Logarithmic improvements in $L^{p}$ bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature

TL;DR

This paper proves improved $L^p$ bounds for Laplacian eigenfunctions at the critical exponent on manifolds with nonpositive curvature, showing a logarithmic gain over classical bounds in high-frequency regimes.

Contribution

It establishes logarithmic improvements in $L^p$ bounds for eigenfunctions at the critical exponent on nonpositively curved manifolds, extending understanding of eigenfunction concentration.

Findings

Achieved inverse logarithmic gain in eigenfunction bounds

Established bounds at the critical exponent $p_c$

Applicable to manifolds without conjugate points

Abstract

We consider the problem of proving $L^p$ bounds for eigenfunctions of the Laplacian in the high frequency limit in the presence of nonpositive curvature and more generally, manifolds without conjugate points. In particular, we prove estimates at the "critical exponent" $p_c = \frac{2(d+1)}{d-1}$, where a spectrum of scenarios for phase space concentration must be ruled out. Our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical $L^p$ bounds of the second author.