On sup-norm bounds part II: GL(2) Eisenstein series
-
Edgar Assing
Abstract
In this paper we consider the sup-norm problem in the context of analytic Eisenstein series for
A Averaging non-unitary Whittaker new vectors
In this appendix we extend [20, Proposition 2.9] to allow non-unitary principal series representations. This is needed to deal with the Whittaker expansion of Eisenstein series for general s. The computations in this appendix rely heavily on the explicit expressions for the constants
All the computations in this appendix are done in a fixed non-archimedean field
For the sake of exposition we consider three cases.
Lemma A.1.
Let
Proof.
By [2, (1.6)] and character orthogonality it is clear that
We insert the expressions for
Case (1).
The easiest situation occurs when
Case (2).
For
Recalling
Case (3).
At last we look at
To estimate the first sum, we write
This yields
Combining these three estimates completes the proof. ∎
Lemma A.2.
Let
Note that this covers also the analogous case
Proof.
For convenience we write
Case (1).
If
Here we used
Case (2).
Similarly, if
Case (3).
Finally, if
Case (4).
Next we consider
Here we used (A.2) to estimate the μ-sum.
Case (5).
The last case to look at is
This covers all cases and the proof is complete. ∎
Lemma A.3.
Let
Proof.
For simplicity we write
The other exceptional contribution comes from a character μ satisfying
At last we deal with the contribution of generic μ. Together with the cases above we have
First, we observe that the generic characters only contribute when
One has the trivial bound
Therefore, by ordering the summation in
The stated inequality follows after putting everything together. ∎
The last three lemmata together imply an extension of [20, Proposition 2.10].
Proposition A.1.
Let
Then we have:
If
If
Proof.
First we note that there is
By the transformation behavior of
With this at hand the proposition follows from the previous lemmata. ∎
Acknowledgements
I would like to thank the referee for carefully reading the manuscript, which has led to a significant improvement of this work.
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- On sup-norm bounds part II: GL(2) Eisenstein series
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Articles in the same Issue
- Frontmatter
- Characteristic functions of semigroups in semi-simple Lie groups
- Damping estimates for oscillatory integral operators with real-analytic phases and its applications
- Formality properties of finitely generated groups and Lie algebras
- Toric actions in cosymplectic geometry
- On classification of typical representations for GL3(F)
- Integrable generators of Lie algebras of vector fields on ℂn
- Expanding phenomena over matrix rings
- On sup-norm bounds part II: GL(2) Eisenstein series
- Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces
- Partial and full boundary regularity for non-autonomous functionals with Φ-growth conditions
- Radial averaging operator acting on Bergman and Lebesgue spaces
- The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem
