# An extension problem related to the fractional Branson-Gover operators

## TL;DR

This paper extends the theory of fractional Branson-Gover operators, showing they can be realized as Dirichlet-to-Neumann maps for boundary value problems, generalizing fractional Laplacian results to differential forms.

## Contribution

It introduces a new boundary value problem framework for fractional Branson-Gover operators, including explicit kernels and solution properties, expanding conformal geometry tools.

## Key findings

- Fractional Branson-Gover operators are Dirichlet-to-Neumann maps.
- Explicit integral kernels for solutions are derived.
- Unique solutions exist in Sobolev spaces.

## Abstract

The Branson-Gover operators are conformally invariant differential operators of even degree acting on differential forms. They can be interpolated by a holomorphic family of conformally invariant integral operators called fractional Branson-Gover operators. For Euclidean spaces we show that the fractional Branson-Gover operators can be obtained as Dirichlet-to-Neumann operators of certain conformally invariant boundary value problems, generalizing the work of Caffarelli-Silvestre for the fractional Laplacians to differential forms. The relevant boundary value problems are studied in detail and we find appropriate Sobolev type spaces in which there exist unique solutions and obtain the explicit integral kernels of the solution operators as well as some of its properties.