I'm trying to understand Sogge's book Lectures on Non-Linear Wave Equations, the part where he proves global existence for semilinear equations. There is one part he uses the following inequality:

$\|u\|_{L^4 L^{12}(K)} \lesssim \|\nabla u(0)\|_{L^2} + \|\partial_t u(0)\|_{L^2} + \|F\|_{L^1 L^2 (K)},$

where $K$ is the cone $\{(t,x) : |x-x_0| \leq t_0 -t\}$ and $u$ solves $\square u = F$. If $K$ was $[0,T]\times \mathbb{R}^d$, then this would just be Strichartz's inequality for the wave equation. He says that we can put a cone instead of $[0,T]\times \mathbb{R}^d$ by Huygens' principle, but I don't know exactly how he's able to do that.

I tried localizing the initial data to the ball $B(x_0,t_0)$, which by finite speed of propagation will yield the inequality, but I can only localize **smoothly**, so I end up with a larger cone on the right hand side. How can I use Huygens' principle to prove this?

Thanks

a prioriestimate. That is, the estimate does not care whether $u$ solves $\Box(\cdot) = (\cdot)^5$ or it solves $\Box (\cdot) = G$ where $G$ just happens to be numerically equal to $(u)^5$. – Willie Wong Mar 26 '12 at 11:56