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

@article{Meichsner2019OnTH, title={On the harmonic extension approach to fractional powers in Banach spaces}, author={Jan Meichsner and Christian Seifert}, journal={Fractional Calculus and Applied Analysis}, year={2019}, volume={23}, pages={1054 - 1089} }

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).

## 8 Citations

### A Note on the Harmonic Extension Approach to Fractional Powers of non‐densely defined Operators

- MathematicsPAMM
- 2019

We show to what extend fractional powers of non‐densely defined sectorial operators on Banach spaces can still be described by the harmonic extension approach.

### Besov spaces associated with non-negative operators on Banach spaces

- Mathematics
- 2020

Motivated by a variety of representations of fractional powers of operators, we develop the theory of abstract Besov spaces $B^{ s, A }_{ q, X }$ for non-negative operators $A$ on Banach spaces $X$…

### On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients

- Mathematics
- 2021

. We consider fractional operators of the form H s , ( x, t ) ∈ R n × R , where s ∈ (0 , 1) and A = A ( x,t ) = { A i,j ( x, t ) } ni,j =1 is an accretive, bounded, complex, measurable, n × n…

### A series representation of the discrete fractional Laplace operator of arbitrary order

- MathematicsArXiv
- 2021

### Functional Calculus via the extension technique: a first hitting time approach

- Mathematics
- 2021

In this article, we present a solution to the problem: Which type of linear operators can be realized by the Dirichlet-to-Neumann operator associated with the operator −∆ − a(z) ∂ 2 ∂z on an…

### On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients

- Materials ScienceJournal of Evolution Equations
- 2022

We consider fractional operators of the form Hs=(∂t-divx(A(x,t)∇x))s,(x,t)∈Rn×R,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}…

### Harmonic extension technique for non-symmetric operators with completely monotone kernels

- ArtCalculus of Variations and Partial Differential Equations
- 2022

We identify a class of non-local integro-differential operators K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

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