2025/04/23 by Xueting Han, Ji Li, Han, Xueting +3 · 1 citation
Computer Science · Mathematics · #42B20 #42B25 #42B35 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2504.16827
openalex publication_date 2025/04/23 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
Let L = -Δ+ V be a Schrödinger operator acting on L2(ℝn) , where the nonnegative potential V belongs to the reverse Hölder class RHq for some q ≥ n/2 . This article is primarily concerned with the study of endpoint boundedness for classical singular integral operators in the context of the space CMOL(ℝn) , consisting of functions of vanishing mean oscillation associated with L . We establish the following main results: (i) the standard Hardy--Littlewood maximal operator is bounded on CMOL(ℝn) ; (ii) for each j = 1, …, n, the adjoint of the Riesz transform ∂j L-1/2 is bounded from C0(ℝn) into CMOL(ℝn) ; and (iii) the approximation to the identity generated by the Poisson and heat semigroups associated with L characterizes CMOL(ℝn) appropriately. These results recover the classical analogues corresponding to the Laplacian as a special case. However, the presence of the potential V introduces substantial analytical challenges, necessitating tools beyond the scope of classical Calderón--Zygmund theory. Our approach leverages precise heat kernel estimates and the structural properties of CMOL(ℝn) established by Song and the third author in [J. Geom. Anal. 32 (2022), no. 4, Paper No. 130, 37 pp].