I am trying to deduce the dispersive estimate for the free wave equation in $\mathbb{R}^{d+1}\equiv\{(x,t) : x\in\Bbb R^d \wedge t\in\Bbb R\}$ $$u_
{tt}-\Delta_xu=0$$ The fundamental solutions of this equation are given by 
$$
\begin{align}
K_t^0(x) &=\cos(t|\nabla|)\delta(x), \\
\\
K_t^1(x)&=\frac{\sin(t|\nabla|)}{|\nabla|}\delta(x).
\end{align}$$ 
According to Tao's dispersive pde textbook, given a Scharwtz function $\phi(x)$, we have the following time decay
$$|K^0_t*\phi_\lambda(x)|\lesssim_{\phi,d}\lambda^d\langle\lambda t\rangle^{-\frac{d-1}{2}}\;,$$
where $\langle y\rangle=(1+|y|^2)^\frac{1}{2}$.
I am trying to deduce the above estimate using stationary phase method. Rewrite the LHS with Fourier transform, we get
$$\text{LHS}=\int_{\mathbb{R}^d}e^{it|\xi|+ix\cdot\xi}\hat\phi(\lambda^
{-1}\xi) d\xi$$
Since $D^2|\xi|$ degenerates in radial direction, in order to apply stationary phase I rewrote it in polar coordinates, where $\xi=rw, w\in S^{d-1}$
$$\int_0^\infty e^{itr}r^{d-1}\hat\phi(\lambda^{-1}r)\int_{S^
{d-1}}e^{irw\cdot x}dw dr$$
But I am not sure how to proceed from here.