PreprintCC BY 4.0arXiv
ℛ-boundedness of Riesz transforms and spectral multipliers of Schrödinger operators
Abstract
Let LL = −Δ+ V be a Schrödinger operator with a non-negative potential. It is known that its associated Riesz transform ∇ LL−sfrac12 is bounded on Lp(RRd) for p ∈ (1,2], with a bound independent of V. We prove more, namely that the set { ∇ LL−sfrac12, 0 ≤ V ∈ L1loc(RRd) } is ℛ-bounded on Lp(RRd) for p ∈ (1,2]. We also prove the ℛ-boundedness on Lp(RRd), for p ∈ (1,∞), of the spectral multipliers m(LL), 0 ≤ V ∈ L1loc(RRd) for any m:[0, ∞) → CC satisfying a Hörmander type condition. Our approach is based on the analysis of Schrödinger operators −bf Δ + bf V on vector-valued spaces Lp(RRd,ℓ2). In particular, we prove heat kernel estimates, Riesz transform bounds, and spectral multiplier results.
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BibTeX
@misc{ouhabaz2026ℛboundedness,
title = {{$\mathcal R$-boundedness of Riesz transforms and spectral multipliers of Schrödinger operators}},
author = {El Maati Ouhabaz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19463},
eprint = {2610.19463},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19463},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - ℛ-boundedness of Riesz transforms and spectral multipliers of Schrödinger operators
AU - El Maati Ouhabaz
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19463
UR - https://arxiv.org/abs/2610.19463
AB - Let LL = −Δ+ V be a Schrödinger operator with a non-negative potential. It is known that its associated Riesz transform ∇ LL^−sfrac12 is bounded on L^p(RR^d) for p ∈ (1,2], with a bound independent of V. We prove more, namely that the set { ∇ LL^−sfrac12, 0 ≤ V ∈ L^1_loc(RR^d) } is ℛ-bounded on L^p(RR^d) for p ∈ (1,2]. We also prove the ℛ-boundedness on L^p(RR^d), for p ∈ (1,∞), of the spectral multipliers m(LL), 0 ≤ V ∈ L^1_loc(RR^d) for any m:[0, ∞) → CC satisfying a Hörmander type condition. Our approach is based on the analysis of Schrödinger operators −bf Δ + bf V on vector-valued spaces L^p(RR^d,ℓ^2). In particular, we prove heat kernel estimates, Riesz transform bounds, and spectral multiplier results.
ER -