text:
\section{Introduction and generalities}
The massless Dirac equation with an electric Coulomb potential reads
\begin{equation}\label{diraccoul}
\begin{cases}
\displaystyle
 iu_t+\mathcal{D}_nu-\frac{\nu}{|x|} u=0,\quad u(t,x):\mathbb{R}_t\times\mathbb{R}_x^n\rightarrow\mathbb{C}^{N}\\
u(0,x)=f(x)
\end{cases}
\end{equation}
where  the massless Dirac operator $\mathcal{D}_n$ is defined in terms of the Pauli matrices
\begin{equation}
\sigma_1=\left(\begin{array}{cc}0 & 1 \\1 & 0\end{array}\right),\quad
\sigma_2=\left(\begin{array}{cc}0 &-i \\i & 0\end{array}\right),\quad
\sigma_3=\left(\begin{array}{cc}1 & 0\\0 & -1\end{array}\right)
\end{equation}
as
$$
\mathcal{D}_2=-i(\sigma_1\partial_{x}+\sigma_2\partial_{y})=
\left(\begin{array}{cc}0 & -i\partial_z \\-i\partial_{\overline{z}} & 0\end{array}\right)
$$
(we denote $\partial_z=\partial_x-i\partial_y$ and $\partial_{\overline{z}}=\partial_x+i\partial_y$) in dimension $n=2$ with $N=2$, and
$$
\mathcal{D}_3=-i\displaystyle\sum_{k=1}^3\alpha_k\partial_k=-i(\alpha\cdot\nabla)
$$
where the $4\times 4$ Dirac matrices are given by
\begin{equation}
\alpha_k=\left(\begin{array}{cc}0 & \sigma_k \\\sigma_k & 0\end{array}\right),\quad k=1,2,3
\end{equation}
in dimension $n=3$, with $N=4$.

The Dirac equation is widely used in physics to describe relativistic particles of spin $1/2$ (see e.g. \cite{landlif2}, \cite{thaller}). In particular, the $2D$ massless equation is used as a model for charge carriers in graphene, a layer of carbon atoms arranged in a honeycomb lattice (see e.g. the physics survey \cite{neto} and the mathematical papers \cite{fef}, \cite{ha-le-sp}, \cite{ma-sie}). Note that there have been many rigorous results in the recent years on the stationary (massive) Dirac equation, essentially in dimension 3 (see e.g. \cite{lewser} for references). Comparatively, its dynamics has been less investigated up to now.

The $\sigma_j$ and $\alpha_j$ matrices were introduced in view of making the Dirac operator a square root of the Laplace operator: therefore, they satisfy by construction the following anticommutating relations
\begin{equation*}
\sigma_j\sigma_k+\sigma_k\sigma_j=2\delta_{ik}\mathbb{I}_2,\quad j,k=1,2;
\end{equation*}
\begin{equation*}
\alpha_j\alpha_k+\alpha_k\alpha_j=2\delta_{ik}\mathbb{I}_4,\quad j,k=1,2,3,
\end{equation*}
(we recall that the spectrum of $\mathcal{D}_n$ is the whole line $\mathbb{R}$). These conditions ensure that
$$
(i\partial_t-\mathcal{D}_n)(i\partial_t+\mathcal{D}_n)=(\Delta-\partial^2_{tt})I_{N},
$$
which strictly links the dynamics of the \emph{free} massless Dirac equation to a system of $N$ decoupled wave equations. Therefore, dynamical properties for the free Dirac equation can directly be derived from their wave counterparts: dispersive properties of the flow can be mainly encoded in the celebrated family of \emph{Strichartz estimates}, which are given by
\begin{equation}\label{freestr}
\| e^{it\mathcal{D}_n}f\|_{L^p_t\dot H^{\frac{1}{q}-\frac{1}{p}-\frac{1}{2}}_x}\lesssim
\| f\|_{L^2}
\end{equation}
where the exponents $(p,q)$ are wave admissible, i.e. satisfy
$$
\frac{2}{p}+\frac{d-1}{q}=\frac{d-1}2,\quad 2\leq p\leq\infty,\quad\frac{2(d-1)}{d-3}\geq q\geq 2
$$
and $e^{it\mathcal{D}_n}$ represents the propagator for the solutions to \eqref{diraccoul} with $V=0$ for $n=2,3$.
We stress the fact that the estimate corresponding to the couple $(p,q)=(2,\infty)$ in dimension $3$, the so called \emph{endpoint}, that is
\begin{equation}\label{end}
\| e^{it\mathcal{D}_3}f\|_{L^2_tL^\infty_x}\lesssim\| f\|_{\dot{H}^1}
\end{equation}
is known to fail as the corresponding one for the 3D wave flow. In \cite{machi} a refined version of this estimate involving angular spaces was given. As a consequence, the well-posedness of the celebrated cubic nonlinear Dirac equation was established, assuming initial data in $\dot{H}^1$ with slight additional regularity on the angular variable. The problem was afterwards completely solved in \cite{bejher}. See also \cite{bejher2} for the $2D$ case.

Another family of {\it a priori} estimates encoding informations about dispersion is given by the so called \emph{local-smoothing type estimates}, which turn to be particularly useful to handle potential-type perturbations. Different versions of these estimates are available: to the best of our knowledge the most general one in this setting is
\begin{equation}\label{freemor}
\|w_\sigma^{-1/2}e^{it\mathcal{D}_3}f\|_{L^2_tL^2_x}\leq C\|f\|_{L^2}
\end{equation}
where $w_\sigma(x)=|x|(1+|\log|x||)^\sigma$, $\sigma>1$. Both \eqref{freestr} and \eqref{freemor} can be found in \cite{danfan} (see also \cite{esc-ve}, \cite{bouss}). We are not aware of any corresponding result for the $2D$ case; nevertheless, the strategy developed in \cite{danfan} can be suitably adapted to this context to obtain similar results.

In recent years a lot of effort has been spent in order to investigate what happens to dispersive flows when potential perturbations come into play. In \cite{danfan} both \eqref{freestr} and \eqref{freemor} were proved for the flow $e^{it\mathcal({D}_3+V)}$ under the assumption
$$
|V(x)|\leq\frac{\delta}{w_\sigma(x)},\qquad \sigma>1,\qquad n=3,
$$
for some $\delta$ sufficiently small, which by the way is a sufficient condition to guarantee that the operator $\mathcal{D}_3+V$ is selfadjoint (see \cite{thaller}). Similar results are available for magnetic potentials as well (see e.g. \cite{boussdanfan}, \cite{caccia3}). The angular endpoint estimates proved in \cite{machi} have been generalized in \cite{cac1}, \cite{cac2} to small potential perturbations, by introducing some new mixed Strichartz-smoothing estimates.  At the same time, some effort has been spent in order to find examples of potentials such that the correspoding flows do \emph{not} disperse (in the sense discussed above). This problem has been tackled for the magnetic Dirac equation in \cite{arrizfan} in $3D$, essentially showing that for vector potentials of the form $A(x)\cong |x|^{-\delta} Mx$ with $\delta\in (1,2)$ most of the mass of the solution is localized around a non-dispersive function.
These arguments suggest that, heuristically, the degree of homogeneity of the operator works as a threshold for the validity of dispersive estimates: therefore, for the Dirac equation, the Coulomb potential is to be thought of as a \emph{critical} case (we stress the fact that the restriction to the massless case seems unavoidable now). In the very last years the problem of understanding dispersive estimates for scaling invariant potentials has been approached for other dynamics: in \cite{burq1} and \cite{burq2} the authors have proved indeed that Morawetz and Strichartz estimates can be recovered for both the Schr\"odinger and wave equations with inverse square potentials, and in \cite{fan} the $L^1\rightarrow L^\infty$ time decay estimate is proved in the electromagnetic case.

The aim of this paper is to adapt the techniques of \cite{burq1} to the massless Dirac Coulomb equation, and thus to prove local smoothing estimates for solutions to \eqref{diraccoul}. The strategy of proof of \cite{burq1} is fairly straightforward: the basic idea is to use a spherical harmonics decomposition to reduce the equation to a radial one and then rely on the Hankel transform to diagonalize it. Several problems arise when trying to adapt these ideas to the Dirac setting. The first difficulty is given by the fact that the Dirac operator does not preserve radiality, and therefore one needs a more complex, but still classical, spherical harmonics decomposition related to the $SU_2$ group action (see \cite{thaller}, section 4.6). Another issue is that we cannot use the Hankel transform, and we have to build up a relativistic analogue. To do this, we need to study the continuous spectrum of the Dirac-Coulomb model, both in $2$ and $3$ dimensions. The corresponding generalized eigenfunctions are well known (see for instance \cite{landlif2} and \cite{nov}), but their rich structure will raise some additional technical difficulties. We stress the fact that, to the best of our knowledge, this is the first result concerning dispersive properties of Dirac-Coulomb, an operator which plays a fundamental role in relativistic quantum chemistry. We hope that our result will be useful in the study of the dynamics of nonlinear models involving, for instance, several relativistic electrons in a molecule.

Before stating our main result, we introduce some notations that will be used throughout the paper.
\\\\
\textbf{Notations.}

We shall use the standard notation $\dot H^s$ for the homogeneous Sobolev space with the norm $\| f\|_{\dot H^s}=\| |D|^sf\|_{L^2}$ where $|D|=(-\Delta)^{1/2}$, and $L^p_tL^q_x=L^p(\mathbb{R}_t; L^q(\mathbb{R}^n_x))$ for the mixed space-time Strichartz spaces.
We denote with $\Omega^s$ the operator
\begin{equation*}
(\Omega^s\phi)(x)=|x|^s\phi(x)
\end{equation*}
and with a little abuse of notation we will use the same symbol to indicate the operators which are pointwise equal for all times,
\begin{equation*}
(\Omega^s\psi)(t,x)=|x|^s\psi(t,x).
\end{equation*}
We introduce the following sets of indices
\begin{equation}\label{B2}
B_2(k):=\{k:k=1/2+m,m\in\mathbb{Z}\}
\end{equation}
\begin{equation}\label{B3}
B_3(k):=\{k:k\in \mathbb{Z}\backslash\{0\}\}.
\end{equation}
As most of the forthcoming objects will be significantly different in $2D$ or $3D$, we will often use the apex $n$ to distinguish the dimensions; this will not have to be confused with standard powers, but interpretation will be clear from time to time.
For the definition of the angular spaces $\mathcal{H}_{k}^n$, $n=2,3$ we refer to forthcoming Proposition \ref{thaller} and subsequent Remark \ref{rknot}.
\\\\

We are now ready to state our main Theorem.

\begin{theorem}\label{thmor}
Let $n=2,3$, $\nu_n\in\left(-\frac{n-1}2,\frac{n-1}2\right)$, and let $k_n\in B_n(k)$. Let $u$ be a solution of \eqref{diraccoul}. Then, for any
$$1/2<\alpha<\sqrt{k_n^2-\nu_n^2}+1/2$$
 and any $f\in L^2((0,\infty))\otimes \mathcal{H}_{\geq {k_n}}^n$ there exists a constant $C=C(\nu,\alpha,k)$ such that the following estimate holds
\begin{equation}\label{mor}
\displaystyle
\left\|\Omega^{-\alpha}\left|\mathcal{D}_n-\frac{\nu_n}{|x|}\right|^{1/2-\alpha} u\right\|_{L^2_tL^2_x}\leq C \|f\|_{L^2}.
\end{equation}
\end{theorem}

\begin{remark}
The dependence on the dimension of $\nu$, which will be neglected in the rest of the paper, has been added in the statement of the Theorem to allow us to remark the different ranges of admissible charges in dimensions $2$ and $3$. These ranges are in fact imposed by the condition on the Dirac Coulomb operator to be self-adjoint, which is indeed guaranteed in those intervals (see Remark \ref{rkself}). We stress also the fact that for $\nu$ approaching the endpoints, respectively $\pm1$ in $3D$ and $\pm1/2$ in $2D$, the range of admissible exponents for the lowest value of $k_n$ in estimate \eqref{mor} shrinks to zero.
\end{remark}

\begin{remark}
It seems to be possible to generalize our proof of Theorem \ref{thmor} to any space dimension $n\geq 2$. As it will be clear, the key ingredients are given indeed by the construction of a suitable integral operator which transforms the problem into an ODE, and then by the careful analysis of some interaction integrals between generalized eigenstates of the Dirac-Coulomb equation that naturally appear. Therefore, the crucial step consists in finding the generalized eigenstates of the Dirac-Coulomb equation: this has been performed in \cite{gu-ma} in higher space dimension $n\geq4$ by generalizing the spherical wave decomposition (see Proposition \ref{thaller}), i.e. by carefully analyzing the symmetries of the $SO(n)$ group. Anyway, we prefer not to deal with this problem here both because technicalities would lead us too far and because of the scarce physical interest of it.
\end{remark}

\begin{remark}
The endpoint case $\alpha=1/2$ in  \eqref{mor} would permit to recover with a standard argument the full set of Strichartz estimates for the perturbed flow (see e.g. \cite{burq1}); even though we don't have a concrete counterexample, this estimate seems to fail. Forthcoming remark \ref{fail} gives more details on the issue.
\end{remark}

\begin{remark}\label{rkself}
In order to have a unitary flow for equation \eqref{diraccoul} one needs the operator $\mathcal{D}_n-\frac{\nu}{|x|}$ to be selfadjoint; many papers have been devoted to the study of this property  (see e.g. \cite{Schmincke-72}, \cite{Wust-73}, \cite{nenciu}, \cite{Klaus-Wust-78}, \cite{estlos}, \cite{estloss2}, \cite{arrveg}) in the $3D$ setting (note that the methods introduced in these papers are quite general and can be adapted to other dimensions). In $3D$ the essential self-adjointness is in fact guaranteed if $|\nu|\leq\frac{\sqrt{3}}2$, while in the interval $\frac{\sqrt3}2<|\nu| \leq 1$ it is still possible to build a distinguished selfadjoint extension. In the $2D$ case, as soon as $\nu$ is nonzero, the Dirac-Coulomb operator $\mathcal{D}_2-\frac{\nu}{|x|}$ defined on $C^\infty_0(\mathbb{R}^2)$ is not essentially selfadjoint (this is an immediate consequence of Theorem 4.16 in \cite{thaller}). But C. Warmt (\cite{wa}, Satz 2.2.6) recently proved the existence of a distinguished self-adjoint extension if $|\nu| < 1/2$. Therefore, in the range of $\nu_n$ for which we prove our Theorem \ref{thmor}, the operator $\mathcal{D}_n-\frac{\nu}{|x|}$ is self-adjoint (in the sense just discussed).
\end{remark}

The plan of the paper is the following: in section \ref{partial} we build the setup, reviewing the theory of partial wave subspaces and the spectrum of the Dirac-Coulomb operator in $2D$ and $3$D; then we define our analogue of the Hankel transform used in \cite{burq1}.
Section \ref{proof} is devoted to the proof of Theorem \ref{thmor}.
A short appendix containing some generalities on special functions  is included at the end for the reader's convenience.
\\\\
\textbf{Acknowledgments.} The first named author is supported by the FIRB 2012 "Dispersive dynamics, Fourier analysis and variational methods" funded by MIUR (Italy). The second named author acknowledges support from the ANR ``NoNAP'' of the french ministry of research.

\section{The setup.}\label{partial}

We devote this section to recall some classical facts about the radial Dirac operator and his spectrum, including a short discussion on how to explicitly calculate the generalized eigenstates of the continuous spectrum. For the sake of brevity we will omit several details, referring to \cite{landlif2} and \cite{thaller} for a complete discussion and for more informations on the topic. Also, we construct an integral transform that will be crucial in our proof.

\subsection{Partial wave decomposition and continuous spectrum}
The main ingredient we need to introduce is the so called \emph{partial wave decomposition}. We collect in Proposition \ref{thaller} the tools we will need in what follows that is, essentially, the fact that the Dirac-Coulomb operator can be seen as a radial operator with respect to some suitable decomposition, both  in dimensions $2$ and $3$. We will denote for brevity with $\mathcal{A}_2$ and $\mathcal{A}_3$ the following families of indices
\begin{equation}\label{j2}
\mathcal{A}_2:=\left\{k\in\mathbb{Z}+\frac12\right\};
\end{equation}
\begin{equation}\label{j3}
\mathcal{A}_3:=\left\{j=\frac12,\frac32,\dots, \: m_{j}=-j,-j+1,\dots,+j, \: k_{j}=\pm(j+1/2)\right\}
\end{equation}
Moreover, we define the $4$-vectors
\begin{equation}\label{Xi}
\Xi^+_{m_j,\mp(j+1/2)}=
\left(\begin{array}{cc}i \Omega^{m_j}_{j\mp1/2}\\
0\end{array}\right),\qquad
\Xi^-_{m_j,\mp(j+1/2)}=
\left(\begin{array}{cc}0\\ \Omega^{m_j}_{j\pm1/2},
\end{array}\right)
\end{equation}
where
\begin{equation*}
\Omega^{m_j}_{j-1/2}=\displaystyle\frac{1}{\sqrt{2j}}
\left(\begin{array}{cc}\sqrt{j+m_j}\:Y^{m_j-1/2}_{j-1/2}\\
\sqrt{j-m_j}\:Y^{m_j+1/2}_{j-1/2}
\end{array}\right)
\end{equation*}
\begin{equation*}
\Omega^{m_j}_{j+1/2}=\displaystyle\frac{1}{\sqrt{2j+2}}
\left(\begin{array}{cc}
\sqrt{j+1-m_j}\:Y_{j+1/2}^{m_j-1/2}\\
-\sqrt{j+1-m_j}\:Y_{j+1/2}^{m_j+1/2}
\end{array}\right)
\end{equation*}
and $Y^m_l$ are the usual spherical harmonics.

\begin{proposition}\label{thaller}
Let $n=2,3$. Then, we can define unitary isomorphisms between the Hilbert spaces
$$L^2(\mathbb{R}^2)^{2} \cong L^2((0,\infty),dr)\bigotimes_{k\in\mathcal{A}_2} h^2_k,$$
$$L^2(\mathbb{R}^3)^{4} \cong L^2((0,\infty),dr)\bigotimes_{{j,m_j,k_j}\in\mathcal{A}_3} h^3_{j,m_j,k_j}$$
by the following decompositions
\begin{equation}\label{iso2}
\Phi_2(x)=\sum_{k\in\mathcal{A}_2}\frac{1}{2\sqrt{\pi}}\left(\begin{array}{cc} F_{k}(r)e^{i(k-1/2)\theta}\\
 G_{k}(r)
e^{i(k+1/2)\theta}
\end{array}\right),
\end{equation}
\begin{equation}\label{iso3}
\Phi_3(x)=\sum_{j,m_j,k_j\in\mathcal{A}_3}\frac{1}{2\sqrt{\pi}}\left(\begin{array}{cc} F_{j,m_j,k_j}(r)\Xi^+_{j,m_j,k_j}(\theta,\phi)\\
 G_{j,m_j,k_j}(r)
\Xi^-_{j,m_j,k_j}(\theta,\phi)
\end{array}\right)
\end{equation}
which hold for any $\Phi_n\in L^2(\mathbb{R}^n,\mathbb{C}^N)$.
 Moreover, the Dirac-Coulomb operator leaves invariant the partial wave subspaces $C^\infty_0((0,\infty))\otimes h^n_{k}$ and, with respect to the basis $\left\{e^{i(k-1/2)\theta},e^{i(k+1/2)\theta}\right\}$ and $\left\{\Xi^+_{j,m_j,k_j},\Xi^-_{j,m_j,k_j}\right\}$ is respectively represented by the radial matrices
\begin{equation}\label{raddir}
\displaystyle
d^2_{k}=\left(\begin{array}{cc}-\frac{\nu}{r}& -\frac{d}{dr}+\frac{k}{r}
\\  \frac{d}{dr}+\frac{k}{r} & -\frac{\nu}{r}\end{array}\right),\qquad
d^3_{j,m_j,k_j}=\left(\begin{array}{cc}-\frac{\nu}{r}& -\frac{d}{dr}+\frac{1+k_j}{r}
\\  \frac{d}{dr}-\frac{1-k_j}{r} & -\frac{\nu}{r}\end{array}\right).
\end{equation}

The Dirac-Coulomb opeator $\mathcal{D}_n-\frac{\nu}{r}$ on $C^\infty_0(\mathbb{R}^n)^N$ is unitary equivalent to the direct sum of "partial wave Dirac operators" $d_{j,m_j,k_j}$,
\begin{equation*}
\mathcal{D}_2-\frac{\nu}{r}\cong \bigoplus_{k}d_{k}^2,\qquad
\mathcal{D}_3-\frac{\nu}{r}\cong \bigoplus_{j,m_k,k_k}d^3_{j,m_k,k_k}
\end{equation*}
For fixed $\overline{k}_2\in B(k_2)$ and $\overline{k}_3\in B(k_3)$ we denote with
\begin{equation}\label{HH}
\mathcal{H}_{\geq \overline{k}_2}^2=\bigoplus_{k:|k|\geq|\overline{k}_2|}h_{k},\qquad
\mathcal{H}_{\geq \overline{k}_3}^3=\bigoplus_{j:j\leq|k_3|-1/2,m_{j},k_{j}:}h_{j,m_{j},k_{j}}.
\end{equation}
\end{proposition}
\begin{proof}
See \cite{thaller}.
\end{proof}

\begin{remark}\label{rknot}
Notations in Proposition \ref{thaller} are very heavy; this is partly due to the richness of the Dirac operator itself and partly  to the fact that the spherical coordinates decomposition changes in dimensions $2$ and $3$. We should stress the fact that formulas \eqref{raddir} only depend on $k$: therefore, in what follows we will abuse notations neglecting the dependence on the other parameters in the case $n=3$ in order to have a unified presentation.

\end{remark}

We now show how the functions of the continuous spectrum for the Dirac-Coulomb operator are obtained. We want to deal with the eigenvalue equation
\begin{equation}\label{eigenpro}
\left(\mathcal{D}_n-\frac{\nu}{|x|}\right)\Psi^n=\varepsilon\Psi^n,\qquad \varepsilon>0,\quad n=2,3.
\end{equation}
The application of Proposition \ref{thaller} suggests to seek for solutions in the form, for a fixed value of $\varepsilon$,
\begin{equation}\label{eigpsi2}
\Psi_{k,\varepsilon}^2(r,\hat{x})=
\left(\begin{array}{cc}F^2_{k,\varepsilon}(r)e^{i(k-1/2)\theta}\\\:G^2_{k,\varepsilon}(r)e^{i(k+1/2)\theta}
\end{array}\right),\qquad n=2
\end{equation}
\begin{equation}\label{eigpsi3}
\Psi_{k,\varepsilon}^3(r,\hat{x})=
\left(\begin{array}{cc}F^3_{k,\varepsilon}(r)\Xi^+_{k}(\hat{x})\\\:G^3_{k,\varepsilon}(r)\Xi^-_{k}(\hat{x})
\end{array}\right),\qquad n=3
\end{equation}
so that equation \eqref{eigenpro} will in fact reduce to a sytem of ODE for the radial components $F^n_k(r)$, $G^n_k(r)$ which is the same in both dimensions, apart for some constants, and can be expliticly solved (for the details on these calculations we refer to \cite{landlif2} section IV for the $3D$ and to \cite{nov} for the $2D$ case). We thus have the following formulas, for fixed values of $k$ and $\varepsilon>0$, and $n=2,3$,
\begin{equation}\label{eigen1}
F^n_{k,\varepsilon}=\frac{\sqrt{2}|\Gamma(\gamma+1+i\nu)|}{\Gamma(2\gamma+1)}e^{\pi\nu/2}(2\varepsilon r)^{\gamma-\frac{(n-1)}2}
 {\rm Re}
\left\{e^{i(\varepsilon r+\xi)}
{_1F_1}(\gamma-i\nu,2\gamma+1,-2i\varepsilon r)\right\}
\end{equation}
\begin{equation}\label{eigen2}
G^n_{k,\varepsilon}=\frac{i\sqrt{2}|\Gamma(\gamma+1+i\nu)|}{\Gamma(2\gamma+1)}e^{\pi\nu/2}(2\varepsilon r)^{\gamma-\frac{(n-1)}2}
 {\rm Im}
\left\{e^{i(\varepsilon r+\xi)}
{_1F_1}(\gamma-i\nu,2\gamma+1,-2i\varepsilon r)\right\}
\end{equation}
where $_1F_1(a,b,z)$ are \emph{confluent hypergeometric functions} (see Appendix for the definition and some properties), $\gamma=\sqrt{k^2-\nu^2}$  and $e^{-2i\xi}=\frac{\gamma-i\nu}{k}$ is a phase shift. We will denote with
\begin{equation}\label{radeigpsi}
\psi_{k,\varepsilon}^n(r)=
\left(\begin{array}{cc}F^n_{k,\varepsilon}(r)\\\:G^n_{k,\varepsilon}(r)
\end{array}\right),\qquad n=2,3
\end{equation}
the vector of radial coordinates of the generalized eigenfunctions.

Notice that with this choice the following normalization condition holds for \eqref{eigpsi2}-\eqref{eigpsi3}
\begin{equation}\label{norm}
\int_0^{+\infty}\overline{\psi^n}_\epsilon(r)\psi^n_{\epsilon'}(r)r^{n-1}dr=\frac{\delta(\epsilon-\epsilon')}{\epsilon^{n-1}},\qquad n=2,3
\end{equation}
meaning with that that the function
$$
K(r,r')=r^{n-1}\int e^{-l\varepsilon}\overline{\psi^n}_\epsilon(r)\psi^n_{\epsilon'}(r)\varepsilon^{n-1}d\varepsilon
$$
is a $\delta$-approximation for $l\rightarrow0$.

\begin{remark}
We point out that, due to our choice of normalization \eqref{norm}, the homogeneity of the functions $\psi^n(r)$ with respect to $\varepsilon$ and $r$ is the same; this fact, that is not true anymore in the massive case, will be crucial in what follows, and therefore prevents the application of our strategy to the massive case.
\end{remark}

\begin{remark}
The sign of the Coulomb potential plays in fact a virtually unessential role in what is above, and the wave functions for repulsive fields can be obtained simply by changing the sign of $\nu$ in \eqref{eigen1}-\eqref{eigen2}.
\end{remark}

\begin{remark}\label{negative}
Formulas \eqref{eigen1}-\eqref{eigen2} give the "eigenstates" of the continuous spectrum corresponding to positive energies $\varepsilon>0$. Using a charge conjugation argument (or directly working on the explicit radial equations), it is possible to obtain the corresponding formulas for the radial components \eqref{eigpsi2} for negative energies: it turns indeed out that the function $\psi_{-k,\varepsilon}(r)$ solves
\begin{equation}\label{negativeig}
\tilde{d}_k\psi_{-k,\varepsilon}(r)=-\varepsilon \psi_{-k,\varepsilon},\qquad n=2,3,
\end{equation}
where the operator $\tilde{d}_k$ is the one defined in \eqref{raddir} with opposite sign of the charge $\nu$. This shows that the "eigenstates" corresponding to negative energies for equation \eqref{eigenpro} can be understood by mean of \eqref{negativeig} to be "eigenstates" corresponding to positive energies, with opposite value of $j$ and opposite charge. Therefore, denoting with $\tilde{F}$, $\tilde{G}$ the functions obtained by \eqref{eigen1}-\eqref{eigen2} by changing the sign of $\nu$, we obtain for the radial coordinates of the "eigenstates" corresponding to negative energies as in \eqref{radeigpsi} the following relation
$$
\psi^n_{k,-\varepsilon}(r)=
\left(\begin{array}{cc}F^n_{k,-\epsilon}(r)\\\:G^n_{k,-\epsilon}(r)\end{array}\right)=
\left(\begin{array}{cc}\tilde{F}^n_{-k,\epsilon}(r)\\\:\tilde{G}^n_{-k,\epsilon}(r)
\end{array}\right).
$$

\end{remark}

\subsection{The integral transform}
We now introduce the crucial integral transform that will be used in the proof of the main result, consisting in a projection on the continuous spectrum of the Dirac-Coulomb operator. Throughout this subsection we will omit the dependance on the dimensions of the functions, as the calculations carried out will be the same.
\begin{definition}\label{hanktras}
Let $\Phi\in L^2((0,\infty),dr)\otimes h_{k}$ for some fixed $k$ and let $\varphi(r)=(\varphi_1(r),\varphi_2(r))$ be the vector of its radial coordinates in decomposition \eqref{iso2}-\eqref{iso3}.  We define the following integral transform
\begin{equation}\label{H}
\mathcal{P}_k\varphi(\epsilon)=
\left(\begin{array}{cc}\mathcal{P}^+_k\varphi(\epsilon)\\
\mathcal{P}^-_k\varphi_k(\epsilon)
\end{array}\right)=
\left(\begin{array}{cc}\int_0^{+\infty}\psi_{k,\varepsilon}(r)\varphi(r)r^{n-1}dr\\
\mathcal{C}\left(\int_0^{+\infty}\psi_{k,-\varepsilon}(r)\varphi(r)r^{n-1}dr\right)
\end{array}\right)
\end{equation}
\begin{equation*}
=\int_0^{+\infty}H_{k}(\varepsilon r)\cdot\varphi(r)r^{n-1}dr
\end{equation*}
where we have introduced the matrix (keep in mind Remark \ref{negative})
\begin{equation}\label{matra}
H_{k}=\left(\begin{array}{cc}F_{k,\varepsilon}(r)\;G_{k,\varepsilon}(r)\\
F_{k,-\varepsilon}(r)\;G_{k,-\varepsilon}(r)
\end{array}\right)
\end{equation}
\end{definition}

We collect in the following proposition some important properties of the operator $\mathcal{P}_k$.
\begin{proposition}\label{properties}
For $n=2,3$ and any $\Phi\in L^2((0,\infty),dr)\otimes h_{k}$ the following properties hold:
\begin{enumerate}
\item
$\mathcal{P}_k$ is an $L^2$-isometry.
\item
$
\mathcal{P}_kd_k=\sigma_3\Omega\mathcal{P}_k.
$
\item
The inverse transform of $\mathcal{P}_k$ is given by
\begin{equation}\label{H-1}
\mathcal{P}_k^{-1}\varphi(r)=\int_0^{+\infty}H_{k}^{*}(\epsilon r)\cdot\varphi(\epsilon)\epsilon^{n-1}d\epsilon
\end{equation}
where $H_{k}^{*}=\left(\begin{array}{cc}F_{k,\varepsilon}(r)\;G_{k,-\varepsilon}(r)\\
G_{k,\varepsilon}(r)\;F_{k,-\varepsilon}(r)
\end{array}\right)
$.
\item
For every $\sigma\in\mathbb{R}$ we can define the fractional operators
\begin{equation}\label{fractiondef}
A_k^\sigma\varphi_k(r)=\mathcal{P}_k\sigma_3\Omega^\sigma\mathcal{P}_k^{-1}\varphi_k(r)=\int_0^{+\infty}S_k^\sigma(r,s)\cdot \varphi_k(s)s^{n-1}ds.
\end{equation}
where the integral kernel $S_k(r,s)$ is the $2\times2$ matrix given by
\begin{equation}\label{mattrix}
S_k(r,s)=\int_0^{+\infty}H_{k}(\epsilon r)\cdot H_{k}^{*}(\epsilon s)\epsilon^{n-1+\alpha}d\epsilon
\end{equation}
\end{enumerate}
\end{proposition}

\begin{remark}
Property $(1)$ allows, by standard arguments, to extend the definition of the operator $\mathcal{P}_k$ to functions in $L^2$.
\end{remark}

\begin{remark}\label{fractpow}
When summing on $k$, property \eqref{fractiondef} defines in a standard way fractional powers of the operator $|\mathcal{D}_n-\frac{\nu}{|x|}|$, which are used in the statement of Theorem \eqref{thmor}.
\end{remark}

\begin{proof}Property $(1)$ and $(2)$ come from the definition of $\mathcal{P}_k$, once noticed that
$$
\mathcal{P}^\pm_k(d_k\varphi)=\langle \psi_{k,\pm\epsilon},d_k\varphi\rangle=\langle d_k \psi_{k,\pm\epsilon},\varphi\rangle=\pm\epsilon\langle  \psi_{k,\pm\epsilon},\varphi\rangle
$$
and from normalization relation \eqref{norm}.
Property $(3)$ is a consequence of $(1)$ ($H^*_{k} $ is the adjoint of $H_{k}$).

To prove property $(4)$ we use the definition of $A_k$ to write
\begin{eqnarray*}
A_k^\sigma\varphi_k(r)&=&\mathcal{P}_k\Omega^\sigma\mathcal{P}_k^{-1}\varphi_k(r)
\\
&=&
\int_0^{+\infty}H_{k}(\epsilon r)\epsilon^{n-1+\sigma}\left(\int_0^{+\infty}H^*_{k}(\epsilon s)\varphi_k(s)s^{n-1}ds\right)d\epsilon.
\end{eqnarray*}
Exchanging the order of the integrals yields $(4)$.
\end{proof}

\section{Proof of Theorem \ref{thmor}}\label{proof}

We here prove our main result. Following \cite{burq1}, we rely on partial wave decomposition and on Proposition \ref{properties} to explicitly write the solutions on the single subspaces; then, a detailed and careful analysis of the kernel $S_k(r,s)$ of the operator $A_k$ defined in \eqref{fractiondef} will be performed to sum back.

\subsection{The strategy}
We start with decomposition \eqref{iso2}-\eqref{iso3} and work on a single spherical space which will be identified by $k\in\mathcal{A}_2$ as in \eqref{j2} and by the triple $\{j,m_k,k_j\}\in\mathcal{A}_3$ as in \eqref{j3} in the case $n=3$. With a slight abuse of notations, we will again neglect the dependence on everything but $k$ in the  $3D$ case (as noticed in Remark \ref{rknot}, the radial Dirac-Coulomb operator in dimension $3$ only depends on $k=k_j$, see \eqref{raddir}), as well as the dependence on the dimension for the various objects, in order to perform much readable and unified calculations. We thus rely on Proposition \ref{thaller} aiming to prove estimate \eqref{mor} on a fixed partial wave subspace with a constant $C$ bounded with respect to $k$, and eventually sum back. We thus take an initial condition $f$ with angular part in $h_k$ for some fixed $k$, and denote with $L_k f$ the solution to the initial value problem
\begin{equation}\label{diraccoulmod}
\begin{cases}
\displaystyle
 iu_t+d_ku=0,\\
u(0,x)=f(x)
\end{cases}
\end{equation}
with $d_k$ as in \eqref{raddir}.
We apply operator $\mathcal{P}_k$, which is an $L^2$-isometry, to the LHS side of estimate \eqref{mor} and use Proposition \ref{properties} to obtain (the application of the matrix $\sigma_3$ does not alter the $L^2$ norm)
\begin{equation*}
\|\mathcal{P}_k\Omega^{-\alpha}|d_k|^{1/2-\alpha}L_k f\|_{L^2_tL^2_x}=\|A_k^{-\alpha}\Omega^{1/2-\alpha}\mathcal{P}_kL_kf\|_{L^2_tL^2_x}
\end{equation*}
where we have used \eqref{fractiondef}. Now, $\mathcal{P}_kL_kf$ solves (see $(2)$ of Proposition \ref{thaller})
\begin{equation}\label{diraccoulbis}
\begin{cases}
\displaystyle
 i\partial_t\mathcal{P}_kL_kf+\sigma_3\Omega \mathcal{P}_kL_kf=0,\\
\mathcal{P}_kL_kf(0,\xi)=\mathcal{P}_kf(\xi),
\end{cases}
\end{equation}
so that the solution to this problem is explicitly given by
\begin{equation*}
\mathcal{P}_kL_kf(t,\xi)=e^{it\sigma_3\xi}\mathcal{P}_kf(\xi).
\end{equation*}
Fourier transforming in time (since $\mathcal{F}_{t\rightarrow\tau}$ is an $L^2$-isometry which commutes with $A_k^\sigma$) gives (with a minor abuse, we will neglect the $\sigma_3$ matrix from now on as we will only need to pick the $L^2$ norm and thus the sign will be unessential)
\begin{equation*}
(\mathcal{F}_t\mathcal{P}_kL_kf)(\tau,\xi)=(\mathcal{P}_kf)(\rho)\delta(\tau+\xi).
\end{equation*}

We can then write
\begin{eqnarray*}
(A_k^{-\alpha}\Omega^{1/2-\alpha}\mathcal{F}_t\mathcal{P}_kL_kf)(\tau,\xi)&=&
\int_0^{+\infty}S_k^{-\alpha}(\xi,s)\delta(\tau+s)\mathcal{P}_kf(s)s^{\frac{2n-2\alpha-1}2}ds
\\
&=&
-S_k^{-\alpha}(\xi,\tau)\mathcal{P}_kf(\tau)\tau^{\frac{2n-2\alpha-1}2}.
\end{eqnarray*}
In view of proving \eqref{mor} we now need to calculate the $L^2$ norm in time and space of the quantity above which then gives (notice that, as the angular part in decomposition \eqref{iso2}-\eqref{iso3} is $L^2$-unitary, we only need to consider the radial integrals)
\begin{equation}\label{1}
\int_0^{+\infty}\int_0^{+\infty}((\mathcal{P}_kf)^*(\tau)S_k^{-\alpha}(\rho,\tau)^T)\cdot(S_k^{-\alpha}(\rho,\tau)(\mathcal{P}_kf)(\tau))\tau^{2n-2\alpha-1}\rho^2 d\rho.
\end{equation}
Since $S_k^p(\rho,\tau)^T=S_k^p(\tau,\rho)$, the integral in $d\rho$ yields $S_k^{-2\alpha}(\tau,\tau)$ and we are therefore left with
\begin{equation}\label{2}
\int_0^{+\infty}(\mathcal{P}_kf)^*(\tau)S_k^{-2\alpha}(\tau,\tau)(\mathcal{P}_kf)(\tau)\tau^{2n-2\alpha-1} \:d\tau.
\end{equation}

To conclude the proof of the estimate on the $k$-th space we need to bound \eqref{2} with the $L^2$-norm of $\mathcal{P}_kf$ (which we recall to be the $L^2$ norm of $f$). Using forthcoming Proposition \ref{tecint} we thus have
\begin{equation}\label{laststep}
\int_0^{+\infty}(\mathcal{P}_kf)^*(\tau)S_k^{-2\alpha}(\tau,\tau)(\mathcal{P}_kf)(\tau)\tau^{2n-2\alpha-1} \:d\tau
\end{equation}
\begin{eqnarray*}
&\leq& \int_0^{+\infty}{\rm Tr}(S_k^{-2\alpha}(\tau,\tau))|(\mathcal{P}_kf)(\tau)|^2\tau^{2n-2\alpha-1} \:d\tau
\\
&\leq& C_k \int_0^{+\infty}|(\mathcal{P}_kf)(\tau)|^2(\tau)\tau^{n-1}\:d\tau
\\
\nonumber
&=&C_k\|f\|_{L^2}.
\end{eqnarray*}
with $C_k$ a bounded constant of $k$. Expanding $f$ in partial waves and using the triangle inequality together with the orthogonality of spherical harmonics concludes the proof.

\subsection{The main integral}

The crucial step consists thus in evaluating
\begin{eqnarray}\label{trace}
\nonumber
{\rm Tr}(S^{-2\alpha}(\tau,\tau))&=&{\rm Re}\left[\int_0^{+\infty}F_{k,\varepsilon}(\tau)^2+
G_{k,\varepsilon}(\tau)^2\epsilon^{n-1-2\alpha} d\epsilon\right]
\\
&+&{\rm Re}\left[\int_0^{+\infty}F_{k,-\varepsilon}(\tau)^2+G_{k,-\varepsilon}(\tau)^2\epsilon^{n-1-2\alpha} d\epsilon\right]
\\
\nonumber
&=&{\rm Re}\big[I_1(\tau)+I_2(\tau)\big].
\end{eqnarray}
More than giving an explicit solution of such integrals, which seems to be possible but significantly complicated, we are more interested in proving the following
\begin{proposition}\label{tecint}
Let $n=2,3$ $k_n\in B(k)$ as defined in \eqref{B2}-\eqref{B3} and $\nu_n$ as in Theorem \ref{thmor}. Then for every $$1/2<\alpha<\sqrt{k_n^2-\nu_n^2}+1/2$$
there exists a constant $C_{k_n}$ such that
$$
\rm{Tr}(S_k^{-2\alpha}(\tau,\tau))=C_{k_n} \tau^{-n+2\alpha}.
$$
Moreover,  $\displaystyle\sup_{k_n} |C_{k_n}|<\infty$
\end{proposition}

\begin{proof}
By recalling the structure of the generalized eigenstates \eqref{eigen1}-\eqref{eigen2}, we have that integral \eqref{trace} takes the form
\begin{equation}\label{thein}
I_1(\tau)=C_{k,\nu}\tau^{2\gamma-(n-1)}
\end{equation}
$$
\times\int_0^{+\infty}\epsilon^{2\gamma-2\alpha}e^{2i\epsilon\tau}{_1F_1}(\gamma- i\nu,2\gamma+1,-2i\epsilon \tau){_1F_1}(\gamma- i\nu,2\gamma+1,-2i\epsilon \tau)d\epsilon
$$
with $$C_{k,\nu}=\frac{2^{2\gamma+2}}{\pi}e^{\pi\nu}\frac{|\Gamma(\gamma+1+ i\nu|^2}{\Gamma(2\gamma+1)^2}e^{2i\xi}$$
where we recall that $\gamma=\sqrt{k^2-\nu^2}$. As the only difference between $I_1$ and $I_2$ is encoded in the sign of $\nu$ (see Remark \ref{negative}) which is virtually unessential, we will limit our discussion to $I_1$, the other case being completely analogous.

Integrals like \eqref{thein} have already been object of study in literature (see e.g. \cite{landlif}, \cite{gargoyle}, \cite{rey}) since they are connected with some different physical problems, as for instance matrix multipole elements for the Dirac Coulomb model, and they present some difficulties being not convergent at infinity as well as in the origin (see \eqref{ashyp}). Singularity at the origin can be dealt with by a standard integration by parts argument (as the one used to define $\Gamma$ function for negative real parts) and is discussed in \cite{sunwright} provided $2\gamma-2\alpha+1$ is not zero or a negative integer, while singularity at infinity is slightly more difficult to be handled.
To make things work, we need first of all to define $I_k(r,s)$ off-diagonal and then use a limiting procedure. Also, the introduction of a term of the form $e^{-\delta\epsilon}$ with $\delta\in\mathbb{R}^+$ in the integral is needed in order to make it convergent; we will then denote with $I^\delta_k$ the integral
$$
I^\delta_k(r,s)=C_{k,\nu}\tau^{2\gamma-(n-1)}
$$
$$
\times\int_0^{+\infty}\epsilon^{2\gamma-2\alpha}e^{\epsilon(i(r+s)-\delta)}{_1F_1}(\gamma- i\nu,2\gamma+1,-2i\epsilon r){_1F_1}(\gamma- i\nu,2\gamma+1,-2i\epsilon s)d\epsilon.
$$
This modified integral can be explicitly computed, and its solution is written in terms of a double Appel series, which unfortunately turns out to be not convergent for the values we are interested in. To deal with this problem, it is necessary to introduce an analytic continuation of such a series: we don't include the proofs of these results here as they can be found in corresponding references.

We begin with the following
\begin{lemma}\label{formint}
Let $\alpha$, $\beta$ such that $|\alpha|+|\beta|<|h|$ and $d$ not a negative integer nor zero. Then
\begin{equation}\label{intapp}
\int_0^{+\infty}t^{d-1}e^{-ht}F(a_1,b_1,\alpha t)F(a_2,b_2,\beta t)dt=h^{-d}\Gamma(d)F_2(d;a_1,a_2;b_1,b_2;\frac{\alpha}h,\frac{\beta}h)
\end{equation}
where $F_2$ is defined in \eqref{f2}.
\end{lemma}
\begin{proof}
See e.g. Lemma 1 in \cite{saadhall}.
\end{proof}

We should now point out that the ray of convergence of $F(\alpha;a_1,a_2;b_1;b_2;x,y)$ is $|x|+|y|<1$, which provides a first obstacle to the application of this result to our case. To overcome this difficulty, we rely on the following analytic continuation of $F_2$.
\begin{lemma}\label{rey}
If Re$(b_1-a_1)>0$, Re$(a_1)>0$
\begin{equation}\label{analcontin}
F_2(d;a_1,a_2,b_1,b_2;1,1)
\end{equation}
\begin{equation*}
=e^{-i\pi(b-a-d)}\frac{\Gamma(b_1)\Gamma(d-b_1+1)}{\Gamma(a_1)\Gamma(d-a_1+1)}{_3F_2}(d,b_2-a_2,d-b_1+1;b_2,d-a_1+1;1)
\end{equation*}
\begin{equation*}
+e^{i\pi a}\frac{\Gamma(b_1)\Gamma(d-b_1+1)}{\Gamma(b_1-a_1)\Gamma(d+a_1+1-b_1)}
{_3F_2}(d,a_2,d+1-b_1;b_2,d+a+1-b_1;1)
\end{equation*}
\end{lemma}

\begin{proof}
See \cite{rey}.

\end{proof}

For the sake of generality we have stated this Lemma with general values of the parameters; we list here for convenience the necessary substitutions to treat our case:
\begin{eqnarray}\label{values}
d&=&2\gamma+1-2\alpha;
\\
\nonumber
a_1=a_2&=&\gamma-i\nu;
\\
\nonumber
b_1=b_2&=&2\gamma+1.
\end{eqnarray}

Since necessary conditions of Lemma \ref{rey} are fullfilled with these values, this result allows us to pass to the limit $I_k^\delta(r,s)$ for $\delta\rightarrow 0$, $s\rightarrow r=:\tau$, and thus leaves us with
$$
I_1(\tau)=C_{k,\nu}\tau^{2\gamma-(n-1)}(2i\tau)^{-(2\gamma+1-2\alpha)}\Gamma(2\gamma+1-2\alpha)
$$
$$
\times F_2(2\gamma+1-2\alpha;\gamma-i\nu,\gamma-i\nu;2\gamma+1,2\gamma+1;1,1)
$$
with
$$C_{k,\nu}=\frac{2^{2\gamma+2}}{\pi}e^{\pi\nu}\frac{|\Gamma(\gamma+1+i\nu|^2}{\Gamma(2\gamma+1)^2}e^{2i\xi}.$$ Using now representation \eqref{analcontin} for $F_2$ and plugging in values \eqref{values}, we are led to
\begin{equation}\label{Itau}
I_1(\tau)=\tau^{2\alpha-n}C_{k,\nu}(2i)^{-(2\gamma+1-2\alpha)}\Gamma(2\gamma+1-2\alpha)(A+B)
\end{equation}
where
$$
A=e^{i\pi(\gamma-2\alpha-i\nu)}\frac{\Gamma(2\gamma+1)\Gamma(1-2\alpha)}{\Gamma(\gamma-i\nu)\Gamma(\gamma+2-2\alpha+i\nu)}
$$
$$
\times\sum_m\frac{(2\gamma+1-2\alpha)_m(\gamma+1+i\nu)_m(1-2\alpha)_m}{(2\gamma+1)_m(\gamma+2-2\alpha+i\nu)_mm!},
$$
$$
B=e^{i\pi(\gamma-i\nu)}\frac{\Gamma(2\gamma+1)\Gamma(1-2\alpha)}{\Gamma(\gamma+1+i\nu)\Gamma(\gamma+1-2\alpha-i\nu)}
$$
$$
\times\sum_m\frac{(2\gamma+1-2\alpha)_m(\gamma-i\nu)_m(1-2\alpha)_m}{(2\gamma+1)_m(\gamma+1-2\alpha-i\nu)_mm!}.
$$
Both the $_3F_2$ functions above turn to be convergent: we can indeed explicitly check convergence condition \eqref{convcond} to give in both cases $\alpha>0$. Notice also that assumption $\alpha<\gamma+1/2$ guarantees boundedness on the term $\Gamma(2\gamma+1-2\alpha)$. To conclude with, we now need to take the real part of \eqref{Itau} and show that it is a bounded function of $\gamma$. After some simplifications, we have
\begin{eqnarray}\label{reref}
\nonumber
{\rm Re}(I_1(\tau))&=&\frac{2^{1+2\alpha}e^{2\pi\nu}}\pi\frac{\Gamma(2\gamma+1-2\alpha)\Gamma(1-2\alpha)}{\Gamma(2\gamma+1)}
\\
\nonumber
&\times&{\rm Re}\Big[e^{-i\frac\pi2(2\alpha+1)}\frac{\Gamma(\gamma+1+i\nu)(\gamma-i\nu)}{\Gamma(\gamma+2-2\alpha-i\nu)}\sum_m\frac{(2\gamma+1-2\alpha)_m(\gamma+1+i\nu)_m(1-2\alpha)_m}{(2\gamma+1)_m(\gamma+2-2\alpha+i\nu)_mm!}
\\
\nonumber
&+&e^{i\frac\pi2(2\alpha-1)}\frac{\Gamma(\gamma+1-i\nu)}{\Gamma(\gamma+1-2\alpha-i\nu)}
\sum_m\frac{(2\gamma+1-2\alpha)_m(\gamma-i\nu)_m(1-2\alpha)_m}{(2\gamma+1)_m(\gamma+1-2\alpha-i\nu)_mm!}\Big]
\end{eqnarray}
where we have written $i^{-(2\gamma+1-2\alpha)}=e^{-i\frac\pi2(2\gamma+1-2\alpha)}$. Using now the basic property of the Pochhammer symbol $(x)_m=\Gamma(x+m)/\Gamma(x)$ many other terms simplify, leaving us with
$$
{\rm Re}(I_1(\tau))=\frac{2^{1+2\alpha}e^{2\pi\nu}}\pi
$$
$$
\times{\rm Re}
\Big[e^{-i\frac\pi2(2\alpha+1)}(\gamma-i\nu)\sum_m\frac{\Gamma(2\gamma+1-2\alpha+m)\Gamma(\gamma+1+m+i\nu)\Gamma(1-2\alpha+m)}{\Gamma(2\gamma+1+m)\Gamma(\gamma+2-2\alpha+m-i\nu)m!}
$$
$$
+e^{i\frac\pi2(2\alpha-1)}(\gamma-i\nu)
\sum_m\frac{\Gamma(2\gamma+1-2\alpha+m)\Gamma(\gamma+m-i\nu)\Gamma(1-2\alpha+m)}{\Gamma(2\gamma+1+m)\Gamma(\gamma+1-2\alpha+m+i\nu)m!}\Big].
$$
Stirling formula for Gamma function
$$
\Gamma(z)=\displaystyle\sqrt{\frac{2\pi}z}\left(\frac{z}e\right)^z\left(1+O\left(\frac1z\right)\right)
$$
shows that both the series above are asymptotic (in $\gamma$) to $\gamma^{-1}$, and therefore the proof is concluded.

\end{proof}

\begin{remark}\label{fail}
It should be noticed that when $\alpha\rightarrow 1$ the term $\Gamma(1-2\alpha)$ produces a singularity of order 1, as it is seen by applying well known property $\Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z)$. Anyway, it can be checked that this singularity vanishes when taking the real part. Calculations are very easy in the unperturbed case (i.e. when $\nu=0$): in this case indeed everything is real except for the exponential terms, which become respectively $e^{-i\frac32\pi}$ and $e^{i\frac12\pi}$ which are purely imaginary. In the perturbed case things are a bit more complicated, but the problem has already been dealt with in literature by some limiting argument (see \cite{rey} and references therein). On the other hand, as $\alpha$ approaches $1/2$ the singularity produced by the term $\Gamma(1-2\alpha)$ is not balanced anymore (exponential terms have a nonzero real part): this rules out the case $\alpha=1/2$ and therefore the chance of recovering Strichartz estimates with standard arguments from our proof. The analysis of this problem will be the object of forthcoming works.
\end{remark}

\section{Appendix: special functions, properties and integrals.}

For reader's convenience we include a short appendix in which we recall definitions and basic properties of the special functions coming into play. We refer to \cite{slater} and \cite{erd} for further details and for a deeper insight of the argument.
\begin{definition}
Given $a_1,\dots ,a_p$ and $b_1,\dots, b_q$ complex numbers, we define the \emph{generalized hypergeometric function} as
\begin{equation}\label{hyp}
_pF_q(a_1,\dots ,a_p;b_1,\dots, b_q;z):=\displaystyle\sum_{n=0}^\infty
\frac{(a_1)_n\dots(a_p)_n}{(b_1)_n\dots(b_q)_n}\frac{z^n}{n!}
\end{equation}
provided the sum of the series is finite, where we are using the \emph{Pochhammer symbols}
\begin{equation*}
(a)_0=1,\qquad (a)_n=a(a+1)\dots(a+n-1),\quad n\geq1.
\end{equation*}
\end{definition}
The first thing to investigate is, of course, under which conditions these series converge. As a very general framework, we can summarize the situation as follows
\begin{itemize}
\item if $p\leq q$ the series is convergent for all values of $z$;
\item if $p=q+1$ the series is convergent for $|z|<1$, and for the special values $z=1$ if
\begin{equation}\label{convcond}{\rm Re}\left(\sum_{n=1}^qb_n-\sum_{m=1}^pa_m\right)>0\end{equation}
and $z=-1$ if
$${\rm Re}\left(\sum_{n=1}^qb_n-\sum_{m=1}^pa_m\right)>-1;$$
\item
if $p> q+1$ the series never converges except for $z=0$.
\end{itemize}

In this work we are especially interested in the \emph{confluent hypergeometric function} $_1F_1(a,c,z)$, which is thus the case $p=q=1$. This series is therefore known to be convergent for every finite value of $z$ (if $c$ is not zero nor a negative integer). Moreover, if $a$ is a negative integer or zero, $_1F_1$ reduces to a polynomial of degree $|a|$. Confluent hypergeometric functions can be thought of as a limiting case of the celebrated \emph{Gauss hypergeometric serie}, that namely is case $p=2$, $q=1$ in \eqref{hyp}, via the relation
\begin{equation*}
\displaystyle
_1F_1(a,c;z)=\lim_{b\rightarrow\infty}\:_2F_1(a,b;c;z/b).
\end{equation*}
These functions satisfy the differential equation
\begin{equation*}
zu''+(c-z)u'-au=0.
\end{equation*}
 The asymptotic behaviour of $_1F_1$ can be derived by writing
\begin{equation}\label{ashyp}
_1F_1(a,b,z)\cong\frac{\Gamma(b)}{\Gamma(b-a)}(-z)^{-a}G(a,a-b+1,z)+\frac{\Gamma(b)}{\Gamma(a)}e^zz^{a-b}G(b-a,1-a,z)
\end{equation}
where $G$ has the asymptotic series representation
\begin{equation*}
G(a,b,z)=1+\frac{ab}{1!z}+\frac{a(a+1)b(b+1)}{2!z^2}+...
\end{equation*}
Among the properties that confluent hypergeometric functions satisfy we recall the following one
\begin{equation}\label{twist}
F(a,c,z)=e^zF(c-a,c,-z).
\end{equation}
Among the possible generalizations of hypergeometric function a very useful one is given by the so called \emph{Appell functions} (see \cite{appel}). Historically, the motivation for introducing series of this type is connected to investigating different kind of products of two different Gauss functions. In particular, we are here interested in the following, which is sometimes called Appell function of the second type,
\begin{equation}\label{f2}
F_2(d;a_1,a_2;b_1;b_2;x,y)\displaystyle=\sum_{m=0}^\infty\sum_{n=0}^\infty\frac{(d)_{m+n}(a_1)_m(a_2)_n}{(b_1)_m(b_2)_nm!n!}x^my^n.
\end{equation}
 It can be seen that convergence condition for $F_2$ is given by $|x|+|y|<1$.

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Lewin Honored by NY Bar Association
For decades, Professor Travis H.D. Lewin's name has been synonymous with the College of Law's trial advocacy program. He has dedicated his life to teaching young professionals the art of advocacy, as well as the lessons of professionalism and character. This year, Professor Lewin was recognized at the NTC-TYLA Regional Trial Tournament in Buffalo, where he was honored by the Trial Lawyers Section of the New York State Bar Association with a plaque that set forth his past service to the competition. The Trial Lawyers section announced that they were creating a new award of $1,000 that will be presented annually to the coach of the winning New York law school and which will be named the Travis H. D. Lewin award.
Events Recap
College of Law alumni joined Dean Hannah R. Arterian and members of the College of Law staff for luncheons in San Diego, CA, at the University Club on March 7, and Phoenix, AZ, at the Capital Grille on March 23. Dean Arterian provided an update on academic initiatives, student life and the new law building, Dineen Hall.
Dean Arterian and the College of Law will host alumni events in various cities, including Westchester County, NY on May 3, Albany, NY on May 4, Los Angeles, CA on May 18, Northern NJ on June 22 and Philadelphia, PA on June 23. These events are complimentary, but require advanced registration.
For the full calendar of events and registration, please click here.
New Student Housing Opening Fall 2012
State-of-the-art student housing is just one example of our on-going commitment to students through long-term improvements in our physical environment. Our new spacious, fully-furnished apartment homes—opening in fall 2012—will be located just steps away from the law school. Amenities will include: a fully-outfitted fitness center; washers, dryers and a full kitchen in each unit; all-inclusive utilities, including internet and cable TV; and a computer lab. Multiple floor-plans will be available, ranging from studio apartments to units with four bedrooms and four bathrooms.
Predators, Porn and the Law: America's Children in the Internet Era April 9 at College of Law
From chat rooms to sexting, how do we keep our children safe from online predators? The Syracuse Law Review and Syracuse College of Law community will present a groundbreaking conference, "Predators, Porn and the Law: America's Children in the Internet Era," at the College of Law on Saturday, April 9, which will bring together legal scholars, children's advocates and law enforcement personnel from across the United States for discussions regarding the safety of America's children in the Internet age. The free conference begins at 9 a.m. and parking is available in the Irving Garage.
"This conference will be a unique opportunity for academics, legal professionals and local community members to come together for a critical examination and timely discussion about the safety of our children in the Internet era," says Jennifer M. Haralambides, third-year law student and editor in chief of the Syracuse Law Review.
Marc Klaas, founder of the KlaasKids Foundation for Children and national advocate for child safety, will be the keynote speaker at 3 p.m. in Hendricks Chapel. Klaas started his foundation after the 1993 kidnap, rape and murder of his 12-year-old daughter.
In addition to Klaas' keynote, the daylong conference will feature presentations by the National Center for Missing and Exploited Children; the U.S. Attorney's Office; the Federal Public Defender's Office; and leading scholars, including Ilya Shapiro from the CATO Institute, Professor Amy Adler from New York University and Professor Mary G. Leary from The Catholic University of America.
CLE credits, including Ethics credits, will be offered during the conference.
A complete agenda, registration form and CLE breakdown is available at http://law.syr.edu/go/predators_porn_and_the_law
For more information, email lawreview@law.syr.edu or call the Syracuse Law Review at (315)-443-3680.
Syracuse Law Review Annual Banquet
The Syracuse Law Review proudly honors William J. Brodsky L'68 as its 2011 Alumni Achievement Award Recipient. Brodsky, chairman and CEO of the Chicago Board Options Exchange, will be honored at an event on April 9, following the law review conference.
Mark Your Calendar for Reunion 2011 - September 22-25, 2011
All College of Law alumni are welcome back for Reunion weekend, with special emphasis on classes ending in "1" or "6"! College of Law Reunion events include the Annual Law Dinner, the College of Law alumni tailgate and the Syracuse vs. Toledo Homecoming football game. Save the date!
Spotlight on...Jennifer L. Dzwonczyk L'95
Jennifer L. Dzwonczyk L'95, a partner in Venable's Intellectual Property Litigation practice in Washington D.C., recently authored a book for the American Bar Association, "The Patent Litigator's Job: A Survival Guide."
Click here to read more about Jennifer and to view more profiles of Syracuse Law alumni.
Social Media Corner
Hire an SU College of Law Student
Syracuse Law students are applying for Financial Aid now. Make a gift online and make a difference!
Newsletter of the
Office of Clinical Legal Education
Syracuse Law Magazine Fall 2010
Browse online or on your mobile device!
Starting a career transition or interested in networking? Looking for the latest news and events for the College of Law? You're only one click away from your goal! Join the hundreds of alumni that are connecting with the College of Law on Facebook, Twitter and LinkedIn. Click the icons above to get started!
Looking for a safe investment? Consider hiring a College of Law student or recent graduate to work at your firm or business.
If your organization has hiring needs for summer, part-time, or permanent employment, please list your job announcement with the Office of Professional & Career Development. We are eager to post jobs and/or collect resumes for you. Job announcements are listed on our password-protected site for College of Law students and alumni. For more information, contact the office at 315-443-1941 or register as an employer on our job search site by visiting www.law.syr.edu/professional-career-development/employers/.
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redpajama_set_name: RedPajamaCommonCrawl


This magazine has been around for many years under the name of Synthesis/Regeneration. We wanted the name to be more indicative of the kind of writing we publish so we have changed the name but the magazine is the same.
For years, past issues were available on the site greens.org which is still accessible for older issues.
Come back and visit from time to time as we launch our new, more useful website.
Hello, this site looks very old, is Green Social Thought still around?
I don't understand your comment. The latest articles are from 2014. Why would you say old?
Green Social Thought | Powered by Mantra & WordPress.


text:
from iris_sdk.models.base_resource import BaseData, BaseResourceList, \
    BaseResourceSimpleList
from iris_sdk.models.data.file_data import FileData
from iris_sdk.models.maps.loas import LoasMap

class LoasData(LoasMap, BaseData):

    def __init__(self):
        self.file_data = BaseResourceList(FileData)
        self.file_names = BaseResourceSimpleList()

redpajama_set_name: RedPajamaGithub


Connecting unapproved devices to phone lines may be illegal in your area.
for transmitting. Use at your own risk.
antennas built into most portables.
several times around the receiver's telescopic whip antenna instead.
probably be better than the whip antenna by itself.
use if you experience frequent thunderstorms.
ringing, so that they will not harm the receiver.
ground is available, you can connect it to this terminal.
purchased from some shortwave dealers.
A low-pass filter can be used to remove signals above the shortwave bands.
connect the antenna in place of the phone line.
end & appear in SW, especially with a long antenna.
intend to try to increase my antenna length again.
Anywho, here is the filter, for interested parties.
The outer capacitors are 1500 pf ceramic disks.
The inner capacitor is 820 pf ceramic disk.
The squiggly things are coils (two total). Each one is 2.7 uh.
(a close value will do).
instead of .36, and .66 uH instead of .42, but it works fine.
going to put them both into one box when I get the time.
The proper way to connect them is in series, i.e.
For small wire, you can assume C = 0.
The cost of these parts is less than $1 each.
CAUTION: Don't look into laser beam with remaining eye.


text:
Home / Electronics & Semiconductor
Delivery Robots Market Share, Size Growth Opportunities, Outlook, Statistics, Market Scope, Revenue, Research, Trends Analysis & Global Industry Forecast Report, 2022-2029
Delivery Robots Market Size, Share & Analysis Report by Load Carrying Capacity (Up to 10 KG, More than 10 kg Up to 50 KG and More than 50.00 KG), By Component (Hardware and Software), By Number of Wheels (3 Wheels, 4 Wheels and 6 Wheels), By Speed Limit (3 KPH, Higher than 3 KPH up to 6 KPH, and Higher than 6 KPH) By End-User Industry (Food & Beverages, Retail, Healthcare, Postal and Others) and By Region
Published Date: Jul-2021 | Report Format: Excel/PPT | Report Code: UP2159-001001
GMI Research analysis indicates that the Delivery Robots Market is slated to register a double digit CAGR of 31.5 over the period, 2022-2029.
To have an edge over the competition by knowing the market dynamics and current trends of "Delivery Robots Market", request for Sample Report here
Rapid industrialization, increasing adoption of delivery robots by online retailers and the medical sector, and evolving need for faster delivery services with an expansion in the e-commerce industry are some of the major factors projected to drive the global delivery robots market growth. In addition, the growing adoption of delivery robots across food & beverage industry to offer doorstep delivery to customers is projected to expand the market growth. Moreover, the leading food industry players have partnered with delivery robot producers to provide food without any human interference. For example, in 2019, Coca-Cola used self-driving robots to offer drinks to visitors at Alton Towers.
Technological development in delivery robots, such as the incorporation of AI and ML, have improved robot functionality, thereby stimulating market growth. Other factors such as robotics venture allowance, intensified investment by leading companies, and increasing application of drones in the commercial sector will contribute to the market growth. However, strict norms associated with the operation of delivery robots might hamper the delivery robots market size over the upcoming years.
Number of Wheels – Segment Analysis
Based on the Number of Wheels, the 4 Wheels segment is expected to grow at a higher CAGR during the forecast period. Companies provide four-wheeled robots for delivering groceries, food orders, flowers, packages, and boxes, among other things. Moreover, lowering the distance between the front and rear set of wheels will result in substantial zero-radius turning in any desired direction with four-wheeled robots.
End-User Industry – Segment Analysis
Based on the End-User Industry, the Retail segment is expected to grow at a higher CAGR during the forecast period. The rising use of the internet offers growth to online shopping. The ever-increasing trend of online shopping is expected to intensify the need for delivery robots for the e-commerce application in the retail industry in the upcoming years. Furthermore, once the rules and regulations for operating autonomous delivery robots are determined worldwide, these robots would also be adopted for cargo deliveries, which is also an important factor that is likely to propel the growth delivery robots market.
Based on the region, North America is projected to dominate the market during the forecast period. Many major market vendors are based out of here, and also the region has a high number of startups and manufacturers working toward the growth of autonomous delivery robotic technology. Moreover, the penetration of autonomous delivery robots across various end-users in the region is higher as compared to other regions of the world. In the hospitality and retail, and logistics segments, the demand for these robots is rising; many retail and hospitality players are collaborating with manufacturers to have a first-hand experience of the prototypes. For instance, delivery robots are helping many prime hospitality organizations to gain significant attraction from media, which is an important channel in the region that influences tourists' decision to book a hotel or resort. North America, being the second largest e-commerce market after Asia-Pacific, has a high number of shoppers. Such a widespread e-commerce market in the region is encouraging several retailers to invest and develop their supply chain facilities, resulting in considerable investments in the autonomous delivery robots sector. Also, many of the end-user companies are shifting to autonomous robots, which is significantly expanding the growth of the ADR market in the region. For example, in 2019, Amazon officially rolled out its last-mile delivery robots in Southern California. The delivery robot is designed to autonomously deliver parcels from urban distribution points to Amazon Prime customers, removing the need for vehicles in last-mile delivery.
Various notable players operating in the market include Starship Technologies, JD.com, Inc., Panasonic Corporation, Nuro LLC, Robby, Boston Dynamics., Robomart.com, Eliport, Inc., Piaggio Fast Forward, TeleRetail., among others.
In 2021, Domino's, a global pizza restaurant chain entered in a partnership with Nuro, Inc., an autonomous delivery robot producer, to offer driverless deliveries. This collaboration helped Nuro, Inc. deliver domino's orders to customers through its autonomous delivery robot.
In 2020, Starship Technologies collaborated with Save Mart Companies to launch an on-demand grocery delivery service to its consumers. Save
In 2020, JD.com, Inc. procured a controlling stake in Chinese express transport company Kuayue Express Group. As a result of this acquisition, JD Logistics, the logistics arm of JD.com, will operate with Kuayue Express, an organization regarded as a front-runner in the limited-time express service, which is supportive of the idea of same- and next-day delivery of JD.com.
In 2020, Nuro LLC partnered with CVS Pharmacy to start delivering prescriptions through driverless vehicles.
The global Delivery Robots market has been segmented on the basis of Load Carrying Capacity, Component, Number of Wheels, Speed Limit, End-User Industry and regions. Based on Load Carrying Capacity, the market is segmented into Up to 10 KG, More than 10 kg Up to 50 KG and More than 50.00 KG. Based on Component, the market is segmented into Hardware and Software. The Hardware market is further segmented into Ultrasonic/ LIDAR Sensors, Chassis and Motors, Control Systems, Radars, GPS, Cameras and Others. The Software market is further segmented into Fleet Management Software and Computer Vision. Based on Number of Wheels, the market is segmented into 3 Wheels, 4 Wheels and 6 Wheels. Based on Speed Limit the market is segmented into 3 KPH, Higher than 3 KPH up to 6 KPH and Higher than 6 KPH. Based on End-User Industry, the market is segmented into Food & Beverages, Retail, Healthcare, Postal and Others.
For detailed scope of the "Delivery Robots Market" report request a Sample Copy of the report
At what pace is global Delivery Robots market growing? What will be the growth trend in the future?
What are the key drivers and restraints in Delivery Robots market? What will be the impact of drivers and restraints in the future?
What are the regional revenues and forecast breakdowns? Which are the major regional revenue pockets for growth in the global Delivery Robots market?
Which Load Carrying Capacity generated maximum revenues in 2021 and identify the most promising Load Carrying Capacity during the forecast period?
What are the various Component areas of global Delivery Robots market and how they are poised to grow?
Market Revenues (2021) USD Billion
Market Base Year 2021
Market Forecast Period 2022-2029
Revenues (USD Billion)
Market Segment By Load Carrying Capacity, By Component, By Number of Wheels, By Speed Limit, By End-User Industry, By Region
Companies Profiled Starship Technologies, JD.com, Inc., Panasonic Corporation, Nuro LLC, Robby, Boston Dynamics., Robomart.com, Eliport, Inc., Piaggio Fast Forward, TeleRetail., among others; a total of 10 companies covered.
Global Delivery Robots Market by Load Carrying Capacity
Up to 10 KG
More than 10 kg Up to 50 KG
More than 50.00 KG
Global Delivery Robots Market by Component
Ultrasonic/ LIDAR Sensors
Chassis and Motors
Global Delivery Robots Market by Number of Wheels
Global Delivery Robots Market by Speed Limit
3 KPH
Higher than 3 KPH up to 6 KPH
Higher than 6 KPH
Global Delivery Robots Market by End-User Industry
Global Delivery Robots Market by Region
North America Delivery Robots Market (Option 1: As a part of the free 25% customization)
By Load Carrying Capacity
By Number of Wheels
By Speed Limit
By End-User Industry
Europe Delivery Robots Market (Option 2: As a part of the free 25% customization)
Asia-Pacific Delivery Robots Market (Option 3: As a part of the free 25% customization)
RoW Delivery Robots Market (Option 4: As a part of the free 25% customization)
Major Players Operating in the Delivery Robots (Option 5: As a part of the Free 25% Customization – Profiles of 5 Additional Companies of your Choice)
Starship Technologies
JD.com, Inc.
Nuro LLC
Boston Dynamics.
Robomart.com
Eliport, Inc.
Piaggio Fast Forward
TeleRetail
SKU: UP2159-001001 Categories: Electronics & Semiconductor, Published, Report Store Tags: Delivery Robots Market, Delivery Robots Market Dynamics, Delivery Robots Market Growth, Delivery Robots Market Key players, Delivery Robots Market Opportunity, Delivery Robots Market Size, Delivery Robots Market Trends


DUBAI: Somali-American, hijab-wearing model Halima Aden is set to touch down in Dubai as she will be launching the Max Mara capsule collection in The Dubai Mall on Monday night.
Aden was just here less than a month ago, with superstar Rihanna, for the Fenty Beauty Masterclass that took place at the end of September.
Aden has become known in the fashion industry, having walked for Max Mara, Yeezy, Alberta Ferretti and many others.
She also is known for her charity work as an ambassador for UNICEF and uses her voice and platforms to advocate for children's rights.
Aden is a former child refugee, who came to the US during the Somali Civil War in the early 1990s.


text:
Finance + Stress = Suicide
May 11, 2016 Posted by Raúl Ilargi Meijer at 3:39 pm Finance Tagged with: Angus Deaton, dairy, New Zealand, stress, suicide, Trump 10 Responses »
Irving Underhill Irving Trust Building, Trinity Church, Wall Street, New York 1931
Last night, I made it back to Athens, still half a cripple, but there must be someone in this city who knows how to stick needles in the appropriate muscles, right?!, paid the rent for the Social Kitchen big house/nerve center late this morning, a tough 1 mile walk for my leg muscles -they kill me!-, still, that's done, and hoping to get back to writing articles very soon, but having an ouzo right now just to make sure I blend in with the Romans. One can never be too sure.
Ergo: first here is, once again, our dearly beloved New England-raised friend from New Zealand, Nelson Lebo III, touching on a theme that will be found to have legs once the world sees Janet Yellen has no clothes on (and I DO understand the problem with that visual) :
Nelson Lebo: "Our already horrendous suicide rate hit a new record high last year." The news of New Zealand's suicide rate did not surprise me when I heard it on the radio earlier this week. Anyone who pays attention to global trends could see this coming. "Psychotherapists say we need a wide-ranging review into the mental health system before there are more preventable deaths" reported Newstalk ZB.
At lighter moments I joke that the best thing about living in New Zealand is that you can see worldwide trends that are heading this way, but the worst part is that no-one believes you. This is not a lighter moment. Suicide is a serious issue and one that is growing dramatically among my peer group: white middle-aged men.
The first people to notice the emerging pattern in the United States were Princeton economists Angus Deaton and Anne Case. The New York Times reported on 2nd November, 2015 that the researchers had uncovered a surprising shift in life expectancy among middle-aged white Americans – what traditionally would have been considered the most privileged demographic group on the planet.
The researchers analyzed mountains of data from the Centers for Disease Control and Prevention as well as other sources. As reported by the Times, "they concluded that rising annual death rates among this group are being driven not by the big killers like heart disease and diabetes but by an epidemic of suicides and afflictions stemming from substance abuse: alcoholic liver disease and overdoses of heroin and prescription opioids. The mortality rate for whites 45 to 54 years old with no more than a high school education increased by 134 deaths per 100,000 people from 1999 to 2014."
The most amazing thing about this discovery is that the Princeton researchers stumbled across these findings while looking into other issues of health and disability. But as we hear so often, everything is connected. A month before releasing this finding Dr. Deaton was awarded the Nobel Prize in Economics based on a long career researching wealth and income inequality, health and well-being, and consumption patterns.
The Royal Swedish Academy of Sciences credited Dr. Deaton for contributing significantly to policy planning that has the potential to reduce rather than aggravate wealth inequality. In other words, to make good decisions policy writers need good research based on good data. Too often this is not the case. "To design economic policy that promotes welfare and reduces poverty, we must first understand individual consumption choices. More than anyone else, Angus Deaton has enhanced this understanding."
Days before hearing the news about New Zealand's rising suicide rate I learned of another major finding from demographic researchers in the United States. For the first time in history the life expectancy of white American women had decreased, due primarily to drug overdose, suicide and alcoholism. This point is worth repeating as it marks a watershed moment for white American women. After seeing life expectancies continually extend throughout the history of the nation, the trend has not only slowed but reversed. Data show the slip is only one month, but the fact that it's a decrease instead of another increase should be taken as significant milestone.
Please note that the following sentence is not meant in the least to make light of the situation, but is simply stating a fact. The demographic groups that are experiencing the highest rates of drug overdose, suicide and alcoholism are also the most likely to be supporters of Donald Trump in his campaign for the U.S. Presidency. It does not take a Nobel Laureate to observe a high level of distress among white middle-class Americans. Trump simply taps into that angst.
As reported by CBS News, "The fabulously rich candidate becomes the hero of working-class people by identifying with their economic distress. That formula worked for Franklin D. Roosevelt in the 1930s. Today, Donald Trump's campaign benefits from a similar populist appeal to beleaguered, white, blue-collar voters – his key constituency."
I don't blame most Americans for being angry. That the very architects of the global financial crisis have only become richer and more powerful since they crashed the world economy in 2008 is unforgivable. The gap between rich and poor continues to widen and the chasm has now engulfed white middle-aged workers. As the Pope consistently tells us, wealth and income inequality is the greatest threat to humanity alongside climate change.
Instead of going down the Trump track for the rest of this piece, I'd rather wrap it up by bringing the issue back to Aotearoa (New Zealand) and my small provincial city of Whanganui. To provide some background for international readers, the NZ economy relies significantly on dairy exports and many dairy farmers hold large debts. Dairy prices are known for their volatility, and recently the payouts have dropped below break-even points for many farmers.
Earlier this month Primary Industries Minister Nathan Guy announced that the government would invest $175,000 to study innovative, low cost, high performing farming systems already in place in New Zealand. Stuff.co.nz reported, "The government is set to pick the brains of New Zealand's top dairy farmers in an effort to help those struggling with the low dairy payout."
That is great news, but the government's investment in researching the best of the best farmers is a pittance when compared with what is spent addressing issues of depression and suicide prevention among Kiwi farmers. Isn't this a case of putting the cart ahead of the horse, or treating symptoms instead of causes?
Research shows that financial stress contributes significantly to the increasing suicide rates here and abroad. We know that innovative farmers who use low-input/high-performance systems are more profitable that their conventional farming brethren. Would it then be a stretch to conclude that depression and suicide is much lower among these innovative and profitable farmers? At the same time, research shows that wealth and income inequality in our more urban centres contribute to anti-social behaviours such as crime, domestic abuse and illegal drug usage.
Angus Deaton, the Nobel-winning economist, would argue that in order for policy planners to address these issues effectively they must understand the underlying causes and resultant costs. Thankfully, we do see glimmers of that from central government instead of the usual neoliberal claptrap. Credit must be given to Finance Minister Bill English for his actuarial approach to some social issues rather than the inaccurate dogmatic position often adopted by the right.
But closer to home for me, such enlightened policy planning has yet to reach our city by the awa (river). To start off, the Council's rates structure is stunningly regressive, clearly taking significantly higher proportions of household wealth from low-income families than from high-income families. If we believe the research in this field (ie, The Spirit Level, etc) wouldn't we expect the widening gap between rich and poor to result in even more anti-social behavior in our city that already suffers from reputation problems nationwide?
Secondly, the council's vision documents and long-term plan are nearly devoid of intelligent strategies to address the underlying issues of anti-social behaviour, depression, poor health, and domestic problems that afflict our community. The Council pours mountains of money into an art gallery and arts events while providing token services and events for low-income families.
Will it take our own Trump or Sanders running for office to stimulate a populist revolt against regressive policies that potentially do more harm than good to our community? What will it take for us to finally get it? I first wrote about these issues in our city's newspaper, the Chronicle, two and a half years ago… but, apparently, no one believed me. Welcome to provincial New Zealand!


text:
Bharath Reddy is an Indian actor who appears in Telugu and Tamil language films. A cardiologist by profession, Reddy made a breakthrough portraying a police officer in Chakri Toleti's Telugu–Tamil bilingual film Eenadu / Unnaipol Oruvan (2009).

Career
Bharath Reddy was born in Chandragiri, a suburb of Tirupathi, Andhra Pradesh. He was brought up in Chennai, completing his schooling from La Chateline Residential Junior College. He completed his medical degree from Yerevan State Medical University, in Armenia, and then went on to complete his diploma in Cardiology. He started working as a Junior Consulting Cardiologist at Apollo Hospitals, Elbit Medical Diagnostics, and Century Hospital in Hyderabad.

He did close to a dozen films in walk-in roles and in characters as a villain's henchman, before being noticed for his work in J. D. Chakravarthy's Siddham (2009). After being impressed by his performance in that film, Chakri Toleti selected him to play a police officer in his bilingual film Unnaipol Oruvan, where Bharath was able to act with veteran actors Kamal Haasan, Mohanlal and Venkatesh. Since then he has appeared in notable Telugu films including the Nagarjuna-starrers Ragada and Greeku Veerudu, as well as playing a role in Mahesh Babu's Businessman. He has also worked in Tamil films including as a police officer in Radha Mohan's Payanam, Ragalaipuram and most recently, Idhu Kathirvelan Kadhal as Udhayanidhi Stalin's brother-in-law.

Filmography

References

External links

Indian male film actors
Male actors in Tamil cinema
Living people
Male actors in Telugu cinema
People from Tirupati
Yerevan State Medical University alumni
Male actors from Chennai
21st-century Indian male actors
Male actors from Andhra Pradesh
Telugu male actors
1978 births

redpajama_set_name: RedPajamaWikipedia


text:
Vol. LXVIII, No. 4
How Exercise, Fewer Calories, Plant Toxins May Protect Our Brains
Planning to Handle the Unplanned Like a Pro
'Military' Uses of Neuroscience Have Tended to Stumble, Moreno Says
New NIH Toolbox App Captures Reliable Research Data
NINDS Investigator, Friends Honored for Rescue at Sea
Understanding the Metabolism of Bacteria Could Lead to Therapies
Estes To Deliver NIAID Chanock Lecture
Schizophrenia's Strongest Known Genetic Risk Deconstructed
More Than 1 in 20 U.S. Children Have Dizziness, Balance Problems
Uncorrected Farsightedness Linked to Literacy Deficits in Preschoolers
Moderate hyperopia, if not treated, may affect reading ability and grade school readiness among preschoolers.
Photo: Joe Balintfy
A study funded by NEI has shown that uncorrected farsightedness (hyperopia) in preschool children is associated with significantly worse performance on a test of early literacy.
The results of the Vision in Preschoolers-Hyperopia in Preschoolers study, which compared 4- and 5-year-old children with uncorrected hyperopia to children with normal vision, found that children with moderate hyperopia did significantly worse on the Test of Preschool Early Literacy than their normal-vision peers.
"This study suggests that an untreated vision problem in preschool, in this case one that makes it harder for children to see things up-close, can create literacy deficits that affect grade school readiness," said Dr. Maryann Redford, a program director in collaborative clinical research at NEI.
In most children with hyperopia, the condition is mild and has little impact on vision. A small number of preschool children have high hyperopia that is corrected with eyeglasses. It's estimated that 4-14 percent have moderate hyperopia, which often goes undiagnosed and untreated.
"Prior studies have linked uncorrected hyperopia and reading ability in school-age children," said Dr. Marjean Taylor Kulp, distinguished professor in the College of Optometry at Ohio State University and lead author of the study. "But large-scale investigations looking at reading readiness skills hadn't been conducted in preschool children. This study was necessary to determine whether or not, at this age, there was a link between the two."
The results were published Jan. 27 in Ophthalmology.


Revoke-Obfuscation is a PowerShell v3.0+ compatible PowerShell obfuscation detection framework.
In the Fall of 2016 and Spring of 2017, Daniel Bohannon (@danielhbohannon) released Invoke-Obfuscation and Invoke-CradleCrafter, two open-source PowerShell obfuscation frameworks. The goal of this research and these frameworks was to highlight the limitations of a purely signature-based approach to detecting attackers' usage of PowerShell. The core message to defenders has been to focus on detecting Indicators of Obfuscation in addition to known suspicious syntax.
However, the extreme levels of randomization in Invoke-Obfuscation and Invoke-CradleCrafter paired with the token-layer obfuscation options that are not deobfuscated in PowerShell's script block logging have led defenders to look for a new, scalable means of generically detecting both known and unknown obfuscation techniques.
A few weeks after the release of Invoke-Obfuscation, Lee Holmes (@Lee_Homes) authored a blog posted entitled More Detecting Obfuscated PowerShell in which he highlighted statistical analysis techniques to detect anomalous features found in heavily obfuscated scripts, specifically those produced by Invoke-Obfuscation.
Since this exchange, Daniel and Lee became good friends and shared many common interests/obsessions -- namely, a love of fine coffee and the pursuit of creating new ways to thoroughly detect obfuscated PowerShell.
The amount of time both Blue Teamers spent pouring over research and POC code would equate to several thousand cups of Chemex-brewed coffee assuming the proper 4-minute target brew time (assuming at least one other coffee enthusiast picked up on this "pour over" pun).
Revoke-Obfuscation is the final hand-crafted product of these efforts.
Revoke-Obfuscation is an open-source PowerShell v3.0+ framework for detecting obfuscated PowerShell commands and scripts at scale. It relies on PowerShell's AST (Abstract Syntax Tree) to rapidly extract thousands of features from any input PowerShell script and compare this feature vector against one of several pre-defined weighted feature vectors computed through an automated learning process conducted against a corpus of 408K+ PowerShell scripts. This full corpus can be downloaded from (https://aka.ms/PowerShellCorpus). Stay tuned for the publishing of the labeled data and the tooling used for training.
Since Revoke-Obfuscation relies on feature extraction and comparison instead of pure IOCs or RegEx matching, it is more robust in its ability to identify unknown obfuscation techniques even when attackers attempt to subdue their obfuscation by padding it with unobfuscated script contents to overthrow basic checks like character frequency analysis.
Revoke-Obfuscation can easily measure most input PowerShell scripts within 100-300 milliseconds. This level of performance allows an organization to measure the obfuscation of (at worst) 12K+ PowerShell scripts per hour without the need to index verbose PowerShell script block logs in a SIEM.
Lastly, Revoke-Obfuscation supports easy whitelisting functionality along with the ability to ingest PowerShell Operational event log records and reassemble script blocks that are recorded across numerous script block EID 4104 records. It can easily become a one-stop shop for ingesting an environment's PowerShell Operational event logs, reassembling and unique'ing all scripts within those logs, and then identifying obfuscated PowerShell scripts that deserve manual inspection.
The source code for Revoke-Obfuscation is hosted at Github, and you may download, fork and review it from this repository (https://github.com/danielbohannon/Revoke-Obfuscation). Please report issues or feature requests through Github's bug tracker associated with this project.
Revoke-Obfuscation will provide a detailed tutorial as well as a few other fun surprises. But if you are not into the lulz then you can simply run Get-Help Measure-RvoObfuscation to see usage syntax or just continue reading.
If you need to reassemble and extract script block logs from PowerShell Operational logs then Get-RvoScriptBlock is your function of choice. It automatically returns only unique script blocks and excludes certain default script block values deemed not malicious. This can be overridden with the -Deep switch.
Get-RvoScriptBlock also supports MIR/HX audit results as well as PowerShell Operational logs retrieved via Matt Graeber's (@mattifestation) CimSweep project (https://github.com/PowerShellMafia/CimSweep). For CimSweep there is a minor registry tweak required to trick WMI into querying a non-classic event log. Details can be found in the NOTES section of Get-RvoScriptBlock.
The -OutputToDisk switch will automatically output all obfuscated scripts to .\Results\Obfuscated. Regardless, all results will be returned as PSCustomObjects containing the script content along with metadata like an obfuscation score, measurement time, whitelisting result, all extracted script features, etc.
.\Whitelist\Scripts_To_Whitelist\ -- All scripts placed in this directory will be hashed and any identical scripts will be whitelisted. This whitelisting method is preferred above the next two options.
Lastly, if looking for a platform for creating indicators (IOCs) that harness the power of PowerShell's AST (Abstract Syntax Tree) -- which we would highly recommend for identifying malicious PowerShell activity that is NOT obfuscated -- then PS Script Analyzer is an excellent framework designed to handle such tasks.
v1.0 - 2017-07-27 Black Hat USA & 2017-07-30 DEF CON: PUBLIC Release of Revoke-Obfuscation.