Characterization of temperatures associated to Schrodinger operators with initial data in BMO spaces

TL;DR

This paper characterizes functions in the BMO space associated with Schrödinger operators as traces of solutions to a heat equation with a Carleson condition, extending classical BMO results.

Contribution

It extends the characterization of BMO functions as traces of solutions to a Schrödinger heat equation with a Carleson condition, generalizing classical results.

Findings

BMO_L functions are traces of solutions satisfying a Carleson condition.

The Carleson condition characterizes all L-carolic functions with BMO_L traces.

Extension of classical BMO characterization to Schrödinger operators.

Abstract

Let L be a Schr\"odinger operator of the form L=-\Delta+V acting on L^2(\mathbb R^n) where the nonnegative potential V belongs to the reverse H\"older class B_q for some q>= n. Let BMO denote the BMO space associated to the Schr\"odinger operator L. In this article we will show that a function f in BMO_L is the trace of the solution of u_t+L u=0, u(x,0)= f(x), where u satisfies a Carleson-type condition. Conversely, this Carleson condition characterizes all the L-carolic functions whose traces belong to the space BMO_L. This result extends the analogous characterization founded by Fabes and Neri for the classical BMO space of John and Nirenberg.