Surjectivity of Convolution Operators on Harmonic NA Groups
E. Papageorgiou, The Journal of Geometric Analysis 35 (2024).
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Journal Article
| Published
| English
Author
Papageorgiou, Effie
Abstract
<jats:title>Abstract</jats:title>
<jats:p>Let <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mu $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>μ</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula> be a radial compactly supported distribution on a harmonic <jats:italic>NA</jats:italic> group. We prove that the right convolution operator <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$c_{\mu }:f \mapsto f* \mu $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>μ</mml:mi>
</mml:msub>
<mml:mo>:</mml:mo>
<mml:mi>f</mml:mi>
<mml:mo>↦</mml:mo>
<mml:mi>f</mml:mi>
<mml:mrow/>
<mml:mo>∗</mml:mo>
<mml:mi>μ</mml:mi>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula> maps the space of smooth <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathfrak {v}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>v</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>-radial functions onto itself if and only if the spherical Fourier transform <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\widetilde{\mu }(\lambda )$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mover>
<mml:mi>μ</mml:mi>
<mml:mo>~</mml:mo>
</mml:mover>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>λ</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>, <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\lambda \in \mathbb {C}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>λ</mml:mi>
<mml:mo>∈</mml:mo>
<mml:mi>C</mml:mi>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>, is slowly decreasing. As an application, we prove that certain averages over spheres are surjective on the space of smooth <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathfrak {v}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>v</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>-radial functions.</jats:p>
Publishing Year
Journal Title
The Journal of Geometric Analysis
Volume
35
Issue
1
Article Number
7
LibreCat-ID
Cite this
Papageorgiou E. Surjectivity of Convolution Operators on Harmonic NA Groups. The Journal of Geometric Analysis. 2024;35(1). doi:10.1007/s12220-024-01837-w
Papageorgiou, E. (2024). Surjectivity of Convolution Operators on Harmonic NA Groups. The Journal of Geometric Analysis, 35(1), Article 7. https://doi.org/10.1007/s12220-024-01837-w
@article{Papageorgiou_2024, title={Surjectivity of Convolution Operators on Harmonic NA Groups}, volume={35}, DOI={10.1007/s12220-024-01837-w}, number={17}, journal={The Journal of Geometric Analysis}, publisher={Springer Science and Business Media LLC}, author={Papageorgiou, Effie}, year={2024} }
Papageorgiou, Effie. “Surjectivity of Convolution Operators on Harmonic NA Groups.” The Journal of Geometric Analysis 35, no. 1 (2024). https://doi.org/10.1007/s12220-024-01837-w.
E. Papageorgiou, “Surjectivity of Convolution Operators on Harmonic NA Groups,” The Journal of Geometric Analysis, vol. 35, no. 1, Art. no. 7, 2024, doi: 10.1007/s12220-024-01837-w.
Papageorgiou, Effie. “Surjectivity of Convolution Operators on Harmonic NA Groups.” The Journal of Geometric Analysis, vol. 35, no. 1, 7, Springer Science and Business Media LLC, 2024, doi:10.1007/s12220-024-01837-w.
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