Barceló, Juan AntonioFolch Gabayet, MagaliLuque Martínez, Teresa ElviraPérez Esteva, SalvadorVilela, María de la Cruz2024-02-072024-02-072020Barceló JA, Folch-Gabayet M, Luque T, Pérez-Esteva S, Vilela MC. 2021 The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 151, 1768–1789. (doi:10.1017/prm.2020.80)0308-210510.1017/prm.2020.80https://hdl.handle.net/20.500.14352/99754The 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 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.engjournal article1473-7124https://doi.org/10.1017/prm.2020.80https://www.cambridge.org/core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematicsrestricted accessElliptic equations and systemsHarmonic analysis in several variablesLinear function spaces and their rulesAnálisis matemático12 Matemáticas