SPARSE POINTWISE BOUNDS FOR MAXIMAL TRUNCATIONS OF ROUGH SINGULAR INTEGRALS AND SOBOLEV-TYPE INEQUALITIES
Abstract
Let 1 < rho < n and let Omega be in Lrho(S^(n-1)) with vanishing mean. We prove that the maximal truncation T*_Omega of the rough singular integral T_Omega is pointwise dominated by finitely many sparse potentials of the form: sum over Q in S of l(Q) * ( (1/|Q|) * integral over Q of |nabla f|^p )^(1/p), where 1/rho~ = 1/rho' + 1/n and rho~ <= p < n. This estimate is uniform in the truncation parameter and extends the subcritical bound of Hoang, Moen, and Perez for T_Omega to T*_Omega. Since the argument does not require the boundedness of T*_Omega on the target space, it yields two-weight Sobolev inequalities: norm(T*_Omega f, L^q(u)) <= C * norm(nabla f, L^p(v)), under joint two-weight conditions, while the target weight u itself is only required to belong to A_infinity. Additionally, we prove a Hedberg-type estimate involving a Morrey norm of the gradient and apply it to weighted grand Lebesgue spaces. Further consequences are obtained in weighted Lebesgue, Orlicz, and variable Lebesgue spaces.
Related
- Weighted Norm Inequalities for Maximal Multi-sublinear Rough Singular Integrals
- On pointwise convergence of singular integrals of Christ-Journé type
- Pointwise Riesz potential bounds for rough singular integrals
- Weak type (1,1) estimates for noncommutative maximal truncated rough singular integrals
- Maximal directional singular integrals
BibTeX
@misc{chamorro2026sparse,
title = {{SPARSE POINTWISE BOUNDS FOR MAXIMAL TRUNCATIONS OF ROUGH SINGULAR INTEGRALS AND SOBOLEV-TYPE INEQUALITIES}},
author = {Diego Chamorro and Anca-Nicoleta Marcoci and Liviu-Gabriel Marcoci},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19185},
eprint = {2609.19185},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05746501v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - SPARSE POINTWISE BOUNDS FOR MAXIMAL TRUNCATIONS OF ROUGH SINGULAR INTEGRALS AND SOBOLEV-TYPE INEQUALITIES
AU - Diego Chamorro
AU - Anca-Nicoleta Marcoci
AU - Liviu-Gabriel Marcoci
PY - 2026
DA - 2026/09/11
PB - HAL
DO - 10.48550/arxiv.2609.19185
UR - https://hal.science/hal-05746501v1
AB - Let 1 < rho < n and let Omega be in Lrho(S^(n-1)) with vanishing mean. We prove that the maximal truncation T*_Omega of the rough singular integral T_Omega is pointwise dominated by finitely many sparse potentials of the form: sum over Q in S of l(Q) * ( (1/|Q|) * integral over Q of |nabla f|^p )^(1/p), where 1/rho~ = 1/rho' + 1/n and rho~ <= p < n. This estimate is uniform in the truncation parameter and extends the subcritical bound of Hoang, Moen, and Perez for T_Omega to T*_Omega. Since the argument does not require the boundedness of T*_Omega on the target space, it yields two-weight Sobolev inequalities: norm(T*_Omega f, L^q(u)) <= C * norm(nabla f, L^p(v)), under joint two-weight conditions, while the target weight u itself is only required to belong to A_infinity. Additionally, we prove a Hedberg-type estimate involving a Morrey norm of the gradient and apply it to weighted grand Lebesgue spaces. Further consequences are obtained in weighted Lebesgue, Orlicz, and variable Lebesgue spaces.
ER -