[Paper Review] Spherical harmonics with maximal Lp (2<p<=6) norm growth

Xiaolong Han|arXiv (Cornell University)|Apr 20, 2014

Advanced Mathematical Modeling in Engineering14 references3 citations

TL;DR

This paper constructs a positive-density subsequence of orthonormal spherical harmonics on the 2-sphere that achieves maximal Lp norm growth for 2 < p ≤ 6, resolving an open question by Sogge and Zelditch. Using a perturbation of Gaussian beams via orthogonalization, the authors prove the existence of such eigenfunctions with explicit lower bounds on density, demonstrating that maximal Lp growth can occur in a generic orthonormal basis, not just sparse sequences like Gaussian beams.

ABSTRACT

In this paper, we show that there exists a positive density subsequence of orthonormal spherical harmonics which achieves the maximal Lp norm growth for 2<p<=6, therefore giving an example of a Riemannian surface supporting such subsequence of eigenfunctions. This answers the question proposed by Sogge and Zelditch (arXiv:1011.0215). Furthermore, we provide an explicit lower bound on the density in this example.

Motivation & Objective

• To resolve a question posed by Sogge and Zelditch on whether maximal Lp norm growth for 2 < p ≤ 6 can occur in a positive-density orthonormal eigenfunction basis on a Riemannian surface.
• To construct an explicit orthonormal sequence of spherical harmonics that achieves the sharp Sogge Lp estimate growth rate σ(p) = (1/2 - 1/p)/2 for 2 < p ≤ 6.
• To provide an explicit lower bound on the density of such a subsequence, showing it is not sparse like Gaussian beams.
• To analyze the localization properties of the constructed eigenfunctions and show they retain near-Gaussian-beam concentration around great circles.

Proposed method

• Perturbing a set of M Gaussian beams (Q±k) on the sphere to produce an orthonormal basis via Gram-Schmidt-like orthogonalization, preserving their Lp norm growth.
• Using the Wigner D-matrix formalism to compute matrix entries Fij of the transformation matrix between Gaussian beams and the orthonormalized functions ui.
• Applying matrix norm estimates |||F||| ≤ 1 + 6r and row sum bounds R′i(F) ≤ 6r to control the L2 mass distribution of the ui's.
• Establishing that the L2 mass of ui remains concentrated in the equatorial tube Gw1 for w ~ k^{-1/2} by choosing small density D and small r.
• Using asymptotic analysis of Riemann sums to approximate integrals involving (sin φ)(cos φ/2)^{2k}, showing convergence to 2/(k+1) as k → ∞.
• Deriving quantitative bounds on the Lp norms of ui by comparing them to the known norms of Gaussian beams Q±k, yielding ∥ui∥p ≳ k^{1/4 - 1/(2p)} for p > 6.

Experimental results

Research questions

• RQ1 Can a positive-density subsequence of orthonormal eigenfunctions achieve maximal Lp norm growth for 2 < p ≤ 6 on a Riemannian surface?
• RQ2 Is it possible to construct such a subsequence explicitly, with a lower bound on its density?
• RQ3 Do the constructed eigenfunctions inherit the localization properties of Gaussian beams, such as concentration near great circles?
• RQ4 How does the density D of the subsequence affect the L2 mass distribution and norm growth of the eigenfunctions?

Key findings

• The paper constructs an orthonormal basis of spherical harmonics with a positive-density subsequence achieving ∥uj∥p ≳ λσ(p)j for 2 < p ≤ 6, confirming maximal Lp norm growth in a dense subsequence.
• An explicit lower bound on the density is provided: for any D > 0, a subsequence of density at least D exists with the desired norm growth, and the construction works as D → 0.
• The L2 mass of the orthonormalized functions ui remains uniformly bounded below in the equatorial tube Gw1 (width w ~ k^{-1/2}), with ∥u1∥L2(Gw1) ≥ 1/8, showing strong localization.
• The Lp norm of the constructed eigenfunctions satisfies ∥ui∥p ≳ k^{1/4 - 1/(2p)} for p > 6, which is weaker than the zonal harmonics' ∥Zk∥p ≳ k^{1/2 - 2/p}, indicating the construction does not maximize for p > 6.
• The method ensures that the L2 mass outside the equatorial tube is small: ∥u1∥L2(S2\Gw1) ≤ ε for small ε, by choosing sufficiently small density D.
• The construction is robust under small perturbations, as the matrix transformation from Gaussian beams to orthonormal functions has small off-diagonal row sums and bounded operator norm.