$L^q$-norm bounds for arithmetic eigenfunctions via microlocal Kakeya-Nikodym estimate

TL;DR

The paper proves a sublocal power-saving for the global $L^6$-norm of Hecke–Maass forms on compact arithmetic hyperbolic surfaces by establishing a microlocal Kakeya–Nikodym estimate. The authors reduce the $L^6(X)$ problem to a microlocal, frequency-localized bound via spectral projections and a microlocal decomposition, and then derive an improved microlocal estimate using arithmetic amplification in conjunction with oscillatory integral techniques. The main result is a bound of the form $\norm{\psi}_{L^6(X)} \lesssim_\varepsilon \lambda^{\frac{1}{6}-\frac{1}{144}+\varepsilon}$ for large spectral parameter $\lambda$, representing a power-saving over the local Sogge bound. The approach combines quaternion-algebra–based arithmetic, pretrace-formula amplification, and refined bilinear oscillatory estimates, with potential generalizations to Eichler orders and higher-rank or higher-dimensional locally symmetric spaces.

Abstract

Let $X$ be a compact arithmetic hyperbolic surface, and let $ψ$ be an $L^2$-normalized Hecke-Maass form on $X$ with sufficiently large spectral parameter $λ$. We give a new proof to obtain some power saving for the global $L^6$-norm $\|ψ\|_{L^6(X)}\lesssim_ελ^{\frac{1}{6}-\frac{1}{144}+ε}$ over the local bound $\|ψ\|_{L^6(X)}\lesssimλ^{\frac16}$ of Sogge. Using the $L^\infty$-norm bound $\|ψ\|_{L^\infty(X)}\lesssim_ελ^{\frac{5}{12}+ε}$ of Iwaniec and Sarnak and harmonic analysis tools, we reduce the $L^6$-norm problem to a microlocal $L^6$ Kakeya-Nikodym estimate for $ψ$. Finally, we establish an improved microlocal $L^6$ Kakeya-Nikodym estimate via arithmetic amplification developed by Iwaniec and Sarnak.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Proposition 5
• proof : Proof of Proposition \ref{['locprop']}
• Lemma 6
• proof
• Lemma 7