论文标题
$ l^p $估算具有特定$ c^{0,1} $系数的波动方程
$L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients
论文作者
论文摘要
Peral/Miyachi在固定时间$ l^{p} $估算的庆祝定理，波动方程的衍生物损失指出，操作员$（i-Δ）^{ - \fracα{2}} \ exp（i \ sqrt（i \ sqrt { - δ}）$ bon上$ l^{p} $ l^{p}（\ s）只有$α\ geq s_ {p}：=（d-1）| \ frac {1} {p} - \ frac {1} {2} {2} | $。我们将此结果扩展到表单$ \ Mathcal {l} = - \ sum \ limits _ {j = 1} ^{d} a_ {j+d} \ partial_ { $ a_ {j+d} $仅取决于$ x_ {j} $，在上方和下方是限制的，但仅是Lipschitz的连续。这在$ c^{1,1} $规则性下方，对于dimension $ d \ geq 2 $中的strichartz估算是必不可少的。我们的证明是基于Hassell，Rozendaal和第二作者最近开发的傅立叶积分运营商的界限方法的方法。我们构造了一个适应性的耐寒空间的比例，$ \ exp（i \ sqrt {\ mathcal {l}}）$通过举起$ l^{p} $函数到帐篷空间$ t^{p，2}（\ mathbb {\ mathbb {r}^d}^{r}^{d} met tronfuse tose tose tose tose tose tose tose tose upt tose tose tose tose tose tose tose upt tons tose upt tose posect，系数$ a_j $。然后，结果来自这些空间的Sobolev嵌入性能。
Peral/Miyachi's celebrated theorem on fixed time $L^{p}$ estimates with loss of derivatives for the wave equation states that the operator $(I-Δ)^{- \fracα{2}}\exp(i \sqrt{-Δ})$ is bounded on $L^{p}(\mathbb{R}^{d})$ if and only if $α\geq s_{p}:=(d-1)|\frac{1}{p}-\frac{1}{2}|$. We extend this result to operators of the form $\mathcal{L} = -\sum \limits _{j=1} ^{d} a_{j+d}\partial_{j}a_{j}\partial_{j}$, such that, for $j=1,...,d$, the functions $a_{j}$ and $a_{j+d}$ only depend on $x_{j}$, are bounded above and below, but are merely Lipschitz continuous. This is below the $C^{1,1}$ regularity that is known to be necessary in general for Strichartz estimates in dimension $d \geq 2$. Our proof is based on an approach to the boundedness of Fourier integral operators recently developed by Hassell, Rozendaal, and the second author. We construct a scale of adapted Hardy spaces on which $\exp(i\sqrt{ \mathcal{L}} )$ is bounded by lifting $L^{p}$ functions to the tent space $T^{p,2}(\mathbb{R}^{d})$, using a wave packet transform adapted to the Lipschitz metric induced by the coefficients $a_j$. The result then follows from Sobolev embedding properties of these spaces.
