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An RKHS approach to the indefinite Schwarz-Pick inequality on the bidisk

Kenta Kojin

Abstract

In this short note, we will give a generalization of the indefinite Schwarz-Pick inequality due to Seto [8]. Our approach is based on a connection between complex geometry and the geometry of reproducing kernel Hilbert spaces, which was crucially used in our previous work [6].

An RKHS approach to the indefinite Schwarz-Pick inequality on the bidisk

Abstract

In this short note, we will give a generalization of the indefinite Schwarz-Pick inequality due to Seto [8]. Our approach is based on a connection between complex geometry and the geometry of reproducing kernel Hilbert spaces, which was crucially used in our previous work [6].

Paper Structure

This paper contains 3 sections, 8 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Theorem

Key Result

Theorem 1.1

If $F:\mathbb{D}^2\rightarrow\mathbb{D}^2$ is a holomorphic map on $\mathbb{D}^2$, then $F$ satisfies for all $(z_1, z_2), (w_1, w_2)\in\mathbb{D}^2$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2: $Kob$
  • Definition 2.3
  • Lemma 2.4: Koj
  • Remark 2.5
  • Example 3.1
  • Example 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 6 more