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The strong type endpoint estimate for the local spherical maximal function
Abstract
Let pd=(d2+1)/(d(d−1)) and qd=(d2+1)/(d−1). We prove that the local spherical maximal operator sup1≤ t≤ 2|f*dσt| is bounded from Lpd(ℝd) to Lqd(ℝd) for every d≥ 2. Previously only the restricted weak type estimate was known at this endpoint. Together with known results, this completes the characterization of the strong type Lp→ Lq bounds for the local spherical maximal operator. We also prove the maximal estimate |supt∈ℝ|eit√(−Δ)f||L2(d+1)/(d−1)(ℝd) ≲|f|Ḣd/(d+1)(ℝd) for the wave equation at the critical exponent. Both results are obtained by exploiting the gain in bilinear estimates that comes from the interaction between high and low frequencies.
Related
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BibTeX
@misc{lee2026strong,
title = {{The strong type endpoint estimate for the local spherical maximal function}},
author = {Sanghyuk Lee and Shuijiang Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27803},
eprint = {2610.27803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27803},
}
RIS
TY - UNPB
TI - The strong type endpoint estimate for the local spherical maximal function
AU - Sanghyuk Lee
AU - Shuijiang Zhao
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27803
UR - https://arxiv.org/abs/2610.27803
AB - Let p_d=(d^2+1)/(d(d−1)) and q_d=(d^2+1)/(d−1). We prove that the local spherical maximal operator sup_1≤ t≤ 2|f*dσ_t| is bounded from L^p_d(ℝ^d) to L^q_d(ℝ^d) for every d≥ 2. Previously only the restricted weak type estimate was known at this endpoint. Together with known results, this completes the characterization of the strong type L^p→ L^q bounds for the local spherical maximal operator. We also prove the maximal estimate |sup_t∈ℝ|e^it√(−Δ)f||_L^2(d+1)/(d−1)(ℝ^d) ≲|f|_Ḣ^d/(d+1)(ℝ^d) for the wave equation at the critical exponent. Both results are obtained by exploiting the gain in bilinear estimates that comes from the interaction between high and low frequencies.
ER -