[Paper Review] On the $L^p$-estimates of Riesz transforms on forms over complete Riemanian manifolds

Xiang‐Dong Li|arXiv (Cornell University)|Apr 10, 2013

Advanced Harmonic Analysis Research4 references3 citations

TL;DR

This paper corrects a gap in the probabilistic representation of Riesz transforms on differential forms over complete Riemannian manifolds, establishes a time-reversed martingale transform formula, and extends $L^p$-norm estimates to Riesz transforms on Euclidean vector bundles under curvature conditions. The key result is dimension-free $L^p$ bounds for $p>1$ with sharp dependence on $p$ via martingale inequalities and curvature-controlled stochastic parallel transport.

ABSTRACT

In our previous paper \cite{Li2010}, we proved a martingale transform representation formula for the Riesz transforms on forms over complete Riemannian manifolds, and proved some explicit $L^p$-norm estimates for the Riesz transforms on complete Riemannian manifolds with suitable curvature conditions. In this paper we correct a gap contained in \cite{Li2010} and prove that the main result obtained in \cite{Li2010} on the $L^p$-norm estimates for the Riesz transforms on forms remain valid. Moreover, we prove a time reversal martingale transform representation formula for the Riesz transforms on forms. Finally, we extend our approach and result to the Riesz transforms acting on Euclidean vector bundles over complete Riemannian manifolds with suitable curvature conditions.

Motivation & Objective

• To correct a gap in the probabilistic representation of Riesz transforms on differential forms over complete Riemannian manifolds identified by Bañuelos and Baudoin.
• To establish a time-reversal martingale transform representation for Riesz transforms on forms.
• To extend $L^p$-norm estimates of Riesz transforms to Euclidean vector bundles over complete Riemannian manifolds with curvature conditions.
• To provide dimension-free $L^p$ bounds for $d(a + oxdot_{ ext{F}, ext{ extbackslash phi}})^{-1/2}$ and $d_{ ext{ extbackslash phi}}^{*}(a + oxdot_{ ext{F}, ext{ extbackslash phi}})^{-1/2}$ with explicit dependence on $p$.

Proposed method

• Derives a corrected martingale transform representation for Riesz transforms using time-reversed stochastic processes and adapted stochastic integrals.
• Applies the corrected representation to derive $L^p$-norm estimates via martingale inequalities, particularly leveraging the inequality from Bañuelos and Baudoin.
• Uses stochastic parallel transport $U_t$ and curvature-controlled evolution $M_{t,k}$ along $L$-diffusion processes $X_t$ to model the Riesz transform action.
• Introduces the Poisson semigroup $Q_a\omega(x,y) = e^{-y\sqrt{a + \square_{\phi}}}\omega(x)$ to represent resolvent operators in the probabilistic framework.
• Applies duality arguments to extend results from $d(a + \square_{\phi})^{-1/2}$ to its adjoint $d_{\phi}^{*}(a + \square_{\phi})^{-1/2}$ on vector bundles.
• Establishes bounds under curvature assumptions $W_i + d\Lambda^i\nabla^2\phi \geq -a$ for $i = k \pm 1$.

Experimental results

Research questions

• RQ1 Can the probabilistic representation of Riesz transforms on forms over complete Riemannian manifolds be corrected to address the non-adaptedness issue in the original formula?
• RQ2 Does the corrected martingale representation allow for valid $L^p$-norm estimates under curvature conditions?
• RQ3 Can the approach be extended to Riesz transforms acting on Euclidean vector bundles with non-trivial connection and curvature?
• RQ4 What is the precise dependence of the $L^p$-bound constants on $p$, especially near $p=1$ and $p=\infty$?

Key findings

• The main result of [4] on $L^p$-norm estimates for Riesz transforms on forms remains valid after correcting the probabilistic representation formula.
• A time-reversal martingale transform representation is established for $R_a^1(\square_\phi)$ and $R_a^2(\square_\phi)$, resolving the original gap.
• For $p>1$, the $L^p$-norm of $d^F(a + \square_{F,\phi})^{-1/2}\omega$ is bounded by $C_p \|A\| \|\omega\|_p$, where $C_p = \|A\|^{-1}$ for $p=2$.
• The constant $C_p$ satisfies $C_p = O((p^*-1)^{3/2})$ as $p \to 1^+$ and $p \to \infty$, indicating sharp growth near the endpoints.
• The results extend to Riesz transforms on Euclidean vector bundles with flat connection and curvature-controlled Witten Laplacian, under the condition $W_i + d\Lambda^i\nabla^2\phi \geq -a$.
• The $L^p$-bounds are dimension-free and depend only on $p$ and the operator norm $\|A\|$ of the connection-induced endomorphism $A$.