Sparse bounds for maximal rough singular integrals

@misc{wu2026sparse,
  title = {{Sparse bounds for maximal rough singular integrals}},
  author = {Yuhao Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03751},
  eprint = {2609.03751},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03751},
}

TY  - UNPB
TI  - Sparse bounds for maximal rough singular integrals
AU  - Yuhao Wu
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.03751
UR  - https://arxiv.org/abs/2609.03751
AB  - Let Ω∈ L^1(S^d−1) have vanishing average, and let T_Ω^∗ be the maximal truncation of the associated rough homogeneous singular integral. We prove quantitative sparse bounds for T_Ω^∗. If Ω∈ L^∞(S^d−1), then, for every 1<p<∞, |T_Ω^∗|_(1,p)−sparse ≲_d p'|Ω|_L^∞(S^d−1). For unbounded angular kernels, if 1<q<∞ and Ω∈ L^q,1log L(S^d−1), then the same estimate holds for q'≤ p<∞, with the right-hand side replaced by C_d,qp' |Ω|_L^q,1log L(S^d−1). These estimates retain a genuine L^1 average in the first entry of the sparse form. In the bounded-kernel case, the upper bound has the same linear growth in p' as the known sparse bound for the nonmaximal operator. The unbounded-kernel estimate includes the critical exponent p=q'. As a consequence, we obtain weighted weak-type (1,1) estimates for all A_1 weights in the bounded-kernel case and for weights in A_1∩ RH_q' in the unbounded-kernel case. The proof combines physical-space linearization and microlocal decomposition with localized sparse testing. An amplitude decomposition of the second input, together with the Rademacher--Menshov inequality, yields the quantitative dependence on p.
ER  -