AI 中文总结
该研究针对带$C^r$系数的粗糙波动方程，利用双线性限制估计证明局部光滑化估计，在二维和三维的$C^{1,1}$系数情形恢复了光滑系数下的最优界，并推导出非平凡的$L^p$-$L^p$局部光滑化估计。
AI 中文摘要
我们针对带有$C^{r}$系数的波动方程证明了局部光滑化估计，该估计依赖于与粗糙波传播相关的双线性限制估计。在二维和三维空间中，对于$C^{1,1}$系数，我们得到了此前针对光滑系数建立的最优$L^p$-$L^q$局部光滑化界。这些结果由带有端点导数损失的更强的$L^{p}$-$L^{q}$解耦不等式推导得出，作为推论，我们还得到了非平凡的$L^{p}$-$L^{p}$局部光滑化估计。
英文摘要
We prove local smoothing estimates for wave equations with $C^{r}$ coefficients, relying on bilinear restriction estimates associated with rough wave propagation. In two and three dimensions, for $C^{1,1}$ coefficients we recover the sharp $L^p$-$L^q$ local smoothing bounds previously established for smooth coefficients. These are implied by stronger $L^{p}$-$L^{q}$ decoupling inequalities with endpoint derivative loss. As a corollary, we obtain non-trivial $L^{p}$-$L^{p}$ local smoothing estimates as well.
Comments57 pages, 3 figures
