L^1-L^∞ Decay Estimates for the Odd-Dimensional Wave Equations

We will show the following estimates.

For $n = 1,3,\cdots$, let $\phi \in C_0^{n} (\Bbb R^n), \; \psi \in C_0^{n-1} (\Bbb R^n) $. And let $v$ solve
\begin{equation}\partial_t^2 v – \Delta v = 0 \;\text{in}\; [0,\infty) \times \Bbb R^n \;\; \text{ with } \;\; v(t=0) = \phi, \; \partial_t v (t=0) = \psi.\end{equation} Then there exists $c = c(n)>0$ such that for $t \geqslant 0$,
\begin{equation} \| v(t) \|_\infty \leqslant c(1+t)^{- \frac{n-1}{2}} ( \| \phi \|_{n,1} + \| \psi \|_{n-1,1} ).
\end{equation}

In fact, for the even dimensions, the similar estimates hold. I will post this later with the case of non-vanishing mass for the Klein-Gordon equation.

Wave_L1-Linfty_decay_estimate_odd.pdf

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