Interpolation Theorems for Self-adjoint Operators

TL;DR

This paper establishes interpolation theorems for function spaces linked to self-adjoint operators, broadening applicability to elliptic and Schrödinger operators without requiring gradient estimates.

Contribution

It proves complex and real interpolation theorems for Besov and Triebel-Lizorkin spaces associated with self-adjoint operators, without assuming spectral kernel gradient estimates.

Findings

Interpolation theorems apply to elliptic and Schrödinger operators

Results do not require gradient estimates for spectral kernels

Broadens the scope of function space analysis for differential operators

Abstract

We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a selfadjoint operator $L$, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where $L$ is a uniformly elliptic operator or a Schr\"odinger operator with electro-magnetic potential.