Euler--Lagrange Flow Monotonicity for the Radon Transform
Abstract
We present a new gradient-like flow that targets critical points of the inequality for the planar Radon transform ℛ proven by Oberlin and Stein, partially answering a question of Christ, who gave a complete characterisation of the extremisers of the inequality via rearrangement methods. We show that for non-negative radial initial data near the extremiser manifold, solutions to the flow exist globally and there exists a rescaled subsequence of the flow that converges to a critical point of the Radon functional |ℛ f|3/|f|3/2 locally in L3/2(ℝ2). We also obtain a quantitative result for the rate of decay of critical points, which complements a result of Christ arXiv:1106.0719 in the special case of radial functions. We prove a novel localised smoothing estimate for the operator ℛ*[(ℛ·)2] restricted to radial functions. This approach is complementary to that of Christ and provides a robust framework for studying extremal problems associated to Radon-like operators.
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BibTeX
@misc{galanos2026eulerlagrange,
title = {{Euler--Lagrange Flow Monotonicity for the Radon Transform}},
author = {Yannis Galanos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39818},
eprint = {2610.39818},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39818},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Euler--Lagrange Flow Monotonicity for the Radon Transform
AU - Yannis Galanos
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39818
UR - https://arxiv.org/abs/2610.39818
AB - We present a new gradient-like flow that targets critical points of the inequality for the planar Radon transform ℛ proven by Oberlin and Stein, partially answering a question of Christ, who gave a complete characterisation of the extremisers of the inequality via rearrangement methods. We show that for non-negative radial initial data near the extremiser manifold, solutions to the flow exist globally and there exists a rescaled subsequence of the flow that converges to a critical point of the Radon functional |ℛ f|_3/|f|_3/2 locally in L^3/2(ℝ^2). We also obtain a quantitative result for the rate of decay of critical points, which complements a result of Christ arXiv:1106.0719 in the special case of radial functions. We prove a novel localised smoothing estimate for the operator ℛ^*[(ℛ·)^2] restricted to radial functions. This approach is complementary to that of Christ and provides a robust framework for studying extremal problems associated to Radon-like operators.
ER -