&\sum^{J_{sn}}_{j=1} { sn \choose j}\left(x^{sn-j}y^{j} + x^{j} y^{sn-j}\right) < \! 2^{J_{sn}}\left(x^{sn-j}y^{j} + x^{j} y^{sn-j}\right). \nonumber
\end{align}

Lemma~\ref{BGP04-bino-sum} and its follow up estimates \eqref{binsum-2} provide a  natural tool to study the $sn^{th}$-moments of collisional operator form  $Q(f,f)(v,t)$ describing a binary interaction framework, that arises
in the calculation  of  upper bounds  terms from  its positive contribution, as shown in the sequel.

The following energy identity is crucial for the development of moment estimates by means of the scattering angular averaging Theorem~\ref{BGP-JSP04}.  It is obtain writing the local energy  conservation written for the Lebesgue weight functions in the post collisional velocities, that is  $ \langle v' \rangle$  and $ \langle v'_* \rangle$   by means  of  center of mass and relative velocity coordinates expressed in the scattering direction $\sigma$.

\begin{lem}[\bf{Energy identity in the scattering  direction coordinates}]\label{PovznerI}
For any $(v',v'_*)$ and $(v,v_*)$ satisfying \eqref{weightuv2}, let  $E:= \langle v \rangle^2 + \langle v_*\rangle^2$,  and
$\psi(x)$ be  any function defined in $\R^+$,
then, the following identity holds
\begin{equation}\label{Pov1}
\begin{split}
&\psi\big(\langle v'\rangle^2\big)   +  \psi\big(\langle v'_* \rangle^2\big)  = \psi\bigg( E\Big( \tfrac{1-\xi\, \hat V\cdot\sigma}2\Big)\bigg) +  \psi\bigg( E\Big( \tfrac{1+\xi\, \hat V\cdot\sigma}2\Big)\bigg)\,,\\
\noalign{\vskip6pt}
&\ \text{ with }\ \ \  \xi = \frac{2|u|\, |V|}{E}\in [0,1]  \qquad \text{and}  \qquad  \hat V=\frac{V}{|V|}\,.
\end{split}
\end{equation}
\end{lem}

\begin{proof} Setting the center of mass $V=\frac{v+v_{*}}{2}$ and relative velocity $u=v-v_*$, we recall from \eqref{weightuv2} that
\begin{equation*}
v' = V-\frac12 |u| \sigma \quad \ \ \text{and } \quad
v'_* = V+\frac12 |u| \sigma\,.
\end{equation*}
As a consequence, we can compute as
\begin{equation*}%\label{povzner2.2}
\begin{split}
\langle v' \rangle^{2} = 1+|v'|^2=1+\Big| V-\frac12|u| \sigma\Big|^2 & = 1+|V|^2 - |u| V\cdot \sigma +\frac{1}{4} |u|^2  = \frac{E}{2} - |u||V|\hat{V}\cdot\sigma\, ,\\
\langle v'_{*} \rangle^{2} = 1+|v'_*|^2=1+\Big| V+\frac12|u| \sigma\Big|^2 & = 1+|V|^2 + |u| V\cdot\sigma+\frac{1}{4} |u|^2 =  \frac{E}{2} + |u||V|\hat{V}\cdot\sigma\, .
\end{split}
\end{equation*}
\begin{equation*}
\langle v' \rangle^{2} = E\bigg(\frac{1 - \xi \, \hat{V}\cdot\sigma}{2}\bigg)\,,\qquad \langle v'_{*} \rangle^{2} = E\bigg(\frac{1 + \xi\, \hat{V}\cdot\sigma}{2}\bigg)\,,\qquad \text{where}\quad \xi:=\frac{2\,|u||V|}{E}\,.
\end{equation*}
In addition, note that
\begin{equation*}
0\leq \xi=\frac{2|u|\, |V|}{E} \le \frac{|v|^{2} + |v_{*}|^{2}}{E}\leq 1\,,
\end{equation*}
which proves the lemma.
\end{proof}
\subsection{Angular averaging regularity} This is a profound result, independent of the solution of the  Cauchy problem for the Boltzmann flow posed in Theorem~\ref{CauchyProblem}. It  provides the basic component  not only a coercive condition  that yields global stability of solutions but also the control of high energy tails or exponential moments to be discussed in Subsection~\ref{exponential-tails} .
It has its roots in the work of  Povzner [94], Desvillettes [51] and Wennberg[109], but it was the ground breaking work of Bobylev [31]  who introduced,  by means of the energy identity \eqref{Pov1}, the concept of angular averaging  that produces a broad and natural argument to  control the integral of the sum of convex weight functions evaluated in post-collisional velocities  proportional to the  angular transition function $\b(\hat u\cdot\sigma) \in L^1(\mathbb{S}^{d-1} )$, without any need to omit point of singularities.  While the original work in [31] was developed  for the case of hard potentials an three dimensions (i.e. $\gamma=1$ and $\b(\hat u\cdot\sigma)$ being constant), we
present in this review
ideas techniques   that are slight modifications of those  from [37, 64, 5]. These new  angular averaging approaches have been recently developed for a system of multi-component gas mixtures [66], and  more recently for a broad class of cross sections and transition probabilities for the case of polyatomic gases [67].
\begin{thm}[\bf{ Angular Averaging Lemma}]\label{BGP-JSP04}
Assume that the collision cross-section splits into $B(|u|, \hat u\cdot\sigma)= \Phi(|u|) \, b(\hat u\cdot\sigma)$ and let $\psi_{r} (x) = \langle x\rangle^{2r}$, with any real value $r$.  Then, there exists a positive constant $\mu_r$, referred by the {\em $r^{th}$-contractive factor}, such that  the positive component of the weight function  \eqref{weightuv2} in the weak formulation of the Boltzmann equation satisfies
\begin{equation}\label{povzner0}
G^+_{\psi_{r}}(v,v_*)  =  \int_{ \mathbb{S}^{d-1} } \big( \psi_{r} (v') + \psi_{r} (v'_*) \big)
B(|u|, \hat u\cdot\sigma)\, d\sigma \le \mu_{r}\,\Phi(|u|)\,\Big( \langle v\rangle^{2} + \langle v_{*}\rangle^{2} \Big)^{r}
\end{equation}
\begin{equation}\label{mu_r}
\mu_r  := \sup_{\{\hat{V}\in \mathbb{S}^{d-1} ,\hat{u} \in \mathbb{S}^{d-1}\}}\int_{ \mathbb{S}^{d-1} }  \bigg(\left| \tfrac{1 - \hat V\cdot\sigma}2 \right|^{r} +  \left| \tfrac{1 + \hat V\cdot\sigma}2 \right|^{r} \bigg)  b(\hat u\cdot\sigma)\, d\sigma \le \|b \|_{L^1(\mathbb S^{d-1})}\,, \   \ r>1\,.
\end{equation}
$\mu_{1}=\|b \|_{L^1(\mathbb S^{d-1})}$, and $\mu_r\searrow0$ strictly as $1<r \nearrow \infty.$

Furthermore, if $b\in L^{p}(\mathbb{S}^{d-1})$, with $1<p\leq \infty$, the following estimate holds
\begin{equation}\label{povzner0.1}
\mu_r\leq 2^{(2p-1)/p} \|b\|_{L^p(\mathbb{S}^{d-1})}|\mathbb{S}^{d-2}|^{(p-1)/p}   \bigg(\frac{p-1}{p(1+r)-1}\bigg)^{(p-1)/p}  =\mathcal{O}(r^{-\frac{p-1}p}) \,.
\end{equation}
\end{thm}

\begin{remark}\label{coercive-factor}  The factor $\mu_r$   generates  constants $0< \|b \|_{L^1(\mathbb S^{d-1})}-\mu_r < 1 $, for all $r>1$,  for the weak form  of Boltzmann flow when tested by the polynomial forms $\psi_{r}$. In particular, the main result from this Theorem~\ref{BGP-JSP04}, is  to show a priori global in time $L^1_k$ estimates to unique solutions of the Boltzmann  initial value problem, where  factor   $\|b \|_{L^1(\mathbb S^{d-1})}-\mu_r >0 $ is associated to coerciveness for the dynamics of the Boltzmann equation in a suitable Banach space and norm. See Remark~\ref{coercive-remark}  for further comments in coerciveness.
\end{remark}

\
\begin{proof}
Using Lemma \ref{PovznerI} and the homogeneity of $\psi_{k}$, it follows that the positive part of the weight function \eqref{weightuv2} is controlled by
\begin{equation*}%\label{povzner4}
\begin{split}
G^+_{\psi_{k}}(v_*,v) & = \Phi(|u|) \,  \int_{ \mathbb{S}^{d-1} }\big( \psi_{r}(v')   +  \psi_{r}(v'_*)  \big) b(\hat u\cdot\sigma)\, d\sigma \\
&= \Phi(|u|)\,\Big(\langle v \rangle^{2} + \langle v_{*}\rangle^{2}\Big)^{r} \, \int_{ \mathbb{S}^{d-1} }  \bigg(\left| \tfrac{1-\xi \hat V\cdot\sigma}2 \right|^{r} +  \left| \tfrac{1+\xi \hat V\cdot\sigma}2\right|^{r}\bigg)  b(\hat u\cdot\sigma)\, d\sigma\,.
\end{split}
\end{equation*}
%The derivative of $|\cdot|^{k}$, for $k\geq1$, is nondecreasing.
%Due to the symmetry of the angular transition rate $b(\hat u\cdot\sigma)$
{Noting that  the derivative of the integral termr $\bigg(\left| \tfrac{1-\xi \hat V\cdot\sigma}2 \right|^{r} +  \left| \tfrac{1+\xi \hat V\cdot\sigma}2\right|^{r}\bigg)$ containing  $\hat V\cdot\sigma$  are non-negative, as their factor $\pm \xi$ is bounded by unity,   then}
\begin{equation}\label{Pov2}
\left| \tfrac{1-\xi \hat V\cdot\sigma}2 \right|^{r} +  \left| \tfrac{1+\xi \hat V\cdot\sigma}2 \right|^{r} \leq \left| \tfrac{1 - \hat V\cdot\sigma}2 \right|^{r} +  \left| \tfrac{1 + \hat V\cdot\sigma}2 \right|^{r}\,.
\end{equation}
\begin{align}\label{Pov0}
G^+_{\psi_{r}}(v_*,v) &\leq \Phi(|u|)\,\Big( \langle v\rangle^{2} + \langle v_{*}\rangle^{2}\Big)^{r} \int_{ \mathbb{S}^{d-1} }  \bigg(\left| \tfrac{1 - \hat V\cdot\sigma}2 \right|^{r} +  \left| \tfrac{1 + \hat V\cdot\sigma}2 \right|^{r}\bigg)  b(\hat u\cdot\sigma)\, d\sigma\\
&\leq \mu_r\,\Phi(|u|)\,\Big( \langle v \rangle^{2} + \langle v_{*} \rangle^{2}\Big)^{r}\,,\qquad r>1\,.\nonumber
\end{align}
Thus, we define the {\sl $r^{th}$-contractive factor }
\begin{equation*}
\mu_r  := \sup_{\{\hat{V}\in \mathbb{S}^{d-1} ,\hat{u} \in \mathbb{S}^{d-1}\}}\int_{ \mathbb{S}^{d-1} }  \bigg(\left| \tfrac{1 - \hat V\cdot\sigma}2 \right|^{r} +  \left| \tfrac{1 + \hat V\cdot\sigma}2 \right|^{r} \bigg)  b(\hat u\cdot\sigma)\, d\sigma < \|b \|_{L^1(\mathbb S^{d-1})}\,,\qquad  \text{for any} \ r>1\,.
\end{equation*}
%It is easy to verify  that $\mu_k$ has the aforementioned properties.
\begin{equation*}
\mu_{1} =  \sup_{\{\hat{V}\in \mathbb{S}^{d-1} ,\hat{u} \in \mathbb{S}^{d-1}\}} \int_{ \mathbb{S}^{d-1} }  \bigg(\left| \tfrac{1 - \hat V\cdot\sigma}2 \right| +  \left| \tfrac{1 + \hat V\cdot\sigma}2 \right|\bigg)  b(\hat u\cdot\sigma)\, d\sigma = \|b \|_{L^1(\mathbb S^{d-1})} % \int_{ \mathbb{S}^{d-1} } b(\hat u\cdot\sigma)\, d\sigma % = 1\,.
\end{equation*}
In addition, for all $r>1$, the function
\begin{equation*}
\mu_r(\hat{V},\hat{u}):= \int_{ \mathbb{S}^{d-1} }  \bigg(\left| \tfrac{1 - \hat V\cdot\sigma}2 \right|^{r} +  \left| \tfrac{1 + \hat V\cdot\sigma}2 \right|^{r}\bigg)  b(\hat u\cdot\sigma)\, d\sigma\,,
\end{equation*}
is continuous in the vectors $\hat{V}$ and $\hat{u}$. Such continuity can be easily verify by  using polar coordinates with zenith $\hat{u}$.  Moreover, the integrand  \eqref{Pov0}being smaller or equal than unity,  with respect to the measure $\left(b(\hat u\cdot\sigma)  d\sigma\right)$,   is strictly decreasing for $r>1$ up to a set of measure zero (namely at $\sigma=\{\pm\hat{V}\}$).
As a consequence $\mu_{r_{1}}(\hat{V},\hat{u})>\mu_{r_{2}}(\hat{V},\hat{u})$ for any $r_{1}<r_{2}$.  Thus, the
$\sup_{(V,\hat u)}\left(  \mu_{r_{1}}(\hat{V},\hat{u})-\mu_{r_2}(\hat{V},\hat{u})\right)> 0$, when   taken over any unitary vector pairs $(\hat{V}, \hat{u} )\in \mathbb{S}^{d-1} \times \mathbb{S}^{d-1} $, which, by continuity,  implies $\mu_{r_{1}}-\mu_{r_{2}}>0$.  Finally, monotonic convergence gives that $\mu_r\searrow0$, with  $\mu_r<\|b \|_{L^1(\mathbb S^{d-1})}$, as soon as $r>1$.\\

In the case that the scattering kernel $b$ is integrable in $L^{p}(\mathbb{S}^{d-1})$ with $p>1$,  estimate \eqref{povzner0.1} follows by H\"{o}lder's inequality
\begin{align}\label{Povp1}
\mu_r\leq \int_{ \mathbb{S}^{d-1} }\left| \tfrac{1 \pm \hat V\cdot\sigma}2 \right|^{r} &b(\hat u\cdot\sigma) d\sigma \leq \|b\|_{L^{p}(\mathbb{S}^{d-1})}\bigg( \int_{ \mathbb{S}^{d-1} } \left| \tfrac{1 \pm \hat V\cdot\sigma}2 \right|^{p'r}  d\sigma \bigg)^{1/p'} \leq  2  \|b\|_{p}  \left(\frac{|\mathbb{S}^{d-2}|}{1+rp'}\right)^{1/p'}\\
=   2^{(2p-1)/p}  &\|b\|_{L^p(\mathbb{S}^{d-1})}|\mathbb{S}^{d-2}|^{(p-1)/p}   \bigg(\frac{p-1}{p-1+pr}\bigg)^{(p-1)/p}\!\!\!= \mathcal{O}\left((1+\frac{p}{p-1}r)^{-\frac{p-1}p}\right),  \nonumber
\end{align}
\begin{align}\label{Povp2}
\mu_r\leq  4  \|b\|_{L^{\infty}(\mathbb{S}^{d-1})}| \mathbb{S}^{d-2}| ((1+r)^{-1}), \qquad \text{for } p\equiv \infty. \qquad \qquad
\end{align}
\end{proof}

Recall  that, since the unit sphere is a bounded set in $\mathbb R^d$, then   the      $L^p$-integrability of the transition probability function $b(\hat u\cdot \sigma)$ on the unit sphere $\mathbb S^{d-1}$, with any  $p\in[1,\infty] $, is control from below from the $L^1$ integrability norm, that is
$ \|b\|_{L^1(\mathbb{S}^{d-1})} \le  \|b\|_{L^p(\mathbb{S}^{d-1})}  |\mathbb{S}^{d-2}|^{(p-1)/p} $.

\subsection{Moment inequalities for binary interactions}\label{sec:mom}

The sharp form of Povzner Lemma in Theoren~\ref{BGP-JSP04} has a profound impact in the theory of existence and uniqueness of the Boltzmann equation in the space homogeneous case because it allows to propagate and generate statistical moments in the case of hard potentials.

\medskip

\begin{lem}[\bf{Angular Averaging for the Binary Weight Function }]\label{PovznerII}
The Boltzmann weight function \eqref{weightuv2} associated to the weak form \eqref{bina-weak2} for  $\varphi(v)=\langle v\rangle ^{sk}$  for $1<k\in\mathbb{N}$ and $s\in(0,1]$, now denoted by $G_{sk}(v_*,v)$, satisfies the  following inequality for any $sk\geq1$,
\begin{align}\label{Pov2}
\begin{split}
G_{sk}(v_*,v) &= \int_{S^{d-1}}\Big( \langle v' \rangle^{2sk}   +   \langle v'_*\rangle^{2sk}  - \langle v\rangle^{2sk} - \langle v_*\rangle^{2sk}\Big) B(|u|, \hat u\cdot\sigma)\, d\sigma\\
&\le  \Phi(|u|)\Big( \mu_{sk}\Big( \langle v \rangle^{2} + \langle v_{*}\rangle^{2} \Big)^{sk}  -\|b\|_{L^1(\mathbb{S}^{d-1})}\left( \langle v\rangle^{2sk} + \langle v_*\rangle^{2s k}\right) \Big)\\[3pt]
&\hspace{-1cm}\le  \Phi(|u|)\Big( 2^{J_{2sk}}\mu_{sk}\Big(\langle v \rangle^{2}\langle v_{*}\rangle^{2(sk-1)} + \langle v\rangle^{2(sk-1)}\langle v_{*}\rangle^{2} \Big) - (\|b\|_{L^1(\mathbb{S}^{d-1})}- \mu_{sk})\Big( \langle v\rangle^{2 sk} +  \langle v_*\rangle^{2sk}\Big) \Big)\,.
\end{split}
\end{align}
\end{lem}
\begin{proof}
It follows by a trivial application, first, of Theorem \ref{BGP-JSP04} and, then, of Lemma \ref{BGP04-bino-sum} with $x=\langle v\rangle^{2}$ and $y=\langle v_{*}\rangle^{2}$ and $J_{2sk}= sk-1$.
\end{proof}

%\begin{lem}[\bf{Angular averaging lemma}]\label{PovznerII}
%The following identity holds for any $k\geq1$,
%G_{k}(v_*,v) &= \int_{S^{d-1}}\Big( \langle v' \rangle^{2k}   +   \langle v'_*\rangle^{2k}  - \langle v\rangle^{2k} - \langle v_*\rangle^{2k}\Big) B(|u|, \hat u\cdot\sigma)\, d\sigma\\
%&\le \Phi(|u|)\Big( \mu_k\Big( \langle v \rangle^{2} + \langle v_{*}\rangle^{2} \Big)^{k}  - \langle v\rangle^{2k} -  \langle v_*\rangle^{2 k} \Big)\\[3pt]
%&\hspace{-1cm}\le  \Phi(|u|)\Big( 2^{k-1}\mu_k\Big(\langle v \rangle^{2}\langle v_{*}\rangle^{2(k-1)} + \langle v\rangle^{2(k-1)}\langle v_{*}\rangle^{2} \Big) - (1- \mu_k)\Big( \langle v\rangle^{2 k} +  \langle v_*\rangle^{2k}\Big) \Big)\,.

\subsection{Moment  estimates - Version $\mathrm I$}\label{L^1-MM}

In order to obtain moment's estimates  associated to the solution of the Boltzmann equation that will also serve to calculate the summability of moments as a function of the potential rate $\gamma$ from \eqref{pot-Phi_1},
defined the $sk^{th}$-moment of  the probability density $f(v,t)$ by  $m_{sk}[f](t) = \int f(v,t) \langle v\rangle^{2sk}  dv,$  as defined in \eqref{k-moment}, for any  $sk\geq0$, with $0<s\le1.$

\begin{lem}[\bf{Lebesgue moment's estimates}]\label{mom-lemma-I}
Each  $sk^{th}$ moment of the Boltzmann equation associated to the Cauchy problem Theorem~\ref{CauchyProblem}
satisfies the following Ordinary Differential Inequality (ODI)

\begin{align}\label{momentineq}
\frac {d}{dt} m_{sk}[f](t) &\leq 2^{sk-1+\frac\gamma 2}\mu_{sk}  C_\Phi  \Big(m_{1+\gamma/2}[f](t) m_{sk-1}[f](t)  + m_{1}[f](t) m_{sk-1+\gamma/2}[f](t) )\Big)\\
&\ \ \ \  + \|b\|_{L^1(\mathbb{S}^{d-1})}\,m_{sk}[f](t) m_{\gamma/2}[f](t)  - 2^{-\frac{\gamma} 2}c_{\Phi} (\|b\|_{L^1(\mathbb{S}^{d-1})}-\mu_{sk}) m_{0}[f](t) m_{sk+\gamma/2}[f](t),  \nonumber\\[6pt]
& \ \ \ \   \ \text{for any }  0<s\le 1, \ 0<\gamma \le 2,\  \text{and} \  sk> s\kc\ge 1, \   \nonumber
\end{align}

with the {$sk^{th}$} contracting factor  $\mu_{sk} < \|b\|_{L^1(\mathbb{S}^{d-1})}$, and the lower bound constant  $2^{-\frac{\gamma}2}c_{\Phi }$ from the intermolecular potential function $\Phi(u)$.
\end{lem}

\begin{proof}
%From now on, we denote by $  \|b\|_1:= \|b\|_{L^1(\mathbb{S}^{d-1})}   $.
A direct application of Theorem \ref{BGP-JSP04} and Lemma \ref{lem:binom} leading to  Lemma \ref{PovznerII}, enable estimates for any  $sk^{th}$-moments of the collisional form  $Q(f,f)(v,t)$, for any real valued $sk\geq1$,   as defined in \eqref{k-moment},  that is
\begin{align}\label{L^1-MM-1}
\begin{split}
\frac {d}{dt} m_{k}[f](t) = \int_{\real^d}& Q(f,f)(v,t) \, \langle v\rangle^{2sk}  dv   =  \int_{ \real^{2d} } f(v) f(v_*) \, G_{sk}(v_{*},v)\, dv_* dv  \\
&= \int_{ \real^{2d} } f(v) f(v_*) \, \left(G^+_{sk}(v_{*},v) - G^-_{sk}(v_{*},v) \right) \, dv_* dv  \\
&\quad\le  2^{sk  -1 +\frac\gamma 2}C_{\Phi}\mu_{sk}   \int_{ \real^{2d} } f(v) f(v_*)\left(\langle v\rangle^{2}\,\langle v_{*}\rangle^{2(sk-1)} +\langle v\rangle^{2(sk-1)}\langle v_*\rangle^{2}\right ) \Phi(|u|)  dv_* dv    \\
&\qquad  -  ( \|b\|_{L^1(\mathbb{S}^{d-1})} -\mu_{sk})\int_{\real^{d}}f(v)\langle v\rangle^{2sk}\Big(\int_{\real^{d}}f(v_{*})\Phi(|v-v_*|) dv_{*}\Big)dv\,.
\end{split}
\end{align}
Thus, the first binary integral term,  or positive part of  \eqref{L^1-MM-1}  is integrated by Fubinni  with respect to $v$ and $v_*$, which combined with  the upper of $\Phi_\gamma$ from \eqref{pot-Phi_2}  yields the following  upper bound  given by the bilinear forms  of  shifted  moments
\begin{align}\label{momentinequp}
\begin{split}
\int_{\real^{2d} }f(v,t) f(v_*,t) G^+_{sk}(v_{*},v)dv_* dv  \!  \leq \! 2^{sk -1+\frac\gamma 2} C_\Phi \mu_{sk}  \Big(\!&m_{1+\gamma/2}[f]m_{sk-1}[f]  \!+\! m_{1}[f] m_{sk-1+\gamma/2}[f] \!\Big)(t).\end{split}
\end{align}

The   negative part of  \eqref{L^1-MM-1},  $\int_{\real^{2d} }f(v,t) f(v_*,t) G^-_{k}(v_{*},v)dv_* dv$  ,   needs a lower lower bound for the collision frequency factor  that uses the pointwise   bound estimates  for $\Phi_\gamma$
from \eqref{pot-Phi_2}, in terms of the Lebesgue weights,
\begin{align}\label{momentineqlow}
\int_{\real^{d}}f(v_{*})\Phi_\gamma(|u|)dv_{*} &=   m_{0}[f],\qquad  \text{ for}\  \gamma = 0 \ \text{(Maxwell type interactions case)}\,, \nonumber\\
\text{and}\qquad  \qquad \qquad\qquad&\ \\
\int_{\real^{d}}f(v_{*})\Phi_\gamma(|u|)dv_{*}&\geq  c_{\Phi}(2^{-\frac\gamma 2}  m_{0}[f]\langle v\rangle^{\gamma} - m_{\gamma/2}[f])\,, \qquad \text{ for} \ 0 < \gamma \le 2 \ \text{(Hard potentials case)}\,, \nonumber
\end{align}

Combining both estimates for the positive and negative parts of the collisional integrals estimates \eqref{momentinequp} and
\eqref{momentineqlow} yields an upper estimate for the right  hand side the $k^{th}$-moment ODI derived from the corresponding moments inequality \eqref{L^1-MM-1}, and so   \eqref{momentineq} holds.
\end{proof}

\begin{remark}\label{remark-moment-low}
Note that the bound from below of $\Phi_\gamma$ introduces a positive contribution, and so coarsening the negative contribution of the lower bound.   However, this lower bound constants does not need any information about the solutions moment's order.
\end{remark}
\begin{remark}\label{remark-moment-0}
Note that the third term positive term in the righthand side of  \eqref{momentineq} disappears for the case  Maxwell interactions when $\gamma=0$.
\end{remark}

\

It is important to point out  that  the above  Ordinary Differential  inequality    \eqref{momentineq} enables global in time propagation and generation of moments estimates, that are sufficient to prove  the existence and uniqueness theory associated to the initial value problem in the proof of Theorem~\ref{CauchyProblem} in a suitable subspace of the positive cone of $L^1_{2k}(\real^d)$ spaces.  A minor and yet profound impact of the lower bound contribution from \eqref{momentineqlow}, is that requires to revisit the classical $k^{th}$-moment inequalities calculated  with weights of the form $|v|^{2k}$ rather the Lebesgue weight $\langle v\rangle^{2k}$.
Because of this ``wrinkle" in the lower bound, we need  to appeal to a
the following classical interpolation results for weighted moments associated to  probability densities.

\begin{lem}[\bf Moment interpolation inequalities]\label{Lemma:mom-Holder}
The moment $m_{j} :=m_{j}[f]  :=\int_{\real^d} f(v,t) \, \psi_{ j} (v)\, dv $, for either $\psi_{ j}(v)=  |v|^{j}$  or  $ \langle v \rangle^{j}$,  satisfies
\begin{equation}\label{mom-Holder}
m_{j} \leq m_{{j}_1}^\tau m_{{j}_2}^{1-\tau},
\end{equation}
where the positive constants ${j}, {j}_1, {j}_2, \tau$ satisfy $0<{j}_1< {j}< {j}_2,$ $0<\tau<1$, and ${j}=\tau{j}_1+(1-\tau){j}_2$.
\end{lem}
\begin{proof}
The proof of this statement is straightforward since $\psi_{j} (v)$ is convex for $j\ge1$.  Indeed, H\"{o}lder's  and Jensen's inequality imply
\begin{align*}
m_{j}[f] \leq \int_{\mathbb{R}^d}\psi(v)^{j} f(v) dv \ &\leq\
\int_{\mathbb{R}^d} \psi(v)^{ {j}_1 \tau+  {j}_2 (1-\tau) } f(v) dv \\
&\left(\int_{\mathbb{R}^d}\psi(v)^{{j}_1} f(v) dv\right)^\tau\, \left(\int_{\mathbb{R}^d}\psi(v)^{{j}_2} f(v) dv \right)^{1-\tau}= m_{{j}_1}^\tau[f]\, m_{{j}_2}^{1-\tau}[f]\,.
\end{align*}
\end{proof}

\subsection{Moment estimates for the collision operator in $L^1_{\ell}(\R^d)$}\label{Coll-mom-estimates}

This task focus in obtaining   Lebesgue moments estimates  for the collisional integral $Q_\gamma(f,f)](v) $ associated to the Boltzmann flow, by estimating  its $\ell^{th}$-order Lebesgue weighted  integrability in probability space $v\in\R^d$ and $\ell$ a real valued number.  That is,  to estimate $m_k[Q(f,f)]= \int_{\mathbb R^d}Q(f,f)(v) \langle v\rangle^{2k} dv$  independently of time.
This is a delicate point as the goal is not only to find bound depending on $q$ and the problem data, but also give an rather acquire description of the dependance of this bound with respect to the coerciveness factor $\mu_r$ obtained by the Angular Averaging Lemma~\ref{BGP-JSP04}.
Previous results along these pages from Subsections~\ref{sec:mom} and ~\ref{L^1-MM}, namely Lemma~\ref{PovznerII} on Angular Averaging for the Binary Weight Function, Lemma~\ref{mom-lemma-I} on  Lebesgue moment's estimates, Lemma~\ref{Lemma:mom-Holder} on  Moment interpolation inequalities and moments lower bounds \eqref{momentineqlow}, are the starting points of the following Theorem~\ref{mom-coll-op} and its proof  developing the refined  bounds.

\medskip

\begin{thm}[{\bf Lebesgue Moments of the collisional integral for hard potentials}] \label{mom-coll-op}

Let $f(v)\in L^1_{\ell}(\R^d)$ and $Q_\gamma(f,f)](v) $ to the Boltzmann collision operator  as posed in \eqref{collision3} and \eqref{collision3.2}, with the   with the collision kernel or transition probability  $B(|u|, \hat u\cdot\sigma)= \Phi(|u|) b(\hat u\cdot\sigma)$,  with $\Phi(u)$ satisfying conditions \eqref{pot-Phi_1}, \eqref{pot-Phi_2.0} and \eqref{pot-Phi_2} for $\gamma\in (0,2]$ and  the angular transition part $b$  integrable in $L^p(\mathbb{S}^{d-1}), \ p\in[1,\infty)$.

Set $\ell=2sk$, for  $s\in (0,1]$ and integer $k$, to define the parameter
\begin{equation}\label{ck}
\end{equation}
then,   for any reference function $f_0(v)\in L^1_{\ell}(\R^d)$, and any other $f(v)$ satisfying $m_i[f]=m_{i\frac{\ell}2}[f] = m_{i\frac{\ell}2}[f_0]$, $i=0,2$,  the $sk^{th}$-Lebesgue moment of the Boltzmann Collision operator is majorized by
\begin{align}\label{Qmoment-0}
m_{sk}[Q_\gamma(f,f)]  &= \| Q_\gamma(f,f)\|_{L^1_{2sk}(\R^d)}= \int_{\mathbb R^d}Q_\gamma(f,f)(v,t) \langle v\rangle^{2sk} dv \\
&\le \mathcal{L}(m_{sk}[f]):=
\B_{  sk}\  -\  \frac{A_{{  s}\kc}}2\,  {m^{-c }_{1}[f_0]}  m^{1+c}_{  sk}[f],
\ \ \text{for any } \ \ k\ge \kc>1, \text{and}\    sk\ge s\kc >1 . \nonumber
\end{align}

The constants $A_{s\kc}$ and  $B_k$ are determined solely from the  Cauchy problem data associated  for the Boltzmann flow stated in Theorem~\ref{CauchyProblem} and the order of $k$ of the Lebesgue polynomial moment. They are
\begin{align}\label{moment-factors}
& A_{{  s}\kc}=   c_{\Phi } 2^{-\frac{\gamma}2} \,(\|b \|_{L^1(\mathbb S^{d-1})} -\mu_{{  s}\kc}) \,{m_0[f_0]} \quad \text{the coercive constant, }    \nonumber \\
\text{ and }   \qquad\qquad&\ \\
& \B_{  sk}:=
%2\left(   C_{\Phi}  2^{(sk-1 +\frac\gamma 2)}   \mu_{{  sk}}  +\|b\|_{L^1(\mathbb{S}^{d-1})} \right)   \mathcal{C}_{  sk}\\
%\qquad =
C_{\Phi} \left(   {2^{sk+\frac\gamma 2}}   \mu_{{  sk}}  + 2\|b\|_{L^1(\mathbb{S}^{d-1})} \right)   \mathcal{C}_{  sk} :=\beta_{  sk} \, \mathcal{C}_{  sk}   \quad \text{the  upper bound constant,} \nonumber
\end{align}
both depending on   the constant $k^{th}$-contracting factor  $\mu_{{  sk}}\in (0,1)$  calculated in the Angular Averaging Lemma~\ref{BGP-JSP04},  and the data  associated to the Cauchy Problem posed in Theorem~\ref{CauchyProblem}.

In particular,  there exists a rate $\kb=\kb(\|b \|_{L^1(\mathbb S^{d-1})} ,m_0[f_0])$, such that for any  $k\ge  \kmax:=\max\{\kb,\kc\} \ge \kmin:=\min\{\kb,\kc\}>1, $
the quotient
\begin{align}\label{delta-k}
\delta_{  sk}\ &:= \ \frac{ A_{{  s}\kc} }{\beta_{  sk}  } \le \frac{ \|b\|_{L^1(\mathbb{S}^{d-1})}  \,m_0[f_0] }{ 2^{sk +\gamma } \mu_{ sk}  + 2^{1+\frac\gamma2}\|b\|_{L^1(\mathbb{S}^{d-1})}} <1.
\end{align}

Hence, the constant $\mathcal{C}_{  sk}$  estimated by the moments interpolation parameters is estimated by above by the constant $\mathcal{C}^{\text{up}}_{  sk}$, related as follows,
%$\beta_k \left(\frac{ \beta_k}{ A_{s\kc} }\right)^{ \ttp(k)}  \, \ttm^\frac{c}{c+1}_{1,k}[f_0] ,  \  \ \text{for any}  \ k\geq \{\kb;\kc\} >1,$

%\begin{align}\label{e max}
%     e_{\text{max}}:=  e_{\text{max}\{e_{\theta}, e_{\alpha} \} } = \begin{cases}
% e_{\alpha}  \quad \text{for } k \ge \kgamma,\\
%    e_{\theta} \quad \text{for } k < \kgamma.
%    \end{cases}
%    \end{align}
\begin{align}\label{mathcal-C-kc}
\mathcal{C}_{sk}
%:=\max\Big\{        \delta_{  sk}^{- \frac{\theta_{k}}{1-\theta_{k}} }\, {C^{\frac{1}{1-\theta_{k}}}_{1,{k}}[f_0]}\, ,\,  \delta_{  sk}^{ -\frac{\alpha_{k}}{1-\alpha_{k}} }\,{C^{\frac{1}{1-\alpha_{{k}}}}_{2,{k}}[f_0]}  \Big\}\\
:=  \ttm_{1,k}[f_0] \, R_{\C}(\delta^{-1}_{sk})  =  \ttm_{1,k}[f_0] \, R_{\C}(\beta_{sk}/ A_\kc)
%=: \mathcal{C}^{\text{up}}_{  sk}
\end{align}
\begin{align}\label{RateC}
R_{\C}(\delta^{-1}_{sk})& :=   R_{\C}(\beta_{sk}/ A_\kc) :=
\left[ \left(   \frac{ \beta_{  sk}  } { A_{{  s}\kc}} \right)^{e_\alpha}   \mathbb{I}_{k\geq \kgamma} +
\left(  \frac{ \beta_{  sk} } { A_{{  s}\kc}} \right)^{e_\theta}   \mathbb{I}_{k<\kgamma}\right], \quad
\nonumber \\
\text{for}  \ \ e_\alpha=\frac1{c_\gamma(sk)}, \quad  &\text{and}\quad\  e_\theta=\left({sk}-1\right) \left(1+c_\gamma(sk) \right)^2 +c_\gamma(sk),
\quad\text{for }  \  s\kgamma:= \frac{1+(\gamma/2)^2}{1-\gamma/2} >1,  \end{align}
for  the factor  $\kgamma$ depending only on the potential rate $\gamma$ and  on fixed moment order $sk\ge s\max\{\kb,\kc\}=s\kmax >1$, with $\kb$ depending on $m_0[f_0]$ and $b(\hat u\cdot\sigma)$.

The factor $\ttm_{1,k}[f_0] $ depends order of moment $k$ and  on data, namely,   the invariant moments $m_0[f_0]$ and $m_1[f_0]$ and $\gamma$ ,
\begin{align}\label{invariant k factor}
0<\ttm_{1,k}[f_0] :=  \max\left\{   m_1^{2c +1}[f_0] \, ;\,    m^{{  sk}+\frac{\gamma}2}_{1}[f_0]   m^{(1+\frac{\gamma}2)\left( 1+c\right)}_{0}[f_0]  \right\}
\end{align}

\medskip

Finally, combining  \eqref{moment-factors}, \eqref{delta-k} and \eqref{mathcal-C-kc},  set  $\B_{sk}:= \beta_{sk} \, \mathcal{C}_{sk}$,  enables the definition of the unique equilibrium state associated to the operator $ \mathcal{L}(m_{sk}[f])(t)$ from \eqref{Qmoment-0} defined as upper bound of the $k^\th$-Lebesgue moment of the collisional integral as to be the unique root of
\begin{align}\label{equi root}
\mathcal{L}(\mathbb{E }_{k,\gamma}[f]) &= 0,
\end{align}
with $\mathbb{E }_{k,\gamma}[f]$ is the unique equilibrium state, fully characterized by
\begin{align}\label{RateE}
\mathbb{E}_{k,\gamma} &:=  m_1^{\frac{c}{1+c}}[f_0]\left(\frac{\B_{sk}}{A_\kc}\right)^{\frac1{1+c}} = m_1^{\frac{c}{1+c}}[f_0]\left(\frac{\beta_{sk} \C_{sk}}{A_\kc}\right)^{\frac1{1+c}} \nonumber\\
%   &\!\!\!\!\!\!\!\!\!= \! \left( m_1\,\ttm_{1,k}\right)^\frac{c}{c+1}\!\![f_0]\,  %\beta_{  sk}^{\frac{1}{1\!+\!c_\gamma(sk)}}
% \left[ \left( \frac{ \beta_{  sk}}{ A_{{  s}\kc} }\right)^{\frac{1 }{c} } \mathbb{I}_{ k\ge \kgamma }+
%  \left(\frac{ \beta_k}{ A_{{  s}\kc} }\right)^{sk+  (sk-1) c }  \mathbb{I}_{ k< \kgamma } \right]. \nonumber\\
&\, =  \left( m_1\,\ttm_{1,k}\right)^\frac{sk-1}{sk-1+\gamma/2}\![f_0] \, := \ R_{\mathbb E}(\delta^{-1}_{sk}) \ := \,R_{\mathbb E}(\beta_{sk}/ A_\kc)
\\
\text{with} \quad R_{\mathbb E}(\delta^{-1}_{sk}) &:= \,R_{\mathbb E}(\beta_{sk}/ A_\kc) :=\left[ \left( \frac{ \beta_{  sk}}{ A_{{  s}\kc} }\right)^{\e_\alpha} \mathbb{I}_{ k\ge \kgamma } \!+\!
\left(\frac{ \beta_{sk}}{ A_{{  s}\kc} }\right)^{sk+\frac\gamma2} \mathbb{I}_{ k< \kgamma } \right],  \nonumber
\end{align}
after following the analog notation from \eqref{RateC}.

The rate factors  $R_\C(\delta^{-1}_{sk})$ and  $R_\mathbb E(\delta^{-1}_{sk})$ defined in   \eqref{RateC} and \eqref{RateE}, respectively,  depend on mass and energy conserved quantities $m_0[f]$ and  $m_1[f]$, the integrable angular transition $b(\hat u\cdot\sigma)$,  potential rate $\gamma$, and Lebesgue moment order $sk$.
\end{thm}

\smallskip

\begin{remark}
The statement of  Theorem~\ref{mom-coll-op}   replaces the definitions of   $A_{s\kc}$  and  $B_k$  in \eqref{moment-factors},  to  $A_{s\kc}=   c_{lb}\,(\|b \|_{L^1(\mathbb S^{d-1})} -\mu_{{  s}\kc}) \,{m_0[f_0]}$ and
$\B_{  sk}:= C_{\Phi}  {2^{sk+\frac\gamma 2}}   \mu_{  sk} \,  \mathcal{C}_{  sk} :=\beta_{  sk} \, \mathcal{C}_{sk},$ respectively, and so
the new $\beta_{sk}:=C_{\Phi}  {2^{sk+\frac\gamma 2}}   \mu_{  sk}.$
\end{remark}

\smallskip

\begin{proof}[Proof  of Theorem~\ref{mom-coll-op}]

First note that  if $f(v)\in L^1_{2k}(\R^d)$, for $k>1$,  $m_1[f]=m_1[f_0]$, then $f(v)\in L^1_{2sk}(\R^d)$ as $s\in(0,1]$. Equivalently, for $f$ non-negative,  $f(v)\in L^1_{2sk}(\R^d) = m_{sk}[f] $, and so $m_{sk}[f]$ is finite. In addition we assume that $f $ is positive, non-singular measure, so $0<m_0[f]  <m_1[f]$.

The  $sk^{th}$-Lebesgue  moment of the  collisional integral was already estimated
in the right hand side the corresponding moment recursion Ordinary Differential Inequalities (\ref{L^1-MM-1},\ref{momentineq}) and by below in \eqref{momentineqlow} for   any fixed  $k$,  such that  $0<s\le 1< sk <\infty$, after making use of the Angular Averaging
Lemma~\ref{BGP-JSP04}, where the contractive constant $\mu_{sk}$ was introduced.

%{\em Case for $0<\gamma\leq 2$. } %We first prove  estimates   \eqref{mom-prop} for $\gamma>0$.
The proof consists in  estimating just  the moment  of the collisional integral appearing in the right hand side  of the time dependent equations \eqref{momentineq}. These bounds  are obtained  by means of
interpolation  estimates  from \eqref{mom-Holder}, for any  $\gamma\in(0,2]$ and ${\kc}>1$,  that control all its positive terms  bounded by above by the higher order moment $m_{k+\gamma/2}[f](t)$,
while controlling the negative contributions from below by coercive estimates proportional	to $m_{k}^{1+ \gamma/(2(sk-1)}[f]$, both  multiplied by  constants that  only  on the moment order ${\kc}>1$ and its corresponding ${\kc}^{th}$-contracting factor $\mu_{\kc}$, as well as on the mass and energy  $m_0[f]=m_0[f_0]$,  $m_1[f]=m_1[f_0]$,  where $f(v)$ is fixed.

Hence, these collision operator moment estimates are obtained from setting  the non-linear factor
\begin{align}\label{little c}
0< c\ =\ c_\gamma(sk)\ :=\ \frac{\gamma}{2(sk-1)},  \qquad \text{with}\ \gamma \in (0,2]  \ \text{and}\ sk\geq\s\kc>1 \ge s>0,
\end{align}
that controls the maximum super linear growth to be determined by a lower bound of the moments of the loss operator.
Thus,  while a Jensen's   inequality is enough to control part of it,  the use of interpolations enables a good control for all the terms, since  while mower moment of $f$ are associated to the conservation properties of the Boltzmann flow,  all other moments above $sk>1$ will have estimates that are easily obtained from interpolation arguments.

Therefore,  starting from the following interpolation, using the short $c$ notation for $c_\gamma(sk)$,
\begin{align}\label{mom-interpol}
m_{1+\gamma/2}[f]  &\leq m^{\frac{1}{1+c}}_{1}[f_0]\, m^{\frac{c}{1+c}}_{sk+\gamma/2}[f]\, ,\qquad \ \ \ \ \
\text{and, } \\
m_{sk-1}[f] & \le m^{\frac{s+\gamma/2}{sk+\gamma/2}}_{0}[f_0]\, m^{\frac{sk-1}{sk+ \gamma/2}}_{sk+\gamma/2}[f] = m^{\frac{s+\gamma/2}{sk+\gamma/2}}_{0}[f_0]\, m^{\frac{1}{sk/(sk-1)+ c}}_{sk+\gamma/2}[f] \,. \nonumber
\end{align}
With the first term in the right side of the inequality \eqref{momentineq} is controlled as
\begin{align}\label{first-mom-holder}
m_{s+\gamma/2}[f]m_{sk-1} [f]&\leq C_{1,k}[f_0]\,m^{\theta_{k}}_{sk+\gamma/2}[f]\,, \nonumber\\
\text{with}  \qquad  \qquad   \qquad  \qquad  \qquad  &\\
C_{1,k}[f_0]= C_{1,sk}[f_0]\!:=\!  m^{\frac{1}{1+c}}_{1}\![f_0] m^{\frac{s+\gamma/2}{sk+\gamma/2}}_{0}\![f_0],  \quad  &\text{and} \quad \theta_{k}\!:=\!\frac{c}{1\!+\!c}\! +\! \frac{1}{\frac{sk}{(sk-1)}\!+\! c} \!=\!
\frac{\gamma/2}{\gamma/2 \!+\!sk\!-\!1}\! +\! \frac{sk\!-\!1}{sk\!+\!\gamma/2}\!   <\!1.\nonumber
\end{align}

Furthermore, $m_{\gamma/2}[f]\leq m_{1}[f] = m_{1}(f_0)<\infty$ and, interpolating again,
\begin{align}\label{second-mom-holder}
&m_1[f_0] m_{sk-1+\gamma/2}[f]\leq m_1[f_0] \, m_{sk}[f] \leq C_{2,sk} [f_0]  \,m^{\alpha_{k}}_{sk+\gamma/2}[f]\,, \nonumber \\
&  \text{with}
%   \  { C_{2,k}[f_0] :=  m^{\frac{c}{1+c}+1}_{1}[f_0]  \ \ \text{and}\ \      \alpha_{k}:=\frac{c}{1+c}<1}\,.
\ \ { C_{2,sk}[f_0] =C_{2,sk}[f_0] :=  m^{\frac{c}{1+c}}_{1}[f_0]  \ \ \text{and}\ \      \alpha_{k}:=\frac{1}{1+c}<1} \,.
%m_1[f_0] m_{k-1+\gamma/2}[f]\leq m_1[f_0] \, m_{k}[f]\leq C_{2,k} [f_0]  \,m^{\alpha_{k}}_{k+\gamma/2}[f]\,, \nonumber \\
%  \text{with}    \ \ \  C_{2,k}[f_0] :=  m^{\frac{\gamma/2}{k-1+\gamma/2}+1}_{1}[f_0]  \ \ \text{and}\ \      \alpha_{k}:=\frac{k-1}{k-1+\gamma/2}<1\,.
\end{align}

Furthermore,  both $\{m_{\gamma/2}[f]\,;\, \m_s[f]\}\leq m_{1}[f] = m_{1}(f_0)<\infty$,  interpolating again
\begin{align}\label{inter msk}
m_{sk}&< m_1^{\frac{c}{1+c}}\, m_{sk +\gamma/2}^{\frac{1}{1+c}}
\end{align}
\begin{align}\label{second-mom-holder}
&m_1[f_0] m_{sk-1+\gamma/2}[f]\leq m_1[f_0] \, m_{sk}[f]
\leq C_{2,k} [f_0]  \,m^{\alpha_{k}}_{sk+\gamma/2}[f]\,, \nonumber \\
&  \text{with}
%   \  { C_{2,k}[f_0] :=  m^{\frac{c}{1+c}+1}_{1}[f_0]  \ \ \text{and}\ \      \alpha_{k}:=\frac{c}{1+c}<1}\,.
\ \ { C_{2,sk}[f_0] =C_{2,sk}[f_0] :=  m^{\frac{2c+1}{1+c}}_{1}[f_0]  \ \ \text{and}\ \      \alpha_{k}:=\frac{1}{1+c}<1} \,.
%m_1[f_0] m_{k-1+\gamma/2}[f]\leq m_1[f_0] \, m_{k}[f]\leq C_{2,k} [f_0]  \,m^{\alpha_{k}}_{k+\gamma/2}[f]\,, \nonumber \\
%  \text{with}    \ \ \  C_{2,k}[f_0] :=  m^{\frac{\gamma/2}{k-1+\gamma/2}+1}_{1}[f_0]  \ \ \text{and}\ \      \alpha_{k}:=\frac{k-1}{k-1+\gamma/2}<1\,.
\end{align}

Therefore, from \eqref{first-mom-holder}, \eqref{second-mom-holder}, and the right hand side  moment inequality \eqref{momentineq},  the following estimate hold
\begin{align}\label{hardpotmom-1}
m_{sk}[Q(f,f)]  &\leq \left(C_{\Phi}2^{(sk-1)(1+c_\gamma(sk))} \mu_{sk} +\|b\|_{L^1(\mathbb{S}^{d-1})}\right)\left[ C_{1,k}[f_0]\,m^{\theta_{k}}_{sk+\gamma/2}[f] +  C_{2,k}[f_0]\,m^{\alpha_{k}}_{sk+\gamma/2}[f]\right]   \nonumber\\
&\qquad - c_{\Phi}2^{-\gamma/2}( \|b\|_{L^1(\mathbb{S}^{d-1})}   -\mu_{s\kc})\,m_0(f_0)\,m_{sk+\gamma/2}[f],
\end{align}
valid for any $\gamma\in(0,2]$, $ 0<s\le 1$, and $k\geq\kc>1.$

Note that  these $sk^{th}$-moment estimates provide a {\em coercive factor} independent of $k$, defined by
\begin{align}\label{Akc}
A_{s\kc} :=
% c_{\Phi}2^{-\frac{2+\gamma}2}( \|b\|_{L^1(\mathbb{S}^{d-1})}   -\mu_\kc)\,m_0(f_0) = \frac14
c_{\Phi}2^{-\gamma/2}( \|b\|_{L^1(\mathbb{S}^{d-1})}   -\mu_{s\kc})\,m_0(f_0), \qquad \text{for }\  sk>s\kc> 1\ge s>0.
\end{align}

\medskip

Next,  the following   key point enables the control of the right hand side of inequality \eqref{hardpotmom-1} by proving that, after the use of a further interpolation, it is possible to obtain an upper  bound for the positive contribution  only depending on data and $k$.

For such task,  take  the couple of  strictly less that unity exponents $\theta_{k}$ and $\alpha_{k}$ from \eqref{first-mom-holder} and \eqref{second-mom-holder}, respectively,   as well as constants  $C_{1,k}[f_0]$ and $C_{2,k}[f_0]$  defined in    \eqref{first-mom-holder} and  \eqref{second-mom-holder}, respectively,  and invoke the $\delta$-Young's inequality, namely, $|a\,b|\leq \delta^{-p/q} p^{-1} |a|^p + \delta q^{-1} |b|^q$, for $\delta>0$ to be chosen sufficiently small,  and $p^{-1}+q^{-1} =1$, making  the following choice, for $c=c_\gamma(sk)$ and $\theta_k$ from \eqref{little c} and \eqref{first-mom-holder}, respectively,
\begin{align*}
&a=C_{1,k}[f_0], \qquad \qquad\qquad  \ \ b=  m^{\theta_{k}}_{k+\gamma/2}[f]\,,
%&a=\left({C}_{\Phi}2^{(k-1)(1+c)} \mu_k +1 \right)\, C_{1,k}[f_0], \qquad \qquad\qquad  \ \ b=  m^{\theta_{k}}_{k+\gamma/2}[f]\,,
\nonumber\\
\text{with} \ \ &p_{1}= \frac{1}{1-\theta_{k}}     > 1,
\  \text{and its convex conjugate}\ \  q_{1}=\frac{1}{\theta_{k}} >1,
\end{align*}
to obtain, again using the short $c$ notation for $c_\gamma(sk)$,
\begin{align}\label{hardpotmom-22}
&C_{1,k}[f_0]\, m^{\theta_{k}}_{k}[f] \leq  \delta^{-\frac{ \theta_k}{1-\theta_k} } C^{\frac{1}{1-\theta_{k}}}_{1,k}[f_0] +
\delta \, m_{k+\gamma/2}[f],
%= \delta^{-\frac{ \theta_k}{1-\theta_k} } C^{\frac{1}{1-\theta_{k}}}_{1,k}[f_0] \, ,
\nonumber\\
\text{for}\  \ &\frac{1}{1-\theta_{k}} =   (1+ c)(sk+ (sk-1) c) =  \left(1+ c \right) \left(sk+{\gamma}/2 \right), \\
\text{and} \ \
&\frac{ \theta_k}{1-\theta_k}=\left(sk+ (sk-1) c\right) c+ (1+ c)(sk-1) +  c=  (sk-1)(1+ c)^2 + c , \nonumber
\end{align}
which controls the first term  of  in \eqref{hardpotmom-1}.
A similar control for the second term  of the positive contribution in  \eqref{hardpotmom-1}, can be obtained choosing $c$ and $\alpha_k$ from \eqref{little c} and \eqref{first-mom-holder}, respectively, to yield
\begin{align*}
&a=  C_{2,k}[f_0], \qquad \qquad\qquad  \ \ b=  m^{\alpha_{k}}_{k+\gamma/2}[f],  \nonumber\\
\text{with} \ \ & p_{2}= \frac{1}{1-\alpha_{k}}=\frac{1+ c}{ c} >1,\  \text{and convex conjugate}\ \  q_{2}=\frac{1}{\alpha_{k}}=1+c >1,
\end{align*}
to obtain  $\frac{\alpha_{k}}{1-\alpha_{k}} = \frac{1}{ c} = \frac{ sk-1}{\gamma/2} $  ,

Hence, combining with  \eqref{second-mom-holder},
\begin{align}\label{hardpotmom-3}
C_{2,k}[f_0]\, m^{\alpha_{k}}_{{  sk}+\gamma/2}[f]  &\leq  \delta^{-\frac{\alpha_k}{ 1-\alpha_k}}
C^{\frac{1}{1-\alpha_{k}}}_{2,k}[f_0] + \delta \, m_{{  sk}+\gamma/2}[f] \\
&= \delta^{-\frac1c}
m_1^{\frac{c+1}{c}} [f_0] + \delta \, m_{{  sk}+\gamma/2}[f]. \nonumber
\end{align}

These previous calculations from  moment interpolations enable  the formulation os a new upper bound for  $sk^{th}$-moments Inequalities \eqref{hardpotmom-1},     by   estimating  upper bounds of the maximum of estimates  obtained in  \eqref{hardpotmom-22}, and \eqref{hardpotmom-3}, respectively.

For the sake of simplicity of the calculations that follow, we set

\begin{align}\label{betak}
\beta_{  sk}:= 2{C}_{\Phi}\left(  2^{(sk-1)(1+ c)} \mu_{  sk} +\|b\|_{L^1(\mathbb{S}^{d-1})} \right) = {C}_{\Phi}\left(  2^{(sk +\frac\gamma 2)} \mu_{  sk} + 2\|b\|_{L^1(\mathbb{S}^{d-1})} \right) .%= 2\left(  {C}_{\Phi}2^{(k-1)(\frac{2(k-1)+\gamma}{2(k-1)})} \mu_{k} +1 \right).
\end{align}

The corresponding $k^{th}$-moments bounds of the collisional  from \eqref{hardpotmom-1} are majorized the the following Ordinary Differential Inequality
\begin{align}\label{hardpotmom-3.1}
m_k[Q_\gamma(f,f)] &\leq \beta_{  sk} C_{sk} \  -\  \Big( A_{{  s}\kc} - \frac{\beta_{  sk}}2{\delta}\Big)\,m_{{  sk}+\gamma/2}[f]\, .
\end{align}

\medskip

The next step consists in choosing a suitable $\delta=\delta_{  sk}$,  and associated  constant  ${C}_{  sk}$ to be defined below,  that will preserve the coerciveness to secure a strict absorption effect from the higher order moment $m_{{  sk}+\gamma/2}[f]$.  It is very  important to notice that these choices only depend on the data, that are the  potential rate $\gamma\in (0,2]$, the moment order  for any fixed fixed $sk\ge s\kc >1$, and  the initial mass $m_0[f_0]$ and energy $m_1[f_0]$, both time invariants by the Boltzmann flow.

Indeed, this is shown in the following Proposition.

\begin{problem2}\label{prop delta k} There exist a large enough moment order $\kb=\kb(m_0[f_0], b(\hat u\cdot\sigma))$,  such that  time independent factor  $\delta_{sk}$,  defined  for  any $sk\geq {s\kc}>1$,  satisfies
\begin{align}\label{hardpotmom-4}
0 < \delta := \delta_{  sk}\   = \frac{  A_{{  s}\kc} }{ \beta_{sk}} \ <\ 1.
%0 < \delta := \    \frac{  A_\kc }{ 2\left( {C}_{\Phi}2^{(k-1)(1+c)} \mu_{k} +1\right) } \ <\ 1.
\end{align}
\end{problem2}

\begin{proof} This property is easy to prove since  this choice is very natural, after evaluating the above  quotient using \eqref{Akc} and \eqref{betak}, short labels for $A_{{  s}\kc}$ and $\beta_{  sk}$, respectively. Indeed,  the denominator in the above defenition of $\delta$, can be shown to grow as large as needed, but means of the angular averaging Lemma~\ref{BGP-JSP04}. Indeed,   from  \eqref{mu_r},   \eqref{Povp1} and  \eqref{Povp2},
the contracting factor $\mu_{  sk}< \mu_1 =\|b\|_{L^1(\mathbb{S}^{d-1})}$, so on one hand
\begin{align}\label{k0-1}
0< A_{{  s}\kc} =   c_{\Phi}2^{-\frac\gamma2}( \|b\|_{L^1(\mathbb{S}^{d-1})}   -\mu_{{  s}\kc})\,m_0[f_0] <   c_{\Phi}2^{-\frac\gamma2}\|b\|_{L^1(\mathbb{S}^{d-1})}  \,m_0[f_0].
\end{align}
On the other hand, from the definition \eqref{mu_r} for $\mu_{sk}$ after multiplying both sides by $2^{sk+\frac{\gamma}2}$, yield the following estimate
\begin{align}\label{k0-2}
\lim_{sk\to \infty} 2^{sk+\frac{\gamma}2} \mu_{sk} \ge\lim_{sk\to \infty} \,  2^{\frac{\gamma}2} \int_{ \mathbb{S}^{d-1} }
\bigg( \left| 1 - \hat u\cdot\sigma \right|^{sk} +   \left| 1 + \hat u\cdot\sigma  \right|^{sk}
\bigg)  b(\hat u\cdot\sigma)\, d\sigma = \infty \,
\end{align}
%$\mu_k=  4  \|b\|_{L^{\infty}(\mathbb{S}^{d-1})}| \mathbb{S}^{d-2}| ((1+k)^{-1}), $

In particular, since  data parameters  from \eqref{pot-Phi_1} assert  $c_\Phi \leq C_\Phi$,  there exists a
\begin{align}\label{bf kb}
\kb = \kb(m_0[f_0], b(\hat u\cdot\sigma)),
\end{align}
large enough such that, for all  ${sk} \ge s\max\{\kb,\kc\}=:s\kmax>1$,
\begin{align}\label{choose kb}
0 <  \delta_{  sk}\   = \frac{  A_{{  s}\kc} }{ \beta_{sk}} \le\frac{  c_{\Phi} \|b\|_{L^1(\mathbb{S}^{d-1})}  \,m_0[f_0] }{{C}_{\Phi} \left( 2^{sk +\gamma } \mu_{{  sk}} + 2^{\frac\gamma2+1}\|b\|_{L^1(\mathbb{S}^{d-1})}\right) } <
\frac{ \|b\|_{L^1(\mathbb{S}^{d-1})}  \,m_0[f_0] }{ 2^{sk +\gamma  } \mu_{ sk} }< 1.
\end{align}
\end{proof}

\medskip

Finally, invoking the interpolation from \eqref{inter msk},
%one last time
yields the following lower bound  for  $m_{{  sk}+\gamma/2}[f]$, \begin{align}
\label{interpol-2}
\  m^{-c}_{1}[f_0]\, m_{  sk}^{1+c}[f]
\leq  m_{{  sk}+\gamma/2}[f]\, ,
\end{align}
%from below by the super linear form associated to the ${  sk}^{th}$-Lebesgue moment  taking the form $m_{  sk}^{1+c}[f]$,
%%given by
%% \begin{align}\label{interpol-1}
%%m_{k}[f] &\leq m^{\frac{c}{1+c}}_{1}[f_0]\, m^{\frac1{1+c}}_{k+\gamma/2}[f], \quad \text{for}\ k\geq {\kc}>1\ \ \text{and}\ \  \gamma\in(0,2]\,,
%which implies, after recalling from \eqref{little c} that $c({  sk})$

%{\color{red} since,  without loss of generality, we can assume $m_1[f] =m_1[f_0] \ge  1$,  for $k\geq \kc >1$.  Note that if $m_1[f]< 1$ then since $0<m_[f_0] = m_0[f] \le  m_1[f_0]$, it is enough to renormalized the initial data $f_0(v)$ by its positive average to set $\tilde{f}_0(v):= m^{-1}_1[f_0](v)$, implies that the corresponding $m_1[\tilde f] \ge 1$ and so the Cauchy Problem for the Boltzmann flow, Theorem~\ref{CauchyProblem}, is solved for $\tilde f(v,t)$. }
%so that the positive terms in  in the differential inequality \eqref{hardpotmom-1} are controlled by
%\frac1{\delta^{{p_1}/{q_1}}}  C^{k-1+\gamma/2}_{1,k}(f_0)&+\frac1{\delta^{{p_2}/{q_2}}}  C^{\frac{k-1+\gamma/2}{\gamma/2}}_{2,k}(f_0) \le  \left(\frac{(1-\mu_k)}4 c(f_0) \right)^{-\frac{\gamma/2}{k-1} } C_{3,k}(f_0)  \, , \\[7pt]
%\text{with} \qquad &C_{3,k}(f_0) :=\max\{ \frac{1}{\delta^{{p_{1}}/{q_{1}}}} C^{k-1+\gamma/2}_{1,k}(f_0) ; \frac{1}{\delta^{{p_{2}}/{q_{2}}}} C^{\frac{k-1+\gamma/2}{\gamma/2}}_{2,k}(f_0) \}\,.
%since $\min\{ k-2+\gamma/2 ; \frac{\gamma/2}{k-1} \}= \frac{\gamma/2}{k-1} $ for $k>1$.
Which gives a bound from above to the righthand side in\eqref{hardpotmom-3.1},   with constants  depending the moment order $k$, on the potential rate $\gamma$, the first two moments of the initial data $f_0$ and the integrability properties of the angular function $b(\hat u\cdot\sigma)$. In particular, this estimate   yields a super-linear  negative term for the  Lebesgue polynomial moment $m_k[f]$, namely
\begin{align}
\label{hardpotmom-3.11}
m_k[Q(f,f)] &<
\beta_{  sk}  C_{sk}  -\  \frac{A_{{  s}\kc}}2 \,{m^{-c}_{1}[f_0]}\,m^{1+c}_{  sk}[f] \ =:\  \mathcal{L}_k(m_k[f])(t).
\end{align}

Hence,  the  estimates take the simple form
{where} the $k$-independent  coercive factor $A_{{  s}\kc}$, defined on \eqref{Akc}, and  the $sk$-dependent positive factor $ \beta_{  sk}   \mathcal{C}_{  sk}$  takes the product  the factor $\beta_{sk}$ from \eqref{betak}, with $k\ge \kb$, and
$ \mathcal C_{  sk} $  is characterized and estimated by above
\begin{align}\label{hardpotmom-3.2}
&C_{sk}= \max\Big\{ \!       \delta^{- \frac{\theta_{k}}{1-\theta_{k}} }\, {C^{\frac{1}{1-\theta_{k}}}_{1,{k}}[f_0]}\, ,\,  \delta_{sk}^{ -\frac{\alpha_{k}}{1-\alpha_{k}} }\,{C^{\frac{1}{1-\alpha_{{k}}}}_{2,{k}}[f_0]}  \Big\}\nonumber\\[4pt]
\max\left\{ \!{C^{\frac{1}{1-\theta_{k}}}_{1,{k}}[f_0]};  {C^{\frac{1}{1-\alpha_{k}}}_{2,{k}}[f_0]} \right\}
\max\left\{  \delta^{- \frac{\theta_{k}}{1-\theta_{k}} } ;   \delta_{sk}^{- \frac{\alpha_{k}}{1-\alpha_{k}} }  \right\}  \nonumber\\[4pt]
&  = \ttm_{1,k}[f_0]  \max\Big\{ \delta_{sk}^{- [(k-1)(1+c)^2 +c]} \, ;\,  \delta_{sk}^{-\frac1c}   \Big\},
\qquad\  \text{for any }   sk> \max\{\kb,\kc\}=\s\kmax >1,
\end{align}
with the factor  $\ttm_{1,k}[f_0] $ being the one defined at \eqref{invariant k factor},  in the statement of this theorem.

To this end, the right hand  of \eqref{hardpotmom-3.2}, can be further estimated by calculating  $e_{\text{max}}$ to be the exponent controlling   both, $e_\alpha=e_\alpha(\gamma,sk)$  and $e_\theta=e_\theta(\gamma,sk)$, for  fixed $\gamma\in(0,2]$ and $c=c_\gamma({  sk})=\frac{\gamma/2}{sk-1}$,  obtain by setting
\begin{align}\label{e max}
e_{\text{max}}:=  e_{\text{max},\gamma}  &= \begin{cases}
e_{\alpha}:=e_{\alpha}(\gamma, sk) = {\frac{\alpha_k}{1-\alpha_k}} =  \frac{1}{c}, &\ \ \ \quad \text{for }sk \ge s\kgamma>1, \\[3pt]
e_{\theta}:=    e_{\theta}(\gamma,sk) = {\frac{\theta_k}{1-\theta_k}} ={\left(sk-1\right)\left(1+c\right)^2 +c}, &\ \ \ \quad \text{for } 1<sk < s\kgamma
\end{cases} \nonumber\\[6pt]
&=: \left[ e_{\alpha}    \mathbb{I}_{k\geq \kgamma} + e_{\theta}    \mathbb{I}_{k< \kgamma}\right],
\end{align}
with a fixed $\kgamma$ parameter characterized by how  the order $k$ compared to $\kgamma$,    for any $0<\gamma \le 2$. That means, the parameter $\kgamma$ is exactly calculated  from  setting ${\bf c}=c_\gamma(s\kgamma)=\frac{\gamma/2}{s\kgamma-1}$  and finding the value  for both exponents $ e_{\theta}(k,\gamma)$ and $ e_{\alpha}(k,\gamma)$ to coincide. That is,    $s\kgamma$  only depends on  the hard potential rate exponent $\gamma\in (0,2]$, since
\begin{align}\label{compare delta expo}
e_{\theta} &:=  ({s\kgamma}-1) (1+{\bf c})^2 +{\bf c}  =   \frac1{\bf c} =: e_{\alpha} \ \
\iff \ \
%({s\kgamma}-1)  = \frac{(1-{\bf c}^2) }{{\bf c} (1+{\bf c})^2} = \frac{1-{\bf c}}{{\bf c}(1+{\bf c})}
\frac{\gamma}2 =  \frac{(s\kgamma-1) -\frac{\gamma}2}{(s\kgamma-1) + \frac{\gamma}2} \nonumber \\[4pt]
&\qquad\qquad \qquad \ \iff\  \ \ s\kgamma =  \frac{1+(\gamma/2)^2}{1-\gamma/2} >1.
\end{align}

As a consequence,   taking $\delta_{sk}^{-1} :=   \beta_{sk}\,A^{-1}_{s\kc}  \ >\ 1$ from \eqref{hardpotmom-4},   with $\beta_{sk}$ from \eqref{betak},
the second factor  in \eqref{hardpotmom-3.2} defines a coercive rate quotient
\begin{align}\label{coer-rate-0}
R_{\mathcal C}(\delta_k^{-1}) &:=  \max\Big\{ \delta_{sk}^{- [(k-1)(1+c)^2 +c]} \, ;\,  \delta_{sk}^{-\frac1c}   \Big\}\\
& \equiv  \left[   \delta_{sk} ^{-e_{\alpha}}   \mathbb{I}_{k\geq \kgamma} +
\delta_{sk}^{-e_{\theta}}   \mathbb{I}_{k<\kgamma}\right]
\equiv \left[ \left(   \frac{ \beta_{  sk}  } { A_{{  s}\kc}} \right)^{e_{\alpha}}   \mathbb{I}_{k\geq \kgamma} +
\left(  \frac{ \beta_{  sk}  } { A_{{  s}\kc}} \right)^{e_{\theta}}   \mathbb{I}_{k<\kgamma}\right]
\equiv  R_\C (   \beta_{  sk} / A_{{  s}\kc} ) \nonumber.
\end{align}

It follows that  the right hand side from
\eqref{hardpotmom-3.2}  is  majorized by the  upper estimate, now written in terms of the coercive rate quotient
defined in \eqref{coer-rate-0}
\begin{align}\label{Bkc}
\B_{sk}&:=\beta_{sk} \mathcal {C}_{  sk} :=  \ttm_{1,{  sk}}[f_0] \, \beta_{  sk}
R_{\mathcal C} (   \beta_{  sk} / A_{{  s}\kc} )\\
& \equiv \ttm_{1,{  sk}}[f_0] \, \beta_{  sk}
\left[ \left(   \frac{ \beta_{  sk}  } { A_{{  s}\kc}} \right)^{\frac1{c}}   \mathbb{I}_{k\geq \kgamma} +
\left(  \frac{ \beta_{  sk}  } { A_{{  s}\kc}} \right)^{\left({sk}-1\right) \left(1+c\right)^2 +c}   \mathbb{I}_{k<\kgamma}\right],\end{align}
for any ${  sk}\ge {  s}\max\{\kb,\kc\}=s\kmax$, after gathering the exponents from \eqref{e max} depending on $\kgamma$  from \eqref{compare delta expo}.
\\ % given by (\ref{hardpotmom-3.2}-\ref{hardpotmom-3.22}). \\

The final upper bound for the $sk^{th}$-moment of the collision operator   from \eqref{hardpotmom-3.11} is controlled
by the following   operator $\mathcal{L}_{sk} (m_{ks}[f])$, acting on the $sk^{th}$-Lebesgue moment of $f\in L^1_{2sk}(\R^d)$,
\begin{align}\label{equi-1}
m_{sk}[Q(f,f)] < \mathcal{L}_{sk}(m_k[f])=   \beta_{  sk}  \C_{sk}  -\  \frac{A_{{  s}\kc}}2 \,{m^{-c}_{1}[f_0]}\,m^{1+c}_{  sk}[f],
\end{align}
whose unique root
$\mathbb{E}_{k,\gamma}$ is referred as to the equilibrium state, that is
%\mathcal{L}_k(\mathbb{E}_{c}[f]):=   \beta_{  sk}  \C_{sk}  -\  \frac{A_{{  s}\kc}}2 \,{m^{-c}_{1}[f_0]}\,m^{1+c}_{  sk}[f]=0,
%or equivalently
\begin{align}\label{equi-1.1}
\mathbb{E}_{c_\gamma(sk)} &:= m_1^{\frac{c}{1+c}}[f_0]\left(\frac{\B_{sk}}{A_\kc}\right)^{\frac1{1+c}}.
\end{align}

Written in terms of  \eqref{Bkc}, $\mathbb{E}_{k,\gamma}$   can be fully characterized by the data and $m_0[f]$ and $m_1[f]$  moments by
\begin{align}\label{equi-2}
\mathbb{E}_{c_\gamma(sk)} &:= m_1^{\frac{c}{1+c}}[f_0]\left(\frac{\beta_{  sk} \C_{sk}}{A_\kc}\right)^{\frac1{1+c}}
\equiv m_1^{\frac{c}{1+c}}[f_0]  R^{\frac1{1+c_\gamma(sk)}}_\C(     \beta_{  sk} / A_{{  s}\kc} )\nonumber \\
&=\left( m_1\,\ttm_{1,k}\right)^\frac{c}{c+1}\!\![f_0] \, %\beta_{  sk}^{\frac{1}{1+c}} \,
\left[ \left( \frac{ \beta_{  sk}}{ A_{{  s}\kc} }\right)^{\frac{1+e_\alpha}{1+c} } \mathbb{I}_{ k\ge \kgamma } \!+\!
\left(\frac{ \beta_k}{ A_{{  s}\kc} }\right)^\frac{1+e_\theta}{1+c } \mathbb{I}_{ k< \kgamma } \right]\nonumber\\
&\, = \left( m_1\,\ttm_{1,k}\right)^\frac{sk-1}{sk-1+\gamma/2}\![f_0]
\left[ \left( \frac{ \beta_{  sk}}{ A_{{  s}\kc} }\right)^{\frac{2(sk-1)}\gamma } \mathbb{I}_{ k\ge \kgamma } \!+\!
\left(\frac{ \beta_k}{ A_{{  s}\kc} }\right)^{sk+\frac\gamma2} \mathbb{I}_{ k< \kgamma } \right]\nonumber\\
&=:  \left( m_1\,\ttm_{1,k}\right)^\frac{sk-1}{sk-1+\gamma/2}\![f_0] \, R_\mathbb E(\delta^{-1}_{sk}) \,
\quad \text{for any}
\ \ sk\ge s\kmax,
%&=  \left( m_1\,\ttm_{1,k}\right)^\frac{c}{c+1}\!\![f_0] \,  %\beta_{  sk}^{\frac{1}{1+c}} \,
% \left[ \left( \frac{ \beta_{  sk}}{ A_{{  s}\kc} }\right)^{\frac{1}{c(1+c)} } \mathbb{I}_{ k\ge \kgamma } \!+\!
%  \left(\frac{ \beta_k}{ A_{{  s}\kc} }\right)^{(sk-1)(1+c) +\frac{c}{1+c}}  \mathbb{I}_{ k< \kgamma } \right],
% \nonumber
\end{align}
after defining  $R_{\mathbb E} (\beta_{sk}/A_\kc):=  R^{\frac1{1+c_\gamma(sk)}}_\C(     \beta_{  sk} / A_{{  s}\kc} )\equiv
R_\mathbb{E}(\delta^{-1}_{sk})$. These new  exponents  obtained by taking $e_{\alpha}(sk)$ and $e_{\theta}(sk)$   defined in \eqref{e max},
devided by $(1+ c),$ respectively, for $c=c_\gamma(sk)=\gamma/{(2(sk-1))}$,  yields the simpler exponent associated to the the characterization
of  $\mathbb{E}_{c_\gamma(sk)}$ as follows
\begin{align}\label{equi-2.2}
\frac{1+e_\alpha}{1+c}%=  \frac{1+c}{c(1+c)}
= \frac1c= \frac{2(sk-1)}\gamma,\qquad \text{and}
\qquad \frac{1+e_\theta}{1+c}= sk +\frac \gamma 2.
\end{align}

Thus, the proof on Theorem~\ref{mom-coll-op} is now complete.
\end{proof}
\ \\

\begin{remark}\label{exp-max}
Note that, for example,  in the classical case for {  $s=1$},   $e_{\text{max}}=e_{\theta}$  for $\gamma =2$ and    $e_{\text{max}}=e_{\alpha}$ for values of $\gamma \approx 0^+$. In addition,  in the case of hard spheres corresponding to  $\gamma=1$, the exponent may change depending on the $k^{th}$-moment order, as  $e_{\text{max}}=e_{\alpha}$ for  if $k\ge \kgamma= 5/2 $, but   $e_{\text{max}}=e_{\theta}$ if $k<\kgamma=5/2 $.

In general, it follows that  for any $sk>s\max\{\kb,\kc\}>1$,   the constant  $\mathcal {C}_k$, as characterized in \eqref{hardpotmom-3.2}, can be recasted by
\begin{align}\label{new Ck}
{C}_{  sk}   &\le \mathcal {C}_{  sk}:=\ttm_{1,{  sk}}[f_0]
\left[ \left(   \frac{ \beta_{  sk}  } { A_{{  s}\kc}} \right)^{\frac1{c}}   \mathbb{I}_{k\geq \kgamma} +
\left(  \frac{ \beta_{  sk} } { A_{{  s}\kc}} \right)^{\left({sk}-1\right) \left(1+c\right)^2 +c}   \mathbb{I}_{k<\kgamma}\right]
%&  \ttm_{1,k}[f_0]
\end{align}
\end{remark}

\bigskip

\begin{remark}\label{reduced mathcalC} In the case where  the initial mass $m_0[f_0]=1 <m_1[f_0]$, after rescaling the Boltzmann solution by its initial mass, since $\delta=\delta_{  sk}<1$, then  for  $sk>  \max\{\kgamma,\kb,\kc\}>1$   by condition \eqref{choose kb},  the constant $\mathcal C_{  sk} $   is
\begin{align}\label{hardpotmom-3.3}
\mathcal {C}_{  sk} & :=\left( \frac{ \beta_{  sk} }{ A_{s\kc} }\right)^{\frac{1}{c}}
m_1^{ sk+\frac{\gamma}2 + (1+\frac{\gamma}2)\left( 1+c_\gamma(sk)\right)  } [f_0],
\quad  \text{for any}\ \  sk\ge {  s}\max\{\kgamma, \kb,\kc\}>1\,.
\end{align}

\end{remark}

\bigskip

\begin{remark}\label{kc=eps}
This remark is fundamental to proof the existence of solutions to any moment either propagated or generated  for as long the initial data is a non-negative $f_0\in L^1_{2\kc}$, with  $\epsilon:=2(\kc -1) $, for $\epsilon\in (0,\gamma)$. In particular,  setting the factor  $s=1$, in the moments estimates,
the  $sk^{th}$ - moments of the collision operator  remain valid when taking, for example   $\mu_{k}< \mu_{\epsilon+1}< \mu_1=
\|b\|_{ L^1(\S^{d-1})}=1$ and $m_0[f_0](0)=1$.
With this choice, which suitable for the scalar binary elastic interacting Collisional form, whether both the initial mass $m_0[f_0]$  and the angular transition function $b(\hat u\cdot \sigma)\in  L^1(\S^{d-1})$ is renormalized to unit spherical average, naturally yields the  order $\kb$ and $\kgamma,$ defined in \eqref{bf kb} and \eqref{compare delta expo}, respectively, are larger than $\kc=(2+\epsilon)/2.$

This observation will be brought to readers' attention in the proof of Lemma~\ref{Lemma_ODE-extension2}, that extends the concept of solutions to any initial data $f_0\in  L^1_{2+\epsilon}. $
\end{remark}

\bigskip

\begin{remark}\label{existence thm remark}
The estimates obtained in  Theorem~\ref{mom-coll-op}  for the moments of the collision operator are sufficient  to prove the existence and uniqueness  of the Boltzmann flow solutions in Section~\ref{existence-uniqueness}, as posed in  Theorem~\ref{CauchyProblem} on Section~\ref{hardpotsection},  by means of Ordinary Differential Equations dynamics for time dependent flows in  Banach spaces $L^1_k(\R^d)$.

More specifically,  Theorem~\ref{CauchyProblem} has two parts.  The first one  is to show existence and uniqueness  in $L^1_{\ell}(\R^d)$ for $\ell>2 +2\gamma$.
The second part    is shown in Lemma~\ref{Lemma_ODE-extension2} proving  existence  for lower order initial Lebesgue weights, that is in $L^1_{\ell}(\R^d)$ for $2+ \eps \le \ell <2 +2\gamma$.
\end{remark}

\bigskip

\subsection{Moment Ordinary Differential Inequalities for Boltzmann flow solutions in $L^1_{\ell}(\R^d)\times\R^+$}\label{moments-ODE's}

The following results derives   a priori estimates for the Lebesgue polynomial  moment  time dynamics to solutions  of the Boltzmann flow and rather accurate bounds as functions of the Cauchy Problem data.  These estimates  are, not only  invoked  in the second	part of the of the proof of Theorem~\ref{CauchyProblem} as mentioned in the last Remark~\ref{existence thm remark}, but also are critical for the moment summability  properties that yield  propagation and generation  exponential moments  globally in time with exponential rates fully characterized as a function of the coercive	constant $A_{s\kc}$ from \eqref{moment-factors} and the Cauchy problem data as well.   In addition these  global moment estimates and bounds  enable  global in time propagation estimates for solutions in  $L^p_\ell(\R^d)\times R^+$, for $1\le \ell\le \infty$.

\medskip

\begin{thm}[{\bf Lebesgue Polynomial Moments  \textit{A priori} Estimates for hard potentials}]
\label{propagation-generation}

Let $f(v,t)\geq0$ be a solution of the Cauchy problem associated to the initial value problem for the Boltzmann flow  in Theorem~\ref{CauchyProblem}, i.e.  with collision operator   $Q_\gamma(f,f)](v, t) $ as defined in  Theorem~\ref{mom-coll-op}.

Let $0<c:=c_\gamma(sk):=\frac{\gamma/2}{sk-1}$ be the positive parameter defined in Theorem~\ref{mom-coll-op}, identity \eqref{ck}.  Then,  a super-solution to \eqref{momentineq}  is constructed  by solving the following Ordinary Differential Inequality
\begin{equation}\label{ODI-0}
\frac {d}{dt} m_{  sk}[f](t)   \le \B_{  sk}\  -\  \frac{A_{{  s}\kc}}2\,  {m^{-c }_{1}[f_0]}  m^{1+c}_{  sk}[f](t)\,,\qquad sk\ge s\kmax >1,%/{  s}, %\ \ \text{for}\  c(k)=\frac{\gamma}{2({k}-1)},
\end{equation}
whose right hand side is an upper bound to the  $sk^{th}$-Lebesgue moment of the Boltzmann collision operator from  inequality \eqref{Qmoment-0} in the statement of Theorem~\ref{mom-coll-op}.
The $sk$-dependent constants $A_{s\kc}$ and  $B_{sk}$ are determined in Theorem~\ref{mom-coll-op}, equations \eqref{moment-factors} to \eqref{RateE}, which depend
solely from the  Cauchy problem data associated  for the Boltzmann flow stated in Theorem~\ref{CauchyProblem} and the $sk^{th}$-order  Lebesgue polynomial moment under consideration.

Hence, the following estimates and  bounds  hold for any fixed $sk$ value $s\in(0,1]$, and integer $k>1$. The first one is characterized by the following propagation of the initial data, after setting  $c=c_\gamma(sk)$.

{\bf Moment's propagation estimates:}  Let  $f_0\in L^1_{2sk}(\R^d)$ non-negative, or equivalent
$m_{2sk}[f_0] $ is finite  for any $sk\ge s\kmax$. Then
\begin{align}\label{mom-prop}
m_{  sk}&[f](t) \leq \max\ \left\{ m_{  sk}[f_0] \ ; \mathbb{E}_{c_\gamma(sk)}\right\}=m_{  sk}[f_0] \, \mathbb{I}_{m_{  sk}[f_0] < \mathbb{E}_{c_\gamma(sk)}} + \mathbb{E}_{c_\gamma(sk)}\,\, \mathbb{I}_{m_{  sk}[f_0] \ge \mathbb{E}_{c_\gamma(sk)}}  \nonumber \\
&= m_{  sk}[f_0] \, \mathbb{I}_{m_{  sk}[f_0] < \mathbb{E}_{c_\gamma(sk)}}  \nonumber \!\\
&\qquad+\!  \left( m_1\,\ttm_{1,k}\right)^\frac{c}{c+1}[f_0]
\left[ \left(\! \frac{ \beta_{  sk}}{ A_{{  s}\kc} }\!\right)^{\frac{2(sk-1)}{\gamma} }\! \mathbb{I}_{ k\ge \kgamma }
\!+\! \left(\!\frac{ \beta_{sk}}{ A_{{  s}\kc} }\!\right)^{sk+\frac\gamma2}  \mathbb{I}_{ k <  \kgamma } \right] \!  \mathbb{I}_{m_{  sk}[f_0] \ge  \mathbb{E}_{c_\gamma(sk)}},
\end{align}
or, equivalently, for the rate factors from \eqref{RateE}
\begin{align}
m_{  sk}&[f](t) \leq m_{  sk}[f_0] \, \mathbb{I}_{m_{  sk}[f_0] < \mathbb{E}_{c_\gamma(sk)}} \!+\!  \left( m_1\,\ttm_{1,k}\right)^\frac{c}{c+1}[f_0]
\, R_\mathbb{E}(\delta^{-1}_{sk})\,  \mathbb{I}_{m_{  sk}[f_0] \ge  \mathbb{E}_{c_\gamma(sk)}}, \nonumber
\end{align}
uniformly in $ t>0,$\ for any $ sk> s\kmax.$

Global moments generation estimates  are characterized by  the generation of any polynomial $sk^{th}$-order from just an initial data $f_0\in L^1_{2s\kmax}(\R^d)$,   with $s\kmax >1$, are as follows. \\

{\bf Moment's generation estimates:} Let  $f_0\in L^1_{2s\kmax}(\R^d)$  non-negative, or equivalent
$m_{2s\kmax}[f_0] $ is finite, that all moments  $m_{sk}[f](t)$ are finite for $sk>s\kmax>1$, with $\min_{t\to 0} m_{sk}[f](t) =\infty$ satisfying the following  global estimate for positive times, that is
\begin{align}\label{mom-gen}
m_{  sk}&[f](t) \le   \mathbb{E}_{c_\gamma(sk)} + m_1[f_0]\bigg( \frac1{c  A_{{  s}\kc}}\bigg)^{\frac{1}{c}} \, t^{-\frac{1}{c} } \\
& = \left( m_1\ttm_{1,{  sk}}\right)^{\!\frac{sk-1}{sk-1+\frac\gamma2}}\![f_0]
\left[ \left( \frac{ \beta_k}{ A_{s\kc} }\right)^{\frac{2(sk-1)}{\gamma} } \!\!\mathbb{I}_{ k\ge \kgamma }\!+\!
\left(\frac{ \beta_{  sk}}{ A_{s\kc} }\right)^{sk+\frac\gamma2} \! \!\mathbb{I}_{ k< \kgamma } \right]  \! +\! m_1[f_0]\bigg( \frac{2(sk-1)}{\gamma\,  A_{{  s}\kc}}\bigg)^{\!\!\frac{2(sk-1)}{\gamma}} \!\!\! t^{-\frac{2(sk-1)}{\gamma} }\!, \nonumber
\end{align}
or, equivalently, for the rate factors from \eqref{RateE}
\begin{align*}
m_{  sk}[f](t) \leq    \left( m_1\,\ttm_{1,k}\right)^\frac{c}{c+1}[f_0]
\, R_\mathbb{E}(\delta^{-1}_{sk})\,  \mathbb{I}_{m_{  sk}[f_0] \ge  \mathbb{E}_{c_\gamma(sk)}} +\! m_1[f_0]\bigg( \frac{2(sk-1)}{\gamma\,  A_{{  s}\kc}}\bigg)^{\!\!\frac{2(sk-1)}{\gamma}} \!\!\! t^{-\frac{2(sk-1)}{\gamma} }, \nonumber
\end{align*}

uniformly in  time $t>0,$ for any $ sk> \max\{ s\kb,s\kc \}=s\kmax.$

For both statements,  constants  $\beta_{sk},\  \mathcal{C}_{sk} $ and the equilibrium state $\mathbb{E}_{c_\gamma(sk)} $,  are obtained in Theorem~\ref{mom-coll-op}, are  characterized in \eqref{moment-factors},  \eqref{mathcal-C-kc} and  \eqref{RateE}, respectively.
%$\B_k= \beta_k \, \mathcal{C}_{sk}$
%$\mathbb{E}_{c_\gamma(sk)} =  m_1^{\frac{c_\gamma(sk)}{1+c_\gamma(sk)}}[f_0]\left(\frac{\B_{sk}}{A_\kc}\right)^{\frac1{1+c_\gamma(sk)}}$, to yield

\end{thm}

\medskip

\begin{proof}
Starting from  the moments estimates and bounds for the Boltzmann collisional integral \eqref{collision3}, \eqref{collision3.2}  in the Banach space $L^1_q(\R^d)$, with $q=2sk$,  developed in Theorem~\ref{mom-coll-op},
the proof of the current theorem  consists in  estimating  the Ordinary Differential Inequalities  for   $sk^{th}$-Lebesgue moments  associated to the solutions of the Boltzmann flow, as developed  in Lemma~\ref{mom-lemma-I}, whose right hand side of  inequality \eqref{momentineq} was estimated in much detailed by analyzing the $sk^{th}$-Lebesgue moment bounds for  the collisional integral.

%by means of
%interpolation  estimates  from \eqref{mom-Holder}, for any  $\gamma\in(0,2]$ and ${s\kc}>1$,  in order to control all its positive terms  bounded by above by the higher order moment $m_{k+\gamma/2}[f](t)$,
% while controlling the negative contributions from below by coercive estimates  involving $m_{k+\gamma/2}[f](t)$, both  multiplied by  constants that  only  on the moment order ${\kc}>1$ and its corresponding ${\kc}^{th}$-contracting factor $\mu_{\kc}$, as well as on the data associated to the Boltzmann flow.
% \begin{align} \label{ODI--1}
%\frac {d}{dt} m_{sk}[f](t) =  m_{sk}[Q(f,f)](t) &\le
%\B_{  sk}\  -\  \frac{A_{{  s}\kc}}2\,  {m^{-c_\gamma(sk) }_{1}[f_0]}  m^{1+c_\gamma(sk)}_{  sk}[f](t), \nonumber\\
%&\text{for any } \ \ sk\ge s\kc >1.
% \end{align}

More  specifically, starting from  right hand side of the  moments inequality in \eqref{momentineq},  and invoking the estimates from Theorem~\ref{mom-coll-op}, it follows that any solution for the Boltzmann flow  with initial data $f_0(v)\in L^1_{2+\eps}$ satisfies, for any $\eps\ge\max\{s\kb,s\kc\}-1>0$ and  $\gamma\in(0,2]$,
\begin{align}\label{sharp coercive}
\frac {d}{dt} m_{  sk}[f](t) \ &<  \B_{  sk} -  \frac{A_{{  s}\kc}}2\,m^{-c}_{1}[f_0]\,m_{  sk}^{1+c}[f](t)\,,\quad {  sk}\ge \max\{s\kb,s\kc\}>1,
\end{align}
{where} the $k$-independent  coercive factor $A_{s\kc}$ was defined on \eqref{Akc}, and  the $k$-dependent positive factor $ \B_{sk} =\beta_{sk}\,   \mathcal {C}_{  sk}$  defined in \eqref{Bkc} and $\beta_{sk}$ defined in \eqref{betak}.
\\ % given by (\ref{hardpotmom-3.2}-\ref{hardpotmom-3.22}). \\

Hence, the proof of Theorem~\ref{propagation-generation}  needs to be completed by showing the  that, for each $sk$ fixed  satisfying ${  sk}\ge \max\{s\kb,s\kc\}>1$, its corresponding  super linear Ordinary Differential Inequality \eqref{sharp coercive}
admits an upper barrier, that is a super solution with a finite bound independent of time, for each $sk$ fixed, by either propagating or generating  the  initial data, supplemented by a comparison principle to ODI's, under the assumption that $m_{sk}[f](t)$ is finite for any  positive time $t$.

Therefore, the goal is to construct  to upper bound form of the right hand side  the initial value problem associated to the Ordinary  Differential Inequality \eqref{sharp coercive}, after setting $c=c_\gamma(sk)= \gamma/(2(sk-1))$ once again,
\begin{align}\label{ODE1}
\begin{cases}&y'(t) <  \B_{sk}  -  A_{s\kc}\, m_1^{-c}[f_0]y^{1 + c}(t)\,, \quad\text{for}\ c(k)=\frac{\gamma/2}{sk-1}, \quad t>0\,, \\
\end{cases}
\end{align}
where $y(t) = m_{sk}[f](t)$ for $t>0$, and  the corresponding value of initial data is $y_0$ finite that may be any $j$-moment   $m_{j}[f](0)$ finite with $1<j\leq {  sk} $.

In particular, for the case where $s=1$ and $j=k$, the propagation of moment results hold if the initial data is $y(0)=y_0= m_k[f_0](0)$.  The generation of moments result  holds if   just  $j=\max\{s\kb,s\kc\}>1$,   $y(0)=y_0= m_j [f_0](0)$, i.e. only mass, kinetic energy and the $j^{th}$-moments are bounded, while estimating $y(t)=m_{sk} [f](t)$ for any $sk>j$.
Both results follow from comparison arguments for Ordinary Differential Inequalities whose right hand side are continuous in $t$ and Lipschitz in $y=y(t).$

By classical  ODEs comparison arguments (or the maximum principle in ODEs), it is easy to verify that any  $y_*(t)$, the solution of the upper ODE problem to \eqref{ODE1} with the initial data $y_*(0):=y_{*,0} \ge y_0$, that is
\begin{align}\label{upperODE}
\begin{cases}&y_*'(t) = \B_{  sk}  -  A_{{  s}\kc}\,  m_1^{-c} y_*^{1 + c}(t)\,,\qquad t>0\,,  \qquad  \\
\end{cases}\end{align}
controls the solution from above,  i.e. $y(t)<y_*'(t)$ unifomly in time $t\ge0$.

It is relevant to notice identify the equilibrium state  $\mathbb{E}_{c_\gamma(sk)}$ associated to  this upper ODE  \eqref{upperODE} is exactly the unique root calculated for the moment operator $\mathcal L_{sk}(y)$ introduced in \eqref{RateE} and \eqref{equi root}, respectively,  where can be easily verify  from the representation \eqref{equi-1.1}, and the fact
$\lim_{sk\to 1} c_{\gamma}(sk)=\infty$, that
\begin{equation}\label{equi-solution}
\mathbb{E}_{c}:= m_1^{\frac{c}{1+c}}[f_0]\left(\frac{\B_{sk}}{A_\kc}\right)^{\frac1{1+c}}
\longrightarrow_{{  sk}\to 1}\  m_1 [f_0]\, ,
\end{equation}
is the unique $y_*^{\text{eq}}=\mathbb{E}_{c}$  time independent equilibrium solution  of   $y'_*=0$.

In addition, since the ODE \eqref{upperODE}  is autonomous,  clearly satisfies that
the solution  $y_*(t)$ monotonically increases in time $t$  to the equilibrium state $y_*^{\text{eq}}$, if the initial data $y_0 <  y_*^{\text{eq}} $; while monotonically decreases to $y_*^{\text{eq}}$,  provided $y_0 >  y_0^{\text{eq}}$.

Therefore,  the propagation of the $k^{th}$-moment  property follows easily from  such    comparison arguments between the solution of ODI \eqref{ODE1}, and the one for the ODE  \eqref{upperODE},    yields the upper control   to the solution of the initial value problem \eqref{ODE1}, with the following propagation of $y(t)=m_k[f](t)$ associated to the initial data  $y(0)=y_0=m_k[f_0](0)$,   that is
\begin{equation*}
y(t)\le \max\Big\{  y(0)\, ; \, \mathbb{E}_{c} \Big\}\, , \nonumber\\
\end{equation*}
or, equivalently,  for the initial data $y_0=m_{sk}[f_0]$  the global in time bound for each $sk^{th}$-moment
\begin{align}\label{moments-bounds}
m_{sk}[f](t) &\leq \max\Big\{ m_{{  sk}}[f_0] \, ;\,  m_1^{\frac{c}{1+c}}[f_0]\big(\tfrac{\B_{  sk}}{A_{{  s}\kc}}\big)^{\frac{1}{1+c}}\Big\}\,,  \quad\text{for}  \quad {  sk} > 1, \quad t>0,
\end{align}
with the coercive constant factor $A_{{  s}\kc}$ and the upper bound constant $\B_{  sk}$ defined on \eqref{Bkc}.
In particular,   this estimate   \eqref{moments-bounds}  complete the moments propagation property stated  \eqref{mom-prop}  of  Theorem~\ref{propagation-generation}.

\medskip

The part of Theorem~\ref{propagation-generation}'s proof concerning the   ${  sk}^{th}$-moments  generation associated to the Boltzmann flow,
for any $sk$-order with only  initial finite mass and energy and  a moment of order $1+\eps$, with $\eps\ge s\kmax-1.$ This statement   means  that there is an instantaneous regularization, in the sense that it is possible  to find an upper bound to solutions $m_{  sk}[f](t)$,   \eqref{upperODE}, or equivalently,  \eqref{ODE1} for $t>0.$\\

The following comparison arguments for generation of moment argument confronts  the fact that the  expected moment estimate at the initial time is actually defined by the unbounded $m_{sk}[f](0)=+\infty$, for $sk>s\max$.  Yet, the following lemma remains true, as we describe the comparison argument.

\begin{lem}{\bf ODI's Comparison Lemma for Moments Generation } \label{ODI Comparison}
Let $A, B$ and $c$ be positive constants and consider a function $y(t)$ which is absolute continuous in $t\in(0,\infty)$ and satisfies
\begin{equation*}
y'(t) \leq  B  -  Ay^{1 + c}(t)\,, \qquad \text{for} \ \ c>0, \quad t>0\,.
\end{equation*}
\begin{equation*}
y(t)\leq \bar{y}(t):=E\bigg(1+ \frac{K}{t^{\beta}}\bigg)\,,\qquad t>0\,,
\end{equation*}
for the choice
\begin{equation*}
E=\mathbb E_c\bigg(\frac{B}{A}\bigg)^{\frac{1}{1+c}}\,,\qquad \beta=\frac{1}{c}\,,\qquad K=\bigg(\frac{1}{c\,A}\bigg)^{\frac1c}\bigg(\frac{A}{B}\bigg)^{\frac{1}{1+c}}\,.
\end{equation*}
\end{lem}
\begin{proof}
Note that in one hand
\begin{equation*}
\bar y'(t) = -\beta E K t^{-(\beta+1)}\,.
\end{equation*}
On the other hand,
\begin{equation*}
B-A\,\bar y^{1+c}(t) = B - AE^{1+c}\bigg(1+\frac{K}{t^\beta}\bigg)^{1+c}\leq B - AE^{1+c}\bigg(1+\frac{K^{1+c}}{t^{\beta(1+c)}}\bigg)
\end{equation*}
where we invoked the binomial inequality  $( 1+ \frac{K}{t^\beta} )^{1+c} \geq  1+ (\frac{K}{t^\beta})^{1+c}$.  With the choice of $E,\,\beta$ and $K$ as suggested in the statement of the lemma it follows then
\begin{equation}\label{upbd}
B-A\,\bar y^{1+c}(t) \leq \bar y'(t)\,,\qquad t>0\,.
\end{equation}
Next, note that for any $\epsilon>0$, the translation $\bar{y}_\epsilon(t):=\bar{y}(t-\epsilon)$ satisfies \eqref{upbd} for $t>\epsilon$ due to the time invariance of the inequality.  Moreover, since $y(t)$ is absolute continuous in $t\in[\epsilon,\infty)$, there exists $\eta_*>0$ such that $\bar{y}_{\epsilon}(\epsilon+\eta)\geq y(\epsilon+\eta)$ for any $\eta\in(0,\eta_*]$.  Consequently, we conclude the following setting for any $\epsilon>0$ and $\eta\in(0,\eta_*]$:
\begin{align}\label{ODE02}
\begin{cases}
&y'(t) \leq B  -  A\,y^{1 + c}(t)\,,\qquad  t>\epsilon+\eta\,\\
&\bar y'_\epsilon(t) \ge B  -  A\, \bar y_\epsilon^{1 + c}(t)\,,\qquad  t>\epsilon+\eta\,,\\
&+\infty >\bar y_\epsilon(\epsilon+\eta) \geq y(\epsilon+\eta)\,.
\end{cases}
\end{align}
Consequently, a standard comparison in ode implies that $\bar{y}_\epsilon(t)\geq y(t)$ in $t>\epsilon+\eta$ valid for any $\epsilon>0$ and $\eta\in(0,\eta_*]$.  In this particular case this fact can be readily proved denoting
\begin{equation*}
\Phi(t) = \frac{y^{1 + c}(t) - \bar y_\epsilon^{1 + c}(t)}{y(t) - \bar y_\epsilon(t)}\,.
\end{equation*}
Since $ |x^{1+c} - y^{1+c} |\leq (1+c)\max\{x^c , y^c \}|x-y|$, it follows that $\Phi(t)$ is bounded by
\begin{equation*}
\big| \Phi(t) | \leq (1+c)\max\big\{ y^c(t)\,,\,\bar y^c_\epsilon(t) \big\}<+\infty\,.
\end{equation*}
Thus, for $w(t)=y(t) - \bar y_\epsilon(t)$ it follows from \eqref{ODE02} that
\begin{equation*}
w'(t) + A\,\Phi(t)\,w(t)\leq0\,,\qquad w(\epsilon+\eta)\leq0\,,\qquad t>\epsilon+\eta\,,
\end{equation*}
which implies that
\begin{equation*}
w(t)\leq w(\epsilon+\eta)e^{-A\int^{t}_{\epsilon+\eta}\Phi(s){\rm d}s}\leq 0\,.
\end{equation*}
Now, sending first $\eta$ to zero and then $\epsilon$ to zero it follows that $\bar{y}(t)\geq y(t)$ for any $t>0$ which is the statement of the lemma.
\end{proof}
Recall that the moments of $f(v,t)$ satisfy the inequality \eqref{ODE1}, specifically, setting $c=c_\gamma(sk)=\frac{\gamma}{ 2(sk-1)}$
\begin{equation*}
m_{sk}'(t) \leq  \B_{sk}  -  A_{s\kc}\, m_1^{-c}[f_0]\,m_{sk}^{1 + c}(t)\,, \quad  \text{for any}\ \  sk\geq s\kmax >1, \quad  t>0\,.
\end{equation*}
Thus, applying  Lemma~\ref{ODI Comparison} with
\begin{equation*}
B=\B_{sk}\,,\quad A = A_{s\kc}\, m_1^{-c}[f_0]\,,\quad %\text{and} \quad c_\gamma(sk)=\frac{\gamma}{2(sk-1)}\,,
\end{equation*}
it follows that
\begin{align}\label{generation-bound}
m_{  sk}[f](t) \ &\le \ \bigg(\frac{\B_{  sk}}{A_{{  s}\kc}}\bigg)^{\frac{c}{ 1+c}}
m_1^{\frac 1{1+c}} [f_0] +
m_1[f_0]\bigg( \frac{1}{c\, A_{{  s}\kc} \, t}\bigg)^{\frac{1}{c }} \nonumber
\\
& \le \  \mathbb E_{c_\gamma(sk)} +
m_1[f_0]\bigg( \frac{1}{c\, A_{{  s}\kc} \, t }\bigg)^{\frac{1}{c }} \,.
\end{align}

That is, $m_{  sk}[f](t)$ is finite for any $t>0$  and for all ${  s}k\ge { s}\max\{\kb , \kc\} >1$ fixed.  In addition, it becomes uniformly bounded in time for say $ t\in[1,\infty)$ satisfying
\begin{equation} \label{generation-bound-limit}
\lim_{t \to \infty}  \left( m_k[f](t) -  \mathbb E_{c}  \right) \le m_1[f_0]\,  \lim_{t \to \infty}  \left(  \frac{1}{ c \, A_{{  s}\kc}\, t}\right)^{\frac{1}{c}} = 0\, ,
\end{equation}
since the fixed  term $A_{s\kc}>0$ for any choice of  $ {\kc}>1$,  which is  the coercive  factor  defined in \eqref{moment-factors}.

\medskip
The proof of Theorem~\ref{propagation-generation} is completed after  the classification  of the  {\bf Moments bounds estimates in terms of the coercive constant $A_{{  s}\kc}$. }
After combining the definitions  of the coercive constant factor $A_{s\kc}$,  and the upper bound constant $\B_{  sk}$ in \eqref{Bkc},  all depending on ${  sk}\ge \max\{s\kb,s\kc\}>1$, a more explicit expression for the upper bound generation  emerges when specifying the equilibrium state $\mathbb E_{c_\gamma(sk)}$ as a function of $\beta_{sk}$ and $\C_{sk}$ and the initial moments associated to $m_0[f_0]$ and  $m_1[f_0]$,  as characterized in \eqref{equi-1}.
Therefore,   the moment bounds estimates, focusing on the dependance of the coercive constant $A_{{  s}\kc}$ are recasted, after invoking the moment propagation estimates  \eqref{mom-prop},  as stated in Theorem~\ref{propagation-generation},

{\bf Moment's propagation estimates:}
\begin{align}\label{moments-bounds-2}
m_{  sk}[f](t) & \!\leq  \!\max \!\left\{ m_{  sk}[f_0] \, ;
\,  \left( m_1\,\ttm_{1,k}\right)^\frac{ks-1}{sk-1+\frac\gamma2}[f_0] \left[ \left( \frac{ \beta_{  sk}}{ A_{{  s}\kc} }\right)^{\frac{2(ks-1)}{\gamma} } \mathbb{I}_{ k\ge \kgamma } \!+\!
\left(\frac{ \beta_k}{ A_{ s \kc} }\right)^{sk+\frac\gamma2  }  \mathbb{I}_{ k< \kgamma } \right] \right\},
\end{align}
uniformly in $ t>0,$\ for any ${  s}k> {  s}\kmax $.  Analogously, the moments generation estimates \eqref{mom-gen}  are characterized by {\bf Moment's generation estimates:}
\begin{align}\label{generation-bound-2}
m_{sk}[f](t) &\le  % \beta_{sk}^{\frac 1{1+c}} \,
\left( m_1\,\ttm_{1, sk}\right)^{\!\frac{ks-1}{sk-1+\frac\gamma2}}[f_0]
\left[ \left( \frac{ \beta_{sk}}{ A_{s\kc} }\right)^{\frac{2(sk-1)}{\gamma}} \!\! \mathbb{I}_{ k\ge \kgamma }\!+\!
\left( \frac{ \beta_{sk}}{ A_{s\kc} }\right)^{sk+\gamma/2  }
\!\!  \mathbb{I}_{ k< \kgamma } \right] \nonumber\\
&\qquad \qquad \qquad \qquad\qquad
\qquad + m_1[f_0]\bigg( \frac {2(sk-1)}{\gamma\, A_{s\kc}\, t}\bigg)^{\frac{2(sk-1)}{\gamma}} \, .
\end{align}
In addition, if $sk> \max\{\kb,\kc \}>1$, then $\beta_{sk} A^{-1}_{\kc} >1.$

Thus, using the notation \eqref{RateE} as defined earlier, one can write
\begin{align}
\left( \frac{ \beta_{sk}}{ A_{s\kc} }\right)^{\frac{2(sk-1)}{\gamma}} \!\! \mathbb{I}_{ k\ge \kgamma }\!+\!
\left( \frac{ \beta_{sk}}{ A_{s\kc} }\right)^{sk+\gamma/2  } =  R_\mathbb{E}(\delta^{-1}_{sk})= R_\mathbb{E}(\beta_{sk}/A_\kc).
\end{align}
The proof of Theorem~\ref{propagation-generation} is now completed.
\end{proof}

\medskip

\begin{remark}\label{coercive-remark} {\bf Coerciveness.} {\em The control of the upper bound any ${sk}^{th}$-moment that solves the differential inequality \eqref{Akc}, degenerates as the term    $A_{{ s}\kc} := c_{\Phi}2^{-\frac{\gamma}2} (\|b\|_{L_1(\mathbb{S}^{d-1})}-\mu_{\kc})  {m_0[f_0]}$, defined in \eqref{mu_r}, would vanish.  The derivation of this term
depends on the choice of  of the positive term \, $0<\delta_{ sk}<1$ made in \eqref{delta-k}, for any  $sk>s\max\{\kb,\kc \}>1 $. This choice  depends on the decaying property of  the contractive constant $\mu_{s}k$  for any ${ sk}\ge {{s}\kc} >1$ calculated in the Angular Averaging Lemma~\ref{BGP-JSP04},  the pointwise factors  for the potential  from \eqref{pot-Phi_1} and  \eqref{pot-Phi_2}  depending  on the data.
In addition,  that the constant $A_{{s}\kc}$ is calculated without using that $f$ solves the Boltzmann equation, it only depends
on the  spherical integrability (or angular averaging) of the transition probability $B(\Phi(|u|), \hat u\cdot\sigma) \in L^1(\mathbb{S}^{d-1}, d\sigma)$, and   lower constants $c_\Phi$ and $C_\Phi$ \eqref{pot-Phi_1} and \eqref{pot-Phi_2}, associated to the the potential rate $\Phi(v).$
%It only needs to estimate the convolution of $f*\Phi (v)$ under the assumption that $f\in \mathcal{C}$ for   the subset   $\mathcal{C}:=\{ f(v,t)  \in L^1_k(\mathbb{R}^n),  k\ge 0, :   \|f\|_{L^1}(0)=\|f_0\|_{L^1}(t)\ \text{and} \   \|f\|_{L^1_1}(0)=\|f_0\|_{L^1_1}(t), \forall t>0 \}$.

That means   $A_{{ s}\kc}$ can be viewed as the  {coercive constant} related to the Cauchy problem  to Boltzmann flow for elastic interactions with transition probabilities corresponding to  hard potentials   with an integrable angular transition probability, posed in the Banach space  with the natural $k$-Lebesgue weight norm.

Hence, the a priori ${sk}^{th}$-moment estimates obtained in Theorem~\ref{propagation-generation} induced an  upper uniform bound in time for each ${ sk}$ fixed with a  coercivity property for any ${sk} \ge { s}\max\{\kb,{\kc}\} >1$.  One can  view  these a priori estimates   as  analog to significance of the Sobolev-Poincare and coercive constants  associated to elliptic and parabolic equations in divergence form.   The corresponding existence and uniqueness theorem in these Banach spaces follow in the next section.

\end{remark}

\bigskip

We conclude this section by stating the corresponding $L^1_k(\mathbb{R}) $ or $K^{th}$-moment estimates for the Boltzmann flow with Maxwell type collisional forms corresponding to potential rates $\gamma=0$, corresponding to transition probability with  potential  rate $\gamma=0$ and integrable angular factor $(\hat u\cdot \sigma)$

\subsection{\em Case for Maxwell type of interactions corresponding to  $\gamma=0$. } This is the well known case   where the estimate \eqref{momentineq}
is obtained by a moment recursion formula for classical cases where the transition probability of collision kernel  simple depend on an integrable $B(\hat u\cdot\sigma) \in L^1(\mathbb{S}^{d-1}, d\sigma)$.

However, the interpolation estimate techniques developed  in
Theorem~\ref{mom-coll-op} and Theorem~\ref{propagation-generation} extend to the case  $c_{\Phi} b(\hat u\cdot\sigma) \le B(\Phi(|u|), \hat u\cdot\sigma) \le C_{\Phi} b(\hat u\cdot\sigma)$, with $b(\hat u\cdot\sigma)\in  L^1(\mathbb{S}^{d-1}, d\sigma)$ and $c_{\Phi} $ and $C_{\Phi} $ the bounds from \eqref{pot-Phi_1} and \eqref{pot-Phi_2}, respectively.

similar estimates by above for  the gain operator by above using interpolation,  while an exact negative identity for the moments of the loss, which clearly maintains the linearity property of the moments estimates,  to obtain  the  corresponding ODI
\begin{align}\label{moments odi Maxwell}
\frac {d}{dt} m_k[f](t) & =  m_k[Q_0[f,f](t) \leq C_{\Phi}2^{k}\mu_km_{1}[f_0]m_{k-1}[f] - c_{\Phi} (\|b\|_{L^1(\mathbb{S}^{d-1})}-\mu_k)m_{0}[f_0]m_{k}[f] \nonumber \\[3pt]
&\leq K_{1,k}[f_0]m^{\frac{k-2}{k-1}}_{k}[f] - (\|b\|_{L^1(\mathbb{S}^{d-1})}-\mu_k)m_{0}[f_0]m_{k}[f]\\[3pt]
&\leq K_{2,k}[f_0] - \tfrac{c_{\Phi}}{2}(\|b\|_{L^1(\mathbb{S}^{d-1})}-\mu_k)m_{0}[f_0]m_{k}[f] =
K_{2,k}[f_0] - \frac{c_{\Phi}}2 A_\kc m_{0}[f_0]m_{k}[f],\nonumber\\[3pt]
&=: \mathcal L^M_k(m_{k}[f]), \qquad
\quad k\geq \kc\ge1\,, \nonumber
\end{align}
where $\mathcal L^M_k(y)$ is a linear operator  in $y=m_{k}[f]$, with
constants   $A_\kc$  the coercive one for $\kc>1$,  and   $ K_{i,k}[f_0]$, for $I=1,2$, depend on  $C_{\Phi}, \mu_k$ the interpolation factors. This case can only  fulfilled the moment propagation  property of a  given initial data, as the negative term in the $k^{th}$-moment ODI   makes a linear upper ODE  with an absorption term $m_k[f](t)$ and with a bounded in time right hand side.  Hence,  this inequality leads to estimate \eqref{mom-prop} and the is no mechanism that can generate a moment or order $k$ from just an initial data  with bounded mass and energy.  More precisely, for any $f_0(v) \in L^1_{2k}(\R^d)$ the $k^{th}$ Lebesgue moment	 associated to the Maxwell type of interactions  Ordinary Differential Inequality \eqref{moments odi Maxwell} is global in time controlled by
\begin{align}\label{moments odi Maxwell2}
m_k[f](t) &\le  \mathbb E_k   + m_k[f_0] \exp (-\frac{c_{\Phi}}2 A_\kc m_{0}[f_0] \, t), \\[4pt]
\text{with} \qquad \mathbb E_k &:=\frac{2K_{2,k}}{ c_{\Phi} A_\kc m_{0}[f_0] }\ \text{being the unique root of  linear operator}  \ \mathcal L^M_k(m_{k}[f])\, \nonumber
\end{align}
making  $ \mathbb E_k $ the unique equilibrium state of an associated  upper ODE $y'(t)= \mathcal L^M_k(y(t))$.

In particular the existence theorem to be developed in the next sections applies to this case as well, although the classical theory of ODEs, for comparisons to linear theory, since the collision operator can be shown to be a Lipschitz form, and the existence and uniqueness of  $f(v,t)$ solving the Cauchy problem if obtained by a  classical Picard type iteration constructs a fulfilling the global moments bounds \eqref{moments odi Maxwell2}. Much work has been written for the theory of conservative and non-conservative models for Maxwell type of interactions, and a sample of them can be found at [30, 29, 35, 34, 36, 32] and references therein.

\medskip

\bigskip

\section{Proof of  existence and uniqueness Theorem \ref{CauchyProblem}}\label{existence-uniqueness}

All  estimates  for $k^{th}$-Lebesgue polynomial moments for $1<k\in \R$,   obtained in Theorem~\ref{mom-coll-op},  are enough to  prove the  existence and uniqueness of solutions to the initial value problem \eqref{collision3}, for the space homogeneous problem of Theorem~\ref{CauchyProblem} for Maxwell and hard potentials for $\gamma\in[0,2]$ and integrable angular scattering transition probability rate.
This result is based on the following result on Banach spaces, Theorem~\ref{Theorem_ODE}, which was initially proposed in the unpublished note [39] in order  to solve the Boltzmann equation for hard spheres in three dimensions, that is $\gamma=1$, and uniform scattering kernel.  While the ideas to prove this theorem are somehow classical, see for example [86], we revisit  this approach here showing     the use of the a-priori estimates on the $k^{th}$-Lebesgue polynomial moments  and their  impact to verify that  the   sub-tangent condition is satisfied,  for any real valued $1<k\in \ \R$. The proof of the  sub-tangent   is crucial for the selection of the solvability invariant set. This point not covered in the proposed proof  in  [39].   The following argument fully completes the argument sub-tangent condition being a sufficient one.  It also extends the argument for the aforementioned more general models for of collision cross sections for the homogeneous Boltzmann equation for elastic binary interactions, and other broader problems such collisional multi-species gas mixtures, as we well as polyatomic gas models.

Following the notation  used earlier for the Lebesgue norm  $\|g\|_{L^1_\ell(\R^d)}$ for $\ell\in \R$,  and the  $k^{th}$-moment  $m_k=\int_{\R^d} g(v) \langle v\rangle^{2k} dv$, as defined in\eqref{k-norm} and \eqref{k-moment}, respectively,  notice that the norm and moments coincide for a positive $g(v)$ and $2k=\ell$.

Hence, focusing the  proof of initial value problem stated Theorem~\ref{CauchyProblem}, initial data is given by moments of positive probability   density  functions taking the identical notation   $m_0:=m[f_0] =\|f_0\|_{L^1(\mathbb R^{d})}$ and  $m_1:=m_1[f_0] =\|f_0\|_{L^1_{2}(\mathbb R^{d})}$ which are  initial mass and energy invariants of the Boltzmann flow, respectively.  In addition, since $k^{th}$-moments associated to to time dependent probability densities $f(v,t)$,   the short notation $m_k(t) := m_k[f](t)$ will be used, unless it is relevant  for the the Boltzmann solution flow.

%COMPARISON WITH BERNOULLI EQS ESTIMATES
%{\color{blue} In addition,   the proof we present for the following  theorem is valid whether  the $k^{th}$-moments  satisfy an ODI $\partial_t m_k[f_0] (t) = \mathcal{L}(m_k[f_0])(t)$, where the operator $\mathcal{L}(y)$ is strictly convex in $y$  and has at most one positive root, when it changes from positive to negative values.  The relevant applications for the scalar Boltzmann equation we are studying in this manuscript are for any fixed parameter $c>0$, either  $\mathcal{L}(y) := B - A y^{1+c}$  whose unique root is the equilibrium solution $y_{eq}=(B/A)^{1/(1+c)} $,   or the classical Bernoulli equation    $\mathcal{L^B}(y) := By - A y^{1+c}$  whose non-trivial  positive root is $y^B_{eq}=(B/A)^{1/c} $.  This differences only appear in the proof of the sub-tangent condition {\bf ii)} in the incoming theorem, which is written here in a unified form.  }

\begin{thm}\label{Theorem_ODE}
Let $\mathcal{E}:=(\mathcal{E},\|\cdot\|)$ be a Banach space, $\Omega$  be a bounded, convex and closed subset of $\mathcal{E}$, and $Q:\Omega\rightarrow \mathcal{E}$ be an operator  satisfying the following properties:
\begin{itemize}
\item[{\bf i)}]  H\"{o}lder continuity condition
\begin{equation}\label{Holder_C}
\big\|Q[f] - Q[g]\big\| \leq C\|f - g\|^{\beta},\quad \beta\in(0,1), \quad \forall\,f,g\in\Omega\,.
\end{equation}
\item[{\bf ii)}]  Sub-tangent condition
\begin{equation}\label{subtangent}
\liminf_{h\rightarrow0^+}h^{-1}\text{Dist}\big(f+h\,Q[f],\,\Omega\big)=0,\qquad \forall\,f\in\Omega\,,
\end{equation}
where $\text{Dist}(v,\Omega) :=  \inf_{\omega\in\Omega} \|v-w\|$.
\item[{\bf iii)}]   One-sided Lipschitz condition
\begin{equation}\label{one-sided-lip}
\big[ Q[f] - Q[g], f - g \big]_{-}\  \leq C\ \|f - g\|,\qquad \forall\,f,g\in\Omega\,,
\end{equation}
where $\big[ \varphi,\phi \big]_{-}\ : = \  \lim_{h\rightarrow 0^{-}}h^{-1}\big(\| \phi + h\varphi \| - \| \phi \| \big)$.
\end{itemize}
Then, the abstract \text{ode} problem
\begin{equation*}%\label{Theorem_ODE_Eq}
\partial_t f = Q[f] \quad\text{on}\quad [0,\infty)\times \mathcal{E},\qquad f(0)=f_0 \in \Omega,
\end{equation*}
has a unique solution in $\mathcal{C}([0,\infty),\Omega)\cap\mathcal{C}^1((0,\infty),\mathcal{E})$.
\end{thm}

\begin{remark}
In the Banach space $L^{1}_{1}$, the one-sided Lipschitz bracket reduces to
\begin{align*}
&\big[\varphi, \psi\big]_{-}\  = \ \lim_{h\rightarrow 0^{-}}h^{-1}\big(\| \varphi + h\psi \|_{L^1_{2}} - \| \varphi \|_{L^1_{2}} \big)\\
&\hspace{-.5cm}= \lim_{h\rightarrow 0^{-} } \int h^{-1}\Big(\big| \varphi + h \psi \big | - | \varphi | \Big) \ \langle v \rangle^{2} \ dv  =     \  \int \psi \,\text{sign}(\varphi)  \ \langle v \rangle^{2} \ dv \,.
\end{align*}
The last step follows by Lebesgue's dominated convergence because
\begin{equation*}
\big| h^{-1} \big|\,\Big|\big| \varphi + h \psi \big | - | \varphi | \Big| \leq |\psi|\,,
\end{equation*}
\begin{equation*}
h^{-1}\Big(\big| \varphi + h \psi \big | - | \varphi | \Big) = \frac{ 2\,\varphi\,\psi + h\,\psi^{2} }{\big| \varphi + h \psi \big | + | \varphi | } \rightarrow \psi \,\text{sign}(\varphi)\quad\text{as}\quad h\rightarrow0\,.
\end{equation*}
We point out that  the one-sided Lipschitz condition \eqref{one-sided-lip} is sufficient to obtain uniqueness for the Boltzmann problem as noticed by [54].
\end{remark}

The proof of theorem~\ref{Theorem_ODE} is shown in the Appendix,  in the exact version as done by A. Bressan in the unpublished work of [39].  For completeness, it is included in his manuscript.

\medskip

\begin{proof}{\em The Cauchy problem as posed  in  Theorem~\ref{CauchyProblem}.}
without loss of generality  taking just  $s=1$. This proof consists in two stages. The first stage  shows that  conditions {\bf i), ii)} and {\bf iii)} from Theorem~\ref{Theorem_ODE} are fulfilled  for the collision operator $Q:=Q_\gamma(f,f)(v)$ as defined in the statement of  Theorem~\ref{mom-coll-op} with all properties proved in that theorem, after setting Banach space $\E := L^1_2(\R^d)$
and the subset $\Omega \subset L^1_2(\R^d)$ defined as follows.
Set $k=1+\gamma$ be moment $m_{1+\gamma}[f]=\int_{\R^d}f(v)\langle v\rangle^{2(1+\gamma)}dv$ and consider the constant $\mathfrak{h}_{1+\gamma} := \mathbb{E}_{1+\gamma} + \B_{1+\gamma}$, where
$\mathbb{E}_{1+\gamma}$ is the only root of the operator $\mathcal L_{k}(y):=\B_{k} -  \frac{A_{\kc}}2 m^{-c_\gamma(k)}_1[f_0] y^{1+c_\gamma(k)}$ as defined in \eqref{Qmoment-0}.

Hence, define the   bounded convex subset $\Omega\subset L^{1}_{2}({\R}^{d}) $ by
%The set so that the operator satisfies the H\"{o}lder, sub-tangent, and one-sided Lipschitz conditions,   \eqref{Holder_C}, \eqref{subtangent}, and \eqref{one-sided-lip} respectively in $\Omega$ and it is consistent with the conservation laws.
\begin{equation}\label{SetOmega}
\begin{split}
\Omega_{\gamma} := &\Big\{ \ g \in L^{1}_{2} (\real^d) \ : \ g \geq 0,
\int g(v)\,dv \!=\!m_0,  \int g(v) |v|^2 dv \!=\! m_{1}-m_0>0,  \\
&\qquad\qquad  \int  g(v) \langle v \rangle^{2(1+\gamma)} dv =m_{1+\gamma} \!\le\!   K \!<\!\infty \ \Big\},
\end{split}
\end{equation}
for any finite constant $K >  \mathfrak{h}_{1+\gamma}.$
In fact,   conditions {\bf i)} and {\bf iii)} are satisfied with any initial moment to have at least $1+\gamma$ moments, that is  $f_0\in L^{1}_{2+2\gamma}(\mathbb{R}^{d})$,     while the Sub Tangent condition  {\bf ii)} proof  enables the existence  as well characterizes the  bounded, convex, closed subset $\Omega \in \mathcal E,$  by that any element $g\in \Omega$ may have   moment $m_{1+\gamma}[g]\ge  \mathfrak{h}_{1+\gamma}.$

The second stage consists in showing the solution set can be enlarge  to solution set posing just initial data in  $L^{1}_{2^+}(\mathbb{R}^{d})$ by means of relaxing the initial data to the lower moment condition $f_0\in L^{1}_{2+\epsilon}(\mathbb{R}^{d})$, for  some $\epsilon>0$.  It covers two delicate points. The first one if to construct a sequence of approximants  in the set $\Omega$  to the initial data $f_0\in L^{1}_{2+\epsilon}(\mathbb{R}^{d})$ by density and, the second one consists	in  showing that the sequence of solutions is uniformly bounded strongly in  $L^{1}_{2+\epsilon}$ and so has a limit that generates the solution for such  $f_0\in L^{1}_{2+\epsilon}(\mathbb{R}^{d}).$

\begin{remark}\label{moment mathfrakh} It is important to notice that the constant $\mathfrak{h}_{1+\gamma}$   is  fully  determined only by the transition probability, or collision kernel term $B_\gamma(|u|,\hat{u}\cdot \sigma)$, and the initial conserved quantities $m_0$ and $m_{1}$, and so it
only depends on the data associated to the Cauchy problem  to solved in Theorem~\ref{CauchyProblem}.
\end{remark}

Without loss of generality, the binary collisional Boltzmann operator   $Q_{\gamma}(\cdot,\cdot)$ if considered to  have $B_\gamma(|u|,\hat u\cdot \sigma):=|u|^{\gamma}\,b(\hat{u}\cdot\sigma)$, with $\gamma\in[0,2]$, and   the angular transition measure  $b(\hat u\cdot \sigma)$  is in $L^1({\mathbb{S}^{d-1}}).$
The subindex $\gamma\in[0,2]$ on $Q_{\gamma}(\cdot,\cdot)$ denotes the potential rate.  Thus, $Q_{0}(\cdot,\cdot)$ represents the Maxwell Type of  interactions.

\medskip

This following Lemma shows the  H\"older property {\bf i)} hold for any   $g\in L^1_{\ell}(\mathbb{R}^d)$, with $\ell\ge2(1+\gamma)$. It does not need  the conservation property stating that  any pair of  functions must have   the same   $L^1_{\ell}(\mathbb{R}^d)$ for $\ell=1,2$, since $L^1_{\ell}(\mathbb{R}^d) \subset L^1_{2(1+\gamma)}(\mathbb{R}^d)$

\begin{lem}[{\bf H\"older estimate  for elements  $L^1_{\ell}(\mathbb{R}^d)$ for any $\ell\ge2(1+\gamma)$  -- Property i)}]\label{Holder-estimate}

Let $f$ and $g$ functions lying in $L^1_{\ell}(\mathbb{R}^d)$ for any $\ell\ge2(1+\gamma)$, with their  $L^1_{2(1+\gamma)}$-norms are bounded. % the same $ \mathfrak{h}_{1+\gamma}.$
Denote by $h=f-g$ and $H=f+g$, noticing that $H$ is nonnegative whenever $f$ and $g$ are nonnegative.
\begin{equation*}%\label{uni01}
\int_{\real^d}  \big|Q_{\gamma}(f,f) - Q_{\gamma}(g,g)\big| \langle v\rangle^{2} \ dv \leq  \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) }\,C_{\Omega}\,\|f-g\|^{1/2}_{L^{1}_{2}}\,.
\end{equation*}
with constant given by $C_{\Omega}:=6\, K^{3/2}$.
\end{lem}
\begin{proof}
Observe that for $h=f-g$ and $H=f+g$,  the  following identity holds
\begin{equation*}
Q_{\gamma}(f,f) - Q_{\gamma}(g,g) \ =\ \tfrac{1}{2}Q_{\gamma}(H, h) + \tfrac{1}{2}Q_\gamma(h,H) \,.
\end{equation*}
In addition, note that
\begin{equation*}
|u| = |u'| \leq \min\big\{\langle v \rangle \langle v_{*} \rangle  ,  \langle v' \rangle \langle v'_{*} \rangle  \big\}\,, \quad \text{and} \quad \langle v \rangle \leq \langle v' \rangle\langle v'_{*} \rangle\,,
\end{equation*}
where the latter uses local conservation of energy $|v'|^2 + |v'_*|^2 = |v|^2 + |v_*|^2$.

\begin{equation*}
\big|Q_{\gamma}(h,H)\big| \langle v \rangle^{2} \leq Q^{+}_0(|h|\langle v \rangle^{2+\gamma}, H \langle v \rangle^{2+\gamma}) + Q^{-}_0(|h|\langle v \rangle^{2+\gamma}, H \langle v \rangle^{2+\gamma})\,.
\end{equation*}

Moreover, multiplying  $Q^{\pm}_0(|\psi|,|\varphi|)(v)$  by  any $\psi, \varphi \in L^{1}(\mathbb{R}^{d})$,  in $v\in\mathbb{R}^{d}$,   the following identity holds%
\begin{equation*}
\int_{\mathbb{R}^{d}}Q^{\pm}_0(|\psi|,|\varphi|)(v)dv = \|b\|_{\mathbb{S}^{d-1}}\| \psi \|_{L^{1}(\mathbb{R}^{d})} \| \varphi \|_{L^{1}(\mathbb{R}^{d})}\,,
\end{equation*}
that allow us to conclude that $Q_{\gamma} (h,H)$ is control by the $(1+\gamma/2)$-moments of $h$ and $H$, respectively, proportionally to the integral of the angular transition function $b(\hat u\cdot \sigma)$, that is
\begin{equation*}
\int_{ \mathbb{R}^{d} } \big| Q_{\gamma} (h,H) \big| \langle v \rangle^{2} \ dv \leq 2\ \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) } \  \|H\|_{L^{1}_{2+\gamma}} \ \|h\|_{L^{1}_{2+\gamma}}\,.
\end{equation*}
Further, using Lebesgue's interpolation, which uses the $L^1_{2(1+\gamma)}$-norm  of  $H=f+g$,   yields
{\begin{equation}\label{holder 1}
\|h\|_{L^1_{2 + \gamma}} \ \le \|h\|^{1/2}_{L^1_{2(1+\gamma)}} \|h\|^{1/2}_{L^1_{2}} \leq \|H\|^{1/2}_{L^1_{2(1+\gamma)}} \|h\|^{1/2}_{L^1_{2}}\,,
\end{equation}}
where the second inequality follows by observing that for nonnegative functions $|h| \leq H $.  Therefore,
\begin{equation}\label{holder 2}
\int_{ \mathbb{R}^{d} } \big| Q_{\gamma} (h,H) \big| \langle v \rangle^{2} \ dv \leq 2\ \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) } \  \|H\|^{3/2}_{L^{1}_{2(1+\gamma)}} \ \|h\|^{1/2}_{L^{1}_{2}}\leq 6\,K^{3/2} \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) } \|h\|^{1/2}_{L^1_{2}}\,.
\end{equation}
The latter estimate holds since $\|H\|_{L^1_{2+2\gamma}}\le 2m_{1+\gamma}[f]\leq  2K$ %2\,\mathfrak{h}_{1+ \gamma}$
for any $f,g\in\Omega$.

\end{proof}

\begin{lem}[{\bf Sub-tangent condition -- Property ii)}]\label{prop_subtan}
Fix $f\in\Omega$.  Then, for any $\epsilon>0$ there exists $\eta_{*}:=\eta_{*}(f,\epsilon)>0$ such that the ball centered at $f+\eta\,Q_{\gamma}(f,f)$ with radius $\eta\,\epsilon>0$ intersects $\Omega$.  That is,
\begin{equation}\label{non-empty-set}
B(f+\eta\,Q_{\gamma}(f,f),\eta\,\epsilon)\cap\Omega,\;  \text{is non-empty for any}\; 0<\eta\leq \eta_{*}\ .
\end{equation}
In particular,
\begin{equation}\label{invariant region limit}
\lim_{\eta\to 0+} \eta^{-1}\text{Dist}\big(f+\eta\,Q_{\gamma}(f,f),\Omega\big)=0\,.
\end{equation}
\end{lem}

\begin{proof}
The proof consists, for a given $f(v)\in \Omega$, we construct an associated function, denoted by $w_{R,f}(v)$ and first show that it is possible to find a small parameter $\eta(\epsilon, f)$,  such that $w_{R,f}(v)\in \Omega$.
The second part simple shows that, for the given arbitrary $\epsilon>0$,  also $ w_{R,f}(v)\in B(f+\eta\,Q_{\gamma}(f,f),\eta\,\epsilon)$.

\smallskip

\noindent {\em Part 1:} Start by recalling $m_0:=m_0[f](0)$ and  $m_1:=m_1[f](0)$, the conserved in time quantities mass and mass plus energy, respectively for any distribution density  $f(v,t)\in \Omega$.

Next,  consider the angular transition $b(\hat u\cdot \sigma)$  integrable %and normalized to have unitary mass in the sphere,
and define  the following cut off function depending on the fixed $f\in \Omega$  with a couple of positive real valued parameters $R$ and $\eta$, to be chosen later, given by
\begin{align}\label{wRf}
w_{R,f}(v) &:=  f(v) + \eta\,Q(f_{R},f_{R})(v) \\
&\ = f(v) + \eta\Big(Q^{+}_{\gamma}(f_{R},f_{R})(v) - f_{R}(v) \|b\|_{L^1(\S^{d-1})}\big(f_{R}\ast \Phi \big)(v)\Big)\,,
\nonumber
\end{align}
with  $ f_{R}(v):= \text{1}_{\{|v|\leq R\}}\,f(v)\geq 0$,  and    the potential function $\Phi(v)$ satisfies  \eqref{pot-Phi_1} and \eqref{pot-Phi_2}.

The  arguments that follow prove that  $w_{R,f}(v)$ is an element in  the set \eqref{non-empty-set} for $R>0$,  chosen sufficiently large,  and    $\eta>0$  in \eqref{invariant region limit}  chosen sufficiently small,  both parameters only depending on $\epsilon$ and $f$.

Indeed,  observing that $\gamma\in (0,2]$,   and invoking  \eqref{pot-Phi_2}, the collision frequency  is bounded above by,
\begin{align}\label{Cm0}
\big(f_{R}\ast \Phi \big)(v)\leq 2^{\frac{\gamma}2} C_{\Phi}  m_0[f](0) (1+ |v|^{\gamma})  = C(m_0) (1+ |v|^{\gamma})\, ,
\end{align}
then,  choosing $0< \eta \leq 1/\big(\|b\|_{L^1(\S^{d-1})}C(m_{0})\big(1 + R^{\gamma}\big)\big)$, the function $w_{R,f}(v)$ can be estimated from below by
\begin{align}\label{wRf lower bound}
w_{R,f}(v)&\geq f(v) + \eta\,\Big(Q^{+}_{\gamma}(f_{R},f_{R})(v) - \text{1}_{\{|v|\leq R\}} \, f(v) \|b\|_{L^1(\S^{d-1})} \big(f_{R}\ast \Phi \big)(v,t)    \Big)\\
&\geq f(v,t)\Big(1 - \eta \|b\|_{L^1(\S^{d-1})}C(m_{0}) \big(1+R^{\gamma}\big)\Big)\geq0\, , \qquad \text{for} \  C(m_{0}):= 2^{\frac{\gamma}2} C_{\Phi}  m_0[f] .     \nonumber
\end{align}

In addition, since the definition of $w_{R,f}(v,t)$ in \eqref{wRf}  preserves the binary structure of the collisional form
$Q(f_{R},f_{R})(v)$, then by conservation of mass and energy, both such Lebesgue moments of   $w_{R,f}(v)$ coincide with the ones for $f(v)$, that is
\begin{equation}\label{moments-wRf}
\int_{\real^{d}}w_{R,f}(v,t) \left(\!\begin{matrix} 1 \\  |v|^2 \end{matrix}\!\right)  dv \!=\! \int_{\real^{d}}f(v) \left(\!\begin{matrix} 1 \\  |v|^2 \end{matrix}\!\right) dv \!+\!\int_{\real^{d}}Q(w_{R,f}, w_{R,f})(v) \left(\!\begin{matrix} 1 \\  |v|^2 \end{matrix}\!\right) dv  \!=\!
\left(\!\begin{matrix} m_0 \\  m_1-m_0 \end{matrix}\!\right), \  \forall \ R>0\,,
\end{equation}
or, equivalently,
\begin{equation}\label{conserve-wRf}
\int_{\real^{d}}w_{R,f}(v)  - f(v)\, \left ( 1 +  |v|^2\right)  dv = 0.
\quad \forall \ R>0\,.
\end{equation}

Thus, in order to show $w_{R,f}(v,t)$ is an element in  the set \eqref{non-empty-set} for $R>0$, recall first the \textit{a priori} estimates developed for the $k^{th}$-Lebesgue moments of the collisional integral $Q(f,f)(v,t)$  in Theorem~ \ref{propagation-generation}, after setting $s=1$,  namely, for the nonlinear operator  acting on $k$-Lebesgue moments,
\begin{align}\label{Lk operator}
\int_{\mathbb{R}^d} \!Q(f,f) \langle v\rangle^{2k} dv \!\leq \! \B_{k} \!- \!A_{\kc}m_1^{-c}[f_0(0)]m_{k}[f]^{ 1 + c }\! =:\!  \mathcal{L}_{k}\big(m_{k} [f] \big), \  \text{for} \ c\!=\!\frac{\gamma}{2(k-1)}, \ k \!\ge \!{\kc}\! >\!1\,,
\end{align}
maps $\mathcal{L}_{k}(y):[0,\infty)\rightarrow \mathbb{R}$,
with strictly positive  constants $\B_{k}$ and  $A_{\kc}$ (the coercive factor) defined in \eqref{moment-factors}  depending on $f$ only through the initial moments mass and energy initial moments $m_0$ and $m_1$, respectively,  the fundamental constant  $\mu_k $ from the angular averaging  Lemma~\ref{BGP-JSP04},  and the parameter $\gamma$, and the bounds of $\Phi$ from \eqref{pot-Phi_1} depending on the data, for any $k>1$.
In addition, the functional   $\mathcal{L}_k(y)$ is  non-linear form  function acting on $y$,  that is strictly decreasing since it has a super linear negative term due to the fact that $c(k, \gamma)$ is strictly positive for any $\gamma\in(0,2]$ and $k>1.$
As a consequence,   $\mathcal{L}_k(y)$ has {\em at most one positive root} characterized by $\mathcal{L}_{k}(\mathfrak{h}_k^{*})=0 $, depending on  $c= \gamma/(2(k-1))$.

It is worthy to note that the root $\mathfrak{h}_k^{*}$ coincides  with the equilibrium solution associated to the flow $y_t(t)=\mathcal{L}y(t)$, denoted by  $\mathbb{E}_k$ defined in \eqref{RateE},  to the upper ODE problem \eqref{upperODE},  with a positive initial $k$-Lebesgue moment that controls the a priori estimates for the $k$-Lebesgue moments ODE, so the root  $\mathfrak{h}_k^{*}$  is identify with the notation introduced in \eqref{RateE}, namely
%% More specifically, in the case   $\mathcal{L}_{k}(y)$ is defined in \eqref{Lk operator},  it is easily seen that   $\mathcal{L}_{k}(y)$ is a decreasing  function in $y(t)$,  for all $t>0$,   since $\partial_y \mathcal{L}_{k}(y) = - A_{\kc}(1+c) y^c  < 0 $ and $y(t)=m_k[f](t) >0 $ by the definition of Lebesgue moments associated to probability density functions $f(v,t).$
% That means, for each $k$ fixed, there is  the only root for the operator $\mathcal{L}_{k}(\mathfrak{h}_k^{*})=0 $, which coincides  with the unique equilibrium solution to the moments upper ODE
%  $y'(t)= \mathcal{L}_{k}(y)(t)$ calculated in \eqref{RateE}, and so
\begin{equation}\label{Lk 0}
\mathfrak{h}_k^{*}:= \mathbb{E}_k =
m_1^{\frac {c}{1+c}}\Big(
\tfrac{\B_{k}}
{ A_{\kc}}\Big)^{\frac1{1+c}}\,,\qquad  k \ge {\kc} >1\,,
\end{equation}
at which $\mathcal{L}_{k}(y)$ changes from positive to negative,  if $y<\mathbb{E}_k$ or $y>\mathbb{E}_k$ respectively.

In addition, looking at  the explicit formula   \eqref{Lk operator},
the maximum value of the non-linear operator $\mathcal{L}_{k}(y) $ is achieved at $\mathcal{L}_{k}(y)\mid_{y=0} :=\B_k $, since $\partial_y \mathcal{L}_{k}(y) <0$ if $c> 0$, or, equivalently, if $\gamma\ge 0$, and $k>1$.

%\begin{equation}\label{Lk max}
%\mathcal{L}_{k}(y) \leq \max_{0 \leq y \leq \mathfrak{h}_k^{*} }  \mathcal{L}_k( y)  =   \mathcal{L}_k(\bar{\mathfrak{h}}_k)
%=  =:  \bar{\mathcal{L}_{ k}}  \, . % \quad\text{attained at }\quad \bar{\mathfrak{h}}_k =0,.
Therefore,  choosing  $\kc$, such that $k=1+\frac\gamma 2\ge \kc$, not only the exponent factor becomes $c(1+\gamma)=1/2$, but also the sum of  the equilibrium solution $ \mathbb{E}_{1+\gamma} $ plus  the maximum value of the nonlinear operator   $\max_{y\in \Omega}{\mathcal{L}}_{1+\gamma}(y):=  \B_{1+\gamma}$,  provides a characterization to find a sufficient bound  on the moment $m_{1+\gamma}[g] <   {\mathfrak{h}}_{1+\gamma} \le  K,$ that secures the set $\Omega$ to be the invariant region, that is
\begin{align}\label{Lk-frac-h}
{\mathfrak{h}}_{1+\gamma}    &:=  \mathbb{E}_{1+\gamma} + B_{1+\gamma}
= m_1^{\frac13}\Big(
\tfrac{\B_{1+\gamma}}
{ A_{\kc}}\Big)^{\frac 23} + \B_{1+\gamma}= m_1^{\frac13}\Big(\tfrac{\beta_{1+\gamma}}
{ A_{\kc}}\,  \mathcal C_{1+\gamma} \Big)^{\frac 23}+ \beta_{1+\gamma}\mathcal C_{1+\gamma}\\
&= \! m_1^{\frac13} \left[ \left(\! \frac{ \beta_{1+\gamma}}{ A_{\kc} }\!\right)^2 \mathbb{I}_{ k\ge \kgamma }
\!+\! \left(\frac{ \beta_{1+\gamma}}{ A_{{  s}\kc} }\!\right)^{1+\frac\gamma3}  \mathbb{I}_{ k <  \kgamma } \right]
+ \beta_{1+\gamma}\left[ \left(\! \frac{ \beta_{1+\gamma}}{ A_{\kc} }\!\right)^2 \mathbb{I}_{ k\ge \kgamma }
\!+\! \left(\frac{\beta_{1+\gamma} }{ A_{\kc} }\!\right)^{\frac12+\frac94\gamma }  \mathbb{I}_{ k <  \kgamma }\right] \le K,\nonumber \\[4pt]
\text {for} \quad&\beta_{1+\gamma}=2C_\Phi (2^{\frac32\gamma}\mu_\kc +1).  \nonumber
\end{align}

Now, it becomes  clear that for  any  given $f(\cdot,t)\in\Omega$ with $t$ fixed, the constructed it $w_{R,f}$ is an element in the intersection set  defined in \eqref{non-empty-set}, for a choice of $\eta(\eps)$ that makes the limit \eqref{invariant region limit} to hold, so  showing that $\Omega$ is an invariant region for the Boltzmann flow under consideration.

First, observe there are only two options  for the associated moment $y:=m_{1+\gamma}[f]$, namely either $y$ is smaller than  $ \mathbb{E}_{1+\gamma}$,   or larger than it.
%either $m_{1+\gamma}[f] \leq
%\mathfrak{h}_{1+\gamma}^{*}$,  or $m_{1+\gamma}[f] > \mathfrak{h}_{1+\gamma}^{*}$,

Thus for the first case, when $y =m_{1+\gamma}[f] \leq \mathbb{E}_{1+\gamma},$
and for any $\eta<1$  from   \eqref {Lk-frac-h}, it clearly follows that
\begin{align}\label{wRf-case1}
m_{1+\gamma}[w_{R,f}] &= \int_{ \mathbb{R}^{d} } w_{R,f}(v) \langle v\rangle^{2(1+\gamma)}  dv =
\int_{\mathbb{R}^{d}}\big( f(v) + \eta \, Q(f_{R},f_{R})(v)\big) \langle v\rangle^{2(1+\gamma)}  \text{d}v  \nonumber\\
&\leq  \mathbb{E}_{1+\gamma} + \eta\
\max_{y\in\Omega}{\mathcal{L}}_{1+\gamma}   =    \mathbb{E}_{1+\gamma}+ \eta\  \B_{1+\gamma}   \leq \mathfrak{h}_{1+\gamma} \le K,
\end{align}

And for the one when $y:=m_{1+\gamma}[f] >
\mathbb E_{1+\gamma}$,  make the choice  of letting the cut-off parameter $R:=R(f)$  from \eqref{wRf} to be sufficiently large  for  $m_{1+\gamma}[f_{R}]\geq\mathbb E_{1+\gamma}$, implying
\begin{equation*}\label{wRf-case2-0}
\int_{\mathbb{R}^d} Q(f_{R},f_{R}) \langle v\rangle^{2(1+\gamma)} dv \leq \mathcal{L}_{1+\gamma}\big(m_{1+\gamma}[f_{R}]\big)\leq 0\,,
\end{equation*}
and, hence,
\begin{equation}\label{wRf-case2}
m_{1+\gamma}[w_{R,f}] = \int_{\mathbb{R}^{d}} \big(f(v)+\eta \, Q(f_{R},f_{R})\big) \langle v \rangle^{2(1+\gamma)} dv \leq \int_{\mathbb{R}^{d}} f(v) \langle v\rangle^{2(1+\gamma)} dv \leq \mathfrak{h}_{1+\gamma}  \le K.
\end{equation}
Thus, {\em Part 1} is complete, since   for any $f\in\Omega$, we have proven in  \eqref{wRf lower bound}  that  $w_{R,f} \ge  c(\eta, R) \, f(v) \ge 0$,  from  \label{moments-wRf} that the  mass and energy of $w_{R,f}(v)$ coincide with those of $f(v,t)$ and,   from \eqref{wRf-case1} and \eqref{wRf-case1},   that $m_{1+\gamma}[w_{R,f} ] \leq \mathfrak{h}_{1+\gamma} $, and then  the constructed $w_{R,f}$ belongs to  the invariant set $\Omega$.

\smallskip

\noindent {\em Part 2:} In order to show that    the constructed $ w_{R,f}(v)$ in {\em Part 1} belongs to $B(f+\eta\,Q_{\gamma}(f,f),\eta\,\epsilon)$ for the given arbitrary $\epsilon>0$, one needs to calculate the normed distance
\begin{equation*}
\big\|w_{R,f} - f - \eta \, Q_{\gamma}(f,f) \big\|_{L^{1}_{2}(\real^{d})} = \eta\, \big\|Q(f_{R},f_{R}) - Q_{\gamma}(f,f) \big\|_{L^{1}_{2}(\real^{d})}\leq  6\,\eta\,  \mathfrak{h}_{1 + \gamma}^{3/2} \|b\|_{ L^{1}( \mathbb{S}) }\|f_{R} - f\|^{1/2}_{L^{1}_{2}(\real^{d})}\leq \eta \, \epsilon\,,
\end{equation*}
which follows from taking the definition of 	$ w_{R,f} - f = Q(f_{R},f_{R})$ from \eqref{wRf} and invoking  the H\"{o}lder property proven in the previous  Lemma \ref{Holder-estimate}, estimate \eqref{holder 2},  when taking  $R\geq R(\epsilon, f)>0$, sufficiently large such that $\|f_{R} - f\|^{1/2}_{L^{1}_{2}(\real^{d})}\leq \epsilon$.

To end, adjust the choice of  $w_{R,f} \in B(f+\eta\,Q[f],\eta\,\epsilon)$  for $\bar R =\max\{R(f),R(\epsilon,f)\}$  and  a possible smaller   $0< \eta_*(\epsilon,f) \leq 1/\big(C(m_{0})\big(1 + \bar R^{\lambda}\big)\big)$ to obtain,  not only $w_{\bar R,f}\in \Omega$, but also  for an arbitrary $\epsilon$, it follows that
\begin{equation}\label{limit 0+}
\eta^{-1}\text{dist}\big( f + \eta\, Q_{\gamma}(f,f) , \Omega\big) \leq \epsilon\,, \qquad  \forall \ 0<\eta\leq  \eta_{*}\,.
\end{equation}

Therefore  taking  the limit as $\eta\to 0^+$ of \eqref{limit 0+} vanishes, proving that    property \eqref{invariant region limit} holds, and so  the proof of Lemma \ref{prop_subtan} is now complete. \\
\end{proof}

The last condition to check that is satisfied is {\bf iii)},  the One-Sided Lipschitz condition,  which is a sufficient to show that if   the initial $f_0\in L^{1}_{2+2\gamma}(\mathbb{R}^{d})$, solutions of  Theorem~\ref{CauchyProblem} are unique . The original proof was performaned by G. Di Blassio in [54] for the case $\gamma =1$, were polynomial moments were defined with weights $|v|^{2\gamma}$.

\begin{lem}[{\bf one-sided Lipschitz condition -- Property iii)}]\label{one-Lips-prop3}
For any function $f$ and $g$ lying in $\Omega$, it holds that
\begin{equation}\label{osLip}
\big[f-g,Q_{\gamma}(f,f) - Q_{\gamma}(g,g)\big]_{-}\leq \|b\|_{L^{1}(\mathbb{S}^{d-1})}\,C_{\Omega}\,\|f-g\|_{L^{1}_{2}(\mathbb{R}^{d})}\,,
\end{equation}
with $C_{\Omega}:= 8K$.
\end{lem}
\begin{proof} Recall that in $L^{1}_{2}(\real^{d})$, the Lipschitz bracket simply reads
\begin{equation}\label{LipBracket}
\big[\varphi, \psi\big]_{-} =    \int_{\real^{d}} \psi \,\text{sign}(\varphi)  \,\langle v\rangle^2 dv \,.
\end{equation}
Then, using the weak formulation (with the shorthand $\int=\int_{\real^d}\int_{\real^d}\int_{\mathbb{S}^{d-1}}$ and $h=f-g$)
\begin{align}\label{osLip1}
\big[f-g&,Q_{\gamma}(f,f) - Q_{\gamma}(g,g)\big]_{-} =\int_{\real^{d}}\big(Q_{\gamma}(f,f) - Q_{\gamma}(g,g)\big)\,\text{sign}(f-g)\,\langle v \rangle^{2} dv\\
&= \int \big(f\,f_{*} - g \, g_{*}\big)\Delta(v,v_{*})|u|^{\gamma}b(\hat{u}\cdot\sigma) \text{d}\sigma \text{d}v_*\text{d}v =\int \big( h\,f_{*} + g\,h_{*} \big)\Delta(v,v_{*})|u|^{\gamma}b(\hat{u}\cdot\sigma) \text{d}\sigma \text{d}v_*\text{d}v\,.\nonumber
\end{align}
Here, $\Delta(v,v_{*}) = \langle v' \rangle^{2}\,\text{sign}(h') + \langle v'_{*} \rangle^{2}\,\text{sign}(h'_{*}) - \langle v \rangle^{2}\,\text{sign}(h) - \langle v_{*} \rangle^{2}\,\text{sign}(h_{*})$.  Observe that
\begin{align}\label{osLip2}
h\, \Delta(v,v_{*})&\leq |h|\,\Big(\langle v' \rangle^{2} + \langle v'_{*} \rangle^{2} + \langle v_{*} \rangle^{2} - \langle v \rangle^{2}\Big) = 2\,|h|\,\langle v_{*} \rangle^{2}\,,\quad\text{ and }\nonumber\\
h_{*}\, \Delta(v,v_{*})&\leq |h_{*}|\,\Big(\langle v' \rangle^{2} + \langle v'_{*} \rangle^{2} + \langle v \rangle^{2} - \langle v_{*} \rangle^{2}\Big) = 2\,|h_{*}|\,\langle v \rangle^{2}\,.
\end{align}
Combining,  both \eqref{osLip1} and   \eqref{osLip2} estimates, local conservation of energy.  Therefore, (with the shorthand $H=f+g$)
\begin{align}\label{one-sided}
\big[f-g,&Q_{\gamma}(f,f) - Q_{\gamma}(g,g)\big]_{-} \leq 2\,\|b\|_{L^{1}(\mathbb{S}^{d-1})}\int_{\real^{d}}\int_{\real^{d}}\big(|h|\,f_{*}\,\langle v_{*}\rangle^{2} + |h_{*}|\,g\,\langle v \rangle^{2} \big)\,|u|^{\gamma} \, dv_{*}\,dv\\
&= 4\,\|b\|_{L^{1}(\mathbb{S}^{d-1})}\int_{\real^{d}}\int_{\real^{d}}|h|\,H_{*}\,\langle v_{*}\rangle^{2}\,|u|^{\gamma} \, dv_{*}\,dv\leq 4\,\|b\|_{L^{1}(\mathbb{S}^{d-1})}\|H\|_{L^{1}_{2+2\gamma}}\|h\|_{L^{1}_{\gamma}}\,.\nonumber
\end{align}
Since $\|H\|_{L^{1}_{2+2\gamma}}= m_{1+\gamma}[f] + m_{1+\gamma}[g]  \,  \le 2K $,  estimate  \eqref{osLip} follows.\\
\end{proof}

%  \max\{ \delta^{- \frac{\theta_{(1+\gamma)}}{1-\theta_{(1+\gamma)}} },     \delta^{- \frac{\alpha_{(1+\gamma)}}{1-\alpha_{(1+\gamma)}}} \}
% %
% \nonumber \\
%  &\qquad +
% 2^{1+\frac{3\gamma}2} {C}_\Phi \mu_{(1+\gamma)}   \max\{ \delta^{- \frac{\theta_{(1+\gamma)}}{1-\theta_{(1+\gamma)}}},

\bigskip

Hence, Lemmata \ref{Holder-estimate}, \ref{prop_subtan}, and \ref{one-Lips-prop3} ensures that conditions $\bf{i), ii), iii)}$ of Theorem~\ref{Theorem_ODE},  and so  the following existence and uniqueness result,  with
$f_0\in L^1_{2+2\gamma}({\mathbb S^{d-1})}$ to the initial value problem for the space homogeneous Boltzmann equation under consideration holds,
\begin{equation*}
f_t = Q(f,f)\,, \qquad \text{with initial data}\qquad  f(v,0) = f_0(v) \in \Omega\,,
\end{equation*}
has a unique solution $0\leq f(v,t)\in \mathcal{C} \big([0,\infty); \Omega\big)\cap  \mathcal{C}^1\big((0,\infty);L^{1}_{2}\big(\mathbb{R}^{d}\big)\big)$.

\medskip

Hence, the second stage of proof of Cauchy Problem, Theorem~\ref{CauchyProblem}, are fulfilled and, as a consequence, the homogeneous Boltzmann problem, and the new    moment condition on the initial data can be relaxed to having an initial data that belongs to $L^1_{2^+}$ for $\gamma\in(0,2]$.   In fact, under Remark~\ref{kc=eps} conditions, set $\epsilon =  2(\kc-1) \in (0, \gamma)$.

The following Lemma justifies this statement.

\begin{lem}\label{Lemma_ODE-extension2}
Let $\gamma\in(0,2]$ and $0\leq f_0\in L^{1}_{2+\epsilon}(\mathbb{R}^{d})$, with  $\epsilon\in(0,\gamma)$ fixed.
\begin{itemize}
\item[{\it i)}]\  There exists a non negative sequence of initial data $\{f^{j}_{0}\}\subset L^{1}_{2+\gamma}(\mathbb{R}^{d})\cap \Omega$ strongly converging to $f_0$, that is
\begin{equation} \label{fj-convergence}
f^{j}_{0} \longrightarrow f_0\,,\quad\text{strongly in} \; L^{1}_{2+\epsilon}(\mathbb{R}^{d})\,.
\end{equation}

\item[{\it ii)}]\ There is a sequence of solutions of the Boltzmann Cauchy problem $f^j(v,t) \in L^{1}_{2+\gamma}(\mathbb{R}^{d})$,  with initial data $f^{j}_{0}\in L^{1}_{2+\gamma}(\mathbb{R}^{d})$,  converging  to   $f\in \mathcal{C}\big([0,\infty); L^{1}_{2}(\mathbb{R}^{d}) \big)\ \cap\ \mathcal{C}^{1}\big((0,\infty); L^{1}_{\ell}(\mathbb{R}^{d}) \big),$ with $2\le \ell<2(1+\gamma), $ that solves  the Cauchy problem with initial data $f_0\in L^{1}_{2+\epsilon}(\mathbb{R}^{d}).$
\end{itemize}

\end{lem}

\begin{proof}
Let $0\leq f_0\in L^{1}_{2+\epsilon}(\mathbb{R}^{d})$, by density, there exits an approximating sequence   $\{f^{j}_0\} \subset \Omega\subset L^1_2(\R^d)$, with $K$ arbitrarely large,   such that  $\lim_{j\to\infty} f^{j}_0 =f_0$  in $L^{1}_{2+\epsilon}(\mathbb{R}^{d})$ satisfying property {\it i)}, such that  $\|f^{j}_0\|_{L^{1}_{2+\epsilon}(\mathbb{R}^{d}) }< C_{\epsilon}[f_0], $ uniformly in $j$.
Such approximations can be obtained, for instance, by taking $f^j_0 (v):=  f_0(v) \, \mathbb I_{v<j}\in L^1_\ell$ for all $\ell\le 2(1+\gamma)$, which implies $\{f^{j}_0\} \subset \Omega^j_\gamma$ and  clearly $\|f^{j}_0 - f_0\|_{L^{1}_{2+\epsilon}(\mathbb{R}^{d})}$  converges  strongly to $0$.

In particular,  for each $f^{j}_0\in \Omega_\gamma^j\subset L^1_2(\R^d)$,  the first stage of the Cauchy problem  Theorem~\ref{CauchyProblem} secures the existence and uniqueness of solutions, invoking   the tools from Lemmas \ref{Holder-estimate}, \ref{prop_subtan}, and \ref{one-Lips-prop3} which are  sufficient  conditions $\bf{i), ii), iii)}$ for Theorem~\ref{Theorem_ODE} to hold for each $j$ fixed, that is  there exists and $f^j(v,t)\in \C(0,\infty, \Omega^j_\gamma) \ \cap\ \mathcal{C}^{1}\big((0,\infty); L^{1}_{2}(\mathbb{R}^{d}) \big).$
%so we can use  the properties for function on the set $\Omega$.

Next, it is important to verify the uniformity of the $\{ f^j(\cdot,t)\}$  sequence   in the $ L^{1}_{2+\epsilon}(\mathbb{R}^{d}) \big)$ topology.
To this end,  start from invoking the multi linearity structure of the collisional form  and   the  one-sided Liptchitz condition \eqref{one-sided} from  Lemma \ref{one-Lips-prop3},  which clearly remains valid  independently of the pair $f^{j_1}$ and $f^{j_2}$ are in different $\Omega_\gamma^{j_1}$ and  $\Omega_\gamma^{j_2}$, respectively, to obtain
%Thus, when   applied to the difference  $\frac{d}{dt}\left(f^j-f^l\right)$,
% in order to obtain an $L^1_2(\R^d) is a $
the following estimate  for the difference on any pair $f^j$ and $f^l$ of the approximating sequence $\{ f^j(\cdot,t)\}$,
\begin{align}\label{e1Cauchy}
\frac{d}{dt}\|f^{j}(t) - f^{l}(t)\|_{L^1_2}  &= \frac{d}{dt}\int_{\R^d} (f^j(t) - f^l(t)) \,\text{sign}(f^j(t) - f^l(t)) \langle v \rangle^2 dv  \nonumber\\
%&= \int_{\R^d} \left(\left( Q(f^j,f^j) - Q(f^l,f^l)\right) \text{sign}(f^j - f^l)\right)(v,t) \langle v \rangle^2 dv \nonumber\\
&= \int_{\R^d} \left( Q_\gamma(f^j+f^l,f^j-f^l )  \,\text{sign}(f^j - f^l) \right)(v,t) \langle v \rangle^2 dv
\\
&\leq 4\|b\|_{L^{1}(\mathbb{S}^{d-1})}
\|f^{j}(t) \!+\! f^{l}(t)\|_{L^{1}_{2+2\gamma}}\|f^{j}(t) \!-\! f^{l}(t)\|_{L^{1}_{\gamma}} =: \mathcal A(t) \,\|f^{j}(t) \!-\! f^{l}(t)\|_{L^{1}_{\gamma}}, \nonumber
\end{align}
for   $\A(t):= 4\|b\|_{L^{1}(\mathbb{S}^{d-1})}
\|f^{j}(t) \!+\! f^{l}(t)\|_{L^{1}_{2+2\gamma}}$.

Hence,  since $\|f^{j}(t) - f^{l}(t)\|_{L^1_2}= \|f^{j}(0) - f^{l}(0)\|_{L^1_2}\exp(-\int_0^t\A(\tau) d\tau)$,  then, the  uniformity of the  sequence $\{ f^j(\cdot,t)\}\in L^{1}_{2+\epsilon}(\mathbb{R}^{d}) $   follows from the uniform integrability of  $\A(t), $ independently of sequence's index $j$.

%Indeed,  since  the sequence of initial their data $\{f^{j}_{0}\}$ converges in  $L^1_{2+\epsilon}(\mathbb{R}^{d})$, then
%it is enough to show
% $\{f^{j}(t)\}$ uniformly converges and it is bounded in $L^1_{2+\epsilon}(\mathbb{R}^{d})$ as well.
%For such task,
Therefore,   invoking the propagation of  moments and global in time  estimates \eqref{mom-prop} from Theorem~\ref{propagation-generation}   for each $j$-term in the sequence,  it ensures a boundedness of
$\{ f^{j}(t)\}$  in the $L^{1}_{2+\epsilon}(\R^d)$  topology,  with  for all $m_0[f^j_0]=m_0[f_0]$ and  $m_1[f^j_0]=m_1[f_0]$, not only, for $c_\gamma(\kc)=\frac{\gamma}{\epsilon}$,
\begin{align}\label{M1eps}
\sup_{t\geq0}\|f^{j}(t)\|_{L^{1}_{2+\epsilon}}
% &\le \sup_{t\geq0} m_{1+\epsilon}[f^{j}(t)]
\le \sup_{t\geq0} m_{1+\frac\epsilon 2}[f^{j}](t)
&= \max\{  m_{1+\frac\epsilon 2}[f^{j}](0),  \mathbb E^j_{c_\gamma(1+\epsilon)}  \}\nonumber\\
&=: M_{1+\frac\epsilon 2 }(m_0[f_0], m_1[f_0],  b(\hat u\cdot\sigma), \gamma, \epsilon),
\end{align}
but also,  since the convergence $f^{j}_0\to f_0$ in the $L^{1}_{2+\epsilon}(\mathbb{R}^{d})$ also holds in $L^{1}_{2}(\mathbb{R}^{d})$, that means there is a constant $C_{\epsilon}[f^{j}_0]$ that only depends on $m_0[f_0]$ and
$m_1[f_0]$,  converging to the mass and energy of $f_0$,   yielding the uniform bound estimate
\begin{equation}\label{uniform j}
\sup_{t\geq0}\|f^{j}(t)\|_{L^{1}_{2+\epsilon}} \leq \max\big\{M_{1+ \epsilon }[f_0],\, C_{\epsilon}[f_0], \gamma, \epsilon\big\}=: C_0(f_0, b(\hat u\cdot\sigma),\gamma,\epsilon), \ \ \text{ independent of} \ j.
\end{equation}

Furthermore, for each fixed $j$,  invoking the generation  of  moments and its  priori global in time estimates Theorem \ref{propagation-generation},  each solution $f^j(t)\in \C(0,\infty, \Omega^j_\gamma) \ \cap\ \mathcal{C}^{1}\big((0,\infty); L^{1}_{2}(\mathbb{R}^{d}) \big)$ constructed from the initial data  $f^j_0$ es estimated by  invoking  generation global bounds \eqref{mom-gen},   with $k=1+\frac\gamma 2+\frac\epsilon 2$,  so $c^{-1}_\gamma(k)= \frac{\gamma +\epsilon}{\gamma} $ to obtain
\begin{equation}\label{gen j}
\|f^{j}(t)\|_{L^{1}_{(2+\gamma+\epsilon)}}\leq \tilde{C}_0\Big(1 + t^{- \frac{\gamma+\epsilon}{\gamma}}\Big)\,,\quad \text{for any}\ \  \epsilon\in (0,\gamma),
\end{equation}
with constant $ \tilde{C}_0 =\tilde{C}_0(f_0,b(\hat u\cdot\sigma), \gamma,\epsilon)$ independent of $j$, after combining   by the aforementioned estimates \eqref{e1Cauchy} with \eqref{M1eps} and \eqref{uniform j}.
Therefore, by interpolation,
\begin{align}\label{interpol-free}
\|f^{j}(t)\|_{L^{1}_{2+\gamma}(\mathbb{R}^{d})} \ &\leq \ \|f^{j}(t)\|^{\frac{\gamma-\epsilon}{\gamma}}_{L^{1}_{(2+\gamma+\epsilon)}(\mathbb{R}^{d})}\|f^{j}(t)\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2+\epsilon}(\mathbb{R}^{d})}\\
&\le\  \tilde{C}_0^{\frac{\gamma-\epsilon}{\gamma}} \left(1 + t^{- \frac{\gamma+\epsilon}{\gamma}}\right)^{\frac{\gamma-\epsilon}{\gamma}}\, \|f^{j}(t)\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2+\epsilon}(\mathbb{R}^{d})}
\leq \  \tilde C_1 \, \mathcal{O}\!\left(1 + t^{-\theta}\right),
\quad \text{with }\theta = \frac{\gamma^{2}-\epsilon^{2}}{\gamma^{2}}  \in(0,1)\, , \nonumber
\end{align}
and $\tilde C_1 =\tilde{C}_0^{\frac{\gamma-\epsilon}{\gamma}} C_0^{\frac{\epsilon}{\gamma}}$,   uniformly in $j$.

This last  estimate  is crucial: not only shows the control the term in \eqref{e1Cauchy}, but also allows the control of the
$ \|f^{j}(t)\|_{L^{1}_{2+\gamma}(\mathbb{R}^{d})}$ by  $ \|f^{j}(t)\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2+\epsilon}(\mathbb{R}^{d})}$  proportional to  $\kappa(t) := \tilde{C}_0^{\frac{\gamma-\epsilon}{\gamma}}\mathcal{O} \left(1 + t^{- \frac{\gamma+\epsilon}{\gamma}}\right)^{\frac{\gamma-\epsilon}{\gamma}}$  in any time interval $t\in(0, T), \ T>0$, uniformly in the $j$-index.

Hence,  the first listed fact implies the control of $\A(t)$ in \eqref{e1Cauchy}, to obtain
\begin{align}\label{mathcalA(t)-uni j}
\A(t)&:= 4\|b\|_{L^{1}(\mathbb{S}^{d-1})}
\|f^{j}(t) \!+\! f^{l}(t)\|_{L^{1}_{2+\gamma+\epsilon}}
\le \tilde C_1 \mathcal{O}\!\left(1 \!+\! t^{-\theta}\right) \,  \|f^{j}(t)\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2+\epsilon}(\mathbb{R}^{d})}=
\tilde{C}_1\mathcal{O}\!\left(1 \!+\! t^{-{\frac{\gamma^{2}-\epsilon^{2}}{\gamma^{2}}}}\!\right)\!, %\nonumber
\end{align}
where the constants $\tilde C_0$ and $\tilde C_1$ depend only on $f_0, b(\hat u\cdot\sigma), \gamma,\epsilon$, independently of the sequence's index $j$.

Therefore, inserting  this last estimate in \eqref{e1Cauchy}
leads to, after performing the time integration
\begin{equation}\label{uni-j-bound}
\|f^{j}(t) - f^{l}(t)\|_{L^{1}_{2}} \leq \|f^{j}_0 - f^{l}_0\|_{L^{1}_{2}} \exp\left(\tilde C_1\mathcal{O}\!\left(t +\tfrac{\gamma^{2}}{\epsilon^{2}}t^{1-\theta}\right) \right)\,,\quad \text{with}\  \theta \in(0,1), \ \  0< t\le T.
\end{equation}

As a consequence, the sequence  $\{f^{j}(t)\}$ is Cauchy, and so it  converges strongly to a unique $0\le f(v,t) \in L^{\infty}\big([0,T);L^{1}_{2}(\mathbb{R}^{d})\big)$,  for any  arbitrary $T>0$.  And thus, the second estimate from \eqref{interpol-free}, ensures that the limiting function $f(v,t)$ satisfies for any terminal $T>0,$
\begin{equation}\label{f-bound 0 }
\|f(t)\|_{L^{1}_{2+\gamma}(\mathbb{R}^{d})}\ \le\  \tilde{C}_1\mathcal{O} \!\left(1\! +\! t^{- \frac{\gamma+\epsilon}{\gamma}}\right)^{\frac{\gamma-\epsilon}{\gamma}} \|f(t)\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2+\epsilon}(\mathbb{R}^{d})} \quad \text{for any}\  \ \epsilon\in(0,\gamma), \ \ \ 0<t\le T.
\end{equation}

In addition,  the convergence $f^j(v,t) \to f(v,t)$  in  $L^{\infty}\big([0,T);L^{1}_{2}(\mathbb{R}^{d})\big)$ holds. Invoking the interpolation estimates developed the H\"older's estimates     Lemma~\ref{Holder-estimate},  \eqref{holder 1} and \eqref{holder 2} in  $L^1_{\ell}(\mathbb{R}^d), \ell\ge2+\gamma$,
the  collision  operator  $Q_\gamma(f^j,f^j)$ is continuous operator in  from  ${L^1_2(\R^d)}$ into itself,  shown as follows.
% \begin{align}\label{oslim 4}
% \|Q_\gamma(f^j,f^j) \|_{L^1_2(\R^d)}&(t) =2\int_{\mathbb{R}^{d}} Q^\pm_\gamma(f^j,f^j)(v,t)   \langle v\rangle^2 \text{d}v
% \le \int_{\mathbb{R}^{2d}\times\mathbb{S}^{d-1}} f^j_*\,j^j |u|^{\gamma} \left(\langle v'_*\rangle^2 +  \langle v'\rangle^2 \right) b(\hat{u}\cdot\sigma) \text{d}\sigma \text{d}v_*\text{d}v \nonumber\\
%&\le C_\Phi 2^{\frac\gamma 2 +2}\|b\|_{L^{1}(\mathbb{S}^{d-1})}
% \int_{\mathbb{R}^{2d}} f^j\,  f^j_* \left(\langle v\rangle^\gamma+ \langle v_*\rangle^\gamma\right) \left( \langle v_*\rangle^2  + \langle v\rangle^2 \right)
% \text{d}v_*\text{d}v\\ %\nonumber
%C_\Phi 2^{-\frac\gamma 2+2}\  \|b\|_{L^{1}(\mathbb{S}^{d-1})} \|f^j\|_{L^1_{2+\gamma}}  \|f_*^j\|_{L^1_{2+\gamma}}
%C_\Phi 2^{\frac\gamma 2+4}\, \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) } \|f^j\|^{3/2}_{L^{1}_{2+2\gamma}} \, \|f^j\|^{1/2}_{L^{1}_{2}}\nonumber.\\
%%&\le C_\Phi 2^{\frac\gamma 2+2}\, \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) } \, K \tilde{C}_0 \mathcal{O}\left(1 + t^{-\theta}\right)^{\frac{\gamma-\epsilon}{\gamma}} \, \|h\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2}} =:
%%{\bf \it{UB}}(t) \,  \|h\|^{\frac{\epsilon}{\gamma}}_{L^1_2},\nonumber
%&C_\Phi 2^{\frac\gamma 2+4}\, \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) }\, K() \, \|f^j\|^{1/2}_{L^{1}_{2}}\nonumber.\\
% \end{align}

Indeed, recalling the previous argument from \eqref{e1Cauchy}, set
\begin{equation}\label{oslim 1}
\frac{d}{dt}\left( f^j-f^l\right) (v,t) = \left(Q_\gamma(f^j,f^j)-Q_\gamma(f^l,f^l)\right)(v,t)
\end{equation}
for $H=f^j+f^l$ and  $h=f^j-f^l.$
Its $L^1_2(\R^d)$ norm is
\begin{equation}\label{oslim 2}
\int_{\mathbb{R}^{d}} Q^-_\gamma(H,h)(v,t) \sign(h)  \langle v\rangle^2 \text{d}v +  \frac{d}{dt} \| h(t) \|_{L^{1}_{2}}(t) =
\int_{\mathbb{R}^{d}} Q^+_\gamma(H,h)(v,t) \sign( h) \langle v\rangle^2 \text{d}v,
\end{equation}
noticing that both the gain and loss collisional forms associated  $\int_{\mathbb{R}^{d}} Q^\pm_\gamma(H,h)(v,t) \sign(h) \,
\langle v\rangle^2 \text{d}v $ are similar due to the local conservation property $\langle v'_*\rangle^2 +\langle v'_*\rangle^2=\langle v_*\rangle^2 +\langle v_*\rangle^2$.  This difference of gain and loss operators are estimated by
\begin{align}\label{oslim 4}
\int_{\mathbb{R}^{d}} Q^\pm_\gamma(H,h)(v,t) \sign( h) & \langle v\rangle^2 \text{d}v
\le \int_{\mathbb{R}^{2d}\times\mathbb{S}^{d-1}} H_*\,h |u|^{\gamma}\left( \sign(h'_*) \langle v'_*\rangle^2 + \sign(h') \langle v'\rangle^2 \right) b(\hat{u}\cdot\sigma) \text{d}\sigma \text{d}v_*\text{d}v \nonumber\\
&\le C_\Phi 2^{\frac\gamma 2 +1}\|b\|_{L^{1}(\mathbb{S}^{d-1})} \int_{\mathbb{R}^{2d}} \left(\langle v\rangle^\gamma+ \langle v_*\rangle^\gamma\right) \left( \langle v_*\rangle^2  + \langle v\rangle^2 \right)
|h|\,  |H_*| \text{d}v_*\text{d}v \nonumber \\
%C_\Phi 2^{-\frac\gamma 2+2}\  \|b\|_{L^{1}(\mathbb{S}^{d-1})} \|h\|_{L^1_{2+\gamma}}  \|H\|_{L^1_{2+\gamma}}
C_\Phi 2^{\frac\gamma 2+2}\, \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) }  \|H\|^{\frac{\gamma-\epsilon}{\gamma}}_{L^{1}_{2+\gamma+\epsilon}} \, \|h\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2}}\\
&\le C_\Phi 2^{\frac\gamma 2+4}\, \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) } \,  \tilde{C}_1 \mathcal{O}\!\left(1\! + \!t^{-\theta}\right)^{\frac{\gamma-\epsilon}{\gamma}} \, \|h\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2}} =:
{\bf \it K}_0   \|h\|^{\frac{\epsilon}{\gamma}}_{L^1_2},\nonumber
\end{align}
%Then,  an analog argument  works to obtain lower bounds for the   loss operator,
% \begin{align}\label{oslim 3}
% \int_{\mathbb{R}^{d}} Q^-_\gamma(H,h)(v,t) \sign( h)  &\langle v\rangle^2 \text{d}v =
%  \int_{\mathbb{R}^{2d}\times\mathbb{S}^{d-1}} H_*\,h |u|^{\gamma}\left( \sign(h_*) \langle v_*\rangle^2 + \sign(h) \langle v\rangle^2 \right) b(\hat{u}\cdot\sigma) \text{d}\sigma \text{d}v_*\text{d}v \nonumber\\
%&\ge - \|b\|_{L^{1}(\mathbb{S}^{d-1})} \int_{\mathbb{R}^{2d}}(C_\Phi 2^{\frac\gamma 2} \langle v\rangle^\gamma+ \langle v_*\rangle^\gamma)
%|h|\,  |H_*| (\langle v_*\rangle^2 +\langle v\rangle^2) \text{d}v_*\text{d}v \nonumber\\
%%-\|b\|_{L^{1}(\mathbb{S}^{d-1})}\left(c_\Phi 2^{-\frac\gamma 2} \|h\|_{L^1_{\gamma}}  \|H\|_{L^1_{2}} +\|h\|_{L^1}  \|H\|_{L^1_{2+\gamma}}
%%\right) \nonumber\\
%%C_\Phi 2^{-\frac\gamma 2+2}\  \|b\|_{L^{1}(\mathbb{S}^{d-1})} \|h\|_{L^1_{2+\gamma}}  \|H\|_{L^1_{2+\gamma}}
%C_\Phi 2^{\frac\gamma 2+2}\, \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) } \  \|H\|^{\frac{\gamma+\epsilon}{\gamma}}_{L^{1}_{2+\gamma+\epsilon)}} \, \|h\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2}}\\
%&-\ge C_\Phi 2^{\frac\gamma 2+2}\, \|b\|_{ L^{1}( \mathbb{S}^{d-1} ) } K \, \tilde{C}_0^{\frac 12 } \mathcal{O}\left(1 + t^{- \theta}\right)^{\frac{\gamma-\epsilon}{\gamma}} \, \|h\|^{\frac{\epsilon}{\gamma}}_{L^{1}_{2}} =:
%{\bf \it{UB}}(t) \,  \|h\|^{\frac{\epsilon}{\gamma}}_{L^1_2},\nonumber
% \end{align}