Weighted and Sobolev Estimates for Fractional Schrödinger Operators with Reverse Hölder Potentials
Abstract
We study weighted and Sobolev estimates for two fractional Schrödinger operators on ℝn, where n≥3 and 0<s<1. For the additive operator L=(−Δ)s+V, with a nonnegative, nontrivial potential V belonging to a suitable reverse Hölder class, RHq, we establish a critical-radius form inequality and decay estimates for the semigroup mass. These yield a heat-kernel bound combining polynomial off-diagonal decay with a fixed positive exponent of decay relative to the fractional critical radius ρs. A decomposition at this scale gives weighted strong-type estimates for the semigroup maximal operator and weighted Hardy--Littlewood--Sobolev estimates for L−α/2, 0<αs<n, together with the corresponding endpoint weak-type estimates. The admissible weights satisfy explicit growth restrictions determined by the kernel decay, and the fractional integration order is αs. We also consider the spectral operator Q=(−Δ+V)s. Assuming V∈ RHq1 with n/(1+s)≤ q1<n and classical critical radius ρ1(x)≥1, we prove that ∇ Q−1/2 maps H1−s,p(ℝn) into Lp(ℝn) for 1<p≤ p0, where 1/p0=1/q1−1/n. By duality, its adjoint maps Lp'(ℝn) into Hs−1,p'(ℝn) for p0'≤ p'<∞.
Related
- Bilinear Strichartz Estimates and Linear Morawetz Estimates for Schrödinger Operators with Potentials
- Resolvent bounds and eigenvalue estimates of generalized Schrödinger operators with complex potentials on compact manifolds
- Schrödinger operators with accretive potentials in weighted spaces
- Orthonormal maximal estimates for fractional Schrödinger operators along Hölder curves
- KATO POTENTIALS AND SCHRÖDINGER OPERATORS ON LIE GROUPS
BibTeX
@misc{dasgupta2026weighted,
title = {{Weighted and Sobolev Estimates for Fractional Schrödinger Operators with Reverse Hölder Potentials}},
author = {Aparajita Dasgupta and Drishti Garg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16274},
eprint = {2610.16274},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16274},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Weighted and Sobolev Estimates for Fractional Schrödinger Operators with Reverse Hölder Potentials
AU - Aparajita Dasgupta
AU - Drishti Garg
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16274
UR - https://arxiv.org/abs/2610.16274
AB - We study weighted and Sobolev estimates for two fractional Schrödinger operators on ℝ^n, where n≥3 and 0<s<1. For the additive operator L=(−Δ)^s+V, with a nonnegative, nontrivial potential V belonging to a suitable reverse Hölder class, RH_q, we establish a critical-radius form inequality and decay estimates for the semigroup mass. These yield a heat-kernel bound combining polynomial off-diagonal decay with a fixed positive exponent of decay relative to the fractional critical radius ρ_s. A decomposition at this scale gives weighted strong-type estimates for the semigroup maximal operator and weighted Hardy--Littlewood--Sobolev estimates for L^−α/2, 0<αs<n, together with the corresponding endpoint weak-type estimates. The admissible weights satisfy explicit growth restrictions determined by the kernel decay, and the fractional integration order is αs. We also consider the spectral operator Q=(−Δ+V)^s. Assuming V∈ RH_q_1 with n/(1+s)≤ q_1<n and classical critical radius ρ_1(x)≥1, we prove that ∇ Q^−1/2 maps H^1−s,p(ℝ^n) into L^p(ℝ^n) for 1<p≤ p_0, where 1/p_0=1/q_1−1/n. By duality, its adjoint maps L^p^'(ℝ^n) into H^s−1,p^'(ℝ^n) for p_0^'≤ p^'<∞.
ER -