Failure of the H1× H1⟶ L1/2,∞ estimate for the lacunary bilinear spherical maximal function and applications
Abstract
In contrast to Christ's theorem that the lacunary spherical maximal operator is bounded from H1(ℝd) to L1,∞(ℝd) (d≥2), we show that the lacunary bilinear spherical maximal operator ℬlac(f,g)(x) =supℓ∈ℤ |∫𝕊2d−1f(x−2ℓ y) g(x−2ℓ z) dσ(y,z)| is not bounded from H1(ℝd)× H1(ℝd) to L1/2,∞(ℝd) for any d≥1. In particular, it is not bounded from L1(ℝd)× L1(ℝd) to L1/2,∞(ℝd). More precisely, for every sufficiently large integer N, we construct two finite linear combinations of smooth, compactly supported, mean-zero H1(ℝd)-atoms for which the ratio of the weak L1/2 quasinorm to the product of the two Hardy norms is at least cdlog(N+1). This lower bound holds already for the supremum over a suitable set of N dyadic radii. For d=1 this answers the case p=q=1 of a question in Borges's survey. For every k≥3 and d≥1, a modification of the construction proves that the lacunary k-linear spherical maximal operator does not map (H1(ℝd))k into L1/k,∞(ℝd). We also indicate extensions to certain hypersurfaces and to horizontal spherical averages on the Heisenberg group.
Related
- Bilinear Korányi spherical maximal function on the Heisenberg group
- The strong type endpoint estimate for the local spherical maximal function
- Strong-type endpoint obstructions for bilinear spherical maximal operators
- Endpoint weighted Fourier estimate with vanishing spherical trace
- On the endpoint estimate for discrete spherical average over sparse sequences
BibTeX
@misc{lee2026failure,
title = {{Failure of the $H^1\times H^1\longrightarrow L^1/2,\infty$ estimate for the lacunary bilinear spherical maximal function and applications}},
author = {Sanghyuk Lee and Ji Li and Chaojie Wen and Liangchuan Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40100},
eprint = {2610.40100},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40100},
}
RIS
TY - UNPB
TI - Failure of the H^1× H^1⟶ L^1/2,∞ estimate for the lacunary bilinear spherical maximal function and applications
AU - Sanghyuk Lee
AU - Ji Li
AU - Chaojie Wen
AU - Liangchuan Wu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40100
UR - https://arxiv.org/abs/2610.40100
AB - In contrast to Christ's theorem that the lacunary spherical maximal operator is bounded from H^1(ℝ^d) to L^1,∞(ℝ^d) (d≥2), we show that the lacunary bilinear spherical maximal operator ℬ_lac(f,g)(x) =sup_ℓ∈ℤ |∫_𝕊^2d−1f(x−2^ℓ y) g(x−2^ℓ z) dσ(y,z)| is not bounded from H^1(ℝ^d)× H^1(ℝ^d) to L^1/2,∞(ℝ^d) for any d≥1. In particular, it is not bounded from L^1(ℝ^d)× L^1(ℝ^d) to L^1/2,∞(ℝ^d). More precisely, for every sufficiently large integer N, we construct two finite linear combinations of smooth, compactly supported, mean-zero H^1(ℝ^d)-atoms for which the ratio of the weak L^1/2 quasinorm to the product of the two Hardy norms is at least c_dlog(N+1). This lower bound holds already for the supremum over a suitable set of N dyadic radii. For d=1 this answers the case p=q=1 of a question in Borges's survey. For every k≥3 and d≥1, a modification of the construction proves that the lacunary k-linear spherical maximal operator does not map (H^1(ℝ^d))^k into L^1/k,∞(ℝ^d). We also indicate extensions to certain hypersurfaces and to horizontal spherical averages on the Heisenberg group.
ER -