[Paper Review] Localized $L^p$-estimates for eigenfunctions: II

Christopher D. Sogge|arXiv (Cornell University)|Oct 21, 2016

Advanced Harmonic Analysis Research11 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for improved $L^p$-norm estimates of $L^2$-normalized eigenfunctions on compact Riemannian manifolds of dimension $n \geq 2$, showing that such improvements occur precisely when $L^p$ or $L^2$ norms over small geodesic balls (radius $\lambda^{-1}$) or tubular neighborhoods of geodesics (radius $\lambda^{-1/2}$) do not saturate universal bounds. The key result is that improved $L^{p_c}$-bounds hold if these localized norms are suboptimal and the half-wave operators have no caustics for $t \neq 0$. The work resolves the structure of $L^p$-growth for eigenfunctions across all $p \neq p_c$, with $p_c = \frac{2(n+1)}{n-1}$ as the critical exponent.

ABSTRACT

If $(M,g)$ is a compact Riemannian manifold of dimension $n\ge 2$ we give necessary and sufficient conditions for improved $L^p(M)$-norms of eigenfunctions for all $2

Motivation & Objective

• To determine necessary and sufficient conditions for improved $L^p(M)$-norm estimates of eigenfunctions on compact Riemannian manifolds for all $p \neq p_c$, where $p_c = \frac{2(n+1)}{n-1}$ is the critical exponent.
• To resolve the structure of $L^p$-growth for eigenfunctions by characterizing when $\|e_\lambda\|_{L^p(M)} = o(\lambda^{\mu(p)})$ for $p > p_c$ and $2 < p < p_c$, in terms of localized $L^p$ or $L^2$ norms over small geodesic sets.
• To show that improved $L^{p_c}(M)$-bounds follow if the localized $L^p$-improvements hold for all $p \in (2,\infty) \setminus \{p_c\}$ and the half-wave operators $U(t) = e^{-it\sqrt{-\Delta_g}}$ have no caustics for $t \neq 0$, thus identifying a sufficient geometric condition.
• To close the gap in understanding the generic behavior of eigenfunctions by showing that saturation of localized norms (over $\lambda^{-1}$-balls or $\lambda^{-1/2}$-tubes) characterizes the failure of improved $L^p$-bounds.

Proposed method

• The analysis uses localized $L^p$ and $L^2$-norms over geodesic balls of radius $\lambda^{-1}$ for $p > p_c$, and over $\lambda^{-1/2}$-tubular neighborhoods of unit-length geodesics for $2 < p < p_c$, to characterize the saturation of universal bounds.
• The proof relies on a recent result from [15] on uniform $L^2$-bounds for spectral projectors over small balls, combined with Hölder’s inequality and the dual form of the $L^2$-restriction estimate (37).
• A key technical tool is the use of spectral projectors $\rho(T(\lambda - P))$ and their associated kernels, with the kernel bounds (37) and (40) derived from the properties of the Schwartz function $|\rho|^2$ with compactly supported Fourier transform.
• The argument involves decomposing level sets into sets $A_j$ of small diameter and positive separation, applying Chebyshev’s inequality and Cauchy-Schwarz to control the measure of the set where $|\rho(T_\varepsilon(\lambda - P))h(x)| > \alpha$, leading to the estimate $|A| \leq \varepsilon \lambda \alpha^{-\frac{2(n+1)}{n-1}}$.
• The method establishes equivalence between global $L^p$-improvement and localized $L^p$ or $L^2$-improvement by showing that failure of the localized bound implies failure of the global one, using contradiction and measure-theoretic estimates.
• The role of caustics is analyzed via the half-wave operator $U(t)$, showing that absence of caustics for $t \neq 0$ ensures improved $L^{p_c}$-bounds when localized improvements hold for all other $p \neq p_c$.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for improved $L^p(M)$-norm estimates of eigenfunctions on compact Riemannian manifolds for $p > p_c$?
• RQ2 How do localized $L^p$ or $L^2$-norms over geodesic balls of radius $\lambda^{-1}$ relate to the saturation of universal $L^p$-bounds for $p > p_c$?
• RQ3 What is the role of tubular neighborhoods of geodesics of radius $\lambda^{-1/2}$ in characterizing improved $L^p$-bounds for $2 < p < p_c$?
• RQ4 Under what geometric conditions on the manifold does the critical $L^{p_c}$-bound improve, given that it implies improvements for all other $p \neq p_c$?
• RQ5 Is the absence of caustics in the half-wave operators $U(t)$ for $t \neq 0$ a sufficient condition for improved $L^{p_c}(M)$-bounds when localized improvements hold for all $p \neq p_c$?

Key findings

• For $p > p_c$, improved $L^p(M)$-bounds $\|e_\lambda\|_{L^p(M)} = o(\lambda^{\mu(p)})$ hold if and only if the localized $L^p$-norms over geodesic balls of radius $\lambda^{-1}$ are suboptimal, i.e., $\sup_x \|e_\lambda\|_{L^p(B_{\lambda^{-1}}(x))} = o(\lambda^{\mu(p)})$.
• For $p > p_c$, the $L^2$-norm over $\lambda^{-1}$-balls must satisfy $\sup_x \|e_\lambda\|_{L^2(B_{\lambda^{-1}}(x))} = o(\lambda^{-1/2})$ for improved $L^p$-bounds to hold, and this condition is equivalent to the $L^p$-improvement.
• For $2 < p < p_c$, improved $L^p(M)$-bounds are equivalent to the localized $L^p$-norms over $\lambda^{-1/2}$-tubular neighborhoods of unit-length geodesics being suboptimal: $\sup_{\gamma \in \Pi} \|e_\lambda\|_{L^p(\mathcal{T}_{\lambda^{-1/2}}(\gamma))} = o(\lambda^{\mu(p)})$, and also equivalent to $\sup_{\gamma} \|e_\lambda\|_{L^2(\mathcal{T}_{\lambda^{-1/2}}(\gamma))} = o(1)$.
• Improved $L^{p_c}(M)$-bounds are guaranteed if the localized $L^p$-improvements hold for all $p \in (2,\infty) \setminus \{p_c\}$ and the half-wave operators $U(t)$ have no caustics for $t \neq 0$, providing a sufficient geometric condition.
• The round sphere $S^n$ saturates the universal $L^p$-bounds for $p \geq p_c$ in $\lambda^{-1}$-balls around poles and for $2 < p \leq p_c$ in $\lambda^{-1/2}$-tubes around the equator, showing the sharpness of the localized conditions.
• The proof establishes that if the localized $L^2$-norms (over $\lambda^{-1}$-balls or $\lambda^{-1/2}$-tubes) do not saturate the universal bounds, then the global $L^p$-norms improve, and this is sharp in the sense that saturation implies no improvement.