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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 $H^\alpha(\mathbb{S}^{d-1}),$ with $\alpha\in \mathbb{R}$. 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 $\alpha>0$ this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for $\alpha<0$ 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.

The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation