Conical square functionals on Riemannian manifolds
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | COMETX, Thomas | |
| dc.date.accessioned | 2024-04-04T02:47:49Z | |
| dc.date.available | 2024-04-04T02:47:49Z | |
| dc.date.created | 2021-01-05 | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/191682 | |
| dc.description.abstractEn | Let $L = \Delta + V$ be Schrödinger operator with a non-negative potential $V$ on a complete Riemannian manifold $M$. We prove that the conical square functional associated with $L$ is bounded on $L^p$ under different assumptions. This functional is defined by $$ \mathcal{G}_L (f) (x) = \left( \int_0^\infty \int_{B(x,t^{1/2})} |\nabla e^{-tL} f(y)|^2 + V |e^{-tL} f(y)|^2 \frac{\mathrm{d}t \mathrm{d}y}{Vol(y,t^{1/2})} \right)^{1/2}.$$For $p \in [2,+\infty)$ we show that it is sufficient to assume that the manifold has the volume doubling property whereas for $p \in (1,2)$ we need extra assumptions of $L^p-L^2$ of diagonal estimates for $\{ \sqrt{t} \nabla e^{-tL}, t\geq 0 \}$ and $ \{ \sqrt{t} \sqrt{V} e^{-tL} , t \geq 0\}$.Given a bounded holomorphic function $F$ on some angular sector, we introduce the generalized conical vertical square functional$$\mathcal{G}_L^F (f) (x) = \left( \int_0^\infty \int_{B(x,t^{1/2})} |\nabla F(tL) f(y)|^2 + V |F(tL) f(y)|^2 \frac{\mathrm{d}t \mathrm{d}y}{Vol(y,t^{1/2})} \right)^{1/2}$$ and prove its boundedness on $L^p$ if $F$ has sufficient decay at zero and infinity. We also consider conical square functions associated with the Poisson semigroup, lower bounds, and make a link with the Riesz transform. | |
| dc.description.sponsorship | Analyse Réelle et Géométrie - ANR-18-CE40-0012 | |
| dc.language.iso | en | |
| dc.title.en | Conical square functionals on Riemannian manifolds | |
| dc.type | Document de travail - Pré-publication | |
| dc.subject.hal | Mathématiques [math]/Equations aux dérivées partielles [math.AP] | |
| dc.subject.hal | Mathématiques [math]/Analyse fonctionnelle [math.FA] | |
| dc.identifier.arxiv | 2101.01922 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | hal-03098420 | |
| hal.version | 1 | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-03098420v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=COMETX,%20Thomas&rft.genre=preprint |
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