论文标题
对称函数的傅立叶延伸估计以及对非线性Helmholtz方程的应用
Fourier-extension estimates for symmetric functions and applications to nonlinear Helmholtz equations
论文作者
论文摘要
We establish weighted $L^p$-Fourier-extension estimates for $O(N-k) \times O(k)$-invariant functions defined on the unit sphere $\mathbb{S}^{N-1}$, allowing for exponents $p$ below the Stein-Tomas critical exponent $\frac{2(N+1)}{N-1}$.此外，在任意封闭的子组$ g \ subset o（n）$和$ g $ invariant函数的更通用设置中，我们研究了加权傅立叶扩展估计值在相应加权Helmholtz分辨率的相应加权Helmholtz Resolvent操作员方面的含义。最后，我们使用这些属性来得出$ g $ invariant解决方案的新生存结果 $$ - ΔU -u = q（x）| u |^{p -2} u，\ quad u \ in w^{2，p}（\ mathbb {r}^{n}），$ q $是$ q $是一个非内部边界和$ g $ g $ invariant and -invariant stranvariant stranvariant权重函数。
We establish weighted $L^p$-Fourier-extension estimates for $O(N-k) \times O(k)$-invariant functions defined on the unit sphere $\mathbb{S}^{N-1}$, allowing for exponents $p$ below the Stein-Tomas critical exponent $\frac{2(N+1)}{N-1}$. Moreover, in the more general setting of an arbitrary closed subgroup $G \subset O(N)$ and $G$-invariant functions, we study the implications of weighted Fourier-extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for $G$-invariant solutions to the nonlinear Helmholtz equation $$ - Δu - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}), $$ where $Q$ is a nonnegative bounded and $G$-invariant weight function.
