Strichartz estimates over cones - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:01:39Z http://mathoverflow.net/feeds/question/92245 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92245/strichartz-estimates-over-cones Strichartz estimates over cones Guillermo 2012-03-26T09:04:28Z 2012-03-26T09:10:55Z <p>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:</p> <p><code>$\|u\|_{L^4 L^{12}(K)} \lesssim \|\nabla u(0)\|_{L^2} + \|\partial_t u(0)\|_{L^2} + \|F\|_{L^1 L^2 (K)},$</code></p> <p>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.</p> <p>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 <strong>smoothly</strong>, so I end up with a larger cone on the right hand side. How can I use Huygens' principle to prove this?</p> <p>Thanks</p>