Surjectivity of Convolution Operators on Harmonic NA Groups <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> 35 1 Springer Science and Business Media LLC