On one Zalcman problem for the mean value operator

Natalia P. Volchkova   ; Vitaliy V. Volchkov

Ural mathematical journal, Tome 9 (2023) no. 1, pp. 187-200

Résumé

Let $\mathcal{D}'(\mathbb{R}^n)$ and $\mathcal{E}'(\mathbb{R}^n)$ be the spaces of distributions and compactly supported distributions on $\mathbb{R}^n$, $n\geq 2$, respectively, let $\mathcal{E}'_{\natural}(\mathbb{R}^n)$ be the space of all radial (invariant under rotations of the space $\mathbb{R}^n$) distributions in $\mathcal{E}'(\mathbb{R}^n)$, let $\widetilde{T}$ be the spherical transform (Fourier–Bessel transform) of a distribution $T\in\mathcal{E}'_{\natural}(\mathbb{R}^n)$, and let $\mathcal{Z}_{+}(\widetilde{T})$ be the set of all zeros of an even entire function $\widetilde{T}$ lying in the half-plane $\mathrm{Re} \, z\geq 0$ and not belonging to the negative part of the imaginary axis. Let $\sigma_{r}$ be the surface delta function concentrated on the sphere $S_r=\{x\in\mathbb{R}^n: |x|=r\}$. The problem of L. Zalcman on reconstructing a distribution $f\in \mathcal{D}'(\mathbb{R}^n)$ from known convolutions $f\ast \sigma_{r_1}$ and $f\ast \sigma_{r_2}$ is studied. This problem is correctly posed only under the condition $r_1/r_2\notin M_n$, where $M_n$ is the set of all possible ratios of positive zeros of the Bessel function $J_{n/2-1}$. The paper shows that if $r_1/r_2\notin M_n$, then an arbitrary distribution $f\in \mathcal{D}'(\mathbb{R}^n)$ can be expanded into an unconditionally convergent series $$ f=\sum\limits_{\lambda\in\mathcal{Z}_{+}( \widetilde{\Omega}_{r_1})}\,\,\, \sum\limits_{\mu\in\mathcal{Z}_+(\widetilde{\Omega}_{r_2})} \frac{4\lambda\mu}{(\lambda^2-\mu^2) \widetilde{\Omega}_{r_1}^{\,\,\,\displaystyle{'}}(\lambda)\widetilde{\Omega}_{r_2}^{\,\,\,\displaystyle{'}}(\mu)}\Big (P_{r_2} (\Delta) \big((f\ast\sigma_{r_2})\ast \Omega_{r_1}^{\lambda}\big) -P_{r_1} (\Delta) \big((f\ast\sigma_{r_1})\ast \Omega_{r_2}^{\mu}\big)\Big) $$ in the space $\mathcal{D}'(\mathbb{R}^n)$, where $\Delta$ is the Laplace operator in $\mathbb{R}^n$, $P_r$ is an explicitly given polynomial of degree $[(n+5)/4]$, and $\Omega_{r}$ and $\Omega_{r}^{\lambda}$ are explicitly constructed radial distributions supported in the ball $ |x|\leq r$. The proof uses the methods of harmonic analysis, as well as the theory of entire and special functions. By a similar technique, it is possible to obtain inversion formulas for other convolution operators with radial distributions.

Détail

Keywords: compactly supported distributions, two-radii theorem
Mots-clés : Fourier–Bessel transform, inversion formulas.

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