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Kakeya-Nikodym norms of Maass forms on $\rm{U}(2,1)$

Jiaqi Hou

TL;DR

The paper investigates Kakeya-Nikodym norms for Hecke-Maass forms on a compact arithmetic complex hyperbolic surface. It employs arithmetic amplification at split finite places to bound the integrated restrictions of Maass forms to tube neighborhoods around geodesics, obtaining power savings over the trivial KN bound and translating these into improved Lp bounds for 2<p<10/3. The approach hinges on an integrated pretrace formula, detailed analysis of oscillatory integrals tied to the Iwasawa A-projection, and a GL(3) amplifier, with near-spectrum and away-from-spectrum regimes handled separately. The results extend the KN-norm framework to complex hyperbolic settings and yield quantitative subconvex-type improvements for Kakeya-type restriction phenomena in the arithmetic context.

Abstract

Let $ψ$ be a Hecke-Maass form with a large spectral parameter on a compact arithmetic complex hyperbolic surface. We apply the amplification method to obtain a power saving over the trivial bound for the Kakeya-Nikodym norm of $ψ$. As a consequence, we obtain power savings over the local bound of Sogge for $\|ψ\|_p$ when $2<p<10/3$.

Kakeya-Nikodym norms of Maass forms on $\rm{U}(2,1)$

TL;DR

The paper investigates Kakeya-Nikodym norms for Hecke-Maass forms on a compact arithmetic complex hyperbolic surface. It employs arithmetic amplification at split finite places to bound the integrated restrictions of Maass forms to tube neighborhoods around geodesics, obtaining power savings over the trivial KN bound and translating these into improved Lp bounds for 2<p<10/3. The approach hinges on an integrated pretrace formula, detailed analysis of oscillatory integrals tied to the Iwasawa A-projection, and a GL(3) amplifier, with near-spectrum and away-from-spectrum regimes handled separately. The results extend the KN-norm framework to complex hyperbolic settings and yield quantitative subconvex-type improvements for Kakeya-type restriction phenomena in the arithmetic context.

Abstract

Let be a Hecke-Maass form with a large spectral parameter on a compact arithmetic complex hyperbolic surface. We apply the amplification method to obtain a power saving over the trivial bound for the Kakeya-Nikodym norm of . As a consequence, we obtain power savings over the local bound of Sogge for when .

Paper Structure

This paper contains 28 sections, 23 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Acknowledgements
  4. Notation
  5. Unitary groups
  6. Lie groups and alegrbas
  7. Complex hyperbolic plane
  8. Hecke algebras
  9. Maass forms
  10. Fourier transforms
  11. Amplification and main theorem
  12. Amplification inequality
  13. Construction of amplifier
  14. Estimates of Hecke returns
  15. Proof of main theorem
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $X$ be a compact arithmetic congruence complex hyperbolic surface and let $\psi$ be an $L^2$-normalized Hecke–Maass form on $X$ with spectral parameter $\lambda>0$ (see § sec: notation for precise definitions). Then for any $\epsilon>0$ we have where the implied constant depends only on $X$ and $\epsilon$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Corollary 1.2
  • Remark
  • Remark
  • Proposition 3.1
  • proof
  • Lemma 3.2: marshall2015restrictions, Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 33 more