rho$.} \par
 To derive the lower bound of the density following Haspot\cite{Haspot 1}'s argument, it is essential to control the norm $\|\rho^{\frac{1}{p+2}} v\|_{L^\infty_T L^{p+2}}$ by a constant independent of $p$. However, the bound for $\|\rho^{\frac{1}{p+2}} v\|_{L^\infty_T L^{p+2}}$ tends to infinity as $p \to \infty$. To overcome this obstacle, Huang-Meng-Zhang \cite{Huang-Meng-Zhang} employed the Moser iteration method to establish a control relationship between $\|v\|_{L^\infty}$ and $\sqrt{\log (e^{\frac{25}{9}}+\|{\rho}^{-1}\|_{L^{\infty}})}$. This critical inequality plays a pivotal role in our analysis. Although their approach can handle the density lower bound in a periodic domain, it fails in the whole space when the density exhibits non-zero far-field behavior at infinity. To address the difficulties arising in this context, we introduce the De Giorgi iteration technique to prove the lower bound of the density by employing a truncation level adapted to the far-field density. Specifically, we construct the following truncation and iteration sequence
 \begin{equation}
\begin{split}
    \rho^{-1}_{(k_n)}&:=\max\{\rho^{-1}-{k_n},0\},
    \\
    U_{n}^{T}&:=\| \rho _{(k_n)}^{-1}\| _{L_{T}^{\infty}L^2}^2+\| \nabla \rho _{(k_n)}^{-1}\| _{L_{T}^{2}L^2}^2.
\end{split}
 \end{equation}
 where $k_n:=M\left( 1-2^{-n} \right)+2\left\| {\rho}^{-1} _0 \right\| _{L^{\infty}}.$ It is worth noting that the initial iteration value $k_0 = 2\|1/\rho_0\|_{L^\infty}$ is chosen specifically to handle the far-field behavior while ensuring that the initial energy $U_0^T$ satisfies the convergence condition
     \begin{equation}
    U_0^T \le K^{-\frac{1}{\nu}} A^{-\frac{1}{\nu^2}}.
\end{equation}
 We choose a suitable $T$ to bound the solution on $[0, T]$ and then employ a shifted iteration sequence for $[T, 2T]$. As long as the time step for each extension is uniform, this procedure proves the boundedness within the maximal lifespan $T^*$, which in turn establishes the time-dependent lower bound of the density.\par
 In Section 2, we introduce some preliminaries, with a particular focus on Littlewood-Paley theory. In Section 3, we first demonstrate that for any $\gamma \in (1, \frac{8}{3})$, there exists a $q>1$ satisfying the condition $\gamma \le \frac{2q+6}{q+2}$ such that the quantity $\sup_{0 \le t \le T} \|\rho^{\frac{1}{q+2}} v\|_{L^{q+2}}$ remains bounded. This allows us to establish the density upper bound by applying maximal regularity estimates for the heat equation. In Section 4, armed with the density upper bound, we extend the range of $q$ for which the boundedness of $\sup_{0 \le t \le T} \|\rho^{\frac{1}{q+2}} v\|_{L^{q+2}}$ holds. Consequently, we employ the Moser iteration method to control $\|v\|_{L^{\infty}}$ by a term involving $\sqrt{\log V_T}$, and subsequently utilize De Giorgi iteration to prove the existence of a density lower bound up to the maximal existence time $T^*$. Finally, in Section 5 and Section 6, we prove the main result by using the established density upper and lower bounds as blow-up criteria.

To prove Theorem~\ref{thm:global_existence_cauchy}, the authors assume the density stays strictly positive and introduce the \emph{effective velocity} $v=u+\nabla\log\rho$, which transforms \eqref{NSK} into the parabolic system \eqref{parabolic_system}. They then “outline the main strategy” in three parts: “establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.”

(1) \textbf{Upper bound of $\rho$.} They “fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation.” Closing the estimate requires bounding $\|\rho^{1/(q+2)}v\|_{L^{q+2}}$ for some $q>1$; in Proposition~\ref{PRO3.4}, “by means of a precise domain decomposition analysis,” they show that for any $\gamma\in(1,\frac{8}{3})$ there exists such a $q$ with $\|\rho^{\frac{1}{q+2}}v\|_{L^{q+2}}$ bounded.

(2) \textbf{Lower bound of $\rho$.} They explain that Haspot’s route would need $\|\rho^{\frac{1}{p+2}}v\|_{L^\infty_T L^{p+2}}$ bounded uniformly in $p$, but here it “tends to infinity as $p\to\infty$.” Instead they use the Huang–Meng–Zhang idea relating “$\|v\|_{L^\infty}$ and $\sqrt{\log( e^{\frac{25}{9}}+\|\rho^{-1}\|_{L^\infty})}$,” but since the periodic-domain method “fails in the whole space” with nonzero far-field density, they “developed a novel truncated De Giorgi iteration method.” They define truncations and energies
\[
\rho^{-1}_{(k_n)}:=\max\{\rho^{-1}-k_n,0\},\qquad
U_n^T:=\|\rho^{-1}_{(k_n)}\|_{L_T^\infty L^2}^2+\|\nabla\rho^{-1}_{(k_n)}\|_{L_T^2 L^2}^2,
\]
with truncation levels $k_n:=M(1-2^{-n})+2\|\rho_0^{-1}\|_{L^\infty}$, choosing $k_0=2\|1/\rho_0\|_{L^\infty}$ so that $U_0^T$ meets a convergence condition $U_0^T\le K^{-1/\nu}A^{-1/\nu^2}$. They “choose a suitable $T$ to bound the solution on $[0,T]$ and then employ a shifted iteration sequence for $[T,2T]$,” and with a uniform time step this yields boundedness up to the maximal lifespan $T^*$, giving a “time-dependent lower bound of the density.”

(3) \textbf{Completion / global existence.} After establishing density upper and lower bounds, they “prove the main result by using the established density upper and lower bounds as blow-up criteria” (Sections 5–6), together with the earlier steps (Sections 3–4: maximal regularity for the upper bound; extending the $q$-range, then Moser to control $\|v\|_{L^\infty}$ by $\sqrt{\log V_T}$, and De Giorgi for the lower bound up to $T^*$).

To prove the main theorem, the authors assume the density stays strictly positive and introduce the \emph{effective velocity} $v=u+\nabla\log\rho$, which transforms
\begin{equation}\label{NSK} \begin{cases} \partial_t \rho + \nabla \cdot(\rho u) = 0,
\\ \partial_t (\rho u) + \nabla \cdot(\rho u \otimes u) + \nabla P(\rho) =\nabla \cdot \mathbb{S} + \nabla \cdot \mathbb{K}. \end{cases} \end{equation}
into the parabolic system \eqref{parabolic_system}. They then “outline the main strategy” in three parts: “establishing the density upper bound, deriving the density lower bound, and finally, improving the regularity of the solution.”

(1) \textbf{Upper bound of $\rho$.} They “fully exploit the parabolic structure of the system and employ maximal regularity estimates for the heat equation.” Closing the estimate requires bounding $\|\rho^{1/(q+2)}v\|_{L^{q+2}}$ for some $q>1$. We first use the following proposition.
\begin{proposition}\label{PRO3.4}
For any $\gamma \in (1, \frac{8}{3})$, there exists $q \in (1, 4)$ such that
\begin{equation}
     \gamma \le \frac{2q+6}{q+2}.
\end{equation}
    For this fixed $q$, there exists a positive constant $C$, depending on $T$, $q$, $\gamma$, $E_0$, and the initial data $\|\rho_0^{1/(q+2)} v_0\|_{L^{q+2}}$, such that
\begin{equation}\label{L^{q+2}}
\sup_{0 \le t \le T} \|\rho^{\frac{1}{q+2}} v\|_{L^{q+2}} \le C.
\end{equation}
\end{proposition}
In particular, for any $\gamma\in(1,\frac{8}{3})$ one can choose such a $q$ so that $\|\rho^{\frac{1}{q+2}}v\|_{L^{q+2}}$ is bounded.

(2) \textbf{Lower bound of $\rho$.} They explain that Haspot’s route would need $\|\rho^{\frac{1}{p+2}}v\|_{L^\infty_T L^{p+2}}$ bounded uniformly in $p$, but here it “tends to infinity as $p\to\infty$.” Instead they use the Huang–Meng–Zhang idea relating “$\|v\|_{L^\infty}$ and $\sqrt{\log( e^{\frac{25}{9}}+\|\rho^{-1}\|_{L^\infty})}$,” but since the periodic-domain method “fails in the whole space” with nonzero far-field density, they “developed a novel truncated De Giorgi iteration method.” They define truncations and energies
\[
\rho^{-1}_{(k_n)}:=\max\{\rho^{-1}-k_n,0\},\qquad
U_n^T:=\|\rho^{-1}_{(k_n)}\|_{L_T^\infty L^2}^2+\|\nabla\rho^{-1}_{(k_n)}\|_{L_T^2 L^2}^2,
\]
with truncation levels $k_n:=M(1-2^{-n})+2\|\rho_0^{-1}\|_{L^\infty}$, choosing $k_0=2\|1/\rho_0\|_{L^\infty}$ so that $U_0^T$ meets a convergence condition $U_0^T\le K^{-1/\nu}A^{-1/\nu^2}$. They “choose a suitable $T$ to bound the solution on $[0,T]$ and then employ a shifted iteration sequence for $[T,2T]$,” and with a uniform time step this yields boundedness up to the maximal lifespan $T^*$, giving a “time-dependent lower bound of the density.”

(3) \textbf{Completion / global existence.} After establishing density upper and lower bounds, they complete the proof of the main theorem by using the established density upper and lower bounds as blow-up criteria (Sections 5–6), together with the earlier steps (Sections 3–4: maximal regularity for the upper bound; extending the $q$-range, then Moser to control $\|v\|_{L^\infty}$ by $\sqrt{\log V_T}$, and De Giorgi for the lower bound up to $T^*$).

\label{thm:global_existence_cauchy}
Let $N=2$ or $N=3$. Assume that $\gamma$ satisfies
\begin{equation}
    \begin{cases}
        \gamma \in [1, \infty) & \text{if } N=2, \\
        \gamma \in [1, 8/3) & \text{if } N=3.
    \end{cases}
\end{equation}
Let $\bar{\rho} > 0$ be the far field behavior of density. Assume that the initial data $(\rho_0, u_0)$ satisfies
\begin{equation}
    0 < \underline{\varrho} \le \rho_0 \le \bar{\varrho}, \quad (\rho_0 - \bar{\rho}) \in H^3(\mathbb{R}^N), \quad u_0 \in H^2(\mathbb{R}^N),
\end{equation}
where $\underline{\varrho}$ and $\bar{\varrho}$ are positive constants. Then the Cauchy problem
\begin{equation}\label{NSK_simplified} \begin{cases} \partial_t \rho + \nabla \cdot(\rho u) = 0, \\ \partial_t (\rho u) + \nabla \cdot(\rho u \otimes u) + \nabla P(\rho) = \nabla \cdot \left(2 \rho \mathcal{D}(u) \right) + \nabla \cdot (\rho \nabla \nabla \log \rho). \end{cases} \end{equation}
\begin{equation}\label{far field}
    \rho \left( x,t \right) \rightarrow \bar{\rho}>0,\quad u\left( x,t \right) \rightarrow 0,\quad \mathrm{as}\, |x|\rightarrow \infty .
\end{equation}
admits a unique global strong solution $(\rho, u)$ satisfying for any $0 < T < \infty$ and $(x,t) \in \mathbb{R}^N \times [0, T]$,
\begin{equation}
    (C(T))^{-1} \le \rho(x,t) \le C(T),
\end{equation}
and
\begin{equation}
\begin{aligned}
    &(\rho - \bar{\rho}) \in C([0, T]; H^3) \cap L^2(0, T; H^4), \quad \rho_t \in C([0, T]; H^1) \cap L^2(0, T; H^2), \\
    &u \in C([0, T]; H^2) \cap L^2(0, T; H^3), \quad u_t \in L^\infty(0, T; L^2) \cap L^2(0, T; H^1),
\end{aligned}
\end{equation}
where the constant $C(T) > 0$ depends on the initial data and $T$.

Question:
Let $N=2$ or $N=3$, and assume
\[
\gamma\in[1,\infty)\quad\text{if }N=2,
\qquad
\gamma\in[1,8/3)\quad\text{if }N=3.
\]
Let $\bar\rho>0$ be the far-field density, and let the initial data $(\rho_0,u_0)$ satisfy
\[
0<\underline\varrho\le \rho_0\le \bar\varrho,
\qquad
\rho_0-\bar\rho\in H^3(\mathbb R^N),
\qquad
u_0\in H^2(\mathbb R^N),
\]
for some positive constants $\underline\varrho,\bar\varrho$. Consider the Cauchy problem for the shallow-water Navier-Stokes-Korteweg system on $\mathbb R^N$:
\[
\begin{cases}
\partial_t\rho+\nabla\cdot(\rho u)=0,\\
\partial_t(\rho u)+\nabla\cdot(\rho u\otimes u)+\nabla P(\rho)
=\nabla\cdot\bigl(2\rho\mathcal D(u)\bigr)+\nabla\cdot\bigl(\rho\nabla\nabla\log\rho\bigr),
\end{cases}
\]
with far-field condition
\[
\rho(x,t)\to\bar\rho>0,
\qquad
u(x,t)\to0,
\qquad \text{as }|x|\to\infty,
\]
and initial data $(\rho,u)|_{t=0}=(\rho_0,u_0)$. Which statement holds about existence and uniqueness of solutions?

There exists a unique global strong solution $(\rho,u)$ to this Cauchy problem such that, for every $0<T<\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with
\[
(C(T))^{-1}\le \rho(x,t)\le C(T)
\quad\text{for all }(x,t)\in\mathbb R^N\times[0,T],
\]
and
\[
\rho-\bar\rho\in C([0,T];H^3(\mathbb R^N))\cap L^2(0,T;H^4(\mathbb R^N)),
\qquad
\rho_t\in C([0,T];H^1(\mathbb R^N))\cap L^2(0,T;H^2(\mathbb R^N)),
\]
\[
u\in C([0,T];H^2(\mathbb R^N))\cap L^2(0,T;H^3(\mathbb R^N)),
\qquad
u_t\in L^\infty(0,T;L^2(\mathbb R^N))\cap L^2(0,T;H^1(\mathbb R^N)).
\]

Choices:
B.
There exists a unique global strong solution $(\rho,u)$ to this Cauchy problem such that there is a constant $C>0$, depending only on the initial data, with
\[
C^{-1}\le \rho(x,t)\le C
\quad\text{for all }(x,t)\in\mathbb R^N\times[0,\infty),
\]
and, for every $0<T<\infty$,
\[
\rho-\bar\rho\in C([0,T];H^3(\mathbb R^N))\cap L^2(0,T;H^4(\mathbb R^N)),
\qquad
\rho_t\in C([0,T];H^1(\mathbb R^N))\cap L^2(0,T;H^2(\mathbb R^N)),
\]
\[
u\in C([0,T];H^2(\mathbb R^N))\cap L^2(0,T;H^3(\mathbb R^N)),
\qquad
\nu_t\in L^\infty(0,T;L^2(\mathbb R^N))\cap L^2(0,T;H^1(\mathbb R^N)).
\]
C.
There exists a global strong solution $(\rho,u)$ to this Cauchy problem such that, for every $0<T<\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with
\[
(C(T))^{-1}\le \rho(x,t)\le C(T)
\quad\text{for all }(x,t)\in\mathbb R^N\times[0,T],
\]
and
\[
\rho-\bar\rho\in C([0,T];H^3(\mathbb R^N))\cap L^2(0,T;H^4(\mathbb R^N)),
\qquad
\rho_t\in C([0,T];H^1(\mathbb R^N))\cap L^2(0,T;H^2(\mathbb R^N)),
\]
\[
u\in C([0,T];H^2(\mathbb R^N))\cap L^2(0,T;H^3(\mathbb R^N)),
\qquad
\nu_t\in L^\infty(0,T;L^2(\mathbb R^N))\cap L^2(0,T;H^1(\mathbb R^N)).
\]
D.
There exists a unique global strong solution $(\rho,u)$ to this Cauchy problem such that, for every $0<T<\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with
\[
(C(T))^{-1}\le \rho(x,t)\le C(T)
\quad\text{for all }(x,t)\in\mathbb R^N\times[0,T],
\]
and
\[
\rho-\bar\rho\in C([0,T];H^3(\mathbb R^N))\cap L^2(0,T;H^4(\mathbb R^N)),
\qquad
\rho_t\in C([0,T];H^1(\mathbb R^N))\cap L^2(0,T;H^2(\mathbb R^N)),
\]
\[
u\in C([0,T];H^2(\mathbb R^N))\cap L^2(0,T;H^3(\mathbb R^N)),
\qquad
\nu_t\in C([0,T];L^2(\mathbb R^N))\cap L^2(0,T;H^1(\mathbb R^N)).
\]
E.
There exists a unique global strong solution $(\rho,u)$ to this Cauchy problem for all
\[

gamma\in[1,\infty)\quad\text{if }N=2\text{ or }N=3,
\]
such that, for every $0<T<\infty$, there is a constant $C(T)>0$ depending on $T$ and the initial data with
\[
(C(T))^{-1}\le \rho(x,t)\le C(T)
\quad\text{for all }(x,t)\in\mathbb R^N\times[0,T],
\]
and
\[
\rho-\bar\rho\in C([0,T];H^3(\mathbb R^N))\cap L^2(0,T;H^4(\mathbb R^N)),
\qquad
\rho_t\in C([0,T];H^1(\mathbb R^N))\cap L^2(0,T;H^2(\mathbb R^N)),
\]
\[
u\in C([0,T];H^2(\mathbb R^N))\cap L^2(0,T;H^3(\mathbb R^N)),
\qquad
\nu_t\in L^\infty(0,T;L^2(\mathbb R^N))\cap L^2(0,T;H^1(\mathbb R^N)).
\]


Question:
Let $N=2$ or $N=3$, and consider the Cauchy problem on $\mathbb{R}^N$ for the compressible Navier--Stokes--Korteweg system with shallow-water viscosity and capillarity
$$
\begin{cases}
\partial_t \rho + \nabla\!\cdot(\rho u)=0,\\
\partial_t(\rho u)+\nabla\!\cdot(\rho u\otimes u)+\nabla P(\rho)=\nabla\!\cdot\bigl(2\rho \mathcal D(u)\bigr)+\nabla\!\cdot\bigl(\rho\nabla\nabla \log \rho\bigr),
\end{cases}
$$
where $P(\rho)=\rho^\gamma$ and $\mathcal D(u)=\tfrac12(\nabla u+\nabla u^{\top})$. Assume
$$
\gamma\in[1,\infty) \text{ if } N=2, \qquad \gamma\in[1,8/3) \text{ if } N=3,
$$
and let the far-field density satisfy $\bar\rho>0$. Suppose the initial data $(\rho_0,u_0)$ satisfy
$$
0<\underline\varrho\le \rho_0\le \bar\varrho,
\qquad (\rho_0-\bar\rho)\in H^3(\mathbb R^N),
\qquad u_0\in H^2(\mathbb R^N),
$$
for some positive constants $\underline\varrho,\bar\varrho$, together with the far-field condition
$$
\rho(x,t)\to \bar\rho>0,\qquad u(x,t)\to 0 \qquad \text{as } |x|\to\infty.
$$
Which existence statement holds for this problem?

There exists a unique global strong solution $(\rho,u)$ such that for every $0<T<\infty$ and every $(x,t)\in\mathbb R^N\times[0,T]$,
$$
(C(T))^{-1}\le \rho(x,t)\le C(T),
$$
where $C(T)>0$ depends on $T$ and the initial data, and moreover
$$
(\rho-\bar\rho)\in C([0,T];H^3)\cap L^2(0,T;H^4),
\qquad \rho_t\in C([0,T];H^1)\cap L^2(0,T;H^2),
$$
$$
u\in C([0,T];H^2)\cap L^2(0,T;H^3),
\qquad u_t\in L^\infty(0,T;L^2)\cap L^2(0,T;H^1).
$$

Choices:
B.
There exists a unique global strong solution $(\rho,u)$ such that for every $0<T<\infty$ and every $(x,t)\in\mathbb R^N\times[0,T]$,
$$
(C(T))^{-1}\le \rho(x,t)\le C(T),
$$
where $C(T)>0$ depends on $T$ and the initial data, and moreover
$$
(\rho-\bar\rho)\in C([0,T];H^3)\cap L^2(0,T;H^4),
\qquad \rho_t\in C([0,T];H^1)\cap L^2(0,T;H^2),
$$
$$
u\in C([0,T];H^2)\cap L^2(0,T;H^3),
\qquad u_t\in L^\infty(0,T;L^2)\cap L^2(0,T;H^1),
$$
and the same conclusion remains valid in dimension $N=3$ for all $\gamma\in[1,3)$.
C.
There exists a unique global strong solution $(\rho,u)$ on $\mathbb R^N\times[0,\infty)$ such that for every $0<T<\infty$,
$$
(C(T))^{-1}\le \rho(x,t)\le C(T)\qquad \text{for all }(x,t)\in\mathbb R^N\times[0,T],
$$
where $C(T)>0$ depends on $T$ and the initial data.
D.
There exists a unique global strong solution $(\rho,u)$ such that for every $0<T<\infty$ and every $(x,t)\in\mathbb R^N\times[0,T]$,
$$
C^{-1}\le \rho(x,t)\le C,
$$
for some constant $C>0$ depending only on the initial data but independent of $T$, and moreover
$$
(\rho-\bar\rho)\in C([0,T];H^3)\cap L^2(0,T;H^4),
\qquad \rho_t\in C([0,T];H^1)\cap L^2(0,T;H^2),
$$
$$
u\in C([0,T];H^2)\cap L^2(0,T;H^3),
\qquad u_t\in L^\infty(0,T;L^2)\cap L^2(0,T;H^1).
$$
E.
For every $T>0$ there exists a unique strong solution $(\rho,u)$ on $\mathbb R^N\times[0,T]$ such that
$$
(C(T))^{-1}\le \rho(x,t)\le C(T)
$$
for all $(x,t)\in\mathbb R^N\times[0,T]$, where $C(T)>0$ depends on $T$ and the initial data, and moreover
$$
(\rho-\bar\rho)\in C([0,T];H^3)\cap L^2(0,T;H^4),
\qquad \rho_t\in C([0,T];H^1)\cap L^2(0,T;H^2),
$$
$$
u\in C([0,T];H^2)\cap L^2(0,T;H^3),
\qquad u_t\in L^\infty(0,T;L^2)\cap L^2(0,T;H^1),
$$
but the theorem does not assert a single global solution on $[0,\infty)$.


\label{thm:intro}
The group of homeomorphisms of the real projective plane $\mathbb{R}\mathrm{P}^2$ and Möbius strip $M$ which are isotopic to the identity is stably unbounded with respect to commutator length in the sense of \cite{BIP}.

\begin{theorem}[Guessing geodesics]\label{thm:bowditch-guessing-geodesics}
    Let $X$ be a graph and $D>0$ a number.
    Suppose that for each pair of distinct vertices $x,y$ of $X$ we have chosen a connected subgraph $P(x,y)$ containing $x$ and $y$.
    Suppose that if $d_X(x,y) = 1$, the diameter of $P(x,y)$ is at most $D$, and in addition we have for all pairwise distinct $x,y,z$:
    \[ P(x,z) \subseteq N_D(P(x,y)\cup P(y,z)). \]
    Then $X$ is $\delta(D)$-hyperbolic, and each $P(x,y)$ is $B(D)$-Hausdorff close to a geodesic joining $x$ to $y$.
\end{theorem}

\subsection{Actions on curve graphs}\label{subsec:actions-on-cg}
One can easily check that the groups $\operatorname{Homeo}_0$ and $\operatorname{Mcg}$ of a surface act on the fine curve graph respectively curve graph of that surface by isometries.

In \cite{Bestvina}, Bestvina and Fujiwara introduced the notion of weak proper discontinuity, which will be the key to finding elements with positive stable commutator length.

First, recall that for two hyperbolic isometries $f$ and $g$ of a $\delta$-hyperbolic space $(X,d)$, we can find invariant quasi-axes $\alpha$ and $\beta$ of quality $L$ respectively $L'$ by taking the orbit of some point, e.g. $\alpha = (a_k) \coloneqq f^k(x_0)$ for $k\in\ZZ$ and $x_0\in X$, and analogously for $\beta = (b_k)$.
We then say that $f$ and $g$ (respectively their axes) are \emph{quasi-equivalent}, if there exists a constant $B = B(L,L',\delta)$ such that:
for any $D>0$, there is an isometry $h$ that sends a segment of length $D$ in $\alpha$ into a $B$-neighbourhood of $\beta$.
If $f$ is quasi-equivalent to its inverse, then we say that $f$ is \emph{quasi-invertible}.

These quasi-geodesics define points in the Gromov boundary of $X$, and we can give an alternative definition of quasi-equivalence in terms of boundary points:
let $\tau_\pm \coloneqq [a_{\pm k}]$ and $\upsilon_\pm \coloneqq [b_{\pm k}]$ as equivalence classes of admissible sequences as in the previous section.
Then $f$ and $g$ (respectively their axes) are quasi-equivalent, if there exists a sequence $h_k$ of isometries of $X$ such that $\lim_{k\rightarrow \infty} h_k(\tau_\pm) = \upsilon_\pm$.

Further, we call two hyperbolic elements \emph{dependent}, if they have quasi-axes which contain rays that are in finite distance of each other, and \emph{independent} otherwise.
If $G$ contains two independent hyperbolic elements, we call the action \emph{non-elementary}.

The central definition in this section is:

\begin{definition}
    Let $G$ be a group acting on the hyperbolic metric space $X$.
    This action is \emph{weakly properly discontinuous (WPD)}, if the following holds: \begin{itemize}
        \item $G$ is not virtually cyclic
        \item $\exists g\in G\colon$ $g$ acts on $X$ as a hyperbolic isometry
        \item $ \forall x\in X\colon \forall g \in G \textrm{ acting hyperbolically}\colon \forall C>0\colon \exists N>0\colon \{\gamma \in G \mid d(x,\gamma(x)) \leq C, d(g^N(x),\gamma(g^N(x)))\leq C \} \textrm{ is finite.} $
    \end{itemize}
    \end{definition}
Note that when we restrict an action $G\curvearrowright X$ to a subgroup $H<G$, we only need to recheck the first two conditions to see that the induced action of $H$ on $X$ satisfies WPD.
The following proposition shows why WPD is a useful property:

\begin{proposition}[Proposition 6 of \cite{Bestvina}]\label{prop:useful-wpd}
    Suppose that $G$ and $X$ satisfy WPD.
    Then \begin{enumerate}
        \item for every hyperbolic $g\in G$, the centraliser $C(g)$ is virtually cyclic,
        \item for every hyperbolic $g\in G$ and every $(K,L)$-quasi-axis $l$ for $g$, there is a constant $M$ depending on $g$, $K$ and $L$ such that if two translates $l_1$ and $l_2$ of $l$ contain (oriented) segments of length $>M$ that are oriented $B(K,L,\delta)$-close, then $l_1$ and $l_2$ are oriented $B(K,L,\delta)$-close and moreover the corresponding conjugates $g_1$ and $g_2$ of $g$ have positive powers which are equal,
        \item the action of $G$ on $X$ is non-elementary,
        \item two elements of $G$ are quasi-equivalent if and only if they have positive powers which are conjugate,
        \item there exist two hyperbolic elements in $G$ which are not quasi-equivalent.
    \end{enumerate}
\end{proposition}

The main result of \cite{Bestvina} is:
\begin{theorem}[Theorem 1 of \cite{Bestvina}]\label{thm:bestvina-fujiwara-main-thm}
Suppose a group $G$ acts on a $\delta$-hyperbolic graph $X$ by isometries.
Suppose also that the action is non-elementary and that there exist independent hyperbolic elements $g_1,g_2 \in G$ which are not quasi-equivalent.
Then, the space of non-trivial quasi-morphisms is infinite dimensional.
\end{theorem}

In this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.
This type of question first appeared in the work of Burago, Ivanov and Polterovich \cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.
Their results have been extended to higher dimensional manifolds in \cite{tsuboi}.
For surfaces, the situation is different:
while the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.
The main result of this article answers this question for the last two remaining cases:

\begin{abstract}
Using a recent result of Bowden, Hensel and Webb, we prove the existence of homeomorphisms with positive stable commutator length in the groups of homeomorphisms of the real projective plane and Möbius strip which are isotopic to the identity.
This completes the answer to a question posed by Burago, Ivanov and Polterovich on the boundedness of diffeomorphism groups of surfaces.
\end{abstract}

\section{Introduction}
In this article, we study the (un-)boundedness of norms on the group of homeomorphisms and diffeomorphisms of the Möbius strip and real projective plane that are isotopic to the identity.
This type of question first appeared in the work of Burago, Ivanov and Polterovich \cite{BIP}, where the authors considered the identity components of groups of compactly supported $C^\infty$-diffeomorphisms of various manifolds, and found that for all compact 3-manifolds and certain compact 4-manifolds these groups only admit conjugation-invariant norms that are bounded.
Their results have been extended to higher dimensional manifolds in \cite{tsuboi}.
For surfaces, the situation is different:
while the sphere again only admits conjugation-invariant norms that are bounded, almost all other surfaces have been found to have unbounded norms.
The main result of this article answers this question for the last two remaining cases:

Therefore, all compact surfaces except the sphere admit unbounded norms.
To prove unboundedness, the central tool is Bavard duality \cite{bavard}, which links a norm, namely commutator length, to the existence of non-trivial homogeneous quasi-morphisms.
An explicit construction of quasi-morphism was given by Brooks in \cite{Brooks} for free groups.
This construction was generalised in \cite{Bestvina} by Bestvina and Fujiwara to groups of isometries of a hyperbolic space, which they applied to (subgroups of) mapping class groups of hyperbolic surfaces.

\paragraph{Organisation of the paper.}
In the next section, we provide some background on the papers of Kuno \cite{kuno-hyperbolicity} and Bowden, Hensel and Webb \cite{BHW1} \cite{BHW2}.
In the third section, we combine results of \cite{kuno-hyperbolicity} and \cite{BHW1} to show that the Möbius strip and real projective plane have hyperbolic and unbounded fine curve graphs.
In the fourth section, we use that result and the methods of \cite{BHW2} to prove that both Möbius strip and real projective plane have stably unbounded $\operatorname{Homeo}_0$.

\paragraph{Bavard duality.}
It is known that the groups $\operatorname{Homeo}_0(M)$ of compact manifolds are perfect \cite{anderson}, i.e. any element of the group can be written as a product of commutators, and the same is true for $\operatorname{Diff}_0$, see e.g. \cite{mann}.
The minimal number of commutators that appear in such a factorisation of a homeomorphism is called \emph{commutator length}, usually denoted by cl, and it is an important example of a conjugation-invariant norm on the group $\operatorname{Homeo}(M)$ as defined in \cite{BIP}.
Such norms have a so-called \emph{stabilisation}, which takes the following form for commutator length:
\[ \operatorname{scl}(g) \coloneqq \lim_{n\rightarrow \infty} \frac{\operatorname{cl}(g^n)}{n} \]
for $g\in \operatorname{Homeo}(M)$.
We call a group \emph{stably unbounded} if the stabilisation of some norm takes positive values.

\subsection{Curve graphs and variations}
Curve graphs and curve complexes were defined by Harvey in \cite{Harvey}.
These are usually defined in terms of essential curves, but for our purposes it will be useful to only work with non-separating curves.
The main definition here is:
the \emph{non-separating curve graph} $\ncg(M)$ of a 2-manifold (possibly with boundary) $M$ is the graph with: \begin{itemize}
 \item vertices corresponding to the homotopy classes of non-separating closed curves in $M$
 \item an edge between two vertices, if the corresponding homotopy classes have disjoint representatives
\end{itemize}
To study homeomorphisms that are isotopic to the identity, we have to get rid of the homotopy classes:
the \emph{fine non-separating curve graph} accomplishes this.
It was introduced in \cite{BHW1} and is defined by:\begin{itemize}
 \item vertices which each represent a non-separating closed curve in $M$
 \item an edge between two vertices, if the corresponding curves are disjoint
\end{itemize}

The main result of \cite{Bestvina} is:
\begin{theorem}[Theorem 1 of \cite{Bestvina}]\label{thm:bestvina-fujiwara-main-thm}
Suppose a group $G$ acts on a $\delta$-hyperbolic graph $X$ by isometries.
Suppose also that the action is non-elementary and that there exist independent hyperbolic elements $g_1,g_2 \in G$ which are not quasi-equivalent.
Then, the space of non-trivial quasi-morphisms is infinite dimensional.
\end{theorem}

\begin{theorem}[cf. Theorem 4.1 in \cite{BHW2}]\label{thm:convergence-criterion}
Suppose $(\mathcal{F}_v,\mathcal{F}_h,d)$ is a BSF structure with ending $\mathcal{F}_v$.
Then there is a point $\tau_{\mathcal{F}_v}$ on the Gromov boundary of $\fncg(N)$ with the following properties: \begin{enumerate}
 \item The boundary point $\tau_{\mathcal{F}_v}$ depends only on the foliation $\mathcal{F}_v$.
 \item A homeomorphism of $N$ fixes $\tau_{\mathcal{F}_v}$ as a point on the Gromov boundary if and only if it preserves the foliation $\mathcal{F}_v$.
 \item A sequence of curves $(\beta_i)$ in $\fncg(N)$ converges to $\tau_{\mathcal{F}_v}$ if and only if
 the following conditions are satisfied: \begin{itemize}
 \item The sizes of the $\beta_i$ diverge to infinity.
 \item For any $B$, $\epsilon$ there is a number $I$ so that if $b\subseteq \beta_i$ with $i>I$ is a segment of size at most $B$, then it $\epsilon$-fellow travels a leaf segment of $\mathcal{F}_v$ with respect to the metric $d$.
 \end{itemize}

Therefore, all compact surfaces except the sphere admit unbounded norms.
To prove unboundedness, the central tool is Bavard duality \cite{bavard}, which links a norm, namely commutator length, to the existence of non-trivial homogeneous quasi-morphisms.
An explicit construction of quasi-morphism was given by Brooks in \cite{Brooks} for free groups.
This construction was generalised in \cite{Bestvina} by Bestvina and Fujiwara to groups of isometries of a hyperbolic space, which they applied to (subgroups of) mapping class groups of hyperbolic surfaces.

In their paper, they consider the action of the mapping class group on the curve graph (introduced by Harvey \cite{Harvey}, shown to be hyperbolic by Masur and Minsky \cite{masur-minsky}), which satisfies a newly identified property, namely weak proper discontinuity (see section \ref{subsec:actions-on-cg}).

An analogous strategy was developed by Bowden, Hensel and Webb in \cite{BHW1}, where the action of the identity component of the homeomorphism group of an orientable surface with positive genus on the newly introduced \emph{fine} curve graph was studied.

This strategy was adapted to non-orientable surfaces of genus at least three in \cite{kimura-kuno-qms}.
In \cite{BHW2}, the action of these groups on the fine curve graph is studied further, and a tool is provided that characterises some of the elements whose powers have increasingly large commutator length.
In addition, \cite{BHW2} provides a connection between certain foliations on surfaces and boundary points of the corresponding fine curve graph.
In this article, we aim to explain that the techniques and results of Bowden, Hensel and Webb generalise to non-orientable surfaces with genus 1.

\paragraph{Organisation of the paper.}
In the next section, we provide some background on the papers of Kuno \cite{kuno-hyperbolicity} and Bowden, Hensel and Webb \cite{BHW1} \cite{BHW2}.
In the third section, we combine results of \cite{kuno-hyperbolicity} and \cite{BHW1} to show that the Möbius strip and real projective plane have hyperbolic and unbounded fine curve graphs.
In the fourth section, we use that result and the methods of \cite{BHW2} to prove that both Möbius strip and real projective plane have stably unbounded $\operatorname{Homeo}_0$.

\paragraph{Acknowledgements.}
The author wants to thank Sebastian Hensel and Javier de la Nuez Gonzalez.

To prove Theorem~\ref{thm:intro}, the introduction says the "central tool is Bavard duality" (linking commutator length to the existence of "non-trivial homogeneous quasi-morphisms"). The strategy follows the quasi-morphism constructions of Brooks and their generalisation by Bestvina--Fujiwara via actions on hyperbolic spaces, and the analogous approach of Bowden--Hensel--Webb using the action of $\operatorname{Homeo}_0$ on a (fine) curve graph. Concretely, the paper is organised to (1) combine results of \cite{kuno-hyperbolicity} and \cite{BHW1} to show the Möbius strip and $\mathbb{R}\mathrm{P}^2$ have "hyperbolic and unbounded fine curve graphs", and then (2) "use that result and the methods of \cite{BHW2}" to prove that both surfaces have "stably unbounded $\operatorname{Homeo}_0$" (hence stably unbounded commutator length).

To prove the main theorem, the introduction says the "central tool is Bavard duality" (linking commutator length to the existence of "non-trivial homogeneous quasi-morphisms"). The strategy follows the quasi-morphism constructions of Brooks and their generalisation by Bestvina--Fujiwara via actions on hyperbolic spaces, and the analogous approach of Bowden--Hensel--Webb using the action of $\operatorname{Homeo}_0$ on a (fine) curve graph. Concretely, the paper is organised to (1) combine results of \cite{kuno-hyperbolicity} and \cite{BHW1} to show the Möbius strip and $\mathbb{R}\mathrm{P}^2$ have "hyperbolic and unbounded fine curve graphs", and then (2) use that result together with the methods of \cite{BHW2} to prove that both surfaces have "stably unbounded $\operatorname{Homeo}_0$" (hence stably unbounded commutator length).

Question:
Let \(\operatorname{Homeo}_0(S)\) denote the group of homeomorphisms of a surface \(S\) that are isotopic to the identity. For \(g\in \operatorname{Homeo}_0(S)\), let \(\operatorname{cl}(g)\) be its commutator length, i.e. the minimal number of commutators whose product is \(g\), and define the stable commutator length by
\[
\operatorname{scl}(g)=\lim_{n\to\infty}\frac{\operatorname{cl}(g^n)}{n}.
\]
Say that \(\operatorname{Homeo}_0(S)\) is stably unbounded with respect to commutator length if there exists some \(g\in \operatorname{Homeo}_0(S)\) with \(\operatorname{scl}(g)>0\). Which statement holds for every surface \(S\) in the class consisting of the real projective plane \(\mathbb{R}\mathrm{P}^2\) and the Möbius strip \(M\)?

For each such surface \(S\in\{\mathbb{R}\mathrm{P}^2,M\}\), the group \(\operatorname{Homeo}_0(S)\) is stably unbounded with respect to commutator length; equivalently, there exists \(g\in \operatorname{Homeo}_0(S)\) such that \(\operatorname{scl}(g)>0\).

Choices:
B. For each such surface \(S\in\{\mathbb{R}\mathrm{P}^2,M\}\), the group \(\operatorname{Homeo}_0(S)\) is unbounded with respect to commutator length; equivalently, for every \(N\in\mathbb{N}\) there exists \(g\in \operatorname{Homeo}_0(S)\) with \(\operatorname{cl}(g)>N\).
C. For each such surface \(S\in\{\mathbb{R}\mathrm{P}^2,M\}\), there exists \(g\in \operatorname{Homeo}_0(S)\) whose commutator lengths \(\operatorname{cl}(g^n)\) are unbounded as \(n\to\infty\).
D. There exists a single element \(g\) such that for every surface \(S\in\{\mathbb{R}\mathrm{P}^2,M\}\), one has \(g\in \operatorname{Homeo}_0(S)\) and \(\operatorname{scl}(g)>0\).
E. For each such surface \(S\in\{\mathbb{R}\mathrm{P}^2,M\}\), every nontrivial element \(g\in \operatorname{Homeo}_0(S)\) satisfies \(\operatorname{scl}(g)>0\), so \(\operatorname{Homeo}_0(S)\) is stably unbounded with respect to commutator length.


\begin{theorem} \label{thm:calderon_intro}
Let $A, P \in \mathrm{GL}(d, \mathbb{R})$.
If $\psi \in \mathcal{B}_{\pi}$ and
$
 \{\pi(A^j P k, A^j) \psi \}_{ j \in \mathbb{Z}, k \in \mathbb{Z}^d }
$
is a Parseval frame for $L^2 (\mathbb{R}^d)$, then $\{ \pi(x, A^j) \psi \}_{x \in \mathbb{R}^d, j \in \mathbb{Z}}$ is a tight frame for $L^2 (\mathbb{R}^d)$, and
\[
\sum_{j \in \mathbb{Z}} |\widehat{\psi} ((A^t)^j \xi)|^2 = |\det (P)| \quad \text{for a.e.} \;\; \xi \in \mathbb{R}^d.
\]
\end{theorem}

\begin{remark} \label{rem:frame_bounds}
 \Cref{cor:quasi-lattice} implies, in particular, that if $\pi(\Lambda) \psi$ is a Bessel sequence for some relatively dense set $\Lambda \subseteq G$, then $\pi$ must be square-integrable, in the sense that $C_{\psi} f \in L^2 (G)$ for all $f \in \Hpi$.
For certain representations of semidirect product groups $G = \mathbb{R}^d \rtimes H$ of $\mathbb{R}^d$ and a matrix group $H \leq \mathrm{GL}(d, \mathbb{R})$, this was shown in \cite{aniello2001discrete}. For that setting, the upper bound provided by \Cref{cor:quasi-lattice} corresponds to \cite[Proposition 1]{aniello2001discrete}. Although the corresponding lower bound is asserted as \cite[Proposition 2]{aniello2001discrete} (even without any additional assumption on $\psi$), the argument provided for the lower bound in \cite{aniello2001discrete} (cf. \cite[Theorem 1]{aniello2001discrete}) is incomplete.
\end{remark}

For $\psi \in L^2 (\mathbb{R}^d)$ and matrices $A, P \in \mathrm{GL}(d, \mathbb{R})$, the associated wavelet system is given by the collection of functions
\begin{align} \label{eq:discrete_wavelet}
\big\{ |\det(A)|^{-j/2} \psi (A^j \cdot - Pk) \big \}_{j \in \mathbb{Z}, k \in \mathbb{Z}^d}.
\end{align}
In higher dimensions, a classical condition in the study of wavelet systems is to assume that the matrix $A$ preserves the integer lattice $\mathbb{Z}^d$, i.e., $A \mathbb{Z}^d \subseteq \mathbb{Z}^d$, and that $A$ is expansive, i.e., all its eigenvalues are strictly greater than one in modulus. Under such conditions, a full characterization of Parseval wavelet frames, or more generally dual wavelet frames, was obtained in \cite{chui2002characterization, bownik2000characterization, calogero2000characterization}, among others. In addition, the existence of wavelet bases for such dilations were shown in \cite{dai1997wavelet}. Both types of results extend classical results from dimension one to arbitrary dimensions.

Beyond the case of expansive dilation matrices, the theory of wavelet systems in higher dimensions is far less complete. Nevertheless, the existence of wavelet bases for nonexpansive dilation matrices has been studied in \cite{speegle2003existence, bownik2017wavelets, ionascu2006simultaneous, wang2002wavelets} and culminated in the recent breakthrough \cite{bownik2021simultaneous} that characterizes the dilation matrices admitting wavelet sets. On the other hand, the aforementioned characterization of Parseval wavelet frames is currently only known for special dilations, such as amplifying dilations \cite{laugesen2002translational}, dilations expanding on a subspace \cite{hernandez2002unified, guo2006some} or dilations satisfying the lattice counting estimate \cite{bownik2017wavelets}. This has lead to the following conjecture \cite[Conjecture 1]{bownik2017wavelets} and open problem \cite[Problem 3.3]{bownik2020open}, which was already implicitly raised in \cite[p. 177]{speegle2003existence}.\footnote{The formulations in \cite{bownik2017wavelets, bownik2020open} are under the implicit assumption that $|\det(P)| = 1$.}

\begin{conjecture}[\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \label{conj:wavelet}
Let $A, P \in \mathrm{GL}(d, \mathbb{R})$ and $\psi \in L^2 (\mathbb{R}^d)$. Suppose that \[ \{|\det(A)|^{-j/2} \psi (A^j \cdot - Pk) \}_{j \in \mathbb{Z}, k \in \mathbb{Z}^d}\]
is an orthonormal basis, or more generally a Parseval frame, for $L^2 (\mathbb{R}^d)$. Then the Calder\'on sum formula holds:
\begin{align} \label{eq:calderon}
\sum_{j \in \mathbb{Z}} |\widehat{\psi} ((A^t)^j \xi)|^2 = |\det(P)| \quad \text{for a.e.} \quad \xi \in \mathbb{R}^d.
\end{align}
\end{conjecture}

The Calder\'on sum formula \eqref{eq:calderon} is part of the aforementioned characterizion of Parseval wavelet frames known under additional assumptions on the dilation matrix. The upper bound for the Calder\'on sum is known to hold for any Bessel sequence with bound $1$ (cf. \cite[Proposition 4.1]{hernandez2002unified}).

In this paper, we present a new approach to \Cref{conj:wavelet} which allows us to prove the Calder\'on sum formula for arbitrary translation and dilation matrices under a mild condition on the wavelet function. Our approach is based on a relation between the frame properties of the discrete wavelet system \eqref{eq:discrete_wavelet} and the semi-continuous wavelet system whose elements are given by
\begin{align} \label{eq:continuous_wavelet}
\pi(x, A^j) \psi := |\det(A)|^{-j/2} \psi(A^{-j} ( \cdot - x)), \quad x \in \mathbb{R}^d, j \in \mathbb{Z}.
\end{align}
The action $\pi$ forms a unitary group representation of the semi-direct product group $G = \mathbb{R}^d \rtimes \langle A \rangle$ of $\mathbb{R}^d$ and the cyclic group $\langle A \rangle := \{ A^j : j \in \mathbb{Z} \}$ generated by $A \in \mathrm{GL}(d, \mathbb{R})$. Observe that the wavelet system \eqref{eq:discrete_wavelet} corresponds to $\{\pi(A^j P k, A^j) \psi : j \in \mathbb{Z}, k \in \mathbb{Z}^d \}$. We will assume a mild condition on the wavelet function. Namely, we assume that $\psi \in L^2 (\mathbb{R}^d)$ is such that $\pi(\Lambda) \psi$ is a Bessel sequence in $L^2 (\mathbb{R}^d)$ for all relatively separated sets $\Lambda $ in $ G = \mathbb{R}^d \rtimes \langle A \rangle$; in notation, $\psi \in \mathcal{B}_{\pi}$. This is a common assumption in the study of frames in the orbit of a group representation, see, e.g., \cite{Gr08, FuGr07, enstad2025dynamical, fuehr2017density, caspers2023overcompleteness}.
We refer to Section \ref{sec:coefficient} for an alternative description of the space $\mathcal{B}_{\pi}$ and further properties.

Using the notation from the previous paragraph, our main result is the following:

\begin{conjecture}[\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \label{conj:wavelet}
Let $A, P \in \mathrm{GL}(d, \mathbb{R})$ and $\psi \in L^2 (\mathbb{R}^d)$. Suppose that \[ \{|\det(A)|^{-j/2} \psi (A^j \cdot -  Pk) \}_{j \in \mathbb{Z}, k \in \mathbb{Z}^d}\]
is an orthonormal basis, or more generally a Parseval frame, for $L^2 (\mathbb{R}^d)$. Then the Calder\'on sum formula holds:
\begin{align} \label{eq:calderon}
\sum_{j \in \mathbb{Z}} |\widehat{\psi} ((A^t)^j \xi)|^2 = |\det(P)| \quad \text{for a.e.} \quad \xi \in \mathbb{R}^d.
\end{align}
\end{conjecture}

\begin{align} \label{eq:discrete_wavelet}
\big\{ |\det(A)|^{-j/2} \psi (A^j \cdot -  Pk) \big \}_{j \in \mathbb{Z}, k \in \mathbb{Z}^d}.
\end{align}

\begin{conjecture}[\cite{bownik2017wavelets, bownik2020open, speegle2003existence}] \label{conj:wavelet}
Let $A, P \in \mathrm{GL}(d, \mathbb{R})$ and $\psi \in L^2 (\mathbb{R}^d)$. Suppose that \[ \{|\det(A)|^{-j/2} \psi (A^j \cdot - Pk) \}_{j \in \mathbb{Z}, k \in \mathbb{Z}^d}\]
is an orthonormal basis, or more generally a Parseval frame, for $L^2 (\mathbb{R}^d)$. Then the Calder\'on sum formula holds:
\begin{align} \label{eq:calderon}
\sum_{j \in \mathbb{Z}} |\widehat{\psi} ((A^t)^j \xi)|^2 = |\det(P)| \quad \text{for a.e.} \quad \xi \in \mathbb{R}^d.
\end{align}
\end{conjecture}

Using the notation from the previous paragraph, our main result is the following:

A combination of \Cref{thm:calderon_intro} with the known fact that tight frames of the form $\{ \pi(x, A^j) \psi \}_{x \in \mathbb{R}^d, j \in \mathbb{Z}}$ exist only when $|\det(A)| \neq 1$ (cf. \cite{larson2006explicit, laugesen2002characterization}) yields the following consequence.

\begin{corollary}
 Let $A, P \in \mathrm{GL}(d, \mathbb{R})$.
If there exists a function $\psi \in \mathcal{B}_{\pi}$ such that
$
\big \{\pi(A^j P k, A^j) \psi \big\}_{ j \in \mathbb{Z}, k \in \mathbb{Z}^d }
$
is a Parseval frame for $L^2 (\mathbb{R}^d)$, then $|\det(A)| \neq 1$.
\end{corollary}

\subsection{Wavelet transform} \label{sec:coefficient}
Let $(\pi, \Hpi)$ be a projective unitary representation of $G$ on a Hilbert space $\Hpi$, i.e., a strongly measurable map $\pi : G \to \mathcal{U}(\Hpi)$ satisfying $\pi(e_G) = I_{\Hpi}$ and
\[ \pi(g_1) \pi( g_2) = \sigma(g_1, g_2) \pi(g_1 g_2), \quad g_1, g_2 \in G ,\]
for some (measurable) function $\sigma : G \times G \to \mathbb{T}$.
For $\psi, f \in \Hpi$, we define the function
\[ C_\psi f(g) = \langle f, \pi(g) \psi \rangle, \quad g \in G. \]
 By \cite[Lemma 7.1, Theorem 7.5]{varadarajan1985geometry}, the absolute value $|C_{\psi} f| : G \to [0, \infty)$ is continuous for arbitrary $f, \psi \in \Hpi$.
We say that a vector $\psi \in \Hpi$ is \emph{admissible} if the associated \emph{wavelet transform} $C_{\psi} : \Hpi \to L^{\infty} (G)$ given by $f \mapsto C_{\psi} f$ is an isometry into $L^2 (G)$, that is, the orbit $\pi(G) \psi$ is a continuous Parseval frame for $\Hpi$. Throughout, we always consider $L^2 (G)$ with respect to the left Haar measure $\mu_G$.

\begin{proof}
Let $\varepsilon > 0$, $f \in \Hpi$ and let $U \subseteq G$ be an open, relatively compact set such that $\Lambda$ is $U$-separated.
Let $\Gamma \in \Omega(\Lambda)$ and let $(\Lambda_n)_{n \in \mathbb{N}}$ be a sequence of left translates $\Lambda_n \subseteq G$ of $\Lambda$ such that $\Gamma = \lim_{n \to \infty} \Lambda_n$. Clearly, each set $\Lambda_n$, $n \in \mathbb{N}$, is $U$-separated and, by \Cref{lem:relsep-reldense-passes-to-hull}, also the set $\Gamma$ is $U$-separated.
\\~\\
(i) Let $\pi(\Lambda)\psi$ be a Bessel sequence with Bessel bound $C_2$. Then $\pi(\Lambda_n)\psi$ is a Bessel sequence with Bessel bound $C_2$ for all $n \in \N$. Let $K \subseteq G$ be a compact set. Enlarging $K$, we may assume that $\Gamma \cap \partial K = \emptyset$. Write $\Gamma \cap K= \{ \gamma^{(1)}, ..., \gamma^{(k)} \} $. Then, by \Cref{lem:convergence_sets}, there exists $n_0 \in \mathbb{N}$ such that $\Lambda_n \cap K = \{\lambda_n^{(1)}, ..., \lambda_n^{(k)} \}$ for all $n \geq n_0$ and $\lambda_n^{(j)} \to \gamma^{(j)}$ for all $1 \leq j \leq k$ as $n \to \infty$. This, combined with the continuity of the function $|C_{\psi} f|$ on $G$, yields that there exists $n_1 \geq n_0$ such that
\begin{align} \label{eq:passes-to-hull_2}
\bigg| \sum_{\gamma \in \Gamma \cap K} |(C_{\psi} f)(\gamma) |^2 - \sum_{\mu \in \Lambda_n \cap K} |(C_{\psi} f)(\mu) |^2 \bigg| \leq \frac{\varepsilon}{2}, \qquad n \geq n_1 .
\end{align}
Thus, since $\varepsilon > 0$ is arbitrary, we conclude that
\[ \sum_{\gamma \in \Gamma \cap K} | (C_\psi f)(\gamma)|^2 \leq \sum_{\mu \in \Lambda_n \cap K} |(C_\psi f)(\mu)|^2 \leq C_2 \| f \|^2 . \]
Since $f \in \Hpi$ and this holds for arbitrarily large compact sets $K \subseteq G$, we deduce that $\pi(\Gamma)\psi$ is a Bessel sequence with Bessel bound $C_2$.
\\~\\
(ii) Suppose $\pi(\Lambda)\psi$ is a frame and $\psi \in \mathcal{B}_{\pi}$. By \Cref{lem:sampling} there exists a compact set $K \subseteq G$ such that
\[
\sum_{\gamma \in \Gamma \cap K^c} |(C_{\psi} f)(\gamma) |^2 < \frac{\varepsilon}{4} \quad \text{and} \quad \sum_{\mu \in \Lambda_n \cap K^c} |(C_{\psi} f)(\mu) |^2 < \frac{\varepsilon}{4}
\]
for all $n \in \mathbb{N}$. Therefore,
\begin{align} \label{eq:passes-to-hull_1}
\bigg| \sum_{\gamma \in \Gamma \cap K^c} |(C_{\psi} f)(\gamma) |^2 - \sum_{\mu \in \Lambda_n \cap K^c} |(C_{\psi} f)(\mu) |^2 \bigg| \leq \frac{\varepsilon}{2}.
\end{align}
By enlarging $K$ if necessary, we may assume that \eqref{eq:passes-to-hull_1} holds for some compact $K \subseteq G$ with $\Gamma \cap \partial K = \emptyset$, see, e.g., the proof of \cite[Theorem 3.9]{enstad2025dynamical}.

\begin{theorem}
Let $A, P \in \mathrm{GL}(d, \mathbb{R})$.
If $\psi \in \mathcal{B}_{\pi}$ and
$
 \{\pi(A^j P k, A^j) \psi \}_{ j \in \mathbb{Z}, k \in \mathbb{Z}^d }
$
is a frame for $L^2 (\mathbb{R}^d)$ with frame bounds $0<C_1\leq C_2 < \infty$, then $\{ \pi(x, A^j) \psi \}_{x \in \mathbb{R}^d, j \in \mathbb{Z}}$ is a continuous frame for $L^2 (\mathbb{R}^d)$ with frame bounds $C_1|\det(P)|$ and $ C_2|\det(P)|$, and
\[
C_1 \leq \frac{1}{|\det(P)|} \sum_{j \in \mathbb{Z}} |\widehat{\psi} ((A^t)^j \xi)|^2 \leq C_2 \quad \text{for a.e.} \;\; \xi \in \mathbb{R}^d.
\]
\end{theorem}
\begin{proof}
By \Cref{lem:quasilattice_affine}, the set $\Lambda = \{ (A^j Pk, A^j) : j \in \mathbb{Z}, k \in \mathbb{Z}^d \}$ is a quasi-lattice in $G$ with complement $C = P [0, 1)^d \times \{I_d\}$, so that $\rho_G (C) = |\det(P)|$.
Hence, if the system $\pi(\Lambda) \psi$
is a frame for $L^2 (\mathbb{R}^d)$ with $\psi \in \mathcal{B}_{\pi}$, then \Cref{cor:quasi-lattice} implies that
\begin{align*} \label{eq:multiple_isometry}
C_1 |\det (P)| \leq \| C_{\psi} f \|^2 \leq C_2 |\det (P)| \| f \|^2
\end{align*}
for all $f \in L^2 (\mathbb{R}^d)$, which shows that
$\{ \pi(x, A^j) \psi \}_{x \in \mathbb{R}^d, j \in \mathbb{Z}}$ is a frame with the claimed bounds.
A standard calculation (see, e.g., \cite{laugesen2002characterization, fuehr2002continuous}) next shows that
\begin{align*}
\| C_{\psi} f \|^2 &= \int_{G} | \langle \widehat{f}, \widehat{\pi(x, A^j) \psi} \rangle |^2 \; d\mu_G(x,A^j) \\
 &= \int_G \bigg| \int_{\mathbb{R}^d} \widehat{f}(\xi) |\det (A) |^{j/2} e^{- 2\pi i \xi \cdot x} \overline{\widehat{\psi}((A^t)^{j} \xi)} \; d\xi \bigg|^2 \; d\mu_G(x, A^j) \\
 &= \sum_{j \in \mathbb{Z}} \int_{\mathbb{R}^d} \bigg| \int_{\mathbb{R}^d} \widehat{f}(\xi) \overline{\widehat{\psi}((A^t)^{j} \xi)} e^{- 2\pi i \xi \cdot x} \; d\xi \bigg|^2 \; dx \\
 &= \sum_{j \in \mathbb{Z}} \int_{\mathbb{R}^d} \big|\widehat{f} (\xi) \widehat{\psi} ((A^t)^j \xi)\big|^2 \; d\xi \\
 &= \int_{\mathbb{R}^d} |\widehat{f} (\xi)|^2 \sum_{j \in \mathbb{Z}} |\widehat{\psi} ((A^t)^j \xi)|^2 \; d\xi,
\end{align*}
where the penultimate step used Plancherel's formula. Hence,
\[
C_1 |\det(P)| \| \widehat{f} \|^2 \leq \int_{\mathbb{R}^d} |\widehat{f} (\xi)|^2 \sum_{j \in \mathbb{Z}} |\widehat{\psi} ((A^t)^j \xi)|^2 \; d\xi
\leq C_2 |\det(P)| \| \widehat{f} \|^2.
\]
Since this holds for arbitrary $f \in L^2 (\mathbb{R}^d)$, it follows that
\[
C_1 |\det(P)| \leq \sum_{j \in \mathbb{Z}} |\widehat{\psi} ((A^t)^j \xi)|^2 \leq C_2 |\det(P)|
\]
for a.e. $\xi \in \mathbb{R}^d$.
\end{proof}

A combination of \Cref{thm:calderon_intro} with the known fact that  tight frames of the form $\{ \pi(x, A^j) \psi \}_{x \in \mathbb{R}^d, j \in \mathbb{Z}}$  exist only when $|\det(A)| \neq 1$ (cf. \cite{larson2006explicit, laugesen2002characterization}) yields the following consequence.

\begin{corollary}
    Let $A, P \in \mathrm{GL}(d, \mathbb{R})$.
If there exists a function $\psi \in \mathcal{B}_{\pi}$ such that
$
\big \{\pi(A^j P k, A^j) \psi \big\}_{ j \in \mathbb{Z}, k \in \mathbb{Z}^d }
$
is a Parseval frame for $L^2 (\mathbb{R}^d)$, then  $|\det(A)| \neq 1$.
\end{corollary}

Our proof of \Cref{thm:calderon_intro} consists of showing lower and upper bounds on the norm of the continuous wavelet transform associated to the semi-continuous system \eqref{eq:continuous_wavelet} in terms of the frame bounds of the discrete wavelet frame \eqref{eq:discrete_wavelet}. In the case of a discrete Parseval wavelet frame \eqref{eq:discrete_wavelet}, the obtained \emph{identity} shows precisely that the semi-continuous system \eqref{eq:continuous_wavelet} is a continuous tight frame as asserted in \Cref{thm:calderon_intro}.
Inequalities relating the continuous wavelet transform and the frame bounds of a discrete frame are classical for isotropic dilations, see, e.g., \cite{daubechies1992ten, chui1993inequalities, fuehr1996wavelet}. For general dilation matrices, only an upper bound of for such type of inequalities is known in general \cite{aniello2001discrete}.\footnote{Although the corresponding inequality involving the lower frame bound is also asserted in \cite{aniello2001discrete}, the claimed proof is incomplete; see also \Cref{rem:frame_bounds}.} We present a new approach towards obtaining both lower and upper bounds by interpreting such estimates as inequalities relating the norm of the wavelet transform, the frame bounds of a frame and the density of its point set. For this, we use a notion of density of a point set (the so-called \emph{covolume} with respect to an invariant measure on the hull of the given point set) recently introduced for unimodular groups in \cite{enstad2025dynamical}; see \Cref{sec:covolume} for precise details. Using this notion in the setting of nonunimodular groups, we derive inequalities (cf. \Cref{cor:quasi-lattice}) relating the covolume of the point set and the frame bounds of the frame that generalize similar inequalities for lattices in unimodular groups obtained in \cite{Rova22}; see also \cite{balan2006density} for such inequalities for Gabor systems.
As in \cite{Rova22, enstad2025dynamical}, the actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand.

The above outlined proof method works naturally for general (projective) unitary representations of locally compact groups. As the auxiliary results might be of interest beyond wavelet systems, we present our results in this general setting. In particular, this allows us to obtain an extension of the main result of \cite{aniello2001discrete} for amenable groups beyond the setting of quasi-regular representations of semi-direct products (see \Cref{rem:frame_bounds}).

The paper is organized as follows. Section \ref{sec:pointsets} is devoted to background on point sets in locally compact groups. In particular, the notion of covolume is discussed here. Inequalities relating the frame bounds, covolume and norm of the wavelet transform are obtained in Section \ref{sec:frame}. Lastly, we show in Section \ref{sec:calderon} how Theorem \ref{thm:calderon_intro} can be derived from the general results of Section \ref{sec:frame}.

To prove Theorem~\ref{thm:calderon_intro}, the authors “show[] lower and upper bounds on the norm of the continuous wavelet transform associated to the semi-continuous system \eqref{eq:continuous_wavelet} in terms of the frame bounds of the discrete wavelet frame \eqref{eq:discrete_wavelet}.” In the Parseval case, “the obtained \emph{identity} shows precisely that the semi-continuous system \eqref{eq:continuous_wavelet} is a continuous tight frame as asserted in \Cref{thm:calderon_intro}.”

The “new approach” is to interpret these estimates “as inequalities relating the norm of the wavelet transform, the frame bounds of a frame and the density of its point set,” using a density notion called “\emph{covolume} with respect to an invariant measure on the hull of the given point set.” With this notion (extended “in the setting of nonunimodular groups”), they “derive inequalities (cf. \Cref{cor:quasi-lattice}) relating the covolume of the point set and the frame bounds of the frame,” generalizing earlier lattice/Gabor-type density results.

The “actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand.” Finally, “Theorem \ref{thm:calderon_intro} can be derived from the general results of Section \ref{sec:frame},” with the derivation carried out in Section~\ref{sec:calderon}.

To prove the main theorem, the authors “show[] lower and upper bounds on the norm of the continuous wavelet transform associated to the semi-continuous system
\begin{align} \label{eq:continuous_wavelet}
\pi(x, A^j) \psi := |\det(A)|^{-j/2} \psi(A^{-j} ( \cdot - x)), \quad x \in \mathbb{R}^d, j \in \mathbb{Z}.
\end{align}
in terms of the frame bounds of the discrete wavelet frame
\begin{align} \label{eq:discrete_wavelet}
\big\{ |\det(A)|^{-j/2} \psi (A^j \cdot -  Pk) \big \}_{j \in \mathbb{Z}, k \in \mathbb{Z}^d}.
\end{align}” In the Parseval case, “the obtained \emph{identity} shows precisely that the semi-continuous system
\begin{align} \label{eq:continuous_wavelet}
\pi(x, A^j) \psi := |\det(A)|^{-j/2} \psi(A^{-j} ( \cdot - x)), \quad x \in \mathbb{R}^d, j \in \mathbb{Z}.
\end{align}
is a continuous tight frame as asserted in establishing the main theorem.”

The “new approach” is to interpret these estimates “as inequalities relating the norm of the wavelet transform, the frame bounds of a frame and the density of its point set,” using a density notion called “\emph{covolume} with respect to an invariant measure on the hull of the given point set.” With this notion (extended “in the setting of nonunimodular groups”), they “derive inequalities (cf. the following corollary) relating the covolume of the point set and the frame bounds of the frame,” generalizing earlier lattice/Gabor-type density results.

We first prove the following corollary.
\begin{corollary} \label{cor:quasi-lattice}
Let $G$ be amenable and let $U \subseteq G$  and $K \subseteq G$ be a nonempty open and compact set, respectively. Let $\Lambda \subseteq G$ be a $U$-separated, $K$-dense set and $\psi \in \Hpi$.

If $\pi(\Lambda)\psi$ is a Bessel sequence in $\Hpi$ with Bessel bound $C_2 > 0$, then
\[ \| C_{\psi} f \|^2 \leq C_2 \mu_G (K) \| f \|^2 \quad \text{for all} \quad f \in \Hpi. \]
 In addition, if $\psi \in \mathcal{B}_{\pi}$ and $\pi(\Lambda) \psi$ also admits a lower frame bound $C_1 > 0$, then
\[ \| C_{\psi} f \|^2 \geq C_1 \mu_G (U) \| f \|^2 \quad \text{for all} \quad f \in \Hpi. \]
In particular, if $\pi(\Lambda) \psi$ is frame with $\psi \in \mathcal{B}_{\pi}$ for some quasi-lattice $\Lambda \subseteq G$ with a relatively compact Jordan measurable complement $C$, then
\[
C_1 \| f \|^2 \leq \frac{\|C_{\psi} f\|^2}{\rho_G (C)} \leq C_2 \| f \|^2
\]
for all $f \in \Hpi$.
\end{corollary}

The “actual proof for such inequalities is based on the simple idea of periodizing the norm of the wavelet transform and applying the corresponding frame inequalities to each integrand.” Finally, the main theorem can be derived from the general results proved later, with the derivation carried out later as well.

Question:
Let $A,P\in \mathrm{GL}(d,\mathbb{R})$, let $\widehat\psi$ denote the Fourier transform of $\psi$, and define
\[
\pi(x,A^j)\psi:=|\det(A)|^{-j/2}\,\psi\big(A^{-j}(\cdot-x)\big),\qquad x\in\mathbb{R}^d,\ j\in\mathbb{Z}.
\]
Assume $\psi\in \mathcal B_\pi$, where $\mathcal B_\pi$ is the class of functions in $L^2(\mathbb{R}^d)$ such that $\pi(\Lambda)\psi$ is a Bessel sequence in $L^2(\mathbb{R}^d)$ for every relatively separated set $\Lambda$ in $G=\mathbb{R}^d\rtimes\langle A\rangle$. Suppose further that the discrete wavelet system
\[
\{\pi(A^jPk,A^j)\psi\}_{j\in\mathbb{Z},\,k\in\mathbb{Z}^d}
\]
is a Parseval frame for $L^2(\mathbb{R}^d)$. Which of the following statements holds?

The semi-continuous system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a tight frame for $L^2(\mathbb{R}^d)$, and
\[
\sum_{j\in\mathbb{Z}}\big|\widehat\psi\big((A^t)^j\xi\big)\big|^2=|\det(P)|
\qquad \text{for a.e. }\xi\in\mathbb{R}^d.
\]

Choices:
B.
The semi-continuous system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a tight frame for $L^2(\mathbb{R}^d)$, and
\[
\sum_{j\in\mathbb{Z}}\big|\widehat\psi\big((A^t)^j\xi\big)\big|^2=1
\qquad \text{for a.e. }\xi\in\mathbb{R}^d.
\]
C. The semi-continuous system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a tight frame for $L^2(\mathbb{R}^d)$.
D.
The semi-continuous system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a Parseval frame for $L^2(\mathbb{R}^d)$, and
\[
\sum_{j\in\mathbb{Z}}\big|\widehat\psi\big((A^t)^j\xi\big)\big|^2=1
\qquad \text{for a.e. }\xi\in\mathbb{R}^d.
\]
E.
For every $\xi\in\mathbb{R}^d$ one has
\[
\sum_{j\in\mathbb{Z}}\big|\widehat\psi\big((A^t)^j\xi\big)\big|^2=|\det(P)|,
\]
and consequently the semi-continuous system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a tight frame for $L^2(\mathbb{R}^d)$.


Question:
Let $A,P\in \mathrm{GL}(d,\mathbb{R})$. For $x\in \mathbb{R}^d$ and $j\in \mathbb{Z}$, define
\[
\pi(x,A^j)\psi:=|\det(A)|^{-j/2}\,\psi\big(A^{-j}(\cdot-x)\big).
\]
Let $\mathcal{B}_{\pi}$ denote the class of functions $\psi\in L^2(\mathbb{R}^d)$ such that, for every relatively separated set $\Lambda\subset G=\mathbb{R}^d\rtimes\langle A\rangle$, the orbit $\pi(\Lambda)\psi$ is a Bessel sequence in $L^2(\mathbb{R}^d)$. Assume that $\psi\in \mathcal{B}_{\pi}$ and that the discrete affine system
\[
\{\pi(A^jPk,A^j)\psi\}_{j\in\mathbb{Z},\,k\in\mathbb{Z}^d}
\]
is a Parseval frame for $L^2(\mathbb{R}^d)$. Here $\widehat{\psi}$ denotes the Fourier transform of $\psi$ and $A^t$ the transpose of $A$. Which of the following conclusions is valid under these hypotheses?

The semi-continuous affine system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a tight frame for $L^2(\mathbb{R}^d)$, and
\[
\sum_{j\in\mathbb{Z}}\big|\widehat{\psi}\big((A^t)^j\xi\big)\big|^2=|\det(P)|
\quad\text{for a.e. }\xi\in\mathbb{R}^d.
\]

Choices:
B.
The semi-continuous affine system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a tight frame for $L^2(\mathbb{R}^d)$, and
\[
\sum_{j\in\mathbb{Z}}\big|\widehat{\psi}\big((A^t)^j\xi\big)\big|^2=1
\quad\text{for a.e. }\xi\in\mathbb{R}^d.
\]
C. The semi-continuous affine system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a tight frame for $L^2(\mathbb{R}^d)$.
D.
The semi-continuous affine system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a Parseval frame for $L^2(\mathbb{R}^d)$, and
\[
\sum_{j\in\mathbb{Z}}\big|\widehat{\psi}\big((A^t)^j\xi\big)\big|^2=1
\quad\text{for a.e. }\xi\in\mathbb{R}^d.
\]
E.
The semi-continuous affine system $\{\pi(x,A^j)\psi\}_{x\in\mathbb{R}^d,\,j\in\mathbb{Z}}$ is a tight frame for $L^2(\mathbb{R}^d)$, and
\[
\sum_{j\in\mathbb{Z}}\big|\widehat{\psi}\big((A^t)^j\xi\big)\big|^2\le |\det(P)|
\quad\text{for a.e. }\xi\in\mathbb{R}^d.
\]

dropped the explicit Calderon sum identity

tight bound equals |det(P)| rather than 1 for the semi-continuous system

identity from two-sided frame estimate weakened to one-sided a.e. inequality