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



The pointwise decay of solutions to the linear wave equation on a
space-time has many applications in terms of behavior of and lifespans
of existence for small data solutions in nonlinear equations,
far-field asymptotics, and stability of a given space-time in the
general relativistic sense.  Here we consider the long-time
asymptotics of solutions of the wave equation on backgrounds with
scale invariant singular potentials.  Specifically, we analyze
the pointwise decay rates in time for waves in the setting of
diffractive and long range potentials.  The models we consider here
are those of the Dirac--Coulomb equation, which the first and second
author considered in \cite{DirCoulRF}, and the wave equation with an
inverse square potential, which is related closely to work of the
first and third authors in \cite{resonances} and \cite{coneradfield}.  In three spatial dimensions, we prove
pointwise decay estimates in time uniformly at any point in space that
vary in an unexpected way from the standard $t^{-3}$ decay rate given
by Price's Law for the wave equation in Minkowski space that has now
been proven in a variety of settings.  




As observed in the previous works of the first author with Wunsch--Vasy in  \cite{BVW1,BVW2}, the pointwise decay rates of waves can be connected to the radiation field asymptotics of the operators considered.  The radiation field for a wave equation on a given space-time is an asymptotic expansion whose 
exponents in the expansion along null infinity are the resonances of
the spectral family of the singularly perturbed Laplacian on the hyperbolic space living\
at the ``northern cap.''  Resonances and radiation field asymptotics for the related problem of the wave equation on conic manifolds were computed in a previous
papers by the first and third authors, \cite{resonances} and
\cite{coneradfield} respectively.  However, for pointwise decay
estimates, we observe that the asymptotics of the radiation field do
not tell the whole story.  The locations of the resonances govern the
time decay as $t$ and $r$ tend to infinity simultaneously, while the
singular behavior of the resonant states lead to modified decay rates
as $t$ tends to infinity with $r$ fixed.

In considering the wave equation on $\mathbb{R} \times \mathbb{R}^{n}$ with an
inverse square potential, we assume $u$ is a solution
of the equation
\begin{align}
  \label{eq:inv-square}
  &\pd[t]^{2}u - \Delta u + \frac{\digamma}{r^{2}}u = 0,\\
  &(u,\pd[t]u)|_{t=0} = (\phi_{0}, \phi_{1}) \in
    C^{\infty}_{c}(\mathbb{R}^{n}\setminus\{ 0\})\times
    C^{\infty}_{c}(\mathbb{R}^{n}\setminus\{0\}), \notag
\end{align}
where $\Delta$ is the negative definite Laplacian on $\mathbb{R}^{n}$ and
\begin{equation*}
  \digamma > -\frac{(n-2)^{2}}{4}.
\end{equation*}
Due to the Hardy inequality, the assumption on $\digamma$ implies that the
Hamiltonian $-\Delta + \frac{\digamma}{r^{2}}$ is essentially self-adjoint with
form domain $H^{1}(\mathbb{R}^{n})$.


For the massless Dirac--Coulomb equation, we consider $\psi : \mathbb{R} \times \mathbb{R}^3 \to \mathbb{C}^4$ a solution to
\begin{align}
  \label{eq:dc}
& \left( \gamma^0 \left( \partial_t + \frac{i Z}{r} \right) + \sum_{j=1}^3 \gamma^j \partial_j \right) \psi = 0, \\
& \psi(0) = \psi_0 \in  (C^{\infty}_{c}(\mathbb{R}^{3}\setminus\{ 0\}))^4 \notag
\end{align}
for $\gamma^{0,1,2,3}$ the gamma (or Dirac) matrices, meaning they satisfy the anticommutation relation
\begin{equation*}
  \gamma^{\alpha}\gamma^{\beta} + \gamma^{\beta}\gamma^{\alpha} = -2
  \eta^{\alpha\beta} Id_{4},
\end{equation*}
where $\eta^{\alpha\beta}$ are the components of the Minkowski metric,
i.e.,
\begin{equation*}
  \eta^{\alpha\beta} =
  \begin{cases}
    -1 & \alpha =\beta=0 \\
    1 & \alpha = \beta \in \{1,2,3\} \\
    0 & \alpha \neq \beta
  \end{cases}.
\end{equation*}
Here $Z$ is a real constant with $|Z| < \frac12$, the bound again
ensuring essential self-adjointness of
the underlying Hamiltonian.



Our main results, given in Corollaries \ref{cor:inv-square-phg} and
\ref{cor:dirac-coulomb-phg} below, can be stated in the following
simplified form.
\begin{theorem}
\label{metathm}
Let $\chi \in C^\infty_{c}(\mathbb{R}^n \setminus \{ 0 \})$.  For $n=3$ and $u$ and $\psi$ as defined in \eqref{eq:inv-square} and \eqref{eq:dc} respectively, we have
\[
\| u (t,\cdot ) \chi(\cdot) \|_{W^{1,\infty}} \lesssim {t^{-2 - \alpha(\digamma )}}, \ \ \| \psi (t, \cdot ) \chi(\cdot) \|_{W^{1,\infty}} \lesssim {t^{-2 - \beta(Z) }}.
\]
where the functions 
\begin{equation*}
\alpha(\digamma) = \sqrt{1 + 4 \digamma} - 1
\end{equation*} 
and 
\begin{equation*}
\beta(Z) =  2\sqrt{1 -  Z^2 } - 1
\end{equation*}
arise from blowing up near the cone point as outlined below in Section~\ref{S2}. In
addition, these decay rates are sharp.

In contrast, for $0 < \gamma < 1$, it follows from the works of \cite{coneradfield,DirCoulRF} that the solutions obey the estimates
\begin{equation*}
  \abs{u(t, \gamma t, \theta)} \lesssim
  t^{-\frac{3}{2}+\sqrt{\frac{1}{4} + \digamma}}, \quad \abs{\psi(t, \gamma t,
    \theta)} \lesssim t^{-2 + \sqrt{1 - \mathbf{Z}^{2}}}. 
\end{equation*}
\end{theorem}

We emphasize that only the first part of Theorem~\ref{metathm} is new;
the second estimate follows from the radiation field asymptotics in
the prior work~\cite{coneradfield,DirCoulRF}.  The important aspect to
note is that the two rates of decay as $t\to \infty$ \emph{differ}.
The singularity of the potential leads to slower time decay in compact
spatial subsets.  

We also note that the results above are written to compare to a slower
base time decay rate of $t^{-2}$ rather than the familiar $t^{-3}$
rate of Price's law as is expected to hold under generic sufficiently
smooth perturbations $\digamma=0$ or $Z=0$.  In fact, our
prior work \cite{DirCoulRF} yields the expected ($t^{-3}$) decay rate
in the Dirac setting with $Z=0$ due to a notable cancellation of terms
that arise in the radiation field asymptotics, which we will discuss
in more detail below.  We also refer the reader to the works of
Tataru~\cite{Tataru} and Morgan \cite{Morgan20} for an overview of the
$t^{-3}$ decay rate in the asymptotically Minkowski setting.



\begin{remark}
  We have stated the results here in terms of inverse-square
  potentials for the wave equation and the Dirac--Coulomb equation,
  but a similar result holds on product cones in $3$ spatial
  dimensions following the first and third author's work on
  \cite{coneradfield}.  Indeed, for a cone with a spherical
  cross-section of radius larger than $\sqrt{\frac83}$, we observe a
  Price's Law with a time decay slower than the typical $t^{-3}$.  By
  taking the radius sufficiently large this decay can be made slower
  than $t^{-2-\epsilon}$ for any $\epsilon > 0$.  In this case, the
  index set from the radiation field alone would give an expected decay rate of $t^{-3}$, but
  the term arising from the first non-constant eigenfunction of the
  cross-section results in the slower decay rates.  This observation
  provides some evidence of the long range nature of the interaction between diffraction and
  wave decay.
\end{remark}


By modifying the approach taken to establish radiation field
asymptotics on product cones by the first and third authors in
\cite{coneradfield}, we can also compute the resonances and radiation
fields for the Laplacian and the wave equation with inverse square
potentials respectively.  The time decay rates will depend explicitly
on these resonances in the case of the Laplacian perturbed by singular
potentials and is to that extent a manifestation of the long-range
nature of such perturbations.  As in the discussion below the main
theorem, one interesting observation following from work of the first
two authors with Booth \cite{DirCoulRF} is that there is an
improvement in decay in time for solutions the Dirac equation in
Minkowski space ($Z= 0$) in \eqref{eq:dc}.  The cancellation we
referred to is due to the absence of a purely $0$ spectral mode for
the Dirac operator, but that gain disappears when you consider the
Dirac--Coulomb equation because of the indicial roots associated with
the lowest mode in that setting.  Similar analysis shows that in the
case $\digamma<0$, the decay rate for the wave equation worsens due
precisely to singular behavior of the corresponding resonant states,
which leads to the differing behavior on compact spatial subsets.

 


The main contributions of this article are to provide a canonical microlocal
framework for proving pointwise wave decay even in the setting of singular potentials, as well as to extend Price's Law to an
interesting physical case where the long-range effects of diffraction dramatically impact the behavior of waves.  The starting point of our work is the
recent work of the first author with Wunsch and Vasy \cite{BVW1,BVW2}.
Hintz \cite{Hintz_PL} took a similarly microlocal approach to Price's
Law calculations by adapting functional analytic tools for wave
equations on compactified space-times using work of Vasy and others
\cite{vasy:microlocal}.  In addition, while the works of Hintz
\cite{Hintz_PL} and Morgan \cite{Morgan20} considered potential
perturbations of flat space, neither of those results could allow
potentials with slow decay at infinity or singularities.  See also the
recent work of Morgan-Wunsch \cite{MorganWunsch21}.  Here we present a
modified Price's Law in exactly both of those cases for this family of
relevant physical examples.  In addition, since the methods come
from a detailed analysis of the scattering matrix for these operators,
one can with relative ease construct quasimodes that prove the
sharpness of our decay estimates as stated in Theorem \ref{metathm}.

\begin{remark}
Here we work with the explicit operators of the form $\pd[t]^{2} - \Delta + \frac{\digamma}{r^{2}}$, but it is natural to ask about equations of the form  $\Box_g + \frac{\digamma}{r^{2}} + V$ for $g$ an asymptotically flat metric and $V$ a non-singular, lower order perturbation.  The effects of these perturbations would most strongly manifest at low frequencies, and hence we conjecture that the decay rates we prove here would still be the expected behavior.  Proving this would require adapting the second microlocalization tools of Vasy \cite{vasy:lag1,vasy:lag2,vasy:lag3} as implemented in the wave decay setting in \cite{Hintz_PL} to the conic setting, and will be an important topic for future work.
\end{remark}


There is a rich history of recent results on Price's Law in the
setting of stationary space times with static perturbations.  There is
a recent survey article by Schlag \cite{Sch21} that gives a much more
thorough overview of the subject than we can do here.  The work
initiated with harmonic by harmonic analysis of
Donninger--Schlag--Soffer \cite{DSS11} and Schlag--Soffer--Staubach
\cite{SSS10-I,SSS10-II}, followed by the pioneering work of Tataru
\cite{Tataru} that relied upon Fourier localization techniques.  The
methods of Tataru have now been generalized to a large number of
settings, \cite{MTT12,MTT17}, and in particular the role of regularity
of the metric was recently studied in the work of Morgan
\cite{Morgan20}.  As mentioned above, as well as the work of Hintz
\cite{Hintz_PL}, in which Price's law on stationary space-times
recently, including potentials using essentially the compactification
machinery we are using, but all potential perturbations there were
assumed to have $r^{-3}$ decay and were smooth.   A pointwise decay
estimate for the wave equation with slow decay in one spatial dimension was proved by Donninger--Schlag in \cite{DS10}, though their result has regular potentials at $x=0$ that decay slightly better than $|x|^{-2}$ at $\infty$.

Dispersive estimates and nonlinear applications for wave and Schr\"odinger equations have been studied quite broadly.  For instance, the wave equation with an inverse square potential has been studied in \cite{BPSS,PSS,MZZ}. The Schr\"odinger equation with an inverse square potential has been studied in \cite{Duy,KMVZZ1,KMVZZ2} among many others.  In particular, the distinction between $\digamma > 0$ and $\digamma < 0$ arises and there is actually a threshold $-1/4 + 1/25 < \digamma^* < 0$ for which local well-posedness theory holds for critical semilinear equations with such potentials.  In the long run, we hope that further control on pointwise behavior of waves will give insight into further nonlinear applications, see \cite{KMVZ,AM} for the most recent literature discussion on this problem to date.  

In Section \ref{S2}, we recall the compactification of our space-times
and discuss the modifications to the case of these singular
potentials. We then recall the nature of the radiation field
expansions derived previously in \cite{coneradfield,DirCoulRF}, and
prove a microlocal lemma that allows us to lift the polyhomogeneous
expansions we have in the radiation fields to prove error estimates in
our time decay estimates.  We then state and prove the main results in
Section \ref{sec:an-asympt-expans}, which follow naturally from
the results in Section \ref{S2} and the corresponding definitions of
certain indexing sets that arise in each of our applications.
Finally, in Appendix \ref{sec:reson-calc}, we provide the resonance
calculations for the case of inverse square potentials that will give
us the explicit decay rates we observe in our Price's Law statements.


\section{Compactifications and partial polyhomogeneity}
\label{S2}




For both the Dirac--Coulomb equation and the wave
  equation on cones, we proceed by defining a compactification of the
  relevant space-time on which the introduced boundaries at ``infinity'' form the
  loci at which various asymptotic behaviors of the solutions take place.  For
  both equations, the space-time decouples into the time factor
  $\mathbb{R}_t$
times a spatial factor which is a cone on a closed manifold $Z$; for
the wave equation on cones, this $Z$ is arbitrary, while for the 
Dirac--Coulomb equation the spatial slice is the polar coordinate
interior $(0, \infty)_r \times \mathbb{S}^{n-1}$.


  



Concretely, suppose $(Z,k)$ is a compact, connected
  $(n-1)$-dimensional Riemannian manifold without boundary. The metric cone on $Z$ is a
  smooth Riemannian manifold on the interior of the open manifold with
  boundary $[0, \infty)_r \times Z $
given by the metric
$$
g_{c} = dr^2 + r^2 k.
$$
The metric $g_c$ therefore
extends to a smooth tensor on $[0, \infty) \times Z$ which is
degenerate (i.e.\ not positive
definite) at the boundary $\{ 0 \} \times Z$, the locus of the conical
singularity.  We let $C(Z)$ denote the interior $(0, \infty)_r \times
Z.$   (This is nonstandard notation for the cone, which typically
includes the cone point, but it is convenient as it accords with our
usage below and in previous work.)



Thus, on the open
manifold $M^{\circ} = \mathbb{R}_{t}\times C(Z)$, we
consider the Lorentzian metric
\begin{equation*}
  g = -dt^{2} \, + dr^{2}\, + r^{2} k.
\end{equation*}
Our compactification will allow us to regard $M^{\circ}$ as the
interior of a compact manifold with corners, as follows.


We compactify $\mathbb{R}_{t}\times (0,\infty)_{r}$ (and thus
$\mathbb{R}_{t}\times (0,\infty)_{r} \times Z$) by stereographic
projection to a quarter-sphere $\mathbb{S}^{2}_{++}$ as depicted in
Figure~\ref{fig:1-d-compact}.  In other words, the map
$\mathbb{R}_{t}\times (0,\infty)_{r} \to \mathbb{S}^{2} \subset \mathbb{R}^{3}$
given by
\begin{equation*}
  (t,r) \mapsto \frac{(t,r,1)}{\sqrt{1+t^{2}+r^{2}}}
\end{equation*}
sends $M^{\circ}$ to the interior of the quarter-sphere given by
\begin{equation*}
  \mathbb{S}^{2}_{++} = \{ (z_{1}, z_{2}, z_{3}) \in \mathbb{S}^{2} \subset
  \mathbb{R}^{3} \mid z_{2} \geq 0, \ z_{3} \geq 0\}.
\end{equation*}
The quarter-sphere $\mathbb{S}^{2}_{++}$ is a manifold with corners and
has two boundary hypersurfaces defined by the boundary defining
functions $z_{2}$ and $z_{3}$.  We let $\mathrm{cf}$ (or the \emph{conic
  face}) be the hypersurface connecting NP to SP defined by the function
\begin{equation*}
  z_{2}= \frac{r}{\sqrt{1+t^{2}+r^{2}}}
\end{equation*}
and we let $\mathrm{mf}$ (or the \emph{main face}) be the face defined by
\begin{equation*}
  z_{3} = \frac{1}{\sqrt{1+t^{2}+r^{2}}}.
\end{equation*}


Equivalently, we can compactify $\mathbb{R}_t \times (0, \infty)_r$
to a half ball $D_+ = \{ w = (w_1, w_2) \in \mathbb{R}^2: |w| \le 1, w_2\ge
0 \}$ via the mapping
$$
(t, r) \mapsto \frac{(t, r)}{1 + \sqrt{1 + t^2 + r^2}}.
$$
We leave it to the interested reader to show that these
compactifications are equivalent, in the sense that they induce
diffeomorphisms of manifolds with corners.  In particular, the
boundary hypersurface (bhs)
$w_2 = 0$ corresponds to $z_2 = 0$ and the bhs $|w| = 1$ corresponds to $z_3 = 0$.



\begin{figure}
  \centering
      \def100mm{100mm}
   \input{compactification.pdf_tex}
  \caption{The compactification $M$ of $\mathbb{R}_t \times (0,\infty)_r \times Z$
    with the $Z$ factor suppressed, mapping on the left with detail on
    the right.  The image of $\mathbb{R}_t \times (0,\infty)_r$ can be thought of either as the right half disk $D_+$ or the quarter sphere
  $\mathbb{S}_{++}$.  $S_\pm$ are, respectively, collapsed future/past
null infinity.  $C_\pm$ and $C_0$ are three subsets of $\mathrm{mf}$}
  \label{fig:1-d-compact}
\end{figure}

There are several convenient coordinate systems valid in
different regions of the compactification.  For notational convenience, we will always use
$\rho$ to denote a defining function for $\mathrm{mf}$ and $x$ to denote a
defining function for $\mathrm{cf}$.  In region I (where we are bounded away
from $\mathrm{mf}$), it is convenient to take $x = r$, while in region II
(where we are bounded away from $\mathrm{cf}$), we can take $\rho = 1/r$.
Finally, in region III (which is the source of most of the new
technical work in this document), it is often convenient to take
$\rho = 1/t$ and $x = r/t$ and write,
  \begin{equation}
    \label{eq:region III decomp}
    \mbox{ region III} \simeq [0, 1)_{\rho} \times [0, c)_x \times Z,
  \end{equation}
where, again, for Dirac--Coulomb, $Z = \mathbb{S}^{n-1}$.  Here $0 < c
< 1$ is arbitrary and can be taken to be $1$, though this makes the
region more awkward to depict, and at any rate the interesting
behavior in region III not covered by region II takes place near $r/t$
small.  \emph{Because polyhomogeneity is
  independent of the choice of (equivalent) boundary defining
  function, we typically use whichever boundary defining functions are
  most convenient at the time.}

\begin{figure}
  \centering
      \def100mm{100mm}
   \input{compactification_blowup.pdf_tex}
   \caption{A schematic view of the space $X$ (on the right) we use to
     study Price's Law.  ``Region III'' is the neighborhood of
     timelike infinity in which the solutions under consideration
     exhibit distinct asymptotic behavior at the three boundary
     hypersurfaces.}
  \label{fig:rf-blowup}
\end{figure}



To understand the detailed asymptotic behavior of solutions we perform
two blow ups of the compactification $M$. The first 
introduces future null infinity, which is the natural domain of
definition of the radiation field, while the second blow up introduces
the precise region needed to distinguish between the limits of
timelike paths with zero and non-zero spatial momentum.  The latter,
as we have said, also distinguishes the two loci of differing
asymptotic behavior of solutions at timelike infinity.

We recall from previous work~\cite{BVW1,BVW2} the first blow up, which
induces the construction of the manifold with corners on which the radiation field
naturally lives.  

Recall that we denote collapsed null infinity by $S$; this is the
intersection of the closure of maximally extended null geodesics with
the introduced boundary $\mathrm{mf}$.  The submanifold is given by $S = \{ v
= \rho = 0\}$ where $v$ is a smooth function on $M$ which vanishes simply on
the sets $x = \pm 1$.  (Such $v$ can be chosen so that $v, \rho$ and
coordinates on the link $Z$ for local coordinate charts near $S$.)
This submanifold naturally
splits into two pieces according to whether $\pm t > 0$ near the
component, which we denote by $S_{\pm}$.  The complement of
$S$ in $\mathrm{mf}$ consists of three regions: the region $C_{0}$ being those
points in $\mathrm{mf}$ where $v < 0$, while the region in $\mathrm{mf}$
where $v>0$ has two components, denoted $C_{\pm}$ according to whether
$\pm t > 0$ nearby.

We now \emph{blow up} $S_+$ in $M$ by replacing it with its inward
pointing spherical normal bundle.\footnote{The reader may wish to
  consult Melrose's book~\cite{Melrose:APS} for more details on the
  blow-up construction.}  In the product cone setting, this is
equivalent to blowing up a pair of points in $\mathbb{S}^{2}_{++}$ and
then taking the product with $Y$.  This process replaces $M$ with a
new manifold $\overline{M} = [M; S_+]$ on which polar coordinates around
the submanifold are smooth; the structure of this manifold with
corners depends only on the submanifold $S_+$ (and not on the particular
choice of defining functions $v$ and $\rho$). 

The space $\overline{M}$ -- which is not yet the final resolution we
consider, as we have yet to blow up the north pole (zero
momentum future timelike infinity) -- is again a manifold with corners and has
four boundary hypersurfaces: the lifts of $C_+$, the union $C_0 \cup C_-$, the lift of $\mathrm{cf}$, again denoted $\mathrm{cf}$, and the
new boundary hypersurface consisting of the pre-image of $S_+$ under
the blow-down map.  This new boundary
hypersurface, which is essentially future null infinity,
is denoted $\mathcal{I}^{+}$.  Moreover, $\mathcal{I}^{+}$ is
naturally a fiber bundle over $S_{+}$ with fibers diffeomorphic to
intervals.  Indeed, the interior of the fibers is naturally an affine
space (i.e., $\mathbb{R}$ acts by translations, but there is no natural
origin).  Figure~\ref{fig:rf-blowup} depicts this blow-up
construction.  Given choices of functions $v$ and $\rho$, the fibers of the interior of
$\mathcal{I}^{+}$ in $\overline{M}$ can be identified with $\mathbb{R} \times Z$ via
the coordinate $s = v/\rho$.

Our focus on the $t > 0$ region of space-time is a matter of
convenience only, it is also possible, indeed very useful, to blow up
$S_-$ and thereby obtain a boundary hypersurface corresponding to past
null infinity.  Indeed, the face obtained by the blow up of $S_-$ is
the domain of the past radiation field.  For Dirac-type equations, one
can see distinct behavior when comparing $\mathcal{I}^+$ to the
corresponding face $\mathcal{I}^-$ over $S_-$, but these details are irrelevant
for our discussion here which treats only the future timelike infinity
behavior.  



Finally, we blow up the ``north pole'' ${\rm NP}= \{x = 0, \rho = 0 \}$.  We
  thereby obtain a manifold with corners $X = [\overline{M}; {\rm NP}]$ with
  an additional boundary hypersurface ${\rm tf}_+$ which separates
  $\mathrm{cf}$ from $C_+$.  Near the intersection of ${\rm tf}_+$ with $C_+$,
  we can use the boundary defining function $\rho_{{\rm tf}_+} = x +
  \rho$, while a full set of coordinates is given by
  $$
\rho_{{\rm tf}_+}, \quad y =  \frac{x -
  \rho}{x + \rho}, \quad \theta,
$$
where $\theta$ are any local coordinates on $Z$.




  

We now come to the main point of this blow up construction, which can
be understood as follows.  The solutions to the differential equations
we consider here are tempered distributions which are conormal on
$\overline{M}$ and admit an asymptotic expansion at $C_+$ with
polyhomogeneous coefficients.  In this
context, conormality \footnote{Here we choose to use $L^\infty$-based
  conormal spaces as opposed to the $H^s_b$-based spaces used in
  \cite{coneradfield} as the $L^\infty$-based spaces make the proof of
Lemma \ref{lemma:phg-lift} slightly clearer.  The equivalence of
$L^\infty$-based conormality and Sobolev space-based conormality
follows from the Sobolev embedding theorem and is discussed in
\cite{Melrose:MWC} in Chapter 4.} is equivalent to the existence of weights $\ell, \ell' \in
\mathbb{R}$ such that for all $j,  k \in \mathbb{N}_0, \alpha \in
\mathbb{N}_0^{\dim Z}$,
$$
(x \partial_x)^j (\rho \partial_\rho)^k \partial_\theta^\alpha
(x^{-\ell} \rho^{-\ell'} u)  \in L^\infty,
$$
while polyhomogeneity at $C_+$ is the condition that $u$ admits an
asymptotic expansion with terms of the form $\rho^s (\log \rho)^k
a(x, \theta)$ with $s \in \mathbb{C}$, $k \in \mathbb{N}_0$ and $k$ bounded
for $\Re s < L$.  The precise
nature of the expansion here is the key to our arguments.
Specifically, given an index set $\mathcal{E}$ (meaning a subset
$\mathcal{E} \subset \mathbb{C} \times \mathbb{N}_0$ such that
$\{ (\sigma, k) \in \mathcal{E} : \Im \sigma > -R \}$ if finite for
each $R$) we assume that for each $L \in \mathbb{R}$, in sets $\{ x < c \}$,
i.e.\ away from $\mathcal{I}_+$,
\begin{equation}\label{eq:first expansion}
  u(x, \rho, \theta) = \sum_{(\sigma, k) \in \mathcal{E}, \Im \sigma > -L} \rho^{i\sigma}(\log
  \rho)^{k}a_{\sigma k}(x, \theta) + \rho^{L} u'(x, \rho, \theta),
\end{equation}
where the remainder $u'(x, \rho, \theta)$ is a bounded conormal
function and the $a_{\sigma k}$ are themselves polyhomogeneous
distributions on $C^+$ with a fixed index set $\mathcal{F}$ at
$\mathrm{cf}$.  (This means $a_{\sigma k} \sim \sum _{(\tau, l) \in
  \mathcal{F}} x^{i \tau} (\log x)^p c_{\tau, p}$ as $x \to 0$.)  We
say that such a distribution is \textbf{partially polyhomogeneous} at
$C^+$ (near $\mathrm{cf}$) with index set $\mathcal{E}$ at $C^+$ and
$\mathcal{F}$ at $\mathrm{cf}$.  \emph{Our main observation is that such a
distribution $u$ pulls back to the blown up space $X$ to be fully
polyhomogeneous at $C^+$ with index set $\mathcal{E}$ and partially
polyhomogeneous at ${\rm tf}$ with index set $\mathcal{F} +
\mathcal{E}$ at ${\rm tf}$ and index set $\mathcal{F}$ at $\mathrm{cf}$.}
This allows us to conclude the distinct asymptotic behavior described
above.  We formalize this pullback statement in Lemma \ref{lemma:phg-lift}.


In contrast with fully polyhomogeneous distributions, which admit
expansions at every face, the partially polyhomogeneous distributions
considered here admit only expansions at $C^+$ (in region III.)

 
 
 
 



This property of partial polyhomogeneity can be seen in the radiation field expansions for both
\eqref{eq:inv-square} and \eqref{eq:dc} using the methods of proof in
\cite{coneradfield} and \cite{DirCoulRF} respectively.  Indeed, this
in the case of product cones, the result is proven directly in
\cite{coneradfield}.  Indeed, there it can be see that we have
conormality relative to the conic singularity in $x$ and
polyhomogeneity in the $\rho$. We will discuss this in more detail
below.









Let
$$
\mathbb{R}_{++}^{2} = \{ (z, w) \in \mathbb{R}^2:  z, w \ge 0 \}
$$
denote the closed upper right quadrant, let $H_1 = \{ z = 0 \}, H_2 =
\{ w = 0 \}$.  Let $v \colon
\mathbb{R}_{++}^{2} \longrightarrow \mathbb{C}$ be a conormal
distribution supported near $0$, and assume that $v$ is partially
polyhomogeneous at $H_1$ with index sets $\mathcal{E}$ at $H_1$ and
$\mathcal{F}$ at $H_{2}$. Let ${\rm ff}$ denote the new face of the blow up
$[\mathbb{R}_{++}^{2} : (0,0) ]$.
\begin{lemma}
  \label{lemma:phg-lift}
  Let $U \subset \mathbb{R}^N$ be an open set. Let $v$ be a
  distribution on $\mathbb{R}_{++}^{2} \times U$ which is partially
  polyhomogeneous at $H_1$ with index set $\mathcal{E}$ at $H_1$ and
  $\mathcal{F}$ at $H_2$.  
  Then the pullback of $v$ to the blown-up space
  $[\mathbb{R}_{++}^{2};\{(0,0)\}] \times U$ is fully polyhomogeneous at
  $H_1$ with index set $\mathcal{E}$ and partially polyhomogeneous at ${\rm ff}
  \times U$ with index set $\mathcal{E} + \mathcal{F}$ at there and index
  $\mathcal{F}$ at $H_2$.
  \end{lemma}

\begin{proof} 
  We suppress the factor $U$ in the domain as the smooth dependence on
  that parameter is irrelevant in the proof.

  By assumption, near $(0,0)$ we can write, for every $L, M \in
  \mathbb{R}$,
  \begin{equation}\label{eq:some expansions}
    \begin{split}
  v &= \sum_{(\sigma, k) \in \mathcal{E}, \Im \sigma > -L} z^{i\sigma}(\log
      z)^{k}a_{\sigma k} + z^{L} v' \\
    &= \sum_{(\sigma, k) \in \mathcal{E}, \Im \sigma > -L} z^{i\sigma}(\log
      z)^{k}\large( \sum_{(\tau, l) \in \mathcal{F}, \Im \tau > -M} w^{i\tau}
      (\log w)^l c_{\sigma k \tau l} + w^M v'' \large) + z^{L} v'
    \end{split}
  \end{equation}
  where $v', v''$ are conormal functions.


  
Blowing up, the functions $w, s = z/w$ define coordinates near the
intersection of ${\rm ff}$ with $H_1$.  The first line of \eqref{eq:some
  expansions}, with $z = sw$, continues to provide a polyhomogeneous
expansions with index set $\mathcal{E}$ but now with index set
$\mathcal{E} + \mathcal{F}$ at ${\rm ff}$ since
  \begin{align*}
  v = \sum_{(\sigma, k) \in \mathcal{E}, \Im \sigma > -L} s^{i\sigma} w^{i \sigma} (\log s +
    \log w)^k
 a_{\sigma k}(w) + (sw)^{L} v'.
\end{align*}
It is easy to check that $v'$ is conormal on
$[\mathbb{R}_{++}^{2};\{(0,0)\}]$.

Near the intersection of ${\rm ff}$ with $H_2$, $z, t = w/z$ define
coordinates, with $z$ a defining function of ${\rm ff}$ and $t$ a defining
function of $H_2$, so using the second line of \eqref{eq:some expansions} we
get
$$
v= \sum_{(\sigma, k) \in \mathcal{E}, \Im \sigma > -L \atop (\tau, l) \in \mathcal{F}, \Im \tau> -M} z^{i\sigma + i \tau}(\log
z)^{k} t^{i\tau}(\log z + \log t)^l c_{\sigma k \tau l} + z^R v'''
$$
where $v'''$ is a conormal distribution and
$$
R < \inf\{ L , M - \Im \sigma - \Im \tau : (\sigma, 0) \in
\mathcal{E}, (\tau, 0) \in \mathcal{F} \}.
$$
This gives the asymptotic expansion with index set $\mathcal{E} +
\mathcal{F}$ in $z$ with terms having expansions in index set
$\mathcal{F}$ in $t$, and choosing $M, L$ sufficiently negative gives
the result.



\end{proof}







\section{Asymptotic
    expansions for equations with inverse square potentials}
\label{sec:an-asympt-expans}



In this section we recall relevant versions of the main results of previous
papers~\cite{coneradfield,DirCoulRF}.  We first turn our attention to the
earlier paper~\cite{coneradfield} in the context of the wave equation on
$\mathbb{R} \times \mathbb{R}^{n}$ with an inverse square potential.  That
paper describes the asymptotics of solutions of the wave equation on
cones, but the techniques apply with no change to the wave equation on
$\mathbb{R}^{n}$ with an inverse square potential.  In particular, the
following theorem is an immediate consequence of those results:
\begin{theorem}
  \label{thm:inv-square}
  Let $u$ denote the solution of the wave equation with inverse square
  potential~\eqref{eq:inv-square} with smooth, compactly supported
  initial data.  If $\chi \in C^{\infty}_{c}([0,\infty))$ is supported
  in $[0, 1)$, then for all $L$, with the local
    coordinates coming from \eqref{eq:region III decomp},
  \begin{equation*}
    \chi(x) u = \sum_{\Im\sigma_{j,k} > -L}
    \rho^{\frac{n-1}{2} + i\sigma_{j,k}}\chi(x) v_{j,k}(x, \theta) +
    \rho^{\frac{n-1}{2} + L}u',
  \end{equation*}
  where $u' \in \conormal{1}$.  Here $\sigma_{j,k}$ are indexed by
  $j,k = 0,1,2,\dots$ and are given by
  \begin{equation*}
    \sigma_{j,k} = -i \left( \frac{1}{2} +
      \sqrt{\left(\frac{n-2}{2}\right)^{2} + \lambda_{j} + \digamma} + k\right),
  \end{equation*}
  when $ \sqrt{\left(\frac{n-2}{2}\right)^{2} + \lambda_{j} + \digamma}  \notin \frac12 + \mathbb{Z}$, where $\lambda_{j}$ are the eigenvalues of the spherical Laplacian.
  Similarly, $v_{j,k}$ are given explicitly in terms of hypergeometric
  functions and eigenfunctions $\phi_{j}$ of the spherical Laplacian:
  \begin{equation*}
    v_{j,k} = x^{-\frac{n-2}{2} + \sqrt{\left(
          \frac{n-2}{2}\right)^{2} + \lambda_{j} + \digamma}} F(a,b,c;
    x) \phi_{j}(\theta),
  \end{equation*}
  and $F(a,b,c;x)$ is Gauss's hypergeometric function with parameters
  \begin{align*}
    a &= \frac{1}{2} + \sqrt{\left( \frac{n-2}{2}\right)^{2} +
        \lambda_{j} + \digamma},\\
    b &= 1+ k + 2 \sqrt{\left( \frac{n-2}{2}\right)^{2} +
        \lambda_{j} + \digamma} ,\\
    c &= 1 + 2 \sqrt{\left( \frac{n-2}{2}\right)^{2} +
        \lambda_{j} + \digamma}.
  \end{align*}
\end{theorem}

Of considerable importance is that the remainder term in
Theorem~\ref{thm:inv-square} lies in a \emph{fixed} conormal space
independent of $L$.  

Thus, to bring Lemma \ref{lemma:phg-lift} to bear on solutions for the
inverse square potential, we define the index sets
\[
\mathcal{E} = \left\{  - i \left( \frac{n}{2} + k + \sqrt{  \left(  \frac{n-2}{2}
    \right)^2 + \lambda_j + \digamma }  \right) : k = 0, 1, 2, \dots \right\}
\]
when $ \sqrt{\left(\frac{n-2}{2}\right)^{2} + \lambda_{j} + \digamma} \notin  \frac12 + \mathbb{Z}$ and 
\[
\mathcal{F} = \left\{ - i \left( \ell  -\frac{n-2}{2}  + \sqrt{  \left(
      \frac{n-2}{2} \right)^2 + \lambda_j + \digamma} \right) : \ell =
0, 1, 2, \dots \right\}
\] 
using the resonance calculation in Appendix \ref{sec:reson-calc}.
(Here there are no logarithms in the expansion so we simply drop the
$\mathbb{N}_0$ factor from the index set definition.)
Theorem \ref{thm:inv-square} states that $u$
on the blown $\overline{M}$ is partially
is partially polyhomogeneous at $C_+$ with index set $\mathcal{E}$ at
$C_+$ and $\mathcal{F}$ at $\mathrm{cf}$.  In particular, from Lemma \ref{lemma:phg-lift}, we conclude
\begin{corollary}
  \label{cor:inv-square-phg}
  Functions $u$ as in the statement of Theorem \ref{thm:inv-square}
  pulled back to the blown up space $X$ from Section \ref{S2} 
  are fully polyhomogeneous at $C_+$ with index set $\mathcal{E}$ and
  partially polyhomogeneous at ${\rm tf}_+$ with index set
  $\mathcal{E} + \mathcal{F}$ at ${\rm tf}_+$ and $\mathcal{F}$ at $\mathrm{cf}$.
\end{corollary}
Looking at the index set $\mathcal{E} + \mathcal{F}$, we have that the leading order decay rate is
\[
1 + \sqrt{ 1 + 4 \digamma},
\]
in particular the part of Theorem \ref{metathm} regarding solutions to
the wave equation with the inverse square potential holds.

Each term in the expansion given by the index set  $\mathcal{E} +
\mathcal{F}$ in both cases above is generically non-vanishing by
arguments similar to those demonstrating sharpness in \cite{Hintz_PL},
Section $5$. The Price's Law decay rates in our theorem correspond to
only the leading order term of the index set, hence the sharpness is
determined by a single, linear condition.  This term is non-zero
precisely when the pairing of the data with the leading order
co-resonant state associated to the leading order term in
$\mathcal{E}$ is non-vanishing, which is a generic condition.







Using the results of \cite{DirCoulRF},we obtain
\begin{theorem}
  \label{thm:dirac-coulomb}
  If $\psi$ is the solution to the massless Dirac--Coulomb
  system~\eqref{eq:dc} with smooth, compactly supported
  initial data.  If $\chi \in C^{\infty}_{c}([0,\infty))$ is
  supported in $[0,1)$, then for all $L$,
  \begin{equation*}
    \chi(x) \psi (\rho, x, \theta) = \sum_{\Im\varsigma_{\kappa, \ell} > -L}
    \rho^{1+i\varsigma_{\kappa, \ell}}\chi(x)\varphi _{\kappa, \ell}(x,\theta) + \rho^{1+L}\psi',
  \end{equation*}
  where $\psi'\in \conormal{1}$.  The numbers $\varsigma_{\kappa, \ell}$ are
  indexed by $\kappa \in \mathbb{Z} \setminus \{0\}$ and $\ell =0, 1,
  2, \dots$ and given by
  \begin{equation*}
    \varsigma_{\kappa, \ell} = - i \left( 1 + \ell + \sqrt{\kappa^{2}
        - \mathbf{Z}^{2}}\right).
  \end{equation*}
  Similarly, the $\varphi_{\kappa, \ell}$ are given explicitly in
  terms of spinor spherical harmonics $\Omega_{\kappa}$ and
  hypergeometric functions:
  \begin{equation*}
    \varphi_{\kappa, \ell} = x^{\sqrt{\kappa^{2}-\mathbf{Z}^{2}}-1}\left( F(a,b,c; x)
      \begin{pmatrix}
        \Omega_{\kappa}(\theta) \\ \Omega _{-\kappa}(\theta)
      \end{pmatrix}
      + F(a+1, b, c; x)
      \begin{pmatrix}
        \Omega _{\kappa}(\theta) \\ -\Omega _{-\kappa}(\theta)
      \end{pmatrix}
    \right)
  \end{equation*}
  with
  \begin{equation*}
    \begin{gathered}
      a = \sqrt{\kappa^{2}-\mathbf{Z}^{2}} - i\mathbf{Z}, \qquad  b= 1 + 2\sqrt{\kappa^{2}-\mathbf{Z}^{2}} + \ell \\
      c = 1 + 2\sqrt{\kappa^{2}-\mathbf{Z}^{2}}.
    \end{gathered}
  \end{equation*}
\end{theorem}

Thus, for the case of Dirac--Coulomb we set
\[
\mathcal{E}' = \left\{  - i \left( 1 + \ell + \sqrt{ \kappa^2 -Z^2} \right) :
  \ell \in \{  0, 1, 2, \dots \}, \kappa \in \mathbb{Z} \setminus \{ 0
\}\right\}
\]
and
\[
\mathcal{F}' = \left\{  - i \left( j + \sqrt{  k^2 - Z^2 } - 1 \right) :
 j \in \{ 1, 2, \dots \},  \kappa \in \mathbb{Z} \setminus \{ 0 \}\right\}.
\] 
Looking at the index set $\mathcal{E}' +
\mathcal{F}'$, we have that the leading order decay rate is
\[
1 + 2\sqrt{ 1 - Z^2}.
\]

Again from Lemma \ref{lemma:phg-lift} we have the following corollary,
which implies the part of Theorem \ref{metathm} related to solutions
to the Dirac--Coulomb equation.
\begin{corollary}
  \label{cor:dirac-coulomb-phg}
    Functions $\psi$ as in the statement of Theorem \ref{thm:dirac-coulomb}
  pulled back to the blown up space $X$ from Section \ref{S2} 
  are fully polyhomogeneous at $C_+$ with index set $\mathcal{E}'$ and
  partially polyhomogeneous at ${\rm tf}_+$ with index set
  $\mathcal{E}' + \mathcal{F}'$ at ${\rm tf}_+$ and $\mathcal{F}'$ at $\mathrm{cf}$.
\end{corollary}


 






