# On the harmonic extension approach to fractional powers in Banach spaces

## TL;DR

This paper demonstrates that fractional powers of sectorial operators in Banach spaces can be derived using the harmonic extension method, establishing existence and uniqueness of solutions for related differential equations.

## Contribution

It introduces the harmonic extension approach for fractional powers of sectorial operators and proves well-posedness of the associated differential equations.

## Key findings

- Fractional powers can be obtained via harmonic extension.
- Existence and uniqueness of bounded solutions are established.
- The approach applies to general sectorial operators.

## Abstract

We show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e. of the harmonic extension).