article published Efthymia Papageorgiou author 100325 <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> Springer Science and Business Media LLC2024 eng 1050-6926 1559-002X10.1007/s12220-024-01837-w 351 E. Papageorgiou, The Journal of Geometric Analysis 35 (2024). Papageorgiou, Efthymia. “Surjectivity of Convolution Operators on Harmonic NA Groups.” <i>The Journal of Geometric Analysis</i> 35, no. 1 (2024). <a href="https://doi.org/10.1007/s12220-024-01837-w">https://doi.org/10.1007/s12220-024-01837-w</a>. E. Papageorgiou, “Surjectivity of Convolution Operators on Harmonic NA Groups,” <i>The Journal of Geometric Analysis</i>, vol. 35, no. 1, Art. no. 7, 2024, doi: <a href="https://doi.org/10.1007/s12220-024-01837-w">10.1007/s12220-024-01837-w</a>. Papageorgiou, E. (2024). Surjectivity of Convolution Operators on Harmonic NA Groups. <i>The Journal of Geometric Analysis</i>, <i>35</i>(1), Article 7. <a href="https://doi.org/10.1007/s12220-024-01837-w">https://doi.org/10.1007/s12220-024-01837-w</a> @article{Papageorgiou_2024, title={Surjectivity of Convolution Operators on Harmonic NA Groups}, volume={35}, DOI={<a href="https://doi.org/10.1007/s12220-024-01837-w">10.1007/s12220-024-01837-w</a>}, number={17}, journal={The Journal of Geometric Analysis}, publisher={Springer Science and Business Media LLC}, author={Papageorgiou, Efthymia}, year={2024} } Papageorgiou E. Surjectivity of Convolution Operators on Harmonic NA Groups. <i>The Journal of Geometric Analysis</i>. 2024;35(1). doi:<a href="https://doi.org/10.1007/s12220-024-01837-w">10.1007/s12220-024-01837-w</a> Papageorgiou, Efthymia. “Surjectivity of Convolution Operators on Harmonic NA Groups.” <i>The Journal of Geometric Analysis</i>, vol. 35, no. 1, 7, Springer Science and Business Media LLC, 2024, doi:<a href="https://doi.org/10.1007/s12220-024-01837-w">10.1007/s12220-024-01837-w</a>. 635022026-01-06T09:39:35Z2026-07-03T12:35:33Z