2025/08/01 by Taryn C. Flock, Flock, Taryn C.
Mathematics · #Advanced Harmonic Analysis Research #Mathematical Analysis and Transform Methods #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2508.00783
The Radon transform is a bounded operator from Lp of Euclidean space to Lq of the manifold of all affine hyperplanes in ℝn 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 \frac1p-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 ℝn to functions on the manifold of all affine k-planes in ℝn by integrating the function over the k-dimensional plane. We show that when q-1 and \frac1p-1 are intergers, all nonnegative critical points of the functional ‖Tn,kf‖Lq(M)/‖f‖Lp(ℝn) are infinitely differentiable, all derivatives are in Lp and exhibit some additional decay measured in a weighted Lp-space.