[{"language":[{"iso":"eng"}],"article_number":"7","doi":"10.1007/s12220-024-01837-w","author":[{"full_name":"Papageorgiou, Efthymia","first_name":"Efthymia","last_name":"Papageorgiou","id":"100325"}],"publication_identifier":{"issn":["1050-6926","1559-002X"]},"title":"Surjectivity of Convolution Operators on Harmonic NA Groups","year":"2024","intvolume":"        35","date_updated":"2026-07-03T12:35:33Z","publication_status":"published","date_created":"2026-01-06T09:39:35Z","type":"journal_article","publication":"The Journal of Geometric Analysis","issue":"1","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>"}],"_id":"63502","publisher":"Springer Science and Business Media LLC","volume":35,"user_id":"100325","status":"public","citation":{"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} }","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>","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>.","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>.","short":"E. Papageorgiou, The Journal of Geometric Analysis 35 (2024).","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>.","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>"}}]