Erasing of singularities of functions with zero integrals over disks

N. P. Volchkova ; Vit. V. Volchkov ; N. A. Ischenko

Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 2, pp. 19-33

Résumé

Let $\mathcal{M}$ and $\mathcal{N}$ be manifolds, let ${\mathcal{D}}$ be a domain in $\mathcal{M}$, and let $E \subset \mathcal{D}$ be a set closed with respect to $\mathcal{D}$. The singularity erasure problem is as follows: find conditions under which any mapping $f:\mathcal{D}\setminus E\rightarrow\mathcal{N}$ from a given class admits a class preserving extension to a mapping $\mathbf{f}: \mathcal{D}\rightarrow \mathcal{N}$. If the indicated extension exists, then the set $E$ is called a removable set in the considered class of mappings. The purpose of this article is to study the singularity erasure problem in the context of the properties of the kernel of the local Pompeiu transform. We study the class $\mathfrak{K}_{+}$ consisting of continuous functions on the complex plane $ \mathbb{C}$ having zero integrals over all circles from $\mathbb{C}$ congruent to the unit disk with respect to the spherical metric. An analogue of the group of Euclidean motions in this case is the group of linear fractional transformations $\mathrm{PSU}(2)$. An exact condition is found under which the functions of the class in question appropriately defined at the infinity have this property on the extended complex plane $\overline{\mathbb{C}}$. The proof of the main result is based on an appropriate description of the class $\mathfrak{K}_{+}$. The central tool in this description is the Fourier series in spherical harmonics. It is shown that the Fourier coefficients of the function $f\in\mathfrak{K}_{+}$ are representable by series in Jacobi functions. The further proof consists in studying the asymptotic behavior of the indicated series when approaching a singular point. The results obtained in this article can be used to solve problems related to spherical means.