The purpose of this paper is to characterize the entire solutions of the homogeneous Helmholtz equation (solutions in ℝd) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha (\mathbb {S}^{d-1}),$ with α ∈ ℝ. We present two characterizations. The first one is written in terms of certain L2-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.

中文翻译：

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球体 Sobolev 空间中分布的傅里叶扩展算子和亥姆霍兹方程

本文的目的是描述齐次亥姆霍兹方程的整个解（ℝd) 由球体 Sobolev 空间分布的傅里叶扩展算子产生$H^\alpha (\mathbb {S}^{d-1}),$和α∈ℝ。我们提出两个特征。第一个是写在某些方面大号2-加权规范涉及球形拉普拉斯算子的实际幂。第二个是本着 P. Hartman 和 C. Wilcox 对 Herglotz 波函数的经典描述的精神。为了α> 0 此表征涉及在亥姆霍兹方程的整个解的向量中评估的多变量平方函数，而对于α< 0 它是根据作为分数积分算子的球积分算子来编写的。最后，我们还描述了球面分布的傅里叶扩展算子的所有解。