---
_id: '63502'
abstract:
- lang: eng
  text: "<jats:title>Abstract</jats:title>\r\n          <jats:p>Let <jats:inline-formula>\r\n
    \             <jats:alternatives>\r\n                <jats:tex-math>$$\\mu $$</jats:tex-math>\r\n
    \               <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mi>μ</mml:mi>\r\n                </mml:math>\r\n              </jats:alternatives>\r\n
    \           </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>\r\n              <jats:alternatives>\r\n                <jats:tex-math>$$c_{\\mu
    }:f \\mapsto f* \\mu $$</jats:tex-math>\r\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:msub>\r\n                      <mml:mi>c</mml:mi>\r\n
    \                     <mml:mi>μ</mml:mi>\r\n                    </mml:msub>\r\n
    \                   <mml:mo>:</mml:mo>\r\n                    <mml:mi>f</mml:mi>\r\n
    \                   <mml:mo>↦</mml:mo>\r\n                    <mml:mi>f</mml:mi>\r\n
    \                   <mml:mrow/>\r\n                    <mml:mo>∗</mml:mo>\r\n
    \                   <mml:mi>μ</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math>\r\n
    \             </jats:alternatives>\r\n            </jats:inline-formula> maps
    the space of smooth <jats:inline-formula>\r\n              <jats:alternatives>\r\n
    \               <jats:tex-math>$$\\mathfrak {v}$$</jats:tex-math>\r\n                <mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>v</mml:mi>\r\n
    \               </mml:math>\r\n              </jats:alternatives>\r\n            </jats:inline-formula>-radial
    functions onto itself if and only if the spherical Fourier transform <jats:inline-formula>\r\n
    \             <jats:alternatives>\r\n                <jats:tex-math>$$\\widetilde{\\mu
    }(\\lambda )$$</jats:tex-math>\r\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mover>\r\n                      <mml:mi>μ</mml:mi>\r\n
    \                     <mml:mo>~</mml:mo>\r\n                    </mml:mover>\r\n
    \                   <mml:mrow>\r\n                      <mml:mo>(</mml:mo>\r\n
    \                     <mml:mi>λ</mml:mi>\r\n                      <mml:mo>)</mml:mo>\r\n
    \                   </mml:mrow>\r\n                  </mml:mrow>\r\n                </mml:math>\r\n
    \             </jats:alternatives>\r\n            </jats:inline-formula>, <jats:inline-formula>\r\n
    \             <jats:alternatives>\r\n                <jats:tex-math>$$\\lambda
    \\in \\mathbb {C}$$</jats:tex-math>\r\n                <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n
    \                 <mml:mrow>\r\n                    <mml:mi>λ</mml:mi>\r\n                    <mml:mo>∈</mml:mo>\r\n
    \                   <mml:mi>C</mml:mi>\r\n                  </mml:mrow>\r\n                </mml:math>\r\n
    \             </jats:alternatives>\r\n            </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>\r\n              <jats:alternatives>\r\n
    \               <jats:tex-math>$$\\mathfrak {v}$$</jats:tex-math>\r\n                <mml:math
    xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">\r\n                  <mml:mi>v</mml:mi>\r\n
    \               </mml:math>\r\n              </jats:alternatives>\r\n            </jats:inline-formula>-radial
    functions.</jats:p>"
article_number: '7'
author:
- first_name: Efthymia
  full_name: Papageorgiou, Efthymia
  id: '100325'
  last_name: Papageorgiou
citation:
  ama: 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>
  apa: 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>
  bibtex: '@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}
    }'
  chicago: 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>.
  ieee: '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>.'
  mla: 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>.
  short: E. Papageorgiou, The Journal of Geometric Analysis 35 (2024).
date_created: 2026-01-06T09:39:35Z
date_updated: 2026-07-03T12:35:33Z
doi: 10.1007/s12220-024-01837-w
intvolume: '        35'
issue: '1'
language:
- iso: eng
publication: The Journal of Geometric Analysis
publication_identifier:
  issn:
  - 1050-6926
  - 1559-002X
publication_status: published
publisher: Springer Science and Business Media LLC
status: public
title: Surjectivity of Convolution Operators on Harmonic NA Groups
type: journal_article
user_id: '100325'
volume: 35
year: '2024'
...