[Paper Review] Littlewood--Paley--Stein Estimates for Non-local Dirichlet Forms

Huaiqian Li, Jian Wang|arXiv (Cornell University)|Apr 10, 2017

Advanced Harmonic Analysis Research17 references3 citations

TL;DR

This paper establishes $L^p$ boundedness of vertical Littlewood–Paley–Stein functions for non-local Dirichlet forms on metric measure spaces. Using a pseudo-gradient for $1 < p \leq 2$ and the Burkholder–Davis–Gundy inequality for $2 \leq p < \infty$, it proves $L^p$ bounds for the associated square function, extending results from symmetric Lévy processes to general non-local generators.

ABSTRACT

We obtain the boundedness in $L^p$ spaces for all $1

Motivation & Objective

• To establish $L^p$ boundedness of vertical Littlewood–Paley–Stein functions for non-local Dirichlet forms in metric measure spaces.
• To address the lack of chain rule for non-local operators by introducing a pseudo-gradient for $1 < p \leq 2$.
• To extend results from pure jump symmetric Lévy processes in Euclidean spaces to general non-local generators on metric measure spaces.
• To unify analytic and probabilistic methods in proving $L^p$ bounds for non-local operators.
• To verify the boundedness of the square function in $L^p$ for all $1 < p < \infty$ under mild structural assumptions on the jumping kernel.

Proposed method

• Introduce a modified gradient $|\widetilde{\nabla}f|$ based on the local ordering of function values to handle non-locality and avoid chain rule issues.
• For $1 < p \leq 2$, employ the pseudo-gradient and Mosco convergence to extend boundedness from finite-jumping-kernel cases to general non-local forms.
• For $2 \leq p < \infty$, apply the probabilistic Burkholder–Davis–Gundy inequality to control the quadratic variation of martingales associated with the semigroup.
• Use the carré du champ operator $\Gamma(f)$ to define the gradient norm $|\nabla f|(x) = \sqrt{\Gamma(f)(x)}$ as the pointwise module of the gradient.
• Leverage the semigroup $(P_t)$ generated by the non-local generator $L$ to define the square function $\mathscr{H}_{\nabla}f$ via time integration of $|\nabla P_t f|^2$.
• Apply Jensen’s inequality and contraction properties of the semigroup to control $L^p$ norms of the square function.

Experimental results

Research questions

• RQ1 Can the Littlewood–Paley–Stein estimate be extended to non-local Dirichlet forms beyond symmetric Lévy processes?
• RQ2 How can one overcome the absence of a chain rule in non-local operators when proving $L^p$ bounds?
• RQ3 What techniques can unify analytic and probabilistic approaches in the study of non-local operators?
• RQ4 Under what conditions on the jumping kernel does the $L^p$ boundedness of the vertical square function hold?
• RQ5 Is the Mosco convergence method effective in passing from finite-jump kernels to general non-local forms?

Key findings

• For all $1 < p < \infty$, the vertical Littlewood–Paley–Stein function $\mathscr{H}_{\nabla}f$ is bounded in $L^p(M,\mu)$, with $\|\mathscr{H}_{\nabla}f\|_p \leq C_p \|f\|_p$.
• For $1 < p \leq 2$, the pseudo-gradient construction enables the proof of $L^p$ boundedness despite the absence of a chain rule.
• For $2 \leq p < \infty$, the Burkholder–Davis–Gundy inequality provides a sharp probabilistic estimate that yields the desired $L^p$ bound.
• The Mosco convergence technique successfully extends the result from finite-jumping-kernel cases to general non-local Dirichlet forms.
• The method applies to regular non-local Dirichlet forms of pure jump type on metric measure spaces under mild assumptions.
• The result generalizes known estimates for symmetric Lévy processes in $\mathbb{R}^d$ to a broader class of non-local generators on metric measure spaces.