Hypoellipticity of Hankel convolution equations on Schwartz distribution spaces (Q1849625)

The authors prove the hypoellipticity of Hankel convolution operators defined by elements of \(\mathcal{E}_{\ast }^{\prime }\) in the space \( \mathcal{D}_{\ast }^{\prime }\), where \(\mathcal{E}_{\ast }^{\prime }\) and \( \mathcal{D}_{\ast }^{\prime }\)\ are the duals of the spaces \(\mathcal{E} _{\ast }\) of smooth even functions and \(\mathcal{D}_{\ast }\)\ of smooth even functions with compact supports on \(\mathbb{R}\), respectively. If \(T\in \mathcal{E}_{\ast }^{\prime },\) the Hankel convolution operator, denoted \(T\# \), is said to be hypoelliptic if, whenever \(T\#S\in \mathcal{E}_{\ast },\) with \( S\in \mathcal{D}_{\ast }^{\prime },\) then the distribution \(S\in \mathcal{E} _{\ast }.\) The main result of the paper is that the authors give sufficient algebraic conditions on \(T\in \mathcal{E}_{\ast }^{\prime }\) such that the Hankel convolution operator \(T\#\) is hypoelliptic.