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PreprintCC BY 4.0arXiv

Weak type (1,1) estimates for Riesz transforms of elliptic operators and magnetic Schrödinger operators

  • El Maati Ouhabaz
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Abstract

Let M be a complete Riemannian manifold and define a magnetic Schrödinger operator LL = (bf d+ iA)*(bf d+iA) + V where A is a real $1$-differential form in L2loc and V ∈ L1loc is a nonnegative electric potential. We prove that the magnetic Riesz transform (bf d + iA) LL−einhalb is of weak type (1,1) with bound at most $2$. It extends to a bounded operator on Lp for p ∈ (1, 2] with norm bounded by a constant cp depending only on p. It is then independent of the dimension of the manifold and independent of A and V. We also prove weak type (1,1) bounds for symmetric elliptic operators subject to various boundary conditions on rough domains of RRd. The proofs are based on a new weak type (1,1) criterion for operators.

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Cite

BibTeXRIS
BibTeX
@misc{ouhabaz2026weak,
  title = {{Weak type $(1,1)$ estimates for Riesz transforms of elliptic operators and magnetic Schrödinger operators}},
  author = {El Maati Ouhabaz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19469},
  eprint = {2610.19469},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19469},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Weak type (1,1) estimates for Riesz transforms of elliptic operators and magnetic Schrödinger operators
AU  - El Maati Ouhabaz
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19469
UR  - https://arxiv.org/abs/2610.19469
AB  - Let M be a complete Riemannian manifold and define a magnetic Schrödinger operator LL = (bf d+ iA)^*(bf d+iA) + V where A is a real $1$-differential form in L^2_loc and V ∈ L^1_loc is a nonnegative electric potential. We prove that the magnetic Riesz transform (bf d + iA) LL^−einhalb is of weak type (1,1) with bound at most $2$. It extends to a bounded operator on L^p for p ∈ (1, 2] with norm bounded by a constant c_p depending only on p. It is then independent of the dimension of the manifold and independent of A and V. We also prove weak type (1,1) bounds for symmetric elliptic operators subject to various boundary conditions on rough domains of RR^d. The proofs are based on a new weak type (1,1) criterion for operators.
ER  -