Endpoint regularity of Radon transforms associated with real analytic phases in the plane
Abstract
In this paper, we consider the endpoint Sobolev regularity of Radon transforms associated with real-analytic functions in the plane. We establish all of the failure of endpoint regularity for real-analytic phases. On the one hand, the critical endpoint regularity Lp(ℝ2)→ Lps(ℝ2), corresponding to extreme points of the reduced Newton polyhedron of phase function, is true if only and only if p=2. For generic extreme points, this failure of endpoint regularity generalizes a result of Christ \cite{christ}. On the other hand, we find that the endpoint Sobolev regularity also fails on a bounded segment of the reduced Newton polyhedron when the bisectrix y=x intersects one of unbounded faces at a relative interior point. In particular, our results imply that the Radon transform is not bounded from Lp into Lp1/n with p=n/(n−1) or p=n, where the phase functions are real-valued homogeneous polynomials S(x,y)=a1xn−1y+a2xn−2y2+⋯+an−1xyn−1 with n≥ 3 and a1an−1≠0. This answers a problem of Phong and Stein in 1994. Moreover, the main results of this paper also give examples which show failure of critical Sobolev regularity for degenerate Fourier integral operators satisfying finite type conditions formulated in \cite{Seeger93,seeger2}. Based on earlier results and our proof of several remaining cases, we conclude that all of other critical endpoint Sobolev estimates are true for Radon transforms associated with real-analytic phases in the plane. Therefore, all failures of endpoint regularity estimates are clearly determined.
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BibTeX
@misc{li2026endpoint,
title = {{Endpoint regularity of Radon transforms associated with real analytic phases in the plane}},
author = {Danglin Li and Zuoshunhua Shi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19854},
eprint = {2610.19854},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19854},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Endpoint regularity of Radon transforms associated with real analytic phases in the plane
AU - Danglin Li
AU - Zuoshunhua Shi
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.19854
UR - https://arxiv.org/abs/2610.19854
AB - In this paper, we consider the endpoint Sobolev regularity of Radon transforms associated with real-analytic functions in the plane. We establish all of the failure of endpoint regularity for real-analytic phases. On the one hand, the critical endpoint regularity L^p(ℝ^2)→ L^p_s(ℝ^2), corresponding to extreme points of the reduced Newton polyhedron of phase function, is true if only and only if p=2. For generic extreme points, this failure of endpoint regularity generalizes a result of Christ \cite{christ}. On the other hand, we find that the endpoint Sobolev regularity also fails on a bounded segment of the reduced Newton polyhedron when the bisectrix y=x intersects one of unbounded faces at a relative interior point. In particular, our results imply that the Radon transform is not bounded from L^p into L^p_1/n with p=n/(n−1) or p=n, where the phase functions are real-valued homogeneous polynomials S(x,y)=a_1x^n−1y+a_2x^n−2y^2+⋯+a_n−1xy^n−1 with n≥ 3 and a_1a_n−1≠0. This answers a problem of Phong and Stein in 1994. Moreover, the main results of this paper also give examples which show failure of critical Sobolev regularity for degenerate Fourier integral operators satisfying finite type conditions formulated in \cite{Seeger93,seeger2}. Based on earlier results and our proof of several remaining cases, we conclude that all of other critical endpoint Sobolev estimates are true for Radon transforms associated with real-analytic phases in the plane. Therefore, all failures of endpoint regularity estimates are clearly determined.
ER -