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Multi-scale bilinear restriction estimates for general phases

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Abstract

We prove (adjoint) bilinear restriction estimates for general phases at different scales in the full non-endpoint mixed norm range, and give bounds with a sharp and explicit dependence on the phases. These estimates have applications to high-low frequency interactions for solutions to partial differential equations, as well as to the linear restriction problem for surfaces with degenerate curvature. As a consequence, we obtain new bilinear restriction estimates for elliptic phases and wave/Klein–Gordon interactions in the full bilinear range, and give a refined Strichartz inequality for the Klein–Gordon equation. In addition, we extend these bilinear estimates to hold in adapted function spaces by using a transference type principle which holds for vector valued waves.

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Notes

  1. This exploits the bound

    $$\begin{aligned} ( 1 + C 2^{-\frac{\alpha }{C}m}) \times ( 1 + C 2^{-\frac{\alpha }{C}(m-1)}) \times \cdots \times ( 1 + C) \lesssim 1\end{aligned}$$

    which follows by taking logs, and recalling the elementary estimate \(\log (1+x) \leqslant x\).

  2. Explicitly, in the region \(|\partial _1\Phi _1 - \partial _1 \Phi _2| \approx |\nabla \Phi _1 - \nabla \Phi _2|\), there exists a function \(\psi (\tau , \xi , \eta '):\mathbb {R}\times \mathbb {R}^n \times \mathbb {R}^{n-1} \rightarrow \mathbb {R}\) such that

    $$\begin{aligned} \Phi _1\big (\xi - (\psi , \eta ')\big ) + \Phi _2(\psi , \eta ') = \tau . \end{aligned}$$

    Thus we can write the surface as a graph \(\Sigma _2({\mathfrak {h}})= \{ (\psi (\eta '), \eta ') \in (h-\Lambda _1) \cap \Lambda _2\}\), and hence the surface measure is then \( d\sigma (\eta ) = \sqrt{1 + |\nabla _{\eta '} \psi |^2} d\eta ' = \frac{|\nabla \Phi _1 - \nabla \Phi _2| }{|\partial _1 \Phi _1 - \partial _1 \Phi _2|} d\eta '\).

  3. Since \(\chi _q\) has compact Fourier support, it can only have countable number of zeros. In particular, \(\chi _q>0\) almost everywhere.

  4. Just use the identities \(X[Q(t,x)] = X[Q(0)] + (t,x)\) and \(-X[Q(0)] = X[-Q(0)] = X[Q(0)]\).

  5. This is simply an application of complex interpolation. In more detail, we just repeat the proof of the Riesz–Thorin interpolation theorem, thus given a functions \(g \in L^2_t(\mathbb {R})\) and \(G \in L^\infty _t L^{r'}_x(\mathbb {R}^{1+n})\), we define the function \(\rho (z)\) for \(z\in \mathbb {C}\) as

    where \(\frac{1}{a_0} = \frac{ \frac{1}{r} + \frac{1}{a}-1}{\frac{2}{r} -1}\) and \(\frac{1}{b_0} = \frac{ \frac{1}{r} + \frac{1}{b}-1}{\frac{2}{r} -1}\). It is easy to check that \(\rho \) is complex analytic when \(0\leqslant \mathfrak {R}(z)\leqslant 1\), is at most of exponential growth, and \(\rho (0)\) can be bounded by the \(L^2_{t,x}\) estimate, \(\rho (1)\) by the \(L^2_t L^1_x\) estimate. Hence interpolated bound follows from the Hadamard Three-Lines Theorem or Lindelöf’s Theorem.

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Acknowledgements

The author would like to thank Sebastian Herr and the referee for a number of helpful comments and corrections.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, Dunedin, 9054, New Zealand

    Timothy Candy

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  1. Timothy Candy
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Correspondence to Timothy Candy.

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Communicated by Loukas Grafakos.

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Candy, T. Multi-scale bilinear restriction estimates for general phases. Math. Ann. 375, 777–843 (2019). https://doi.org/10.1007/s00208-019-01841-4

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