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\begin{document}

\title[Stein's counterexample]{Stein's counterexample and the $n=3$ endpoint
Strichartz estimate for the wave equation}
\author{Terence Tao}
\address{Department of Mathematics, UCLA, Los Angeles, CA 90024}
\email{tao@@math.ucla.edu}
\subjclass{35J05}

\begin{abstract}
In this note we show that Stein's example gives a concrete counterexample
to the false $L^2_t L^\infty_x$ estimate for the wave equation in 3 dimensions.
\end{abstract}

\maketitle

\section{Introduction}

For all test functions $f,g$ in $\R^3$, let $u$ denote the solution
to the homogenous Cauchy problem for the wave equation
$$ \left\{\begin{array}{ll}
(-\frac{\partial^2}{\partial t^2} + \Delta) u(t,x) = F(t,x),& (t,x) \in \R
\times \R^3\\
u(0,\cdot) = f,\ \partial_t u(0,\cdot) = g;&\\
\end{array}\right.$$

One has the explicit formula
$$ u(x,t) = \frac{\partial}{\partial t} t
\int_{S^2} f(x + t\omega)\ d\omega + t \int_{S^2}
g(x + t\omega)\ d\omega
$$
for $u$ (see \cite{sogge:wave}).

We consider all global smoothing (or Strichartz) estimates of the form

\begin{equation} \label{strichartz}
 \|u\|_{L^q_t L^r_x} \lesssim \|f\|_{H^\gamma} + \|g\|_{H^{\gamma -1}}.
\end{equation}

For simplicity we restrict ourselves to the case $f=0$, to avoid derivatives
and cancellation in the explicit solution for $u$.

In the wave equation space and time scale the same way, and so the
left hand side has $\frac{1}{q} + \frac{3}{r}$ dimensions, while the
right-hand side is mixed, with $\frac{3}{2}$ to $\frac{3}{2}-\gamma$
dimensions.  Thus scaling considerations force one to have
$$ \frac{3}{2} - \gamma \leq \frac{1}{q} + \frac{3}{r} \leq \frac{3}{2}.$$
The latter condition will turn out to be redundant, and it is only the
former restriction
\begin{equation} \label{scaling}
\gamma \geq \frac{3}{2} - \frac{1}{q} - \frac{3}{r}
\end{equation}
which is important.  This condition can also be picked up by
considering the focus example
$$ g(x) = \delta(|x| - 1)$$
(or some suitable thickening of this example).

The Knapp example
$$ g(\underline x, x_3) = \psi(2^{k/2} \underline x, 2^k x_3)$$
for any non-negative bump function $\psi$,
gives another restriction on the exponents:
\begin{equation} \label{knapp}
\frac{1}{q} + \frac{1}{r} \leq \frac{1}{2}.
\end{equation}

It turns out that these conditions are (with the exception of $q=2, r=
\infty$) also sufficient to obtain \eqref{strichartz}.  In fact if
\eqref{scaling} is satisfied with equality then one can even replace
the inhomogeneous Sobolev spaces $H^\gamma$, $H^{\gamma - 1}$ with their
homogeneous counterparts.  See ~\cite{lindbladsogge:semilinear},~\cite{kapitanski:variablestrichartz}, ~\cite{mss:localsmoothing},
~\cite{ginebre:summarywave}, ~\cite{sogge:wave}, ~\cite{damiano},
~\cite{tao:keel}.  For $n \neq 3$ the corresponding endpoint estimate is
true, and the corresponding scaling and Knapp conditions are necessary and 
sufficient for
\eqref{strichartz} to be true; see \cite{tao:keel}.

The estimate when $q=2$, $r=\infty$ is known to be false for any
value of $\gamma$, although the usual proofs (\cite{klainerman:nulllocal}, 
~\cite{montgomery-smith}) are somewhat indirect
(e.g. showing that the $TT^*$ version of the estimate fails).  We first
consider the critical case $\gamma = 1$:

$$\|u\|_{L^2_t L^\infty_x} \lesssim \|g\|_2.$$

An adaptation of Stein's counterexample for the spherical maximal function
(see \cite{stein:large}, p. 472) shows that this estimate fails quite
concretely, and the solution can concentrate on the lightray $(t, te_3)$.  

\begin{proposition} If $g$ is the function
$$ g(x) = \frac{\chi_A(x)}{|x|^2 \log|x|}$$
where $A$ is the set
$$A = \{x: |x-e_3| \geq 1, |x-2e_3| \leq 2\}$$,
then $g$ is in $L^2$, but $u(t, te_3)$ is infinite for all $1 < t < 2$.
\end{proposition}

Thus, $g$ is supported on a set which has a cusp at the origin, and grows
to infinity at a carefully controlled rate there.

We leave the proof of this proposition as an easy exercise.  The
the portion of $g$ in the dyadic shell $\{|x| \sim 2^{-k}\}$ lives in
a set of volume $2^{-4k}$ and has a height of $2^{2k}/k$, and the intersection
of this set with any sphere of radius $t$ centered at $te_3$ has surface
area roughly $2^{-2k}$.  Thus
the $L^2$ norm is roughly $1/k$, which sums in $l^2$, but the contribution
to $u$ on the light ray is also roughly $1/k$, which is not summable.

One might think that this failure can be circumvented by placing more
derivatives on the data:
$$\|u\|_{L^2_t L^\infty_x} \lesssim \|g\|_{H^s},\ s \hbox{ large.} .$$
However, the estimate still fails, basically because the $L^2$ estimate
is scale invariant.  By approximating the above counterexample one can
find a Schwarz function $g$ which has bounded
$L^2$ norm, but the $L^2 L^\infty$ norm of $u$ is arbitrarily large.  Now
replace $g$ by 
$$g_\lambda(x) = \lambda^{-3/2} g(x/\lambda)$$
and let $\lambda \to \infty$.  The $L^2$ norm of $g$ and the $L^2 L^\infty$
norm of $u$ remain constant, while the $H^s$ norm of $g$ tends to the
$L^2$ norm of $g$.  Thus the above estimate also fails.


\begin{thebibliography}{10}

\bibitem{damiano}
D. Foschi, \emph{Lecture Notes for S. Klainerman's Graduate Course In
Nonlinear Wave Equations:  Fall 1996, Princeton University},
Private Communication.

\bibitem{ginebre:summarywave}
J. Ginebre, G. Velo, \emph{Generalized Strichartz Inequalities for the
Wave Equation}, Jour. Func. Anal., \textbf{133} (1995), 50--68.

\bibitem{kapitanski:variablestrichartz}
L. Kapitanski, \emph{Some Generalizations of the Strichartz-Brenner
Inequality}, Leningrad Math. J., \textbf{1} (1990), 693--676.

\bibitem{tao:keel}
M. Keel, T. Tao, \emph{Endpoint Strichartz Estimates}, To appear, Amer. Math. J.

\bibitem{klainerman:nulllocal}
S. Klainerman, M. Machedon, \emph{Space-time Estimates for
Null Forms and the Local Existence Theorem}, Comm. Pure Appl.
Math., \textbf{46} (1993), 1221--1268.

\bibitem{lindbladsogge:semilinear}
H. Lindblad, C.~D. Sogge, \emph{On Existence and Scattering with Minimal
Regularity for Semilinear Wave Equations}, Jour. Func. Anal., \textbf{130} (1995
),
357--426.

\bibitem{montgomery-smith}
S.~J. Montgomery-Smith, \emph{Time Decay for the Bounded Mean
Oscillation of Solutions of the Schr\"odinger and Wave Equation},
Preprint, 1996.

\bibitem{mss:localsmoothing}
G. Mockenhaupt, A. Seeger, C.~D. Sogge,  \emph{Local Smoothing of
Fourier Integrals and Carleson-Sj\"olin Estimates},
J. Amer. Math. Soc., \textbf{6} (1993), 65--130.

\bibitem{sogge:wave}
C.~D. Sogge, \emph{Lectures on Nonlinear Wave Equations}, Monographs in Analysis
 II, International Press, 1995.

\bibitem{stein:large}
E.~M. Stein, \emph{Harmonic Analysis}, Princeton University Press, 1993.


\end{thebibliography}

\end{document}
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