2020/04/03 by Juan Antonio Barceló, Magali Folch-Gabayet, Barceló, J. A. +7
Computer Science · Mathematics · #46E35 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical methods in inverse problems #Primary 35J05 #Secondary 42B10
paper · pdf · doi:10.48550/arxiv.2004.01508
openalex publication_date 2020/04/03 · openalex created_date 2020/04/10 · openalex updated_date 2026/07/28
The purpose of this paper is to characterize all 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 Hα(\mathbbSd-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.