Smoothness of extremizers for certain inequalities of the Radon transform

The Radon transform is a bounded operator from $L^p$ of Euclidean space to $L^q$ of the manifold of all affine hyperplanes in ${\mathbb{R}}^n$ for certain exponents depending dimension. Extremizers have been determined for certain values of $q$ and $p$, but most remain open. We show that extremizers are infinitely differentiable whenever the exponents in the associated Euler-Lagrange equation, $q-1$ and $\frac{1}{p-1}$, are integers. The proof adapts the method of Christ and Xue, to the case where the underlying space is a manifold. The proof is carried out in the setting of the $k$-plane transform, which takes functions on ${\mathbb{R}}^n$ to functions on the manifold of all affine $k$-planes in ${\mathbb{R}}^n$ by integrating the function over the $k$-dimensional plane. We show that when $q-1$ and $\frac{1}{p-1}$ are intergers, all nonnegative critical points of the functional $$\mathrm{\parallel }{T}_{n,k}f{\mathrm{\parallel }}_{L^q(M)}\mathrm{/}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}$$ are infinitely differentiable, all derivatives are in $L^p$ and exhibit some additional decay measured in a weighted $L^p$-space.