# Sobolev spaces for multi-black hole initial data

## TL;DR

This paper introduces specialized weighted Sobolev spaces tailored for multi-black hole initial data, proves key elliptic operator results on these spaces, and establishes the existence of initial data for multiple extremal black holes.

## Contribution

It develops a new framework of weighted Sobolev spaces for black hole initial data and proves existence results for extremal black hole configurations.

## Key findings

- Weighted Sobolev spaces are effective for multi-black hole initial data.
- Elliptic operators are well-behaved on these spaces.
- Existence of initial data for many extremal black holes is demonstrated.

## Abstract

In this article we introduce weighted Sobolev spaces that are well suited to treat initial data for multiple black hole systems. We prove general results for elliptic operators on these spaces and give a simple proof of existence of a class of initial data describing many extremal black holes.