[Paper Review] Weighted Nash Inequalities

Dominique Bakry, François Bolley+2 more|Apr 20, 2010

Nonlinear Partial Differential Equations17 references4 citations

TL;DR

This paper introduces a novel method using weighted Nash inequalities to derive non-uniform bounds on heat kernel densities for Markov semigroups that are not ultracontractive. By incorporating weights tied to the potential growth of the invariant measure, the approach yields pointwise estimates and trace-class control for the semigroup, particularly applied to the $&lt;em&gt;\mathbb{R}$-valued Lévy process with density $C_a\exp(-|x|^a)$ for $1 < a < 2$, where uniform bounds fail.

ABSTRACT

Nash or Sobolev inequalities are known to be equivalent to ultracontractive properties of Markov semigroups, hence to uniform bounds on their kernel densities. In this work we present a simple and extremely general method, based on weighted Nash inequalities, to obtain non-uniform bounds on the kernel densities. Such bounds imply a control on the trace or the Hilbert-Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on $\dR$ naturally associated with the measure with density $C_a\exp(-|x|^a)$, with $1

Motivation & Objective

• To develop a general method for obtaining non-uniform bounds on heat kernel densities when ultracontractivity fails.
• To extend classical Nash inequality techniques to weighted settings that capture the decay and growth properties of non-Gaussian, heavy-tailed measures.
• To provide trace and Hilbert-Schmidt norm control for the semigroup associated with the generator $Lf = f'' - a|x|^{a-2}f'$ under the measure $d\mu_a(x) = C_a\exp(-|x|^a)\,dx$.
• To analyze the spectral properties of the generator $-L$ via kernel estimates, particularly the discrete nature of the spectrum and eigenvalue decay.

Proposed method

• The method relies on a weighted Nash inequality of the form $\|f\|_2^{1+n/2} \leq \|f\|_1 \left[ a\mathcal{E}(f,f) + b\|f\|_2^2 \right]^{n/4}$ with a weight function $V$ that captures the measure's tail behavior.
• A Lyapunov function $V = \exp(T^{a}/2)T^{-\beta}$ is used to control the growth of the weight, ensuring integrability and enabling the derivation of non-uniform bounds.
• The core technique applies Theorem 2.5, which links the existence of a weighted Nash inequality with a rate function $\phi(x)$ to pointwise bounds on the semigroup kernel $p_t(x,y)$.
• The method uses the duality between the Dirichlet form and the semigroup to derive bounds of the form $p_t(x,y) \leq C t^{-\delta} \rho_a^{-1/2}(x)\rho_a^{-1/2}(y) (1+|x|^2)^{-\beta/2}(1+|y|^2)^{-\beta/2}$.
• The trace of the semigroup is controlled by integrating the diagonal estimate $p_t(x,x)$, leading to $\sum_n e^{-\lambda_n t} \leq C t^{-\delta} e^{Ct}$, implying discrete spectrum.

Experimental results

Research questions

• RQ1 Can non-uniform bounds on heat kernel densities be derived for non-ultracontractive semigroups using functional inequalities?
• RQ2 How can weighted Nash inequalities be constructed and applied to measures with heavy tails, such as $\exp(-|x|^a)$ for $1 < a < 2$?
• RQ3 What spectral consequences follow from non-uniform kernel bounds, particularly regarding the trace and eigenvalue distribution?
• RQ4 Is the Gaussian case ($a=2$) truly exceptional in terms of kernel bounds, and can it be characterized as a critical case?

Key findings

• For the measure $d\mu_a(x) = C_a \exp(-|x|^a)\,dx$ with $1 < a < 2$, the heat kernel satisfies $p_t(x,y) \leq C t^{-\delta} \rho_a^{-1/2}(x)\rho_a^{-1/2}(y) (1+|x|^2)^{-\beta/2}(1+|y|^2)^{-\beta/2}$ for all $t > 0$, with $\delta > 0$.
• The semigroup $P_t$ is Hilbert-Schmidt, and its trace satisfies $\sum_n e^{-\lambda_n t} \leq C t^{-\delta} e^{Ct}$, implying a discrete spectrum.
• The method fails to recover Orlicz hypercontractivity, indicating that non-uniform bounds cannot imply stronger smoothing properties like hypercontractivity.
• For $a > 2$, the method recovers ultracontractivity via a non-weighted Nash inequality, consistent with prior results.
• The Gaussian case ($a=2$) is exceptional: the optimal diagonal bound is $C(t) \exp(|x|^2/(1+e^{2t}))$, which cannot be captured by any $\rho^{-\lambda}(x)$ with $\lambda < 1/2$, suggesting critical behavior.