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arXiv · 2410.15043

Surjectivity of convolution operators on harmonic $NA$ groups

Effie Papageorgiou
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Abstract

Let $\mu$ be a radial compactly supported distribution on a harmonic $NA$ group. We prove that the right convolution operator $c_{\mu}:f \mapsto f* \mu$ maps the space of smooth $\mathfrak{v}$-radial functions onto itself if and only if the spherical Fourier transform $\widetilde{\mu}(\lambda)$, $\lambda \in \mathbb{C}$, is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth $\mathfrak{v}$-radial functions.

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Effie Papageorgiou. 2024-10-19. Surjectivity of convolution operators on harmonic $NA$ groups. https://arxiv.org/abs/2410.15043

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