arXiv subjects

Jocelyn Magniez

Publications and source records attributed to Jocelyn Magniez.

3 recordsLinked to original sources

arXiv2017

L p -estimates for the heat semigroup on differential forms, and related problems

We consider a complete non-compact Riemannian manifold satisfying the volume doubling property and a Gaussian upper bound for its heat kernel (on functions). Let -- $\rightarrow$ $Δ$ k be the Hodge-de Rham Laplacian on differential k-forms with k $\ge$ 1. By the Bochner decomposition formula -- $\rightarrow$ $Δ$ k = * + R k. Under the assumption that the negative part R -- k is in an enlarged Kato class, we prove that for all p $\in$ [1, $\infty$], e --t -- $\rightarrow$ $Δ$ k p--p $\le$ C(t log t) D 4 (1-- 2 p) (for large t). This estimate can be improved if R -- k is strongly sub-critical. In general, (e --t -- $\rightarrow$ $Δ$ k) t>0 is not uniformly bounded on L p for any p = 2. We also prove the gradient estimate e --t$Δ$ p--p $\le$ Ct -- 1 p , where $Δ$ is the Laplace-Beltrami operator (acting on functions). Finally we discuss heat kernel bounds on forms and the Riesz transform on L p for p > 2.

math.AP↗

arXiv2014

Riesz transforms on non-compact manifolds

Let $M$ be a complete non-compact Riemannian manifold satisfying the doubling volume property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform $dΔ^{-\frac{1}{2}}$ on both Hardy spaces $H^p$ and Lebesgue spaces $L^p$ under two different conditions on the negative part of the Ricci curvature $R^-$. First we prove that if $R^-$ is $α$-subcritical for some $α\in [0,1)$, then the Riesz transform $d^*Δ^{-\frac{1}{2}}$ on differential $1$-forms is bounded from the associated Hardy space $H^p_{\overrightarrowΔ}(Λ^1T^*M)$ to $L^p(M)$ for all $p\in [1,2]$. As a consequence, the Riesz transform (on functions) is bounded on $ L^p$ for all $p\in (1,p_0)$ where $p_0>2$ depends on $α$ and the constant appearing in the doubling property. Second, we prove that if $$\int_0^1 \left\|\frac{|R^-|^{\frac{1}{2}}}{v(\cdot,\ \sqrt{t})^{\frac{1}{p_1}}}\right\|_{p_1}\frac{dt}{\sqrt{t}}+\int_1^\infty \left\|\frac{|R^-|^{\frac{1}{2}}}{v(\cdot,\ \sqrt{t})^{\frac{1}{p_2}}}\right\|_{p_2}\frac{dt}{\sqrt{t}}<\infty,$$ for some $p_1>2$ and $p_2>3$, then the Riesz transform $dΔ^{-\frac{1}{2}}$ is bounded on $L^p$ for all $1 0$, then $dΔ^{-\frac{1}{2}}$ is bounded on $L^p$ for all $1 2$ under conditions on $R^-$ and the potential $V$. We prove both positive and negative results on the boundedness of $dA^{-\frac{1}{2}}$ on $L^p$

math.AP↗

arXiv2014

Riesz transforms of the Hodge-de Rham Laplacian on Riemannian manifolds

Let $M$ be a complete non-compact Riemannian manifold satisfying the doubling volume property. Let $\overrightarrowΔ$ be the Hodge-de Rham Laplacian acting on 1-differential forms. According to the Bochner formula, $\overrightarrowΔ=\nabla^*\nabla+R_+-R_-$ where $R_+$ and $R_-$ are respectively the positive and negative part of the Ricci curvature and $\nabla$ is the Levi-Civita connection. We study the boundedness of the Riesz transform $d^*(\overrightarrowΔ)^{-\frac{1}{/2}}$ from $L^p(Λ^1T^*M)$ to $L^p(M)$ and of the Riesz transform $d(\overrightarrowΔ)^{-\frac{1}{2}}$ from $L^p(Λ^1T^*M)$ to $L^p(Λ^2T^*M)$. We prove that, if the heat kernel on functions $p_t(x,y)$ satisfies a Gaussian upper bound and if the negative part $R_-$ of the Ricci curvature is $ε$-sub-critical for some $ε\in[0,1)$, then $d^*(\overrightarrowΔ)^{-\frac{1}{2}}$ is bounded from $L^p(Λ^1T^*M)$ to $L^p(M)$ and $d(\overrightarrowΔ)^{-\frac{1}{2}}$ is bounded from $L^p(Λ^1T^*M)$ to $L^p(Λ^2T^* M)$ for $p\in(p_0',2]$ where $p_0>2$ depends on $ε$ and on a constant appearing in the doubling volume property. A duality argument gives the boundedness of the Riesz transform $d(Δ)^{-\frac{1}{2}}$ from $L^p(M)$ to $L^p(Λ^1T^*M)$ for $p\in [2,p_0)$ where $Δ$ is the non-negative Laplace-Beltrami operator. We also give a condition on $R_-$ to be $ε$-sub-critical under both analytic and geometric assumptions.

math.AP↗