Sharp decay of the spherical averages of the Fourier transform of open sets
Abstract
For a bounded domain Ω⊂ℝd, the spherical average EΩ(R)=∫𝕊d−1|𝟙Ω̂(Rω)|2dω is never O(R−a) with a>d+1, and for smooth boundaries it is O(R−d−1), as it is for the ball, whether or not the boundary is curved. In 2003, Brandolini, Hofmann and Iosevich extended this to every convex domain and to every domain with C1,1/2 boundary, and this regularity condition has not been improved since. How much regularity does the decay R−d−1 require? We show that among Hölder classes the answer is C1,1/4. Every bounded domain with C1,1/4 boundary satisfies EΩ(R)≤ CΩR−d−1, while for every 0<s<1/4 any smooth domain can be altered near a single boundary point to produce a C1,s domain for which this bound fails along a sequence of radii. More generally, when the boundary has fractional Sobolev regularity W1+s,p, we determine the optimal decay exponent under a Dini condition on the normal, and in a large range of s and p without it. The upper bounds follow from L2 estimates for nondegenerate oscillatory integral operators whose phases are distances between points of two graphs with Dini or fractional Sobolev gradients, and hence barely better than C1. These are far too rough for the integrations by parts techniques from the smooth theory. We replace derivatives by finite differences adapted to the phase, a device that we expect to be useful in a variety of related problems.
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BibTeX
@misc{iosevich2026sharp,
title = {{Sharp decay of the spherical averages of the Fourier transform of open sets}},
author = {A. Iosevich and A. Merlo and M. Mourgoglou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04426},
eprint = {2611.04426},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04426},
}
RIS
TY - UNPB
TI - Sharp decay of the spherical averages of the Fourier transform of open sets
AU - A. Iosevich
AU - A. Merlo
AU - M. Mourgoglou
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04426
UR - https://arxiv.org/abs/2611.04426
AB - For a bounded domain Ω⊂ℝ^d, the spherical average E_Ω(R)=∫_𝕊^d−1|𝟙_Ω̂(Rω)|^2dω is never O(R^−a) with a>d+1, and for smooth boundaries it is O(R^−d−1), as it is for the ball, whether or not the boundary is curved. In 2003, Brandolini, Hofmann and Iosevich extended this to every convex domain and to every domain with C^1,1/2 boundary, and this regularity condition has not been improved since. How much regularity does the decay R^−d−1 require? We show that among Hölder classes the answer is C^1,1/4. Every bounded domain with C^1,1/4 boundary satisfies E_Ω(R)≤ C_ΩR^−d−1, while for every 0<s<1/4 any smooth domain can be altered near a single boundary point to produce a C^1,s domain for which this bound fails along a sequence of radii. More generally, when the boundary has fractional Sobolev regularity W^1+s,p, we determine the optimal decay exponent under a Dini condition on the normal, and in a large range of s and p without it. The upper bounds follow from L^2 estimates for nondegenerate oscillatory integral operators whose phases are distances between points of two graphs with Dini or fractional Sobolev gradients, and hence barely better than C^1. These are far too rough for the integrations by parts techniques from the smooth theory. We replace derivatives by finite differences adapted to the phase, a device that we expect to be useful in a variety of related problems.
ER -