text:
\section{Introduction and main results}
Consider the free Schr\"odinger equation
\begin{equation}\label{Sch}
\begin{cases}i{u_t}(t, x)&=-\Delta u(t, x) \hspace{0.5cm}x\in\mathbb R^d, t\in\mathbb R\\
u(0, x)&=f(x)
\end{cases}
\end{equation}
where $\Delta=\sum_{j=1}^d \frac{\partial ^2}{\partial  x_j^2}$,  Laplacian on $\mathbb R^d$. For $f\in L^2(\mathbb R^d)$, $e^{it\Delta}f$ is the unique solution to the initial value problem (\ref{Sch}). The associated Strichartz inequality reads,
\begin{equation}\label{Ste1}
\int_{\mathbb R}\left(\int_{\mathbb R^d}|\left(e^{it\Delta}f\right)|^{2q}dx\right)^\frac{p}{q}dt\leqslant C\left(\int_{R^d}|f(x)|^2dx\right)^p,
\end{equation}
where $p, q\geqslant1$ satisfy $(p, q, d)\neq(1, \infty, 2)$ and $$\frac{2}{p}+\frac{d}{q}=d,$$
see \cite{RS,KY,LS,KT,TT}.  Rupert L. Frank et. al in \cite{FLLS} and \cite{FS} generalized the above inequality for a system of  orthonormal functions and the authors proved the following:

\begin{theorem}\label{SL}
Let $d\geqslant 1$ and $ p, q\geqslant 1$ be such that
$$\frac{2}{p}+\frac{d}{q}=d,~~~1\leqslant q < \frac{d+1}{d-1}. $$Then for any infinite or finite orthonormal system $(f_j)$ in $L^2(\mathbb R^d)$ and for any sequence $(n_j)$ in ${\mathbb C}$, we have
\begin{equation}\label{Sil}
\int_{\mathbb R}\left(\int_{\mathbb R^d}\left|\sum_{j} n_j \big | e^{it\Delta} f_j(x)\big|^2 \right|^q dx\right)^{p/q}dt \leqslant C_{d, q}^p \Big( \sum_j |n_j|^{\frac{2q}{q+1}}\Big)^{\frac{p(q+1)}{2q}} \end{equation}
 where $C_{d, q}^p$ is a universal constant which only depends on $d$ and $q$.
\end{theorem}
 In \cite{FLLS}, the authors proved that if $q\geqslant \frac{d+1}{d-1}$, then the inequality \eqref{Sil} doesn't hold and the exponent on the right hand side of the inequality \eqref{Sil} is optimal i.e., the exponent $\frac{2q}{q+1}$ can not be increased.
It is mentioned in \cite{FLLS} that the functional inequalities involving a large number of orthonormal functions like the inequality \eqref{Sil} are very useful in the mathematical analysis of large quantum system. Our aim of this article is to prove the inequality analogues to the inequality \eqref{Sil} when we replace Laplacian by Dunkl Laplacian (for the definition, see Section \ref{Pr}).
Let $e^{it\Delta_k}f$ be the unique solution of the free Schr\"odinger equation associated with Dunkl Laplacian:
\begin{equation}\label{Schd}
\begin{cases}i{u_t}(t, x)&=-\Delta_{\kappa} u(t, x) \hspace{0.5cm}x\in\mathbb R^d, t\in\mathbb R\\
u(0, x)&=f(x).
\end{cases}
\end{equation}
Mejjaoli studied the above equation first in the literature for the case of Dunkl Laplacian in \cite{M1} and he proved the following Strichartz inequality:
\begin{equation}\label{Sted}
\int_{\mathbb R}\left(\int_{\mathbb R^d}|e^{it\Delta_{\kappa}}f(x)|^{2q}dx\right)^\frac{p}{q}dt\leqslant C\left(\int_{R^d}|f(x)|^2dx\right)^p,
\end{equation}
where $d\geqslant 1$ and $p, q\geq 1$ are such that
$$\frac{2}{p}+\frac{d+2\gamma_\kappa}{q}=d+2\gamma_\kappa.$$
As applications of study of the Strichartz estimates \eqref{Sted} for the Dunkl Schr\"{o}dinger equation \eqref{Schd}, Majjaoli had studied the global well-posedness and scattering for a class of non linear Dunkl Schr\"{o}dinger equations in \cite{M2}. For all unexplained notations we refer Section \ref{Pr}.  In the same spirit of \cite{FLLS} and \cite{FS}, it is natural to ask the inequality analogues to \eqref{Sil} for the case Dunkl Laplacian. Our main result lies on this direction:

\begin{theorem}[Strichartz inequality for orthonormal functions for Dunkl Laplacian] \label{TSOF}
Let $d\geqslant 1$ and $p,q\geqslant1$ satisfy $$1\leqslant q < {\frac{d+2\gamma_\kappa+1}{d+2\gamma_\kappa-1}}~ \mbox{and}~ \frac{2}{p}+\frac{d+2\gamma_\kappa}{q}=d+2\gamma_\kappa.$$
Then for any (possibly infinite) system $(f_j)$ of orthonormal functions in $L_{\kappa}^2 (\mathbb R^d)$ and any cofficients $(n_j)\subset{{\mathbb C}},$ we have
\begin{equation}\label{Ste2} \int_\mathbb R\left(\int_{\mathbb R^d}\left|\sum_{j}n_j|e^{it\Delta_{\kappa}}f_j(x)|^2\right|^q dx\right)^{\frac{p}{q}}dt\leqslant{{C^{p}_{d,q}}\left(\sum_{j}|n_j|^{\frac{2q}{q+1}}\right)}^\frac{p(q+1)}{2q}
\end{equation}
where $C_{d,q}$ is a universal constant which only depends on $d$ and $q.$
\end{theorem}

\begin{rem}
For $q=1$ and $p=\infty,$ since $e^{it\Delta_\kappa}$ is a unitary operator on $L^2_\kappa(\mathbb R^d),$  using triangle inequality it can be seen that $$\sup_{t\in\mathbb R}\left(\int_{\mathbb R^d}\left|\sum_j n_j|e^{it\Delta_\kappa}f_j(x)|^2\right|h^2_\kappa(x)dx\right)\leqslant\sum_j|n_j|.$$
\end{rem}

Further extension of inequality \eqref{Ste1} has been made for the Schr\"odinger equation of the form $iu_t(t, x)-P u(t, x) - V(x)u(t, x) = 0,$ for a suitable potential $V$ by several authors, see \cite{GV, JSS, KPV} when $P=-\Delta$ and \cite{M3, M4} when $P=-\Delta_{\kappa}$. In particular, when $V(x)=|x|^2$,  the Strichartz inequalities have been studied in the literature see \cite{KT, NR, MJ}. It is natural to ask such inequalites are true when  $\Delta$ is replaced by $\Delta_{\kappa}$. In this case the initial value problem \eqref{Sch} turns out to be an initial value problem for the Schr\"odinger equation associated with the Dunkl Hermite operator $H_{\kappa}=-\Delta_{\kappa}+|x|^2$:
\begin{equation}\label{Schdh}
\begin{cases}i{u_t}(t, x)&=H_\kappa  u(t, x) \hspace{0.5cm}x\in\mathbb R^d, t\in\mathbb R\\
u(0, x)&=f(x).
\end{cases}
\end{equation}
 The author in \cite{M3} proved the Strichartz inequalities for the above problem. If $f\in L^2_\kappa (\mathbb R^d),$ the solution of the initial value problem \eqref{Schdh} is given by $u(t, x) = e^{-itH_\kappa }f(x).$ The Strichartz inequality in this case reads as:

\begin{theorem}\cite{M3} Let $f \in L^2_\kappa (\mathbb R^d).$ If $p, q \geq 1$ satisfying
$$\left( \frac{d+2\gamma_\kappa-2}{d+2\gamma_\kappa}\right)<\frac{1}{q}\leq 1 ~~\text{and}~~1\leq \frac{1}{p} \leq 2, $$ or $$ 0\leq \frac{1}{p} <1 ~~\text{and}~~ \frac{2}{p}+\frac{d+2\gamma_\kappa}{q}\geq (d+2\gamma_\kappa).$$
Then
\begin{equation}\label{Stedh}
\|e^{-itH_\kappa }f\|_{L^{2p}_t L^{2q}_{\kappa , x}(\mathbb T \times \mathbb R^d)} \leq C \|f\|_{L^2_\kappa (\mathbb R^d)},
\end{equation}
where $\mathbb T=(-\pi/4, \pi/4), (-\pi/2, \pi/2)$ or $(-\pi, \pi)$.
\end{theorem}
We are interested to extend the inequality \eqref{Stedh} for orthonormal family of functions. So we prove the following :
\begin{theorem}\label{TSOFdh}
Let $d\geqslant 1$ and $p,q\geqslant1$ satisfy $$1\leqslant q < {\frac{d+2\gamma_\kappa+1}{d+2\gamma_\kappa-1}}~~ \mbox{and}~~ \frac{2}{p}+\frac{d+2\gamma_\kappa}{q}=d+2\gamma_\kappa.$$
Then for any (possibly infinite) system $(f_j)$ of orthonormal functions in $L_{\kappa}^2 (\mathbb R^d)$ and any cofficients $(n_j)\subset{{\mathbb C}},$ we have
\begin{equation}\label{Ste3} \int^\pi_{-\pi}\left(\int_{\mathbb R^d}\left|\sum_{j}n_j|\left(e^{-itH_{\kappa}}f_j\right)(x)|^2\right|^q dx\right)^{\frac{p}{q}}dt\leqslant{{C^{p}_{d,q}}\left(\sum_{j}|n_j|^{\frac{2q}{q+1}}\right)}^\frac{p(q+1)}{2q}
\end{equation}
where $C_{d,q}$ is a universal constant which only depends on $d$ and $q.$
\end{theorem}
\begin{rem}
For $q=1$ and $p=\infty,$ since $e^{-itH_{\kappa}}$ is a unitary operator on $L^2_\kappa(\mathbb R^d),$  using triangle inequality it can be seen that $$\sup_{t\in(-\pi, \pi)}\left(\int_{\mathbb R^d}\left|\sum_j n_j|e^{-itH_{\kappa}}f_j(x)|^2\right|h^2_\kappa(x)dx\right)\leqslant\sum_j|n_j|.$$
\end{rem}
In \cite{RS, FS}, the authors proved  Fourier restriction theorems and they linked these theorems to space time decay estimates for certain evolution equations. We prove Theorem \ref{TSOF} assuming Theorem \ref{TSOFdh} and the relation between the Schr\"odinger kernels associated with $H_{\kappa}$ and $\Delta_{\kappa}$, see Lemma \ref{rel}. In proving Theorem \ref{TSOFdh}, we closely follow the ideas of famous work of Frank-Sabin \cite{FS} and Strichartz \cite{RS}. For better understading of the ideas we need to start with Fourier-Dunkl Hermite transform.

Let $f\in L^1_\kappa ((-\pi, \pi)\times \mathbb R^d)$ and $\mathbb N_0$ denotes the set of all non negative integers.  We define the Fourier-Dunkl Hermite transforms of $f$ by $$ \hat{f}(\mu, \nu) :=\int_{-\pi}^{\pi} \int_{\mathbb R^d} f(t, x)\phi^\kappa _\mu(x) e^{i\nu t} h^2_\kappa (x) dx dt,~~~\text{for}~~(\mu, \nu)\in\mathbb N_0^d\times \mathbb Z,$$
where $\phi^\kappa _\mu$'s are the $d$-dimensional Dunkl Hermite functions (see Section \ref{Pr}). Since the set $\{\phi_\mu^\kappa : \mu \in \mathbb N_0^d\}$ is a complete orthonormal system for $L^2_\kappa (\mathbb R^d)$, if $f\in  L^2_\kappa ((-\pi, \pi)\times \mathbb R^d)$ then $\{\hat{f}(\mu, \nu)\} \in \ell^2(\mathbb N_0^d \times \mathbb{Z})$ and satisfies the Plancheral formula $$\|f\|^2_{ L^2_\kappa ((-\pi, \pi)\times \mathbb R^d)}=\sum_{\mu, \nu}|\hat{f}(\mu, \nu)|^2.$$. The inverse Fourier-Dunkl Hermite transform is given by $$f(t, x) = \sum_{\mu, \nu} \hat{f} (\mu, \nu) \phi^\kappa _\mu(x) e^{-i\nu t}.$$

Given a discrete surface $S$ in $\mathbb N^d_0\times \mathbb Z,$ we define the restriction operator on suitable functions by $\mathcal{R}_s f :=\{{\hat{f} (\mu, \nu)}\}_{(\mu, \nu)\in S}$ and the operator dual to $\mathcal{R}_S$ is called the extension operator defined as $\mathcal{E}_S(\{g(\mu, \nu)\}) := \sum_{(\mu, \nu) \in S} g(\mu, \nu) \phi^\kappa _\mu(x)e^{-i\nu t}$ and ask the following question. For which exponents $1\leq p \leq 2,$  Fourier-Dunkl Hermite transforms of a function $f \in L^p_\kappa ((-\pi, \pi)\times \mathbb R^d)$ belongs to $\ell^2(S)?$ This question can be reframed to the boundedness of the operator $\mathcal{E}_S$ from $\ell^2(S)$ to $L^{p'}_\kappa ((-\pi, \pi)\times \mathbb R^d),$ where $p'$ is the conjugate exponent of $p$ by a duality argument. Since $\mathcal{E}_S$ is bounded from $\ell^2(S)~ \mbox{to}~ L^{p'}_\kappa ((-\pi, \pi)\times \mathbb R^d)$ if and only if $T_S := \mathcal{E}_S (\mathcal{E}_S)^*$ is bounded from $L^p_\kappa ((-\pi, \pi)\times \mathbb R^d)~ \mbox{to}~ L^{p'}_\kappa ((-\pi, \pi)\times \mathbb R^d)$. Theorem of this problem in the literature is known as restriction theorem.

Restriction theorems for the Fourier transforms for certain quadrtic surfaces $S$ of $\mathbb R^N$ have been proved by Stein \cite{ES} and Strichartz \cite{RS} using the Stein's interpolation theorem \cite{ES2}.  As an application of H\"older's inequaltiy, it can be verified that $T_S$ is $L^p-L^{p'}$ bounded if and only if the operator $W_1T_S W_2$ is bounded from $L^2$ to $L^2$ for any $W_1, W_2\in L^{\frac{2p}{2-p}}$. Recently Frank and Sabin \cite{FS} considered the similar problem for the case of Fourier transform and they proved that for any $W_1, W_2 \in  L^{\frac{2p}{2-p}}(\mathbb R^N)$, the operator $W_1T_S W_2$ is not only bounded operator on $L^2$, also belongs to a Schatten class (for definitions see Subsection \ref{Sbss}). The estimate that they proved is
$$\|W_1 T_S W_2\|_{\mathfrak{S}^\alpha(L^2{(\mathbb R^N)})}\leq C \|W_1\|_{L^{\frac{2p}{2-p}}(\mathbb R^N)}  \|W_2\|_{L^{\frac{2p}{2-p}}(\mathbb R^N)}$$ for some $\alpha \geq 1$, where $C>0$ independent of $W_1$ and $W_2$. Using these kind of estimates along with duality argument (Lemma 3 in \cite{FS}), they proved inequality \eqref{Sil}.

We consider a particular discrete surface $S\subset \mathbb N_0^d \times \mathbb Z$  with respect to counting measure for which $\mathcal{E}_S=e^{-itH_{\kappa}}$ and define an analytic family of operators $(T_z)$ defined on the strip $a \leq \text{Re} ~z \leq b$ in the complex plane such that $T_S=T_c$ for some $c\in (a, b)$. Then we show that the operator $W_1T_S W_2$ belongs to a Schatten class for $W_1, W_2$ in suitable mixied norm spaces and applying the duality argument (see Lemma \ref{Ld}) we obtain the Strichartz inequality \eqref{Ste3} for the system of orthonormal functions for Dunkl Hermite operator $H_{\kappa}$. In order to prove Theorem \ref{TSOF}, we will establish a relation between Schr\"odinger kernels associated with Dunkl Laplacian and Dunkl Hermite operator using which along with Theorem \ref{TSOFdh} we obtain the inequality \eqref{Ste2}.

The paper is organized as follows: We give introduction and main results in Section 1. In Section 2, we recall the basic ingradiants in Dunkl setting including Dunkl Hermite functions and their generating function identity and  Schatten class of operators. We also state duality statements. We will establish necessary theorems and proposition to give the proofs of our main results Theorem \ref{TSOF}, \ref{TSOFdh} and Theorem \ref{TSOFD} in Section 3. As an application of inhomogeneous Strichartz inequalty, we prove the global well-posedness for the Dunkl Hermite-Hartree equation in Schatten spaces in Section 4. In Section 5, we will give an alternate proof of Theorem \ref{TSOF} for $1<q \leq 1+\frac{2}{d+ 2\gamma_\kappa}$.    Finally, we shall write $C$ to denote positive constants independent of significant quantities the meaning of which can change from one occurrence to another.

\subsection{Dual Strichartz inequality.}
In order to prove Theorem \ref{TSOFdh} and Theorem \ref{TSOF}, we prove that  the inequalities \eqref{Ste2} and \eqref{Ste3} are  equivalent to some dual inequalities and then we proved the dual inequalities for the range $1<q\leq 1+\frac{2}{d+2\gamma_\kappa}$.

One can verify that Theorem \ref{TSOFdh} (or Theorem \ref{TSOF}) is equivalent to the boundedness of the operator $\gamma \mapsto \rho_{\gamma(t)}$ from $\mathfrak{S}^{\frac{2q}{q+1}}(L^2_\kappa (\mathbb R^d))$ to $L^p_t L^q_{\kappa, x}(\mathbb T \times\mathbb R^d)$, where $\gamma(t)=e^{-itP}\gamma e^{itP}$ with $P=H_{\kappa}$ and $\mathbb T=(-\pi, \pi)$ (or $P=-\Delta_{\kappa}$ and $\mathbb T=\mathbb R$) (See Subsection \ref{Sbss}). Dual of this operator is given by
$$V\in L^{p'}_t L^{q'}_{\kappa , x}(\mathbb T \times \mathbb R^d) \mapsto\int_\mathbb T e^{itP}V(t, x)e^{-itP}dt\in\mathfrak{S}^{2q'}(L^2_\kappa (\mathbb R^d)).$$

Therefore the duality argument shows that Theorem \ref{TSOFdh} and Theorem \ref{TSOF} are equivalent to the following:

\begin{theorem}[Strichartz inequality in Schatten spaces, dual version] \label{TSOFD}
Assume that $p', q', d\geq1$ satisfy $$1+\frac{d+2\gamma_\kappa}{2}\leq q'<\infty~ \mbox{and}~ \frac{2}{p'}+\frac{d+2\gamma_\kappa}{q'}=2.$$
We have
\begin{equation}\label{Sc}
\left\| \int_\mathbb T e^{itP} V(t, x) e^{-itP} dt \right\|_{\mathfrak{S}^{2q'}(L^2_\kappa (\mathbb R^d))}\leq C_{d, q}\|V\|_{L^{p'}_t L_{\kappa , x}^{q'}(\mathbb T \times\mathbb R^d))}
\end{equation}
where $C_{d, q}$ is the same constant as in Theorem \ref{TSOF} when $P=-\Delta_{\kappa}, \mathbb T = \mathbb R$ or as in Theorem \ref{TSOFdh} when $P=H_{\kappa}, \mathbb T = (-\pi, \pi)$.
\end{theorem}

\begin{rem}

For $q' = \infty,$ and $p' = 1,$ it can be easily seen that
$$\left\| \int_\mathbb T e^{itP} V(t, x) e^{-itP} dt \right\| \leq \|V\|_{L^{1}_t L_{\kappa , x}^{\infty}(\mathbb T \times\mathbb R^d)}$$
\end{rem}

Using \eqref{Sc} when $P=H_{\kappa}$, we prove an inhomogeneous Strichartz inequality. Consider the equation given by
\begin{equation}\label{Inh}
\begin{cases}
i\dot{\gamma}(t)&= [-H_\kappa , \gamma(t)] + iR(t)\\
\gamma(t_0)&=0,
\end{cases}
\end{equation}
where $R(t)$ is a self adjoint operator on $L^2_\kappa (\mathbb R^d)$ and is bounded for almost every $t.$ The solution of the system \eqref{Inh} can be written as
\begin{equation}\label{Inh2}
\gamma(t) = \int^t_{t_0} e^{i(t-s)H_\kappa } R(s)  e^{i(s-t)H_\kappa } ds.
\end{equation}
We obtain the following inhomogeneous Strichartz inequality.
\begin{cor}[Inhomogeneous Strichartz inequality]\label{ISI}
Assume that $p, q, d \geq 1$ satisfy
$$1 \leq q < \frac{d + 2\gamma_\kappa +1}{d+2\gamma_\kappa -1}~ \mbox{and}~ \frac{2}{p}+\frac{d+2\gamma_\kappa }{q}=d+2\gamma_\kappa $$
and let $\gamma(t)$ be given by \eqref{Inh2}. Then
$$\|\rho_{\gamma(t)}\|_{L^p_tL^q_{\kappa , x}((-\pi, \pi)\times\mathbb R^d)} \leq C \left\|\int^\pi_{-\pi} e^{isH_\kappa } |R(s)| e^{-isH_\kappa } ds\right\|_{\mathfrak{S}^\frac{2q}{q+1}}$$
for a constant $C$ which is independent of $t_0.$
\end{cor}
\begin{proof} For a trace-class operator $\gamma$ and any bounded function $V$ of compact support,
$$Tr(V(x)\gamma) = \int_{\mathbb R^d} V(x) \rho_{\gamma}(x) h_\kappa ^2(x)dx,$$ where $V(x)$ is identified with the corresponding multiplication operator on $L^2_\kappa (\mathbb R^d)$ and $\rho_\gamma (x)$ is the density of $\gamma$. And for a time-dependent potential $V(t, x) \in L^\infty_c((-\pi, \pi) \times \mathbb R^d),$ we have
 \begin{align*}
\left|\int^\pi_{t_0}\int_{\mathbb R^d} V(t, x) \rho_{\gamma(t)}(x)  h_\kappa ^2(x) dx dt\right|&= \left|\int^\pi_{t_0}\int^t_{t_0} Tr(e^{itH_\kappa } V(t, x) e^{-itH_\kappa } e^{isH_\kappa } R(s) e^{-isH_\kappa }) ds dt\right|\\
&\leq \int^\pi_{t_0}\int^t_{t_0} Tr(e^{itH_\kappa } |V(t, x)| e^{-itH_\kappa } e^{isH_\kappa } |R(s)|\ e^{-isH_\kappa }) ds dt\\
&\leq Tr \left(\left( \int^\pi_{t_0} e^{itH_\kappa } |V(t, x)| e^{-itH_\kappa }\right) \left(\int^\pi_{t_0}e^{isH_\kappa } |R(s)|\ e^{-isH_\kappa }\right)\right).
\end{align*}
In the first inequality we have used the fact that $|Tr(AB)| \leq Tr(|A||B|)$ for self-adjoint operators $A$ and $B.$ Then applying the H\"older's inequality for traces and \eqref{Sc} for the term involving $V(t, x).$ This completes the proof.
\end{proof}

\section{Preliminaries}\label{Pr}
~~~In this section we will define Dunkl operators, some other related operators and function spaces which will be used in this article. Dunkl operators were introduced by Charles Dunkl (1989) to built a framework for a theory of special functions and integral transforms in several variables related to reflection groups. Such operators are relevant in physics, namely for the analysis of quantum many body systems of Calogero-Moser-Sutherland type (see \cite{DV,LV}). From the mathematical analysis point of view, the importance of Dunkl operators lies on the fact that they generalize the theory of symmetric spaces of Euclidean type. There are many developments in harmonic analysis of the operators which are defined in terms of Dunkl opearators in different directions in recent years.  There is a vast literature related to Dunkl operators, see for instance \cite{BRS,CFD,CFD2,DX,H,MR}.

\subsection{The general Dunkl setting}
~~~The basic ingradiants in the theory of Dunkl operators are the root systems and finite reflection groups associated to them. For $\nu\in\mathbb R^d\setminus\{0\}$, we denote by $\sigma _\nu$ the reflection in the hyperplane perpendicular to $\nu,$ i.e.,
$$\sigma _\nu(x)=x-2\frac{\langle\nu, x\rangle}{|\nu|^2}\nu.$$

 Let $O(d)$ be the group of orthogonal matrices acting on $\mathbb R^d.$ Given a root system $R$ associate a finite subgroup $G\subset O(d)$ is the reflection group which is generated by the reflections $\{\sigma_{\nu}: \nu\in R\}.$

A function $\kappa : R\rightarrow {\mathbb C}$ is said to be a multiplicity function on $R,$ if it is invariant under the natural action of $G$ on $R,$ i.e. $\kappa(g\nu)=\kappa(\nu) $ for all $\nu\in R$ and $g\in G.$

Every root system can be written as a disjoint union $R = R_+ \cup (-R_+)$, where $R_+$ and $-R_+$ are separated by a hyperplane through the origin. Such $R_+$ is called the set of all positive roots in $R$. Of course its choice is not unique.

The weight function associated to the root system $R$ and the multiplicity function $\kappa$ is defined by
$$h^2_{k}(x):=\prod_{\nu\in R_+}|\langle x, \nu\rangle|^{2\kappa(\nu)}.$$

Note that $h^2_{\kappa}(x)$ is $G$-invariant and homogenous of degree $2\gamma_\kappa$ where, by definition,
$$\gamma_\kappa:=\sum_{\nu\in R_{+}}\kappa(\nu).$$

We will assume through out the article $2\gamma_\kappa$ is a non negative integer. Let $L^p_{\kappa}(\mathbb R^d)$, $1\leq p\leq \infty,$  stands for the space of $L^p$- functions with respect to the measure $h_\kappa^2(x) dx$, and $L^p_t L^q_{\kappa, x}(\mathbb T \times \mathbb R^d), 1\leq p, q \leq \infty$ stands for the space of all measurable functions $h(t, x)$ on $\mathbb T  \times \mathbb R^d$   for which
$$\|h\|_{L^p_t L^q_{\kappa, x}(\mathbb T \times \mathbb R^d)}:=\|\|h(t, \cdot)\|_{L^q_{\kappa}(\mathbb R^d)}\|_{L^p(\mathbb T)}<\infty.$$
We may consider $\mathbb T=(-\pi, \pi), \mathbb R$ or any interval in $\mathbb R$ with Lebesgue measure. $L^p_\kappa (\mathbb T\times \mathbb R^d):=L^p_t L^p_{\kappa, x}(\mathbb T \times \mathbb R^d)$.

Now we define the difference-differential operators which were introduced and studied by C. F. Dunkl (for $\kappa\geq 0$) see \cite{CFD,CFD2}. These operators are also called Dunkl operators and are analogues (generalization) of directional derivatives. We fix a root system $R$ with a positive subsystem $R_+$ and the associated reflection group $G.$ We also fix a nonnegative multiplicity function $\kappa$ defined on $R.$

For $\xi\in\mathbb R^d,$ the Dunkl operator $T_{\xi}:=T_{\xi}(\kappa)$ is defined by
$$T_{\xi}f(x)=\partial_{\xi}f(x)+\sum_{\nu\in R_{+}}\kappa(\nu)\langle\nu, \xi\rangle\frac{f(x)-f(\sigma_{\nu}x)}{\langle\nu, x\rangle}$$
for smooth functions $f$ on $\mathbb R^d.$ Here $\partial_{\xi}$ denotes the directional derivative along $\xi.$ For the standard coordinate vectors $\xi=e_{j}$ of $\mathbb R^d$ we use the abbreviation $T_{j}=T_{e_j}.$

Let $\mathcal{P}$ be the space of all polynomials with complex coefficients in $d$-variables and $\mathcal{P}_m$ the subspace of homogeneous polynomials of degree $m$. The Dunkl-operators $T_\xi$ and directional derivatives $\partial_\xi$ are closely related and intertwined by an isomorphism on $\mathcal{P}.$ indeed, if the multiplicity function $\kappa$ is nonnegative then by Theorem 2.3 and Proposition 2.3 in R\"osler \cite{MR}, there exist a unique linear isomorphism (intertwining operator) $V_\kappa$ of $\mathcal{P}$ such that $V_\kappa(\mathcal{P}_m)=\mathcal{P}_m, V_\kappa|_{\mathcal{P}_0}=id$ and $T_{\xi} V_\kappa=V_\kappa\partial_\xi$ for all $\xi\in\mathbb R^d.$ It can be checked that $V_\kappa\circ g=g\circ V_\kappa$ for all $g\in G.$

For $y\in{\mathbb C}^d,$ define
$$E_\kappa(x, y):=V_\kappa\left(e^{\langle \cdot, y\rangle}\right)(x),~ x\in\mathbb R^d.$$
The function $E_\kappa$ is called the Dunkl-kernel, or $\kappa$-exponential kernel, associated with $G$ and $\kappa,$ see \cite{CFD4}.

The Dunkl-Laplacian is the second order operator defined by
$$\Delta_{\kappa}=\sum^d_{j=1}T^2_j$$
which can be explicitly calculated, see Theorem 4.4.9 in Dunkl-Xu \cite{DX}.

It can be seen that $\Delta_\kappa=\sum^d_{j=1}T^2_{\xi_j}$ for any orthonormal basis $\{\xi_1, \xi_2,\ldots, \xi_d\}$ of $\mathbb R^d$, see \cite{CFD2}.
We also recall the definition of heat kernel, see \cite{MR2}. We are considering the initial value problem for heat equation associated to the Dunkl Laplacian,
$$u_t(t, x)=\Delta_{\kappa} u(t, x), ~~~~ u(0, x) = f(x), ~~~ t>0,~~ x\in \mathbb R^d.$$
For $f \in L^p_\kappa (\mathbb R^d)$ the solution of this equation is given by
$$u(t, x) =e^{t\Delta_{\kappa}}f(x)= \int_{\mathbb R^d} \Gamma_\kappa (t, x, y) f(y) h^2_\kappa (y) dy, $$
where the heat kernel $\Gamma_\kappa $ associated to the Dunkl Laplacian is explicitly given by

\begin{equation}\label{l4}
\Gamma_\kappa (t, x, y) =\frac{M_\kappa }{ (2t)^{\gamma_\kappa+d/2}}e^{-\frac{|x|^2+|y|^2}{4t}}E_\kappa \Big(\frac{x}{2t}, y\Big),
\end{equation}
where $\displaystyle M_\kappa =\left(\int_{\mathbb R^d} e^{-\frac{|x|^2}{2}} h_\kappa ^2(x) dx \right)^{-1}$.
The kernel of the operator $e^{it\Delta_{\kappa}}$ is given by
\begin{equation}\label{lsk} L_{it}(x, y)=\frac{M_\kappa }{ (2 i t)^{\gamma_\kappa+d/2}}e^{\frac{i(|x|^2+|y|^2)}{4t}}E_\kappa \Big(\frac{x}{2it}, y\Big).\end{equation}

\subsection{Dunkl harmonic oscillator.}
The Dunkl harmonic oscillator (which we also call the Dunkl Hermite operator) is defined by $$H_\kappa  = -\Delta_{\kappa} + |x|^2.$$
Let $\mathbb N_0$ be the set of all non-negative integers and for the multi index $\mu = (\mu_1, . . ., \mu_d) \in \mathbb N^d_0$ R\"osler introduced eigenfunctions $\phi^\kappa _\mu$ of Dunkl harmonic oscillator $H_\kappa $ in \cite{MR2} with eigenvalue $(2|\mu| + d + 2\gamma_\kappa)$, where $|\mu| = \mu_1 + \cdot \cdot \cdot + \mu_d,$ i. e.,
$$H_\kappa  \phi^\kappa _\mu = (2|\mu| + d + 2\gamma_\kappa) \phi^\kappa _\mu,$$
and they form a complete orthonormal system in $L^2_\kappa (\mathbb R^d).$  These functions $\phi^\kappa _\mu$ are called the Dunkel Hermite functions (also called generalized Hermite functions) and they satisfy the following generating function identity:

\begin{equation}\label{Mhef}
\sum_{\mu\in\mathbb N^d_0} \phi^\kappa _\mu(x) \phi^\kappa _\mu(y) w^{|\mu|}= 2^{\frac{d}{2}+\gamma_\kappa} M_\kappa  (1-w^2)^{-(\frac{d}{2}+\gamma_\kappa)} e^{-\frac{1}{2}\left(\frac{1+w^2}{1-w^2}\right)(|x|^2+|y|^2)} E_\kappa \left(\frac{2w x}{1-w^2}, y\right).
\end{equation}

For any $\phi \in L^2_\kappa (\mathbb R^d)$, $\displaystyle \phi=\sum_{\mu\in \mathbb N_0^d} \hat{\phi}(\mu) \phi_\mu^\kappa $, where
$$\hat{\phi}(\mu)=\int_{\mathbb R^d}\phi(x) \phi_\mu^\kappa  (x) h_\kappa ^2(x) dx.$$
 For the $Re(z)>0$, using spectral theory,
$$e^{-zH_{\kappa}}f(x)=\int_{\mathbb R^d} K_z(x, y)f(y) h_\kappa ^2(y) dy$$
for $f\in L^2_\kappa (\mathbb R^d)$, where
\begin{align*}
K_z(x, y) &= \sum_{\mu\in \mathbb N^d_0} e^{-z(2|\mu|+d+2\gamma_\kappa)} \phi^\kappa _\mu(x) \phi^\kappa _\mu(y)\\
&=e^{-(d+2\gamma_\kappa)z}\sum_{\mu\in \mathbb N^d_0} e^{-2|\mu|z} \phi^\kappa _\mu(x) \phi^\kappa _\mu(y).
\end{align*}
In the view of Meheler's formula \eqref{Mhef} for $w=e^{-2z}$, we will get

\begin{align*}
K_z(x, y) &=\frac{M_\kappa }{(\sinh 2z)^{\frac{d}{2}+\gamma_\kappa}} e^{-\frac{1}{2} \coth 2z(|x|^2+|y|^2)}E_\kappa \left(\frac{x}{\sinh 2z}, y\right)
\end{align*}

Let $z = r+it$ with $r\rightarrow 0^+$, we will get the kernel of the operator $e^{-itH_{\kappa}}$:
\begin{equation}\label{hsk}
K_{it}(x,y) = \frac{M_\kappa }{( i \sin 2t)^{\frac{d}{2}+\gamma_\kappa}} e^{\frac{i}{2} \cot 2t(|x|^2+|y|^2)}E_\kappa \left(\frac{x}{i \sin 2t}, y\right).
\end{equation}
It can be easily verified that the following:
\begin{equation}\label{pk1}
K_{-it}(x, y)=\overline{K_{it}(x, y)}~~~\quad~~\text{and}~~~\quad~~~K_{i(t+\frac{\pi}{2})}(x, y)=(-1)^{\frac{d}{2}+\gamma_\kappa}K_{it}(-x, y),
\end{equation}
for all $t\in \mathbb R\setminus (\frac{\pi}{2})\mathbb{Z}$.

\subsection{Schatten Spaces.}\label{Sbss}
Let $\mathcal{H}$ be complex separable Hilbert space and $\gamma$ be a compact operator on $\mathcal{H}$.  We say that $\gamma\in \mathfrak{S}^p(\mathcal{H}),$ Schatten space, for $ 1\leqslant p<\infty,$ if $Tr|\gamma|^p<\infty,$ where $|\gamma|=\sqrt{\gamma^*\gamma}$ and $Tr(\gamma)$ is trace of $\gamma$. For $\gamma\in\mathfrak{S}^p(\mathcal{H}),$ the Schatten $p$-norm of $\gamma$ in $\mathfrak{S}^p(\mathcal{H})$ is defined by
$$\|\gamma\|_{\mathfrak{S}^p(\mathcal{H})}=(Tr|\gamma|^p)^\frac{1}{p}.$$
It can be verified that  $\|\gamma\|_{\mathfrak{S}^p(\mathcal{H})}=\left(\sum_{j}|\lambda_{j}|^p\right)^\frac{1}{p},$ where $\lambda_1\geqslant\lambda_2\geqslant \dots \geqslant\lambda_n\geqslant \dots \geqslant 0$ are the singular values of $\gamma,$ that is the eigenvalues of $|\gamma|=\sqrt{\gamma^*\gamma}.$

When $p=2,$  the Schatten $p$-norm coincides with the Hilbert-Schmidt norm.  Also when $p=\infty,$ we define $\|\gamma\|_{\mathfrak{S}^\infty(\mathcal{H})}$ to be the operator norm of $\gamma$ on $\mathcal{H}.$ For $\gamma \in \mathfrak{S}^p(L^2_\kappa (\mathbb R^d)), \rho_\gamma$ denotes the density of $\gamma$ and it is defined for any finite rank $\gamma$ by duality,
$$\int_{\mathbb R^d} \rho_\gamma (x) V(x) h_\kappa ^2(x) dx := Tr(\gamma V)$$ and extended to all $\gamma \in \mathfrak{S}^p(L^2_\kappa (\mathbb R^d))$ using the density of finite rank operators in $\mathfrak{S}^p(L^2_\kappa (\mathbb R^d)$. We refer \cite{S} for more details of Schatten spaces.

 The following lemma plays an important role in proving the main results.

\begin{lem}[Duality principle]\label{Ld}
Let $p, q \geq 1~ \mbox{and} ~ \alpha \geq 1.$  Let $A$ be a bounded linear operator from $L^2_\kappa (\mathbb R^d)$ to $L^{2p}_t L^{2q}_{\kappa , x}(\mathbb T \times \mathbb R^d)$. Then the following statements are equivalent.

$(1)$ There is a constant $C > 0$ such that
\begin{equation}\label{DI}
\|WAA^*\overline{W}\|_{\mathfrak{S}^\alpha\left(L^2_{\kappa }(\mathbb T\times \mathbb R^d)\right)} \leq C \|W\|^2_{L^{2p'}_t L^{2q'}_{\kappa , x}(\mathbb T \times \mathbb R^d)}
\end{equation}
for all $W\in L^{2p'}_t L^{2q'}_{\kappa , x}(\mathbb T \times \mathbb R^d),$ where the function $W$ is interpreted as an operator which acts by multiplication.\\

$(2)$ For any orthonormal system $(f_j)_{j\in J}$ in $\mathcal{H}$ and any sequence $(n_j)_{j\in J}\subset {\mathbb C},$

\begin{equation}\label{DI2}
\left\|\sum_{j\in J}n_j\left|Af_j\right|^2\right\|_{L^p_t L^q_{\kappa , x}(\mathbb T \times \mathbb R^d)}\leq C' \left(\sum_{j\in J}|n_j|^{\alpha'}\right)^{1/\alpha'},
\end{equation}
where $C'$ is a constant.
\end{lem}
For the proof of above lemma we refer Lemma 3 in \cite{FS} with appropriate modification. We end this section with followiing two important remarks.

\begin{rem}
Let $S$ be the discrete surface $S = \{(\mu, \nu) \in \mathbb N^d_0 \times \mathbb Z : \nu = 2|\mu|+d+2\gamma_\kappa \}$ with respect to the counting measure. Then for all $\{g(\mu, \nu)\} \in L^1(S)$ and for all $(t, x) \in (-\pi, \pi)\times\mathbb R^d,$  the extension operator can be written as
\begin{equation}\label{Eoe}
\mathcal{E}_S(\{g(\mu, \nu)\}) (t, x) = \sum_{\mu, \nu \in S} g (\mu, \nu) \phi^\kappa _\mu(x) e^{-it\nu}.
\end{equation} Choosing
\begin{equation} \label{cg}
g(\mu, \nu) = \begin{cases}\hat{u}(\mu) & \text{if}~ \nu=2|\mu|+d+2\gamma_\kappa,\\
0 &\text{otherwise},
\end{cases}
\end{equation}
for any suitable $u : \mathbb R^d \rightarrow {\mathbb C}$, where $\hat{u}(\mu)=\int_{\mathbb R^d} u(x) \phi_\mu^\kappa  (x) h_\kappa ^2(x) dx$. We will get
\begin{align*}
\mathcal{E}_S(\{g(\mu, \nu)\}) (t, x)  &= \sum_{\mu, \nu \in S} g (\mu, \nu) \phi^\kappa _\mu(x) e^{-it\nu}\\
& = \int_{\mathbb R^d} \left(\sum_\mu \phi^\kappa _\mu (x) \phi^\kappa _\mu (y) e^{-it(2|\mu|+d+2\gamma_\kappa)} \right) u(y) h^2_\kappa (y) dy\\
& =e^{-itH_\kappa } u(x).
\end{align*}
Thus we have for $S = \{(\mu, \nu) \in \mathbb N^d_0 \times \mathbb Z : \nu = 2|\mu|+d+2\gamma_\kappa \}$,
\begin{equation}\label{210}
\mathcal{E}_S(\{g(\mu, \nu)\}) (t, x) = e^{-itH_\kappa } u(x),
\end{equation}
whenever $g$ and $u$ are related by \eqref{cg}.
\end{rem}

\begin{rem}
For any compact operator $\gamma$ on $L^2_\kappa(\mathbb R^d)$ we define
$$\gamma(t) :=e^{-itP}\gamma e^{itP}, ~~t\in\mathbb R,$$where $P=-\Delta_{\kappa}$ or $H_{\kappa}$. If the operator $\gamma$ is of the form $\gamma :=\sum_{j}n_j|f_j\rangle\langle f_j|$ associated with a given orthonormal system $(f_j)_j,$ where $|f \rangle \langle g |$ is Dirac's notation for the rank one operator $\phi\mapsto\langle\phi, g\rangle f$,  for such $\gamma$ one may check that
$$\rho_{\gamma(t)}(x) :=\sum_jn_j|e^{-itP}f_j(x)|^2$$
Using this relation, the inequality \eqref{Ste3} in Theorem \ref{TSOFdh} (or the inequality \eqref{Ste2} in Theorem \ref{TSOF})   can also be written as
\begin{equation}\label{op1}
\|\rho_{\gamma(t)}(x)\|_{L^p_tL^q_{\kappa , x}(\mathbb T \times \mathbb R^d)}\leqslant C_{d, q}\|\gamma\|_{\mathfrak{S}^{\frac{2q}{q+1}}(L^2_\kappa (\mathbb R^d))},
\end{equation}
where $\|\gamma\|_{\mathfrak{S}^{\frac{2q}{q+1}}(L^2_\kappa (\mathbb R^d))} = {\left(\sum_{j}|n_j|^{\frac{2q}{q+1}}\right)}^\frac{q+1}{2q}, P=H_{\kappa}$ and $\mathbb T =(-\pi, \pi)$ (or $P=-\Delta_{\kappa}$ and $\mathbb T=\mathbb R$). The estimate \eqref{op1} is equivalent to say that $\gamma \mapsto \rho_{\gamma(t)}$ is a bounded operator from $\mathfrak{S}^{\frac{2q}{q+1}}(L^2_\kappa (\mathbb R^d))$ to $L^p_tL^q_{\kappa , x}(\mathbb T \times \mathbb R^d)$.

 Similarly in view Lemma \ref{Ld}, the estimates \eqref{DI} and  \eqref{DI2} for $A=e^{-itP}$ are equivalent to the following bound: for any $\gamma\in\mathfrak{S}^{\alpha'}(L^2_{\kappa }(\mathbb R^d))$ we have
$$\|\rho_{e^{-itP}\gamma e^{itp}}\|_{L^p_t L^q_{\kappa , x}(\mathbb T \times \mathbb R^d)} \leq C \|\gamma\|_{\gamma\in\mathfrak{S}^{\alpha'}\left(L^2_{\kappa }(\mathbb R^d)\right)}.$$
\end{rem}

\section{Proofs}

\subsection{Proof of Theorem \ref{TSOF}}\label{pt1}
In this subsection we will give the proof of Theorem \ref{TSOF} by assuming Theorem \ref{TSOFdh} and we will prove Theorem \ref{TSOFdh} in Subsection \ref{pt2}. In order to prove Theorem \ref{TSOF}, we first establish the relation between the kernels of $e^{it\Delta_{\kappa}}$ and $e^{-it H_\kappa }$ then the proof follows from Theorem \ref{TSOFdh}. The following lemma will give the link between the kernels of $e^{it\Delta_{\kappa}}$ and $e^{-it H_\kappa }$:
\begin{lem}\label{rel}
Let $K_{it}(x, y)$ and $L_{it}(x, y)$ be the  same as in \eqref{hsk} and \eqref{lsk} which are the  kernels of $e^{-itH_{\kappa}}$ and $e^{it\Delta_{\kappa}}$ respectively. For $v>0$ we have \begin{equation}\label{kr}
K_{i\frac{\tan^{-1}v}{2}}(x, y)=(1+v^2)^{\frac{d+2\gamma_\kappa}{4}}e^{-\frac{iv}{2}|x|^2}L_{iv/2}(x\sqrt{1+v^2}, y).
\end{equation}
\end{lem}
\begin{proof}
The proof of this lemma can be abstracted from the proof of Proposition 8 in \cite{M3}. To make the article self contained we will only give the main steps. Let $v=\tan{2t}$ for $0<t<\frac{\pi}{4}$ then $t=\frac{1}{2}\tan^{-1}v$ and $\cot(2t)=\frac{1}{v}$ and $\sin(2t)=\frac{v}{\sqrt{1+v^2}}$ which implies
\begin{equation}\label{eq1}K_{\frac{i}{2}\tan^{-1}v}(x, y)=M_\kappa  \Big(\frac{\sqrt{1+v^2}}{i v} \Big)^{\frac{d+2\gamma_\kappa}{2}} e^{\frac{i}{2v}(|x|^2+|y|^2)}E_\kappa \Big(\frac{x\sqrt{1+v^2}}{iv}, y\Big)\end{equation}
 and
\begin{equation}\label{eq2}L_{\frac{iv}{2}}(x\sqrt{1+v^2}, y)= M_\kappa  \Big(\frac{1}{iv} \Big)^{\frac{d+2\gamma_\kappa}{2}} e^{\frac{i}{2v}\big((1+v^2)|x|^2+|y|^2\big)}E_\kappa \Big(\frac{x\sqrt{1+v^2}}{iv}, y\Big)\end{equation}
In view of \eqref{eq1} and \eqref{eq2} we get our required identity which completes the proof of the lemma.
\end{proof}
Now we are ready to prove Theorem \ref{TSOF}.
It can be verified using Lemma \ref{rel} and change of variable that for $p, q\geq 1$
\begin{equation}\label{id1}
\left\|\sum_jn_j|e^{-itH_\kappa }f_j|^2\right\|^p_{L^p_tL^q_{\kappa , x}((0, \frac{\pi}{4})\times \mathbb R^d)} =\left \|\sum_jn_j|e^{-it\Delta_{\kappa}}f_j|^2\right\|^p_{L^p_tL^q_{\kappa , x}((0, \infty)\times \mathbb R^d)}.
\end{equation}
For any orthonormal family of functions $(f_j)$ in $L^2_\kappa (\mathbb R^d)$ and $(n_j)\subset{\mathbb C},$ let us define
\begin{equation*}
\varphi_{\{f_j\}}(t) := \left\|\sum_jn_j|e^{-itH_\kappa }f_j|^2\right\|_{L^q_{ \kappa }(\mathbb R^d)},
\end{equation*}
then the identity \eqref{id1} can be written as
 \begin{equation}\label{id2}\int^{\frac{\pi}{4}}_0|\varphi_{\{f_j\}}(t)|^p dt= \left\|\sum_jn_j|e^{-it\Delta_{\kappa}}f_j|^2\right\|^p_{L^p_tL^q_{\kappa , x}((0, \infty)\times \mathbb R^d)}.\end{equation}
Using the properties of kernel $K_{it}(x, y)$ given in \eqref{pk1} one can easily show that $\varphi_{\{f_j\}}$ satisfies the following:
\begin{equation}\label{rel2}
\varphi_{\{f_j\}}\Big(t+\frac{\pi}{2}\Big) = \varphi_{\{f_j\}}(t)~~ \text{and} ~~\varphi_{\{f_j\}}(-t) = \varphi_{\{\overline f_j\}}(t).
\end{equation}
Using \eqref{rel2} and \eqref{id2}, it can be easily proved that
\begin{equation}\label{id3}
\left(\int^\pi_{-\pi}|\varphi_{\{f_j\}}(t)|^p dt\right) = 4\int^{\frac{\pi}{4}}_{-\frac{\pi}{4}}|\varphi_{\{f_j\}}(t)|^p dt
=4 \int^\infty_{-\infty} \left\|\sum_jn_j|e^{-it\Delta_{\kappa}}f_j|^2\right\|^p_{{L^q_\kappa }(\mathbb R^d)}dt.
\end{equation}
Thus we establish
$$\left\|\sum_jn_j|e^{-itH_\kappa }f_j|^2\right\|^p_{L^p_tL^q_{\kappa , x}((-\pi, \pi)\times \mathbb R^d)}= 4\left \|\sum_jn_j|e^{-it\Delta_{\kappa}}f_j|^2\right\|^p_{L^p_tL^q_{\kappa , x}(\mathbb R \times \mathbb R^d)}.$$
In view of the above indentity Theorem \ref{TSOF} follows from Theorem \ref{TSOFdh}.

\subsection{Proof of Theorem \ref{TSOFdh} and Theorem \ref{TSOFD}}\label{pt2}
Our strategy of the proof of the Theorem \ref{TSOFdh} is as follows: We first prove the estimate \eqref{Sc} in Theorem \ref{TSOFD} for the exponents $(p', q')=(1, \infty)$ and $(p', q')=(1+\frac{d+2\gamma_\kappa}{2}, 1+\frac{d+2\gamma_\kappa}{2})$ (diagonal case) and then apply complex interpolation theorem to complete the proof of Theorem \ref{TSOFD}.

  Using the duality argument we get the inequality \eqref{Ste3} from Theorem \ref{TSOFD} for the range of the exponent $q, 1< q\leq 1+\frac{2}{d+2\gamma_\kappa}.$ Then we make use of Theorem \ref{gse} and Lemma \ref{Ld} to prove the inequality \eqref{Ste3} for the remaining range of the exponent $q$ i.e., $1+\frac{2}{d+2\gamma_\kappa}<q<\frac{d+2\gamma_\kappa+1}{d+2\gamma_\kappa-1}$ which would complete the proof of Theorem \ref{TSOFdh}.

The Schatten norm estimates for the operator $W_1T_SW_2$ which are useful in proving Theorem \ref{Dc} and
Theorem \ref{gse}, are proved using the following proposition. For the proof of the following proposition we refer the reader to see Proposition 1 in \cite{FS} with appropriate modifications.

\begin{prop}\label{Ip}
Let $(T_z)$ be an analytic family of operators on $\mathbb T \times \mathbb R^d$ in the sense of Stein defined on the strip $-\lambda_0 \leq \text{Re}(z) \leq 0$ for some $\lambda_0 > 1.$ Assume that we have the following bounds
\begin{equation}\label{B1}
\|T_{is}\|_{L^2_\kappa (\mathbb T\times\mathbb R^d)\rightarrow L^2_\kappa (\mathbb T\times\mathbb R^d)}\leq M_0 e^{a|s|}, \|T_{-\lambda_0+is}\|_{L^1_\kappa (\mathbb T\times\mathbb R^d)\rightarrow L^\infty_\kappa (\mathbb T\times\mathbb R^d)}\leq M_1 e^{b|s|},
\end{equation}
 for all $s\in \mathbb R,$ for some $a, b\geq 0$ and for some $M_0, M_1 \geq 0.$ Then, for all $W_1, W_2 \in L_\kappa ^{2\lambda_0}(\mathbb T\times \mathbb R^d, {\mathbb C})$ the operator $W_1T_{-1}W_2$ belongs to $\mathfrak{S}^{2\lambda_0}(L^2_\kappa (\mathbb T\times\mathbb R^d))$ and we have the estimate
\begin{equation}\label{B2}
 \|W_1T_{-1}W_2\|_{\mathfrak{S}^{2\lambda_0}(L^2_\kappa (\mathbb T\times\mathbb R^d))}\geq M^{1-\frac{1}{\lambda_0}}_0M^\frac{1}{\lambda_0}_1 \|W_1\|_{L^{2\lambda_0}_tL^{2\lambda_0}_{\kappa , x}(\mathbb T\times\mathbb R^d)} \|W_2\|_{L^{2\lambda_0}_tL^{2\lambda_0}_{\kappa , x}(\mathbb T\times\mathbb R^d)}.
\end{equation}
\end{prop}
To define the analytic family of operators, we first consider the generalized functions
\begin{equation} \label{gz}
G_z(\mu, \nu) = \frac{1}{\Gamma(z+1)}(\nu-(2|\mu|+d+2\gamma_\kappa))^z_+,~ \mbox{where}~ x^z_+ = \begin{cases}
x^z & \mbox{ for } x > 0\\
0 &\mbox{ for }x \leq 0,\end{cases}
\end{equation} where $(\mu, \nu)\in \mathbb N_0^d\times \mathbb Z$. For Schwartz class functions $\phi$ on $\mathbb N^{d}_0\times \mathbb Z ,$ we have $$\displaystyle \langle G_z, \phi\rangle := \frac{1}{\Gamma(z+1)}\sum_{\mu, \nu}(\nu - (2|\mu|+d+2\gamma_\kappa))^z_+\phi(\mu, \nu)$$ and $$\displaystyle \lim_{z \rightarrow -1} \langle G_z, \phi\rangle = \lim_{z \rightarrow -1} \frac{1}{\Gamma(z+1)} \sum_{\mu, \nu}\phi(\mu, \nu)(\nu - (2|\mu|+d+2\gamma_\kappa))^z_+ = \sum_{(\mu, \nu)\in S} \phi(\mu, \nu).$$ For the distributional calculus of $(\nu-(2|\mu|+d+2\gamma_\kappa))^z_+$  see \cite{GS}.  Thus $G_{-1} = \delta _S.$ Define the analytic family of operators $T_z$ by
\begin{equation}\label{tz}T_zg(x, t) = \sum_{\mu, \nu}\hat g(\mu, \nu) G_z(\mu, \nu)\phi_\mu^\kappa (x) e^{-i\nu t}\end{equation}
Then \begin{equation}\label{Afk}
T_zg(x, t) = \int_{\mathbb R^d} (K_z(x, y, \cdot) \ast g(y, \cdot))(t)h_\kappa ^2(y) dy,
\end{equation}
where $\displaystyle K_z(x, y, t) = \sum_{\mu, \nu} \phi_\mu^\kappa (x) \phi_\mu^\kappa  (y) G_z(\mu, \nu) e^{-i\nu t}.$ Using the definition of $G_z(\mu, \nu)$ we have
\begin{equation}
K_z(x, y, t) = \frac{1}{\Gamma(z+1)} \sum_{\mu} \phi_\mu^\kappa (x) \phi_\mu^\kappa (y) e^{-it(2|\mu|+d+2\gamma_\kappa)}\sum^\infty_{k=0} k^z_+ e^{-itk}.
\end{equation}
By using \eqref{hsk}, we get
\begin{equation}\label{kz}
K_z(x, y, t) = \frac{K_{it}(x, y)}{\Gamma(z+1)} \sum^\infty_{k=0} k^z_+ e^{-itk}
\end{equation}
where $\displaystyle  \sum^\infty_{k=0} k^z_+ e^{-itk}\sim \Gamma(z+1) (it)^{-z-1}+b(t)$ for some $b\in C^\infty[-\pi, \pi]$,  see Propostion 4.1 in \cite{MJ}.
\begin{theorem}[Diagonal Case]\label{Dc}
Let $d\geq 1.$ Then for any (possibly infinite) system $(f_j)$ of orthonormal functions in $L^2_\kappa (\mathbb R^d)$ and any $(n_j)\in {\mathbb C},$ we have
\begin{equation}\label{A}
\left\|\sum_jn_j|e^{-itH_\kappa }f_j|^2\right\|_{L^{1+\frac{2}{d+2\gamma_\kappa}}_\kappa ((-\pi, \pi)\times \mathbb R^d)}\leq C \Big(\sum_j|n_j|^\frac{d+2\gamma_\kappa+2}{d+2\gamma_\kappa+1}\Big)^{\frac{d+2\gamma_\kappa+1}{d+2\gamma_\kappa+2}}.
\end{equation}
\end{theorem}
\begin{proof}
To prove \eqref{A}, in view of \eqref{id3}, it is enough to prove that
\begin{equation}\label{B}
\left\|\sum_jn_j|e^{-itH_\kappa }f_j|^2\right\|_{L^{1+\frac{2}{d+2\gamma_\kappa}}_\kappa ((-\frac{\pi}{4}, \frac{\pi}{4})\times \mathbb R^d)}\leq C \Big(\sum_j|n_j|^\frac{d+2\gamma_\kappa+2}{d+2\gamma_\kappa+1}\Big)^{\frac{d+2\gamma_\kappa+1}{d+2\gamma_\kappa+2}}.
\end{equation}
In order to prove \eqref{B}, using Lemma \ref{Ld}, it is enough to prove that
\begin{multline}\label{C}
\|W_1T_SW_2\|_{\mathfrak{S}^{d+2\gamma_\kappa+2}(L^2_{\kappa }((-\frac{\pi}{4}, \frac{\pi}{4}) \times \mathbb R^d))} \\ \leq C\|W_1\|_ {L^{d+2\gamma_\kappa+2}_{\kappa }((-\frac{\pi}{4}, \frac{\pi}{4}) \times \mathbb R^d))}\|W_2\|_{L^{d+2\gamma_\kappa+2}_{\kappa }((-\frac{\pi}{4}, \frac{\pi}{4}) \times \mathbb R^d))}
\end{multline}
for all $W_1, W_2\in L^{d+2\gamma_\kappa+2}_{\kappa }((-\frac{\pi}{4} \frac{\pi}{4}), \times \mathbb R^d),$ where $S = \{(\mu, \nu)\in \mathbb N^d_0\times \mathbb Z : \nu=2|\mu|+d+2\gamma_\kappa\}.$
We now show that the family of operator $(T_z)$ defined in \eqref{tz} satisfies \eqref{B1}.
When $Re (z) = 0,$ in view \eqref{gz}, we have
\begin{multline}\label{D}
\|T_{is}\|_{L^2_{\kappa }((-\frac{\pi}{4}, \frac{\pi}{4}), \times \mathbb R^d)\rightarrow L^2_{\kappa }((-\frac{\pi}{4}, \frac{\pi}{4}), \times \mathbb R^d)} = \|G_{is}\|_{L^\infty_{\kappa }((-\frac{\pi}{4}, \frac{\pi}{4}), \times \mathbb R^d)}\leq \left|\frac{1}{\Gamma(1+is)}\right|\leq C e^{\pi|s|/2}.
\end{multline}
When $z=-\lambda_0+is$,  in view of definition of $T_z,$ see \eqref{tz}, we have
\begin{equation*}
|T_zg(x, t)|\leq \Big(\sup_{\tau \in(-\frac{\pi}{4}, \frac{\pi}{4}), y\in\mathbb R^d} |K_z(x, y, \tau)|\Big)\|g\|_{L^1_\kappa ((-\frac{\pi}{4}, \frac{\pi}{4}) \times\mathbb R^d)},~~ \mbox{for all}~~ t\in (-\frac{\pi}{4}, \frac{\pi}{4}).
\end{equation*}
Therefore, $T_z$ is bounded from $L^1_\kappa ((-\frac{\pi}{4}, \frac{\pi}{4}) \times\mathbb R^d)$ to $L^\infty_{\kappa }((-\frac{\pi}{4}, \frac{\pi}{4})\times \mathbb R^d)$ if and only if $ |K_z(x, y, t)|$ is bounded on $\mathbb R^d\times \mathbb R^d\times (-\frac{\pi}{4}, \frac{\pi}{4})$ and in this case
\begin{equation}\label{F}
 \|T_{-\lambda_0+is}\|_{{L^1_\kappa ((-\frac{\pi}{4}, \frac{\pi}{4}) \times\mathbb R^d)} \rightarrow {L^\infty_{\kappa }((-\frac{\pi}{4}, \frac{\pi}{4}) \times \mathbb R^d)}}\leq \sup_{(x, y, t)\in \mathbb R^d\times \mathbb R^d\times (-\frac{\pi}{4}, \frac{\pi}{4})} |K_{-\lambda_0+is}(x, y, t)|.
\end{equation}
By \eqref{kz} and \eqref{hsk}, we get
\begin{equation}\label{G}
|K_z(x, y, t)| \sim  \frac{C |e^{iz\pi/2}|}{|t|^{Re(z+1+\frac{d+2\gamma_\kappa}{2})}}
\end{equation}
 for $t\in (-\frac{\pi}{4}, \frac{\pi}{4}).$
So $|K_z(x, y, t)|$ is bounded if and only if $\text{Re}(z) = -\Big(\frac{d+2\gamma_\kappa+2}{2}\Big),$ in which case we will have
$$\|T_{-\lambda_0+is}\|_{{L^1_\kappa ((-\frac{\pi}{4}, \frac{\pi}{4}) \times\mathbb R^d)} \rightarrow {L^\infty_{\kappa }((-\frac{\pi}{4}, \frac{\pi}{4}) \times \mathbb R^d)}}\leq C e^{\pi |s|/2}.$$
In view of Proposition \ref{Ip} and $T_{-1} = T_S$,  we get the inequality \eqref{C}, which completes proof of the theorem.
\end{proof}

Now we are ready to prove Theorem \ref{TSOFD} for the case $P = H_{\kappa}.$ We have to show that the operator
\begin{equation}\label{p}
V\in L^{p'}_t L^{q'}_{\kappa , x}(\mathbb T \times \mathbb R^d) \mapsto\int_\mathbb T e^{itH_{\kappa}}V(t, x)e^{-itH_{\kappa}}dt\in\mathfrak{S}^{2q'}(L^2_\kappa (\mathbb R^d)),
\end{equation}
 where $\mathbb T = (-\pi, \pi),$ is bounded under the assumption of Theorem \ref{TSOFD}. By complex interpolation \cite[Chap. 4]{BL}, it is sufficient to prove the boundedness for this operator at the two points $(p', q') = (1, \infty)~ \text{and}~ (p', q')=(1+\frac{d+2\gamma_\kappa}{2}, 1+\frac{d+2\gamma_\kappa}{2}).$   \\

\noindent For $(p', q') = (1, \infty):$ we will get with well known argument that
\begin{align*}
\left\|\int_\mathbb T e^{itH_{\kappa}}V(t, x)e^{-itH_{\kappa}}dt\right\|_{\mathfrak{S}^\infty(L^2_\kappa (\mathbb R^d))}&\leq \int_\mathbb T \|e^{itH_{\kappa}}V(t, x)e^{-itH_{\kappa}}\|_{\mathfrak{S}^\infty(L^2_\kappa (\mathbb R^d))}dt \\
&\leq \int_\mathbb T\|V(t, \cdot)\|_{L^\infty_\kappa (\mathbb R^d)}dt = \|V\|_ {L^1_tL^\infty_{\kappa , x}(\mathbb T\times\mathbb R^d)},
\end{align*}
thus we get the boundedness for $(p', q') = (1, \infty).$ \\

\noindent For $(p', q') = (1+\frac{d+2\gamma_\kappa}{2}, 1+\frac{d+2\gamma_\kappa}{2}):$ As we know the required estimate \eqref{Sc} is equivalent to the inequality \eqref{Ste3}, the boudedness of $\eqref{p}$ for $(p', q') =  (1+\frac{d+2\gamma_\kappa}{2}, 1+\frac{d+2\gamma_\kappa}{2})$ follows from Theorem \ref{Dc}.

Now we are in a position to prove Theorem \ref{TSOFdh}. Using the duality argument the estimate \eqref{Ste3} follows from Theorem \ref{TSOFD} for the range $1 < q\leq 1+\frac{2}{d+2\gamma_\kappa},$ for the remaining range of $q$ i.e., $1+\frac{2}{d+2\gamma_\kappa}<q<\frac{d+2\gamma_\kappa+1}{d+2\gamma_\kappa-1}$, since $e^{-itH_{\kappa}}$ satisfies \eqref{Stedh} and \eqref{210} applying Lemma \ref{Ld} for $A = e^{-itH_{\kappa}}$ and $\alpha'=\frac{2q}{q+1},$ it is enough to prove the following theorem.

\begin{theorem}\label{gse}
Let $S$ be the discrete surface $S = \{(\mu, \nu) \in \mathbb N^d_0\times \mathbb Z : \nu = 2|\mu|+d+2\gamma_\kappa\}$ with respect to the counting measure and $\mathbb T=(-\frac{\pi}{4}, \frac{\pi}{4})$ with Lebesgue measure. Then for all exponents $p, q \geq 1$ satisfying $$\frac{2}{p}+\frac{d+2\gamma_\kappa}{q}=d+2\gamma_\kappa ,~~~~~ \text{and}~~~~1+\frac{2}{d+2\gamma_\kappa}<q < \frac{d+2\gamma_\kappa+1}{d+2\gamma_\kappa-1}$$ we have
$$\|W_1T_SW_2\|_{\mathfrak{S}^{2q'}(L^2(\mathbb T\times\mathbb R^d))} \leq \|W_1\|_{L^{2p'}_tL^{2q'}_{\kappa , x}(\mathbb T\times\mathbb R^d) }\|W_2\|_{L^{2p'}_tL^{2q'}_{\kappa , x}(\mathbb T\times\mathbb R^d)}$$
with $C>0$ independent of $W_1, W_2.$
\end{theorem}
\begin{proof}
 The family of operators defined in \eqref{tz} is an analytic family of operators that coinsides with the operators $T_S$ when $z = -1.$ Notice that the operator $T_{-\lambda_0+is}$ is an integral operator with kernel $K_{-\lambda_0+is}(x, x', t-t')$ defined in $\eqref{Afk}$. An application of Hardy-Littlewood-Sobolev inequality (see \cite{BW}) along with \eqref{D} and \eqref{G} yields
\begin{align*}
&\|W^{\lambda_0-is}_1T_{-\lambda_0+is}W^{\lambda_0-is}_2\|^2_{\mathfrak{S}^2}\\
&=\int_{\mathbb T^2}\int_{\mathbb R^{2d}}|W_1(t, x)|^{2\lambda_0}|K_{-\lambda_0+is}(x, x', t-t')|^2 |W_2(t', x')|^{2\lambda_0}h^2_\kappa (x) h^2_\kappa (x')  dx dx' dt dt'\\
&\leq C_1 \int_{\mathbb T^2}\int_{\mathbb R^{2d}} \frac{|W_1(t, x)|^{2\lambda_0} |W_2(t', x')|^{2\lambda_0}}{|t-t'|^{d+2\gamma_\kappa+2-2\lambda_0}}h^2_\kappa (x) h^2_\kappa (x') dx dx' dt dt'\\
&\leq C_1 e^{\pi|s|} \int_{\mathbb T}\int_{\mathbb T} \frac{\|W_1(t, \cdot)\|^{2\lambda_0}_{L^{2\lambda_0}_{\kappa , x}(\mathbb R^d)}{\|W_2(t', \cdot)\|^{2\lambda_0}_{L^{2\lambda_0}_{\kappa , x}(\mathbb R^d)}}}{|t-t'|^{d+2\gamma_\kappa+2-2\lambda_0}} dt dt'\\
&\leq C_1 e^{\pi|s|} {\|W_1(t, \cdot)\|^{2\lambda_0}_{L^{\frac{4\lambda_0}{2\lambda_0-d-2\gamma_\kappa}}_{t}L^{2\lambda_0}_{\kappa , x}(\mathbb T\times\mathbb R^d)}} {\|W_2(t, \cdot)\|^{2\lambda_0}_{L^{\frac{4\lambda_0}{2\lambda_0-d-2\gamma_\kappa}}_{t}L^{2\lambda_0}_{\kappa , x}(\mathbb T\times\mathbb R^d)}}
\end{align*}
provided we have $0\leq d+2\gamma_k+2-2\lambda_0 < 1,$ that is $\frac{(d+2\gamma_\kappa+1)}{2} < \lambda_0 \leq \frac{d+2\gamma_\kappa+2}{2}.$ By Theorem 2.9 of \cite{BS} we have
$$\|W_1T_{-1}W_2\|_{\mathfrak{S}^{2\lambda_0}(L^2_\kappa (\mathbb T\times\mathbb R^d))} \leq {\|W_1(t, \cdot)\|^{2\lambda_0}_{L^{\frac{4\lambda_0}{2\lambda_0-d-2\gamma_\kappa}}_{t}L^{2\lambda_0}_{\kappa , x}(\mathbb T\times\mathbb R^d)}} {\|W_2(t, \cdot)\|^{2\lambda_0}_{L^{\frac{4\lambda_0}{2\lambda_0-d-2\gamma_\kappa}}_{t}L^{2\lambda_0}_{\kappa , x}(\mathbb T\times\mathbb R^d)}}, $$ for $\frac{d+2\gamma_\kappa+1}{2}<\lambda_0\leq \frac{d+2\gamma_\kappa+2}{2}.$ Since $ \frac{d+2\gamma_\kappa+2}{d+2\gamma_\kappa}<q<\frac{d+2\gamma_\kappa+1}{d+2\gamma_\kappa-1}$, we have $\frac{d+2\gamma_\kappa+1}{2}<q'<\frac{d+2\gamma_\kappa+2}{2}$. Thus the theorem is proved by chosing $\lambda_0=q'$.
\end{proof}

\begin{rem}
In future work we will prove the following:
\begin{enumerate}
\item We will try to prove that the range of the exponents given in Theorem \ref{TSOF} and Theorem \ref{TSOFdh} are necessary to satisfy the estimates \eqref{Ste2} and \eqref{Ste3}.

\item We will try to prove the Schatten exponent $\alpha=\frac{2q}{q+1}$ appear in the right hand side of the estimates \eqref{Ste2} and \eqref{Ste3} is optimal.
\end{enumerate}
\end{rem}
\section{Applications of Strichartz estimates: Global well-posedness for the Dunkl Hermite-Hartree equation in Schatten spaces}
In this section we show the well-posedness results in the sprit of \cite{LS1, LS2} for a system of infinitely many equations (without a trace class assumption) in Schatten spaces for the  Dunkl Hermite-Hartree equation by applying our orthonormal Strichartz inequalities associated with Dunkl Hermite operator. Recall that $f\ast_\kappa  g$ denotes the generalized (Dunkl) convolution of functions $f$ and $g$ (for the definition see in \cite{TX}).

\begin{theorem}
Let $$1\leq q<1+\frac{2}{d+2\gamma_\kappa-1}~~ \mbox{and}~ ~\frac{2}{p}+\frac{d+2\gamma_\kappa}{q} = d+2\gamma_\kappa$$
and $w: \mathbb R^d\rightarrow{\mathbb C}$ such that $f\mapsto w\ast_\kappa  f$ is bounded operator from $L^p_\kappa (\mathbb R^d)\rightarrow L^\infty_\kappa (\mathbb R^d).$ Then for any $\gamma_0\in\mathfrak{S}^{\frac{2q}{q+1}},$ there exists a unique $\gamma\in C^0_t([0, T], \mathfrak{S}^{\frac{2q}{q+1}}(L^2_\kappa (\mathbb R^d))$ satisfying $\rho_\gamma\in L^p_tL^q_{\kappa , x}([0, T] \times \mathbb R^d)$ and
\begin{align*}
i\partial_t\gamma&=[H_\kappa  + w\ast_\kappa  \rho_\gamma, \gamma]\\
&\gamma|_{t=0}=\gamma_0
\end{align*}
\end{theorem}
\begin{proof}
Let $R > 0$ such that $\|\gamma_0\|_{\mathfrak{S}^\frac{2q}{q+1}} = R<\infty.$ Let $T= T(R) (\leq 1$ to be chosen later). Consider the space
$$X_T = \{(\gamma, \rho) \in C^0_t([0, T], \mathfrak{S}^{\frac{2q}{q+1}}(L^2_\kappa (\mathbb R^d))\times L^p_tL^q_{\kappa , x}([0, T] \times \mathbb R^d) : \|(\gamma, \rho)\|_{X_T}\leq 4~ max\{1, C_{Stri}\}R\},$$ where the norm $\|(\gamma, \rho)\|_{X_T}$ is defined by $$\|(\gamma, \rho)\|_{X_T} := \|\gamma\|_{ C^0_t([0, T], \mathfrak{S}^{\frac{2q}{q+1}}(L^2_\kappa (\mathbb R^d))}+\|\rho\|_{L^p_tL^q_{\kappa , x}([0, T] \times \mathbb R^d)}.$$
Consider the map $$\Phi_1(\gamma, \rho)(t) = e^{-itH_\kappa }\gamma_0 e^{itH_\kappa } - i\int^t_0 e^{i(s-t)H_\kappa }  [w\ast_\kappa \rho_\gamma(s), \gamma(s)] e^{-i(s-t)H_\kappa } ds$$
Using the above map, we define the contraction map $\Phi$ by $$\Phi(\gamma, \rho) = (\Phi_1(\gamma, \rho), \rho[\Phi_1(\gamma, \rho)]),$$ where we have used the notation $\rho[\gamma] = \rho_\gamma.$ Now
\begin{align*}
 \|\Phi_1(\gamma, \rho)\|_{ C^0_t([0, T], \mathfrak{S}^{\frac{2q}{q+1}}(L^2_\kappa (\mathbb R^d))}& \leq R + 2 \int^T_0 \|w\ast_\kappa \rho(s)\|_{L^\infty_{\kappa , x} \|\gamma(s)\|_{\mathfrak{S}^\frac{2q}{q+1}}} ds\\
&\leq R + 2T^{1/p'} C_w \|\rho\|_{L^t_pL^q_{\kappa , x}}\|\gamma\|_{C^0_t\mathfrak{S}^\frac{2q}{q+1}}\\
& \leq R + 8T^{1/p'} C_w max (1, C^2_{Stri}) R^2
\end{align*}
and from Corollary \ref{ISI} we also have
$$\|\rho[\Phi_1(\gamma, \rho)]\|_{L^t_pL^q_{\kappa , x}} \leq C_{Stri} R + 8C_{Stri} T^{1/p'} C_w  max (1, C^2_{Stri}) R^2.$$
Choosing $T\leq 1$ small enough so that
$$8C_{Stri} T^{1/p'} C_w  max (1, C^2_{Stri}) R^2 \leq C_{Stri} R$$
and $\Phi$ maps $X$ to itself. Thus $\Phi$ is a contraction mapping and has a unique fixed point on $X$ which is a solution to the Hatree equation on $[0, T].$
\end{proof}

\section{Alternative proof of Theorem \ref{TSOF}}

\subsection{Proof of Theorem \ref{TSOF} and \ref{TSOFD} in case $P=-\Delta_{\kappa}$}
Throughout this section $\mathfrak{S}^p$ stands for $\mathfrak{S}^p(L^2_\kappa (\mathbb R^d))$.
As discussed before, the duality argument shows that Theorem \ref{TSOF} is equivalent to Theorem\ref{TSOFD} for the case $P=-\Delta_{\kappa}$. By assuming $V\in L^\infty_c(\mathbb R\times\mathbb R^d)$ and that $\gamma$ is finite rank, we have to show that the operator
$$ L^{p'}_tL^{q'}_{\kappa , x}(\mathbb R\times \mathbb R^d)\ni V\mapsto\int_\mathbb R e^{-it\Delta_\kappa}V(t, x)e^{it\Delta_\kappa}dt\in\mathfrak{S}^{2q'}$$ is bounded. By using complex interpolation method \cite[Chap. 4]{BL},  it is enough to show the boundedness for the two points $(p', q')=(1, \infty)$ and $(p', q')=(1+\frac{d+2\gamma_\kappa}{2}, 1+\frac{d+2\gamma_\kappa}{2}).$

For the point  $(p', q')=(1, \infty)$: We will get with the well known argument that.
\begin{align*}
\left\|\int_\mathbb R e^{-it\Delta_\kappa}V(t, x)e^{it\Delta_\kappa}dt\right\|_{\mathfrak{S}^\infty}&\leq \int_\mathbb R \|e^{-it\Delta_\kappa}V(t, x)e^{it\Delta_\kappa}\|_{\mathfrak{S}^\infty}dt\\ &=\int_\mathbb R\|V(t, \cdot)\|_{L^\infty_\kappa (\mathbb R^d)}dt=\|V\|_{L^1_tL^\infty_{\kappa , x}(\mathbb R \times \mathbb R^d)}.\end{align*}

Now it is time to go for the case $p'= q'= 1+\frac{d+2\gamma_\kappa}{2}.$ Without any loss of generality, we may assume that $V\geq 0$ and  $V\in L^\infty_c(\mathbb R\times\mathbb R^d).$  Then we have $e^{-it\Delta_\kappa}V(t, x)e^{it\Delta_\kappa} \geq 0$ as an operator on $L^2_\kappa (\mathbb R^d),$ for all $t$ in $\mathbb R.$  It can be easily proved that $e^{-it\Delta_\kappa} x_j e^{it\Delta_\kappa}=x_j-2itT_j$ for each $j=1, 2, \ldots , d,$ using Dunkl transform (for the definition, see \cite{MR}) and its properties. So it can be written as $$e^{-it\Delta_\kappa} x e^{it\Delta_\kappa}=x-2it\nabla_\kappa ,$$ where $\nabla_\kappa =(T_1, T_2, \ldots , T_d)$ and $x$ is identified with the multiplication operator.  By the functional calculus we deduce that
\begin{equation}\label{alt1}
e^{-it\Delta_\kappa} f(x) e^{it\Delta_\kappa}=f(x+2tp),$$ where $p:=-i\nabla_\kappa .$ From this we can get $$e^{-it\Delta_\kappa} V(t, x) e^{it\Delta_\kappa}=V(t, x+2tp).
\end{equation}
Using the fact $V\geq 0,$ we can write the Schatten norm as
\begin{align*}
&\left\|\int_\mathbb R e^{-it\Delta_\kappa}V(t, x)e^{it\Delta_\kappa}dt\right\|^{d+2\gamma_\kappa+2}_{\mathfrak{S}^{d+2\gamma_\kappa+2}}\\
&=Tr\left(\int_\mathbb R e^{-it\Delta_\kappa}V(t, x)e^{it\Delta_\kappa}dt\right)^{d+2\gamma_\kappa+2}=Tr\left(\int_\mathbb R V(t, x+2tp)dt\right)^{d+2\gamma_\kappa+2}\\
&=Tr\left(\int_\mathbb R\cdot\cdot\cdot\int_\mathbb R V(t_1, x+2t_1p)\cdot\cdot\cdot V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)\right)dt_1\cdot\cdot\cdot dt_{d+2\gamma_\kappa+2}
\end{align*}

To exchange the trace and the integral, we need to prove that
\begin{equation}\label{TNE}
\int_\mathbb R\cdot\cdot\cdot\int_\mathbb R \|V(t_1, x+2t_1p)\cdot\cdot\cdot V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)\|_{\mathfrak{S}^1}dt_1\cdot\cdot\cdot dt_{d+2\gamma_\kappa+2}<\infty
\end{equation}
(We are assuming $V\in L^\infty_c(\mathbb R\times\mathbb R^d))$ throughout). In order to estimate the Schatten norm in the integral, we make use of the following.
\begin{lem}\label{LE}
Let $\alpha, \beta , \gamma, \delta  \in \mathbb R$. We have
\begin{equation}\label{l1}
\|f(\alpha x +\beta  p)g(\gamma x+\delta  p)\|_{\mathfrak{S}^r}\leq \frac{M_\kappa ^{\frac{2}{r}}\|f\|_{L_\kappa ^r(\mathbb R^d)}\|g\|_{L_\kappa ^r(\mathbb R^d)}}{|\alpha\delta  -\beta  \gamma|^{(d+2\gamma_\kappa )/r}}
\end{equation}
for all $r\geq 2$.
\end{lem}
 For the value $\alpha=\delta =1, \beta =\gamma=0,$ and the multiplicity function $\kappa =0$, the above estimate is the well-known Kato-Seiler-Simon inequality (\cite{SS} and \cite[Thm. 4.1]{S}).

We refer Subsection \ref{plsb} for the proof of this lemma. We now estimate the trace norm \eqref{TNE}. By using the fact $V\geq 0$ and H\"older's inequality in Schatten space, we can write
\begin{align*}
\|V(t_1, x+2t_1p)\cdots &V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)\|_{\mathfrak{S}^1}\\
&=\|V(t_1, x+2t_1p)\sqrt{V(t_2, x+2t_2p)}\sqrt{V(t_2, x+2t_2p)}\\ &\quad \times \cdot \cdot \cdot \times \sqrt{V(t_{d+2\gamma_\kappa+1}, x+2t_{d+2\gamma_\kappa+1}p)}V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)\|_{\mathfrak{S}^1}\\
&\leq\|V(t_1, x+2t_1p)\sqrt{V(t_2, x+2t_2p)}\|_{\mathfrak{S}^{d+2\gamma_\kappa+1}}\\&\quad\|\sqrt{V(t_2, x+2t_2p)}\sqrt{V(t_3, x+2t_3p)}\|_{\mathfrak{S}^{d+2\gamma_\kappa+1}}\\ &\quad\times\cdot\cdot\cdot \times\|\sqrt{V(t_{d+2\gamma_\kappa+1}, x+2t_{d+2\gamma_\kappa+1}p)}V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)\|_{\mathfrak{S}^{d+2\gamma_\kappa+1}}
\end{align*}

Using \eqref{l1} and the fact $V\in L^\infty_c(\mathbb R\times\mathbb R^d),$ we get

\begin{align*}
\|&V(t_1, x+2t_1p)\cdot\cdot\cdot V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)\|_{\mathfrak{S}^1}\\
&\leq \frac{\|V(t_1, \cdot)\|_{L^{d+2\gamma_\kappa+1}_\kappa } \|V(t_2, \cdot)\|_{L^{({d+2\gamma_\kappa+1})/2}_\kappa }\cdot\cdot\cdot \|V(t_{d+2\gamma_\kappa+1}, \cdot)\|_{L^{({d+2\gamma_\kappa+1})/2}_\kappa }\|V(t_{d+2\gamma_\kappa+2}, \cdot)\|_{L^{d+2\gamma_\kappa+1}_\kappa }}{c^2_\kappa  2^{d+2\gamma_\kappa}|t_1-t_2|^{\frac{d+2\gamma_\kappa}{d+2\gamma_\kappa+1}} \cdot\cdot\cdot |t_{d+2\gamma_\kappa+1}-t_{d+2\gamma_\kappa+2}|^{\frac{d+2\gamma_\kappa}{d+2\gamma_\kappa+1}}}\\
&\leq C \frac{\prod^{d+2\gamma_\kappa+2}_{j=1}\mathbbm{1}(a\leq t_j\leq b)}{c^2_\kappa  2^{d+2\gamma_\kappa}|t_1-t_2|^{\frac{d+2\gamma_\kappa}{d+2\gamma_\kappa+1}} \cdot\cdot\cdot |t_{d+2\gamma_\kappa+1}-t_{d+2\gamma_\kappa+2}|^{\frac{d+2\gamma_\kappa}{d+2\gamma_\kappa+1}}}
\end{align*}
where $(a, b)$ is the support of $V$ in the time variable. Now we will use the multilinear Hardy-Littlewood-Sobolev inequality.

\begin{theorem}[Multilinear Hardy-Littlewood-Sobolev inequality  (See \cite{BW})]\label{HARDY}
Assume that $(\beta_{ij})_{1\leqslant i, j\leqslant N}$ and $(r_k)_{1\leqslant k\leqslant N}$ are real-numbers such that
$$\beta_{ii}=0,~~ 0\leqslant\beta_{ij}=\beta_{ji}<1, r_k>1,~~ \sum^N_{k=1}\frac{1}{r_k}>1, ~~\sum^N_{i=1}\beta_{ik}=\frac{2(r_k-1)}{r_k}.$$
Then there exist a constant $C$ such that
\begin{equation}\label{HINQ}
\left|\int_\mathbb R \cdots \int_\mathbb R\frac{f_1(t_1)\cdots f_N(t_N)}{\prod_{i<j}|t_i-t_j|^{\beta_{ij}}}dt_1\cdots dt_N\right|\leqslant C\prod^N_{k=1}\|f_k\|_{L^{r_k}(\mathbb R)}
\end{equation}
 for all $f_k\in L^{r_k}.$

\end{theorem}

The particular case of Multilinear Hardy-Littlewood-Sobolev inequality where all the $\beta _{ij}$ and the $r_k$ are identical, can be found in [\cite{CM}, Prop. 2.2]. Applying \eqref{HINQ} with taking $r_1=r_{d+2\gamma_\kappa+2}=\frac{2(d+2\gamma_\kappa+1)}{(d+2\gamma_\kappa+2)}$ and $r_2=\cdot \cdot \cdot = r_{d+2\gamma_\kappa+1}=d+2\gamma_\kappa+1,$ then we will get \eqref{TNE}. Hence now we can write
\begin{align*}
Tr&\left(\int_\mathbb R e^{-it\Delta_\kappa}V(t, x)e^{it\Delta_\kappa}dt\right)^{d+2\gamma_\kappa+2}\\
&=\int_\mathbb R\cdot\cdot\cdot\int_\mathbb R Tr\left(V(t_1, x+2t_1p)\cdot\cdot\cdot V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)\right)dt_1\cdot\cdot\cdot dt_{d+2\gamma_\kappa+2}
\end{align*}

By the previous argument we will now obtain a more symmetric estimate of the trace in the integral. Simply using the cyclicity of the trace we get,

\begin{align*}
&\left|Tr\left(V(t_1, x+2t_1p)\cdot \cdot \cdot V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)\right)\right|\\ &\quad \quad \quad \quad
=\left|Tr\left(\sqrt {V(t_1, x+2t_1p)} \cdot \cdot \cdot V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p \sqrt{ V(t_1, x+2t_1p)}\right)\right|\\ &\quad \quad \quad \quad
\leq \|\sqrt {V(t_1, x+2t_1p)} \sqrt {V(t_2, x+2t_2p)}\|_{\mathfrak{S}^{d+2\gamma_\kappa+2}}\\ &\quad \quad \quad \quad \quad \times \cdot \cdot \cdot \times
\|\sqrt {V(t_{d+2\gamma_\kappa+1}, x+2t_{d+2\gamma_\kappa+1}p)} \sqrt {V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)}\|_{\mathfrak{S}^{d+2\gamma_\kappa+2}}\\ &\quad \quad \quad \quad \quad
 \times \|\sqrt {V(t_{d+2\gamma_\kappa+2}, x+2t_{d+2\gamma_\kappa+2}p)} \sqrt {V(t_1, x+2t_1p)}\|_{\mathfrak{S}^{d+2\gamma_\kappa+2}}
\end{align*}

Again using of \eqref{l1} and multilinear Hardy-Littlewood-Sobolev inequality \eqref{HINQ} \\ with $r_1 = \cdot \cdot \cdot = r_{d+2\gamma_\kappa+2} = 1+\frac{d+2\gamma_\kappa}{2}$ we will get our required inequality

$$Tr\left(\int_\mathbb R e^{-it\Delta_\kappa}V(t, x)e^{it\Delta_\kappa}dt\right)^{d+2\gamma_\kappa+2} \leq C \|V\|^{d+2\gamma_\kappa+2}_{L^{1+\frac{d+2\gamma_\kappa}{2}}_\kappa  (\mathbb R\times \mathbb R^d)}$$

\subsection{Proof of  Lemma \ref{LE}}\label{plsb}
 We prove the inequality \eqref{l1} for $r=2$ and $r=\infty$, then the general case follows from the complex interpolation. Since $e^{-it\Delta_\kappa }f(x)e^{it\Delta_\kappa }=f(x+2tp)$ with $p=-i\nabla_\kappa$, we get that
$$e^{-it\Delta_\kappa }f( \alpha x)e^{it\Delta_\kappa }=f( \alpha x+2t\alpha p).$$ If we substitute $t=\frac{\beta }{2\alpha}$ in the above, we get that
\begin{equation}f( \alpha x+\beta  p)=e^{-i\frac{\beta }{2\alpha}\Delta_\kappa }f( \alpha x)e^{i\frac{\beta }{2\alpha}\Delta_\kappa }\end{equation}
where $p=-i\nabla _\kappa .$ Similarly we have that,
\begin{equation}g( \gamma x+\delta p)=e^{-i\frac{\delta}{2\gamma}\Delta_\kappa }g( \gamma x)e^{i\frac{\delta}{2\gamma}\Delta_\kappa }.\end{equation} For $r=\infty,$ by noting that the Schatten $\infty$-norm is the operator norm on $L^2_\kappa (\mathbb R^d)$, the inequality \eqref{l1} can be seen easily, indeed
\begin{align*}
\|f(\alpha x+\beta  p)g(\gamma x+\delta p)\phi\|_{L^2_\kappa (\mathbb R^d)}&=\| e^{-i\frac{\beta }{2\alpha}\Delta_\kappa }f( \alpha x)e^{i\alpha_0\Delta_\kappa }g( \gamma x)e^{i\frac{\delta}{2\gamma}\Delta_\kappa }\phi\|_{L^2_\kappa (\mathbb R^d)}\\
&\leq \|f\|_{L^\infty_\kappa (\mathbb R^d)}\|g\|_{L^\infty_\kappa (\mathbb R^d)}\|\phi\|_{L^2_\kappa (\mathbb R^d)}
\end{align*} where \begin{equation}\label{l2}\alpha_0=\frac{\beta \gamma-\alpha\delta}{2\alpha\gamma}.\end{equation}
Here we used the facts $\|e^{it\Delta_\kappa }\phi\|_{L^2_\kappa (\mathbb R^d)}=\|\phi\|_{L^2_\kappa (\mathbb R^d)}$ and $\|f\phi\|_{L^2_\kappa (\mathbb R^d)}\leq\|f\|_{L^\infty_\kappa (\mathbb R^d)}\|\phi\|_{L^2_\kappa (\mathbb R^d)}$, Hence, $$\|f(\alpha x+\beta  p)g(\gamma x+\delta p)\|_{\mathfrak{S}^\infty}\leq \|f\|_{L^\infty_\kappa (\mathbb R^d)}\|g\|_{L^\infty_\kappa (\mathbb R^d)}.$$

For $r=2,$ we know that $\|T\|_{\mathfrak{S}^2}^2=Tr|T|^2$, we get
\begin{align*}
\|f(\alpha x+\beta  p)g(\gamma x+\delta p)\|_{\mathfrak{S}^2}^2&=Tr[|f(\alpha x+\beta  p)|^2|g(\gamma x+\delta p)|^2]\\
&=Tr\Big[e^{-i\frac{\beta }{2\alpha}\Delta_{\kappa}}|f( \alpha x)|^2e^{i\frac{\beta }{2\alpha}\Delta_{\kappa}}e^{-i\frac{\delta}{2\gamma}\Delta_{\kappa}}|g( \gamma x)|^2e^{i\frac{\delta}{2\gamma}\Delta_{\kappa}}\Big]\\
&=Tr\Big[|f( \alpha x)|^2e^{i\alpha_0\Delta_{\kappa}}|g( \gamma x)|^2e^{-i\alpha_0\Delta_{\kappa}}\Big].
\end{align*}
We now show that the operator $|f( \alpha x)|^2e^{i\alpha_0\Delta_{\kappa}}|g( \gamma x)|^2e^{-i\alpha_0\Delta_{\kappa}}$ is an integral operator with some kernel $k(x, y)$ using which we find the trace.
\begin{align*}
e^{it\Delta_\kappa }f(x) &=\int_{\mathbb R^d}L_{it}(x, y) f(y) h_\kappa ^2(y) dy \\=&\frac{M_\kappa }{ (2it)^{\gamma_\kappa+d/2}}\int_{\mathbb R^d}e^\frac{{i(|x|^2+|y|^2})}{4t}E_\kappa \left(\frac{x}{2it}, y\right)f(y)h^2_\kappa (y)dy.
\end{align*}
So, for a suitable $\phi$ we have
$$|g(\gamma\xi)|^2e^{i\alpha_0\Delta_\kappa }\phi(\xi)=\frac{M_\kappa }{ (2i\alpha_0)^{\gamma_\kappa+d/2}}\int_{\mathbb R^d}e^\frac{{i(|\xi|^2+|y|^2)}}{4\alpha_0}E_\kappa \Big(\frac{\xi}{2i\alpha_0}, y\Big)|g(\gamma\xi)|^2\phi(y)h^2_\kappa (y)dy$$
Which implies
\begin{align*}
|f(\alpha x)|^2e^{-i\alpha_0\Delta_\kappa }|g(\gamma\cdot)|^2e^{i\alpha_0\Delta_\kappa }\phi(x)=&\frac{M_\kappa ^2}{ (2\alpha_0)^{d+2\gamma_\kappa}}\int_{\mathbb R^d}\int_{\mathbb R^d}\left(e^\frac{-i(|x|^2+|\xi|^2)}{4\alpha_0}e^\frac{i(|\xi|^2+|y|^2)}{4\alpha_0}E_\kappa \Big(\frac{-x}{2i\alpha_0}, \xi\Big) \right.\\ &\quad \times \left. E_\kappa \Big(\frac{\xi}{2i\alpha_0}, y\Big)|f(\alpha x)|^2|g(\gamma\xi)|^2\phi(y)  h^2_\kappa (y)h^2_\kappa (\xi)\right) dy d\xi\\
=&\int_{\mathbb R^d}k(x,y)\phi(y)h^2_\kappa (y)dy,
\end{align*}
where $$k(x, y)=\frac{M_\kappa ^2}{(2\alpha_0)^{d+2\gamma_\kappa}}\int_{\mathbb R^d}e^\frac{-i(|x|^2-|y|^2)}{4\alpha_0} E_\kappa \Big(\frac{-x}{2i\alpha_0}, \xi\Big)E_\kappa \Big(\frac{\xi}{2i\alpha_0}, y\Big)|f(\alpha x)|^2|g(\gamma\xi)|^2h^2_\kappa (\xi)d\xi.$$
Therefore,
$$k(x, x)=\frac{M_\kappa ^2}{(2\alpha_0)^{d+2\gamma_\kappa}}\int_{\mathbb R^d}E_\kappa \Big(\frac{-x}{2i\alpha_0}, \xi\Big)E_\kappa \Big(\frac{\xi}{2i\alpha_0}, x \Big)|f(\alpha x)|^2|g(\gamma\xi)|^2h^2_\kappa (\xi)d\xi.$$

Now we can find the required trace,
\begin{align*}
Tr\big[|f( \alpha x)|^2e^{i\alpha_0\Delta_{\kappa}}|g( \gamma x)|^2e^{-i\alpha_0\Delta_{\kappa}}\big]=&\int_{\mathbb R^d}k(x,x)h^2_\kappa (x)dx\\
\leq &\frac{M_\kappa ^2}{(2|\alpha_0|)^{d+2\gamma_\kappa}}\int_{\mathbb R^d}\int_{\mathbb R^d}|f(\alpha x)|^2|g(\gamma\xi)|^2 h^2_\kappa (\xi) h^2_\kappa (x) d\xi dx\\
=&\frac{M_\kappa ^2}{(2|\alpha_0 \alpha\gamma|)^{d+2\gamma_\kappa}}\|f\|^2_{L^2_\kappa (\mathbb R^d)}\|g\|^2_{L^2_\kappa (\mathbb R^d)}.
\end{align*}
The first inequality in the above follows in view of the expression for $k(x, x)$ and the fact $|E_\kappa (ix, y)|\leq1$.

In view of \eqref{l2}, we have that $2|\alpha_0 \alpha \gamma|=|\beta \gamma-\alpha \delta |$ and hence
$$\|f(\alpha x+\beta  p)g(\gamma x+\delta p)\|_{\mathfrak{S}^2}^2\leq \frac{M_\kappa ^2}{|\beta \gamma-\alpha\delta |^{d+2\gamma_\kappa }}\|f\|^2_{L^2_\kappa }\|g\|^2_{L^2_\kappa }.$$
The inequality in $\mathfrak{S}^r$ now follows from the complex interpolation.

\section*{Acknowledgments}
Both the authors are grateful to the Department of Mathematics, BITS Pilani K K Birla Goa Campus for the fecilities utilised. The authors thank Prof. S. Thangavelu, IISc Bangalore for the fruitful discussions.

redpajama_set_name: RedPajamaArXiv


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text:
Home Scores Bracket Full printable bracket Rankings Stats Video History
Trine Athletics | May 17, 2014
DIII SOFT: Home runs lead Trine past Emory into finals
Trine will play in its first NCAA championships next week.
ATLANTA -- A home run barrage in the first led Trine to a 6-2 victory over Emory in the Atlanta Super Regional to earn the Thunder their first berth in the NCAA Division III national championship.
Leah Hall hit a grand slam as part of a five-run first and was named most outstanding player of the Atlanta Super Regional. Catherine Porter homered to cap off the scoring in the big opening frame. Bree Fuller picked up the victory in the clincher, while Justine Robles came out of the bullpen to pick up her first career save. Fuller was named the most outstanding pitcher of the tournament. Emory concludes the season 39-9 overall. Trine will travel to Tyler, Texas, for the NCAA Division III national championships, where they will open play Thursday, May 22 at 7:30 p.m. under the lights against Montclair State.
The Thunder came out with their bats on fire, loading the bases with one out to set the stage for Hall. The junior pummeled the first pitch for a no-doubt grand slam to dead center to make the score 4-0. The homer was Hall's fifth of the season, and the second postseason grand slam of her Trine career.
Porter continued the Thunder power surge, as the freshman lined a solo shot deep into the trees beyond the left field fence to give Trine a 5-0 lead before Emory had a chance to hit. Porter's homer was her fifth of the season.
Melody Carter got Emory on the board in the bottom of the first with a solo home run to left-center to cut the score to 5-1.
Pitchers Fuller and Brittany File shut down each team's respective bats until Emory broke through again in the sixth on a solo home run by UAA most outstanding player Megan Light. The Eagles threatened to do further damage, but a 4-6-3 double play induced by Robles helped keep Trine keep Emory off the board.
Trine got an insurance run in the seventh as Carly Searles scored on an RBI-groundout by Rachel Harris.
Fuller picked up the victory with five strong innings of work, scattering five hits and two earned runs. The junior walked one and struck out three to record her school-record 24th victory of the season. The most outstanding pitcher of the Super Regional award adds to a list of accolades that has included MIAA Pitcher of the Year, Academic All-District, All-Region, and All-Regional Tournament honors this season.
Robles earned the save with two innings of shutout relief. The sophomore allowed three hits and struck out one. Robles allowed just one earned run in 4.1 innings of relief work in the Super Regional.
Hall batted .500 (3-for-6) with a home run, five RBI, two walks and two runs scored to earn most outstanding player honors.
The best college sports traditions you've (maybe) never heard of
You've probably heard of Howard's Rock at Clemson and the dotting of the "i" at Ohio State. Here are some of the best college sports traditions that you've (maybe) never heard of.
Texas Lutheran wins 2019 NCAA DIII softball national championship
Texas Lutheran dominated the DIII softball tournament to claim the first DIII softball national title in program history. The Bulldogs wrapped up tournament play with a series sweep over Emory, 12-4, 6-1.

redpajama_set_name: RedPajamaCommonCrawl


Did you know that there is a cost to caring? Is your workforce exposed to the suffering, traumatised, sick, dying and vulnerable of our society as a function of their duties? If so, then there is a high chance of them developing compassion fatigue.
Compassion fatigue is insidious, and cumulative, and can have far reaching harmful effects, on an individual and organisational level.
Compassion fatigue is a recognised Health, Welfare and Safety Hazard (HWS) to helping organisations, and addressing this condition is an ethical imperative.
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Why are they doing this? The answer is simple: they want you to try out their more involved music streaming service. Or, pick up an Amazon Prime account (which comes bundled with music).
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The work included open cut construction of approximately 1,500 linear feet of reinforced concrete pipe 72 inches in diameter.
Soledad officials recognized the need for expanding its storm drain system to accommodate the small south Monterey County city's population growth.
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Big thanks goes to this team for the attention to detail and safety: Joe Gomes, Gerrod Dewey, Scott Sollid, Jorge Bautista, Jonathan Nieto, Chris Rincon and Martin Canning.


At some point or another, your kids will want to be more involved with dressing themselves. It isn't because you have bad taste, but rather, your kids are growing up. Wanting control over what they wear is inevitable and good as it helps kids express their individuality and independence. Here are some tips for helping your kids get dressed.
Wait, what? Letting your kids dress themselves doesn't mean relinquishing full control. However, you may run into a tantrum or two if you simply pick their options for them. Make it a collaborative effort. When your child picks their clothes, make gentle suggestions. If you think something would look better than what they've selected, don't simply say "that looks bad," but give them an alternative.
If you're not a fan of the sweater they've chosen, provide an alternative. They may make some odd choices, but part of growing up is also learning what works and what doesn't, so don't rob them of that lesson.
A lot of the time, kids will make some questionable choices. They'll pick costumes over practical options suitable for the existing weather conditions, etc. Chances are if you tet them (briefly) wear the light clothes in freezing weather once and they won't do it twice.This may send you into a tizzy but sometimes you just have to let them look a little foolish, to drive those lessons home.
A lot of the first couple tries of kids dressing themselves may end badly. They might not take to your suggestions. They may pick ridiculous options that you can't accept (like a bathing suit in the winter for example), the possibilities are endless. Eventually, they'll get the hang of it, so be patient and embrace the process.
There are certain instances where there might be a dress code or different cultural/social rules that don't have a lot of flexibility. For instance, your kids may be required to wear cultural garb, or there might be an event like a wedding where there is a dress code, or perhaps your kids attend a school with a uniform. These scenarios can be tough as there is not a lot of flexibility in what your kids can wear, so how can you help them stand out?
You can always help them stand out by letting them pick their own accessories, like socks or a bag or a hat. It is a small gesture that goes a long way. Though it may not feel like a lot of choice at first, being able to pick something will make them feel like they had some part in getting dressed.
Watch this kids talk on being oneself and being 'in style," and fitting in. Have them design their own t-shirt with the Anne of Green Gables T-Shirt Painting Kit!
In order to raise open-minded children, here are some tips on teaching kids about diversity.
Rules are important. They keep us in order, and benefit everyone. Here are some tips for establishing clear expectations and setting house rules for kids.
Here some tips on how to make their first day back at school great!


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Q: Intercept traffic from android app I need to intercept requests from the mobile application to take advantage of its internal api. In the application there is a chat, I need to receive the information that the server periodically sends to the application. For this, I installed Genymotion with the Xposed + module to bypass SSLPinning and Fiddler as a proxy (tried also Charles). After that I see some of the traffic from the application (long-polling requests), but the events I need are not displayed, although the application still receives them and displays them.
Can I somehow intercept absolutely all traffic from the application and what should I do to do this? Thank you in advance)

A: *

*On Fiddler side you have to enable in the settings Allow remote computers to connect.

*You also need the IP address of your computer in your home network, e.g. 192.168.0.13
Now start the virtual device and configure the proxy (taken from Genymotion FAQ):

How do I configure proxy settings in my virtual device? In your
  virtual device:

*

*Click Menu > System settings.

*In Wireless & Networks, select Wi-Fi.

*Click WiredSSID and hold it down for two seconds.

*Click Modify Network.

*Check Show advanced options.

*In Proxy settings, select Manual.

*Enter the proxy settings (the noted IP address of step 2 of your PC and port 8888)

*Click Save.

redpajama_set_name: RedPajamaStackExchange


Sometimes the environment feels like an intrusion. Anonymity is appealing.
It's like…if it can sense me, maybe I won't sense it.
Sometimes I want only to experience myself. My own energy, my own voice, my own thoughts and my own feelings.
I need silence to hear myself, clearly.
The day I'm writing this, December 27th, is my birthday, and I am spending most of it alone. Precisely what I want. I really wanted to spend Christmas alone too, but I resisted the deep inner feeling which tells me when something is true for me. And so I had a bunch of plans. I'm okay with it.
It's easier to concentrate. On doing work. To monitor my own thoughts. To focus, and feel. To feel all those things.
But why did I want to be alone this Christmas and birthday?
I felt less obligated. I felt more empowered doing what is true for me. And I didn't want to spend my energy with others. I want to spend it with me.
Because I didn't want any more distractions. I distract myself all the time. There's stuff in there I don't want to deal with.
I don't want to though, not today. Drown myself out. I do want to watch Broad City (yet again), but I want to do it later, after I have listened to myself for a while. Let Abbi & Ilana be the candle on my cake.
As this shift happened on my birthday, I am going to take it as a habitual rebirth, and see that many more days go by with less and less time spent ignoring myself with a bombardment of noises and procrastination.


The children's art program at Sydney Children's Hospital, Randwick, has revealed that exhibitions of children's art, in the context of a child-oriented environment, are at least as significant to their target audience as art by adult professionals.
The children's art program is administered by a company called Identity, Environment and Art, which specialises in developing art and cultural programs, primarily in health facilities. In 1997 Identity, Environment and Art was contracted to draw up and implement an Arts and Cultural Plan linked to the redevelopment of Sydney Children's Hospital. The planning involved patients, families, staff and the local community, and established policies for the arts in the hospital. The plan also established the Sydney Children's Gallery in the main corridors of the hospital, a gallery specialising in children's art. In 1997, the gallery became part of the Cultural Gifts Program of the Australian Taxation Office.
During the implementation of the plan, Identity commissioned artworks from professional artists for the main public areas within the hospital. They included a kinetic sculpture of butterflies for the main entry lobby, murals for the Allied Health Waiting Rooms and Physiotherapy Department, interactive wall panels for the Accident and Emergency Department, and integrated mosaics for the Physiotherapy Courtyard and the Star Café.
The first exhibition the hospital hosted was an art competition for local children. There were over 300 entries and it became very clear that children's artwork was a popular choice for the long, featureless corridors of the hospital. People actually stopped, looked and discussed the artwork on the walls. Even small children in strollers pointed at the works. The decision that we would only have children's artwork in the main corridors was easily made.
Almost all of the children's art displayed in the hospital is on loan. It comes from either the Sydney Children's Hospital School (most children go to school even when they are sick), local schools (both public and private), as well as from children's art schools and holiday camps. At any one time the hospital may have 200 artworks by children up on the walls, both in the corridors and in the wards.
The participating schools are all very supportive of the Arts for Health Program and the students enjoy coming into the Sydney Children's Hospital to see their artworks up on the walls. Many of the children, or their siblings, have been patients here and it helps them to regard the hospital in a different light. It makes the building more user-friendly and less clinical, providing a welcome distraction both for patients and visitors.
The ages of the child artists ranges five to 15 (kindergarten/reception to Year 10). Works range from the very naive to the very sophisticated and I am always amazed at the talent and the ideas that the children display. When a work is very good we ask the child to donate their art to the hospital and we have it framed and hung permanently in the wards or waiting areas. It is genuinely satisfying when a relative rings asking where the work is hanging and if they can come in and see it.
The works that we hang are oil, acrylic, watercolour, collage, pastel, pencil, print and photographic. They nearly always reflect an art topic being studied at school. Quite often we will have works from three different schools on show simultaneously which can be fascinating because each school will interpret and express the curriculum exercise in a different way. It is fun spotting the odd artistic influence in some of the works: Gauguin and Matisse seem to be all-time favourites; recently Brett Whiteley's palette and forms are pervasive.
Lucinda Bartrum, who had been an exchange teacher at St Aloysius College in Bandung, Indonesia, initiated our first touring exhibition. The Indonesian children had made drawings in oil pastels of games that they played at school. Pupils at Maroubra Primary did the same exercise, but in watercolour and collage. Maroubra Senior School students then taught the children in Primary School how to transfer these images to acetate film and print them onto paper. This exhibition toured Australia with a school in each state adding to the exhibition. At the conclusion of the tour the hospital was presented with 15 of the works (five Indonesian, five watercolours and five acetate prints) and they are now part of our permanent collection, hanging in the hospital's Executive Unit.
In 1998 the hospital organised an artist-in-residence program through the British Airways Creative Connection exchange program. A young British artist, Sarah Cole, was sponsored to work at the Museum of Contemporary Art in the Youth Programs Department. As part of the MCA's program Sarah worked in the Adolescent Ward where she and the children created a number of artworks which included two glass murals, x-ray prints (actually photocopies on film), a large painting (which now hangs in the play room) and a 'dream bed'. The 'dream bed' installation was wonderful. It was an old hospital bed that had been bandaged and 'grown' wings. The final touch was a curtain made up of photographic slides which the children had made themselves by drawing holiday scenes on acetate film, which hung around the bed. It really was a bed of dreams.
The 'art gallery' is a working hospital. Patients are wheeled to and from the operating theatre and wards on trolleys and the corridors are also escape routes in the case of an emergency evacuation. This means that art has to be on canvas or behind glass, and that three-dimensional work cannot be exhibited. Works are hung for two or three months and then returned to the school to be given back to the children. Once the parents of a child who died asked if they could have a particular painting, as it was their daughter's favourite and they wanted to have the memory of her enjoying the painting each time they looked at it.
The emphasis at Sydney Children's Hospital is on an environment that is child and family friendly, one that treats the whole person and not just the disease. By lending us their art the children are helping us to create a hospital that projects a caring and friendly image from the moment patients and their families walk through the front door. The extraordinary thing about the children's art is that everybody loves it. At weekends, when life on the wards slows down and school is not open, the artwork becomes especially important. The changing exhibitions provide entertainment for the parents and their child-patients, who often spend hours walking up and down the corridors looking at the art.
Exhibitions of children's artwork are becoming increasingly popular. The (other) Sydney Children's Hospital, at Westmead, has an annual competition and exhibition in conjunction with the Department of Education and the Penrith Panthers Leagues Club called Operation Art. From this competition the best works are hung in the Art Gallery of New South Wales and are on the hospital's and Art Gallery's websites. The art goes on to tour the regional galleries of New South Wales and is then installed permanently at the Children's Hospital, Westmead, where it complements the permanent 'fine art' collection of works by leading Australian artists.
There is also growing corporate support for children's art. Qantas, for example, held an Australia-wide art competition early in 2001, entitled 'Spirit of Australia Kids Competition'. This competition received thousands of poems, paintings, songs and videos and was judged by Elizabeth Ann Macgregor (Director, Museum of Contemporary Art, Sydney), Olympians Susie O'Neill and Cathy Freeman, and Dare Jennings of Mambo. The prize-winning entries are now displayed in QANTAS Club Lounges at all major airports around Australia and can be seen on the QANTAS website.
Children's art does not patronise the patients in the hospital in the way that animated pop imagery or adult art deemed suitable for children can. At Sydney Children's Hospital, Randwick, it is definitely more popular and elicits a greater response than non-child art from all who use the hospital - patients, their families, visitors and hospital staff.


My htaccess was not working after Ubuntu 14.04 upgrade. I realized the issue is with Apache configuration file, because I was allowed to override the Apache configuration while upgrade. So I compared the /etc/apache2/apache2.conf with auto backup file apache2.conf.dpkg-old. So we have to do the following replace in apache2.conf file to restore the htaccess.
Restart your Apache server and check!
Posted on April 23, 2014 by Gowri.
Big thankyou for this post. After searching for hours I found your post and it worked.


Excalibur's new Explore Take-down Utility soft case redefines protecting and transporting your Excalibur crossbow. Due to Excalibur's quick removable front end limb assembly, the Explore enables you to transport your Excalibur in a padded, compact and discrete case. The Explore eliminates the need for bulky, cumbersome, full sized cases allowing you to make better use of space. The Explore's main compartment is padded with two Velcro straps to securely hold your stock in place, while the limb assembly rests in an internal sleeve. Other storage compartments include, three external pockets, three internal pouches and a document sleeve. Constructed of heavy duty 1000D Cordura nylon and ultra strong zippers with rugged bungee style pull cords, two carry handles and shoulder strap to make this your perfect go anywhere, do anything utility case.


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Cuba qualifies for the first time to the Volleyball Nations League
The Cuban men's volleyball team qualified for the League of Nations for the first time, the elite tournament of this sport, by achieving the only place in dispute in the Challenger Cup with a 3-1 victory in the final of the tournament against the team from Turkey.
The Cuban team only lost one set during the entire event, precisely during the final, with a score of 25-17, 23-25, and identical scores of 25-20 in the third and fourth sets. The Caribbean squad was superior in all aspects of the game, except receiving.
In the final against the Turkish team, Cuba lined up with passer Adrián Goide, assistants Miguel Ángel López and Osniel Mergarejo, opposite Jesús Herrera, center backs Roamy Alonso and Javier Concepción and Yonder García as libero, they report official media.
Miguel Ángel López and the opposite Jesús Herrera assumed the offensive weight with 20 and 19 points, respectively, while central defender Roamy Alonso with 12 points and 11 Mergarejo (four aces), were the highest scorers for the winning squad.
WELCOME TO THE #VNL CUBA!
? Congratulations to Cuba on winning the 2022 #ChallengerCup! This victory books them a spot in the 2023 VNL!#volleyball fans can't wait to see the mighty Cubans to #BePartOfTheGame next year!
? Watch the full match replays on VBTV. pic.twitter.com/N49D7Kkmq9
– Volleyball World (@volleyballworld) July 31, 2022
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text:
Q: Why async.map returns multiple copies of array? const async = require('async');

const arr = [
    { name: 'john', id: '1' },
    { name: 'Andrie', id: '2' }]

let collectArr = [];
let data = async.mapLimit(arr, 5, async function (input) {

    collectArr.push({ name: input.name, id: input.id });
    return collectArr;
})

data.then((result) =>{
    console.log('success',result);
}).catch(e => console.log('err'));

So here i am providing array to async.mapLimit without callback and expecting promise here.
Expected Output :- [ { name: 'john', id: '1' }, { name: 'Andrie', id: '2' } ] ,
Got Result :-
[ [ { name: 'john', id: '1' }, { name: 'Andrie', id: '2' } ],
[ { name: 'john', id: '1' }, { name: 'Andrie', id: '2' } ] ]
So my question is why it is creating multiple copies of array, how to deal with this?

A: You are needlessly returning a sub array, and the same array reference each iteration, when all you want is to return the new object.
let data = async.mapLimit(arr, 5, async function (input) {
    return { name: input.name, id: input.id };
});

Not sure why you need this to be async


text:
Karen Harding
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