论文标题
在球体和Helmholtz方程的Sobolev空间中分布的傅里叶扩展运算符
The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation
论文作者
论文摘要
本文的目的是表征同质Helmholtz方程的所有整个解决方案（在$ \ Mathbb {r}^d $中的解决方案）是由Sobolev Sobolev sobolev sobolev sobolev sobolev sobolev sobolev sobolev sobolev sobolev sobolev sobher $ h^α（\ mathbb {s}}^{d-d-1}^{d-d-1}的分布的运算符，$ n和$ a \ n $ n n v in。我们提出两个特征。第一个是用某些$ l^2 $加权的规范来编写的，涉及球形拉普拉斯的真实力量。第二个是本着P. Hartman和C. Wilcox给出的Herglotz Wave功能的经典描述的精神。对于$α> 0 $，此特征涉及在Helmholtz方程的整个解决方案的向量中评估的多变量平方函数，而对于$α<0 $，它是用球形积分运算符编写的，该函数用作分数集成运算符。最后，我们还表征了球体中分布的傅立叶扩展运算符的所有解决方案。
The purpose of this paper is to characterize all the entire solutions of the homogeneous Helmholtz equation (solutions in $\mathbb{R}^d$) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^α(\mathbb{S}^{d-1}),$ with $α\in \mathbb{R}$. We present two characterizations. The first one is written in terms of certain $L^2$-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For $α>0$ this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for $α<0$ it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.
