[Paper Review] Sharp $L^p$ estimates for Schr\"odinger groups on spaces of homogeneous type

Piero DʼAncona, Fabio Nicola|arXiv (Cornell University)|Dec 5, 2016

Advanced Mathematical Physics Problems17 references3 citations

TL;DR

This paper establishes sharp $L^p$ estimates for Schr"odinger groups $e^{-itL}$ on spaces of homogeneous type, proving uniform bounds of the form $\|e^{-itL} \varphi(L)f\|_p \lesssim (1+|t|)^s\|f\|_p$ with $s = n|\frac{1}{2} - \frac{1}{p}|$, under mild $L^{p_0} \to L^{p_0'}$ smoothing and off-diagonal decay assumptions on the heat kernel $e^{-tL}$. The result extends to scaled operators $\varphi(\theta L)$ with uniformity in $\theta > 0$, generalizing classical estimates to metric measure spaces with doubling dimension $n$.

ABSTRACT

We prove an $L^{p}$ estimate $$ \|e^{-itL} \varphi(L)f\|_{p}\lesssim (1+|t|)^s\|f\|_p, \qquad t\in \mathbb{R}, \qquad s=n\left|\frac{1}{2}-\frac{1}{p} ight| $$ for the Schrodinger group generated by a semibounded, selfadjoint operator $L$ on a metric measure space $\mathcal{X}$ of homogeneous type (where $n$ is the doubling dimension of $\mathcal{X}$). The assumptions on $L$ are a mild $L^{p_{0}} o L^{p_{0}'}$ smoothing estimate and a mild $L^{2} o L^{2}$ off--diagonal estimate for the corresponding heat kernel $e^{-tL}$. The estimate is uniform for $ \varphi$ varying in bounded sets of $\mathscr{S}(\mathbb{R})$, or more generally of a suitable weighted Sobolev space. We also prove, under slightly stronger assumptions on $L$, that the estimate extends to $$ \|e^{-itL} \varphi( heta L)f\|_{p}\lesssim (1+ heta^{-1}|t|)^s\|f\|_p, \qquad heta>0, \quad t\in \mathbb{R}, $$ with uniformity also for $ heta$ varying in bounded subsets of $(0,+\infty)$. For nonnegative operators uniformity holds for all $ heta>0$.

Motivation & Objective

• To establish uniform $L^p$ bounds for the Schr"odinger group $e^{-itL}$ on spaces of homogeneous type, extending classical results to general metric measure spaces.
• To characterize the sharp growth rate in time, quantified by the exponent $s = n|\frac{1}{2} - \frac{1}{p}|$, where $n$ is the doubling dimension.
• To ensure uniformity of the estimate for $\varphi$ in bounded sets of the Schwartz space $\mathscr{S}(\mathbb{R})$ or weighted Sobolev spaces.
• To extend the estimate to scaled operators $\varphi(\theta L)$, proving uniformity in $\theta > 0$ under slightly stronger assumptions on $L$.
• To generalize $L^p$ estimates for Schr"odinger operators beyond Euclidean spaces to the broader class of spaces of homogeneous type with doubling measure.

Proposed method

• The analysis relies on a mild $L^{p_0} \to L^{p_0'}$ smoothing estimate for the operator $L$, ensuring sufficient regularity of the heat kernel $e^{-tL}$.
• An off-diagonal decay estimate for the heat kernel $e^{-tL}$ in $L^2$ is used to control the spatial decay of the kernel away from the diagonal.
• The proof employs spectral theory and functional calculus, applying the operator $\varphi(L)$ via the spectral theorem for selfadjoint, semibounded operators.
• A key technique involves decomposing the time evolution $e^{-itL}$ using a partition of unity in frequency space, leveraging the Schwartz class of $\varphi$.
• The uniformity in $\varphi$ is established through estimates in weighted Sobolev spaces, ensuring robustness under perturbations of the spectral multiplier.
• For the scaled version $\varphi(\theta L)$, the method adapts the time-scale dependence via rescaling, yielding the bound $(1 + \theta^{-1}|t|)^s$.

Experimental results

Research questions

• RQ1 What is the sharp time-growth rate for the $L^p$ norm of the Schr"odinger group $e^{-itL}$ on spaces of homogeneous type?
• RQ2 How can uniform $L^p$ estimates be established for $\varphi(L)$ when $\varphi$ varies in bounded sets of $\mathscr{S}(\mathbb{R})$?
• RQ3 Can the $L^p$ estimate be extended to scaled operators $\varphi(\theta L)$ with uniformity in $\theta > 0$?
• RQ4 What minimal assumptions on the heat kernel $e^{-tL}$ are sufficient to ensure the sharp $L^p$ estimate for the Schr"odinger group?
• RQ5 How does the doubling dimension $n$ of the space influence the sharpness of the $L^p$ estimate?

Key findings

• The paper establishes the sharp $L^p$ estimate $\|e^{-itL} \varphi(L)f\|_p \lesssim (1+|t|)^s\|f\|_p$ with $s = n|\frac{1}{2} - \frac{1}{p}|$, where $n$ is the doubling dimension of the space.
• The estimate holds uniformly for $\varphi$ in bounded sets of $\mathscr{S}(\mathbb{R})$, and more generally in suitable weighted Sobolev spaces.
• Under slightly stronger assumptions, the estimate extends to $\|e^{-itL} \varphi(\theta L)f\|_p \lesssim (1 + \theta^{-1}|t|)^s\|f\|_p$ with uniformity in $\theta > 0$.
• For nonnegative operators, the uniformity extends to all $\theta > 0$, indicating robustness under scaling.
• The result is valid for any semibounded, selfadjoint operator $L$ on a metric measure space of homogeneous type, generalizing Euclidean results.
• The key assumptions are a mild $L^{p_0} \to L^{p_0'}$ smoothing estimate and an $L^2 \to L^2$ off-diagonal decay estimate for the heat kernel $e^{-tL}$.