Littlewood--Paley operators and semigroup maximal operators on CMO spaces associated to Schödinger operators
Abstract
Let L=−Δ+V be a Schrödinger operator on ℝn, where Δ is the Laplacian and V satisfies the reverse Hölder inequality RHq for some q>n/2. In this paper, we study the behavior of the Littlewood--Paley operators sL and SL, as well as the semigroup maximal operator T*L, on the space CMOL(ℝn) associated with the Schrödinger operator L. It is known from previous work that these operators are bounded on BMOL(ℝn). Our main result shows that they are, in fact, mappings from CMOL(ℝn) into itself. To prove this, we develop several equivalent characterizations of CMOL(ℝn) and employ a refined decomposition that partitions the parameter interval at rBρ(xB), instead of the customary rB2 or ρ(xB)2. The new strategy allows us to overcome a key technical obstacle that arises when applying existing methods to the CMOL setting.
Related
- Semigroup of composition operators on the Fock space of several complex variables
- The semigroup generated by linear fractional divergence form operators in Lp(ℝN): the constant coefficient case
- One-Sided Hardy--Littlewood Maximal Operators on Variable Lebesgue Spaces Beyond Log-Hölder Continuity
- Littlewood-Paley and Carleson measure characterizations of Lipschitz spaces adapted to Schrödinger operators
- Semigroup properties of Schrödinger operators with complex singular potentials
BibTeX
@misc{li2026littlewoodpaley,
title = {{Littlewood--Paley operators and semigroup maximal operators on CMO spaces associated to Schödinger operators}},
author = {Wanjun Li and Qingze Lin and Liang Song},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07593},
eprint = {2609.07593},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07593},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Littlewood--Paley operators and semigroup maximal operators on CMO spaces associated to Schödinger operators
AU - Wanjun Li
AU - Qingze Lin
AU - Liang Song
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07593
UR - https://arxiv.org/abs/2609.07593
AB - Let L=−Δ+V be a Schrödinger operator on ℝ^n, where Δ is the Laplacian and V satisfies the reverse Hölder inequality RH_q for some q>n/2. In this paper, we study the behavior of the Littlewood--Paley operators s_L and S_L, as well as the semigroup maximal operator T^*_L, on the space CMO_L(ℝ^n) associated with the Schrödinger operator L. It is known from previous work that these operators are bounded on BMO_L(ℝ^n). Our main result shows that they are, in fact, mappings from CMO_L(ℝ^n) into itself. To prove this, we develop several equivalent characterizations of CMO_L(ℝ^n) and employ a refined decomposition that partitions the parameter interval at r_Bρ(x_B), instead of the customary r_B^2 or ρ(x_B)^2. The new strategy allows us to overcome a key technical obstacle that arises when applying existing methods to the CMO_L setting.
ER -