# Complex interpolation with Dirichlet boundary conditions on the half line

## TL;DR

This paper establishes complex interpolation results for vector-valued Sobolev spaces with Dirichlet boundary conditions on the half-line, with applications to evolution equations and fractional domain spaces.

## Contribution

It provides new results on complex interpolation of weighted Sobolev spaces with boundary conditions and introduces a simplified proof for related multiplier theorems.

## Key findings

- Characterization of fractional domain spaces of the first derivative operator
- New simplified proof for pointwise multipliers in Bessel potential spaces
- Application to evolution equations with boundary conditions

## Abstract

We prove results on complex interpolation of vector-valued Sobolev spaces over the half-line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space-valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel potential spaces, for which we present a new and simpler proof as well. We apply the results to characterize the fractional domain spaces of the first derivative operator on the half line.