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On page 4 of this paper of Ibrahim, Majdoub, and Masmoudi, the authors claim in Proposition 2 that solutions to

$\left\{\begin{array}{ll}\square u=F(t,x)\\ u(0,x)=f(x), \partial_tu(0,x)=g(x)\end{array}\right.$

enjoy the estimate

$||u||_{L_T^4(C^{1/4}(\mathbb{R}^2))}\le C[||g||_{L^2(\mathbb{R}^2)}+||f||_{H^1(\mathbb{R}^2)}+||\square u+u||_{L_T^1(L^2(\mathbb{R}^2))}]$

They reference a paper of Ginibre and Velo, which present a series of estimates which do not resemble the estimate cited above. I'm a bit confused on how the authors derived their estimate from Ginibre and Velo, though I recognize they could be using the fact that the $\dot{B}_{\infty,\infty}^{\alpha}$ and $C^{\alpha}$ norms coincide. I've only encountered these estimates in their Sobolev form, and can't see where they're getting the $\square u+u$ term on the right, so I hope you'll forgive me if I'm not seeing the forest for the trees.

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  • $\begingroup$ The $\Box u + u$ term makes me think that they are using the Klein-Gordon version of the Strichartz estimates. You should check whether Ginibre and Velo's paper says something about that. $\endgroup$
    – Willie Wong
    Commented Aug 20, 2014 at 15:51
  • $\begingroup$ Right, the estimate below is the estimate for KG, but the equation is a wave equation. Maybe this is explained in the paper? $\endgroup$
    – Piero D'Ancona
    Commented Aug 21, 2014 at 9:24

1 Answer 1

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Look at Figure 3 in the paper by Ginibre-Velo, point $C_2$ is the estimate of $\|u\|_{L^4\dot B^{1/4}_{\infty,2}}$ in terms of the $L^1L^2$ norm of $F$, and one has $\dot B^{1/4}_{\infty,2}\hookrightarrow \dot B^{1/4}_{\infty,\infty}$.

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  • $\begingroup$ Thank you for your helpful comment. I see now how they're getting the exponents, though I'm still confused about the $\square u+u$ term on the right hand side. $\endgroup$
    – PDELearner
    Commented Aug 20, 2014 at 17:57
  • $\begingroup$ The estimates for Klein-Gordon are more complex. However all cases which are true for the wave equation are also true for KG, only you must replace homogeneous norms with nonhomogeneous ones. $\endgroup$
    – Piero D'Ancona
    Commented Aug 21, 2014 at 9:23

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