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PreprintCC BY 4.0HAL

REVERSE INEQUALITY FOR THE RIESZ TRANSFORMS ON RIEMANNIAN MANIFOLDS

  • Baptiste Devyver
  • Emmanuel Russ
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Abstract

Let M be a complete Riemannian manifold satisfying the doubling volume condition for geodesic balls and L^q scaled Poincaré inequalities on suitable remote balls for some q<2. We prove the inequality ||Δ1/2f||p≲ ||∇ f||p for all p ∈ (q, 2], which generalizes previous results due to Auscher and Coulhon. Our conclusion applies, in particular, when M has a finite number of Euclidean ends. The proof strongly relies on Hardy inequalities, which are also new in this context and of independent interest.

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Cite

BibTeXRIS
BibTeX
@misc{devyver2026reverse,
  title = {{REVERSE INEQUALITY FOR THE RIESZ TRANSFORMS ON RIEMANNIAN MANIFOLDS}},
  author = {Baptiste Devyver and Emmanuel Russ},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  eprint = {2209.05083},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-03773356v2},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - REVERSE INEQUALITY FOR THE RIESZ TRANSFORMS ON RIEMANNIAN MANIFOLDS
AU  - Baptiste Devyver
AU  - Emmanuel Russ
PY  - 2026
DA  - 2026/10/16
PB  - HAL
UR  - https://hal.science/hal-03773356v2
AB  - Let M be a complete Riemannian manifold satisfying the doubling volume condition for geodesic balls and L^q scaled Poincaré inequalities on suitable remote balls for some q<2. We prove the inequality ||Δ^1/2f||_p≲ ||∇ f||_p for all p ∈ (q, 2], which generalizes previous results due to Auscher and Coulhon. Our conclusion applies, in particular, when M has a finite number of Euclidean ends. The proof strongly relies on Hardy inequalities, which are also new in this context and of independent interest.
ER  -