@article{63502,
  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>}},
  author       = {{Papageorgiou, Efthymia}},
  issn         = {{1050-6926}},
  journal      = {{The Journal of Geometric Analysis}},
  number       = {{1}},
  publisher    = {{Springer Science and Business Media LLC}},
  title        = {{{Surjectivity of Convolution Operators on Harmonic NA Groups}}},
  doi          = {{10.1007/s12220-024-01837-w}},
  volume       = {{35}},
  year         = {{2024}},
}