[Paper Review] Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Xuan Thinh Duong, Adam Sikora+1 more|arXiv (Cornell University)|Mar 9, 2010

Advanced Harmonic Analysis Research26 references3 citations

TL;DR

This paper establishes sharp weighted $L^p$-boundedness for spectral multipliers of non-negative self-adjoint operators $L$ on spaces of homogeneous type, under Gaussian heat kernel bounds and $L^2$-estimates of multiplier kernels. The key contribution is a general framework yielding Hörmander-type multiplier theorems in weighted $L^p$ spaces, applicable to Laplacians on Lie groups, irregular domains, compact manifolds, and Schrödinger operators with non-negative potentials.

ABSTRACT

Let $L$ be a non-negative self adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ have Gaussian upper bounds but possess no regularity in variables $x$ and $y$. In this article, we study weighted $L^p$-norm inequalities for spectral multipliers of $L$. We show sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate $L^2$ estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials on complete Riemannian manifolds.

Motivation & Objective

• To extend sharp spectral multiplier theorems to weighted $L^p$ spaces for non-negative self-adjoint operators with Gaussian heat kernel bounds.
• To overcome the lack of kernel regularity in $x$ and $y$ variables, which invalidates standard Calderón-Zygmund theory.
• To establish sufficient conditions on spectral multipliers $F(L)$ ensuring boundedness on weighted $L^p(X,w)$ spaces using $L^2$-estimates and Plancherel-type estimates.
• To generalize results from unweighted to weighted settings, particularly for operators on irregular domains, compact manifolds, and Schrödinger operators.
• To derive quantitative operator norm estimates for holomorphic functional calculi in terms of the spectral function's growth in a sector.

Proposed method

• Use of Gaussian upper bounds on the heat kernel $p_t(x,y)$ of the semigroup $e^{-tL}$, without requiring smoothness in $x$ and $y$.
• Application of Plancherel-type estimates to control the $L^2$-norm of multiplier kernels, enabling control in $L^p$ via interpolation.
• Adaptation of techniques from [18] involving analyticity in $t$ and dyadic decomposition of the multiplier function $F$.
• Introduction of weighted norm inequalities via the $A_p$-Muckenhoupt class, with sharp dependence on the weight characteristic $p$.
• Use of the scaling $\delta_t F(\lambda) = F(t\lambda)$ and the norm $\|\eta \delta_t F\|_{W^{\infty}_s}$ to quantify smoothness requirements.
• Interpolation between $L^2$ and $L^p$ estimates, leveraging the sharp $L^2$-boundedness of spectral multipliers under Gaussian bounds.

Experimental results

Research questions

• RQ1 Under what conditions is a spectral multiplier $F(L)$ bounded on weighted $L^p(X,w)$ spaces when the heat kernel of $L$ satisfies Gaussian bounds but lacks regularity in $x$ and $y$?
• RQ2 Can sharp Hörmander-type multiplier theorems be extended to weighted $L^p$ spaces under only Gaussian heat kernel bounds and $L^2$-estimates of the multiplier kernel?
• RQ3 How does the range of $p$ for which $F(L)$ is bounded depend on the weight $w$ and the smoothness of $F$?
• RQ4 What is the sharp dependence of the operator norm $\|F(L)\|_{L^p(X,w)\to L^p(X,w)}$ on the spectral function $F$ and the angle $\theta$ of its holomorphic extension?
• RQ5 To what extent can these results be applied to Schrödinger operators, Laplacians on irregular domains, and Lie groups of polynomial growth with non-smooth kernels?

Key findings

• Sharp weighted $L^p$-boundedness of spectral multipliers $F(L)$ holds for $p > r_0 = \max\left\{1, \frac{2(n+D)}{2s+D}\right\}$, where $s > n/2$ is the smoothness threshold and $D$ is a dimension-like parameter.
• The operator norm satisfies $\|F(L)\|_{L^p(X,w)\to L^p(X,w)} \leq \frac{C_\epsilon}{\theta^{n/2 + \epsilon}} \|F\|_{\theta,\infty}$ for holomorphic $F$ on a sector of angle $\theta$, with $\epsilon > 0$.
• For the Laplacian $\Delta_\Omega$ on an irregular domain $\Omega$ with Dirichlet boundary conditions, $\|F(\Delta_\Omega)\|_{L^p(\Omega,w)\to L^p(\Omega,w)} \leq C_s \left( \sup_{t>0}\|\eta \delta_t F\|_{W^{\infty}_s} + |F(0)| \right)$.
• For Schrödinger operators $L = -\Delta + V$ with $V \geq 0$, the same weighted estimates hold due to stochastic domination of heat kernels by the Gaussian kernel.
• The results generalize the unweighted sharp multiplier theorems from [18] to weighted spaces, with the same smoothness threshold $s > n/2$.
• The method avoids standard Calderón-Zygmund theory by relying on Plancherel estimates and analyticity in the time parameter $t$ of the heat semigroup.