Boundedness of operators generated by fractional semigroups associated with Schr\"odinger operators on Campanato type spaces via $T1$ theorem

TL;DR

This paper investigates the boundedness of operators generated by fractional semigroups related to Schrödinger operators on Campanato spaces, using the $T1$ theorem to establish key boundedness results.

Contribution

It introduces new boundedness results for fractional heat semigroup operators associated with Schrödinger operators on Campanato spaces, employing the $T1$ theorem approach.

Findings

Boundedness of maximal functions on $BMO^{eta}_{ ext{L}}$ spaces.

Boundedness of Littlewood-Paley $g$-functions associated with $ ext{L}$.

Application of $T1$ theorem to Schrödinger operator semigroup operators.

Abstract

Let $\mathcal{L}=-\Delta+V$ be a Schr\"{o}dinger operator, where the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $B_{q}$. By the aid of the subordinative formula, we estimate the regularities of the fractional heat semigroup, $\{e^{-t\mathcal{L}^{\alpha}}\}_{t>0},$ associated with $\mathcal{L}$. As an application, we obtain the $BMO^{\gamma}_{\mathcal{L}}$-boundedness of the maximal function, and the Littlewood-Paley $g$-functions associated with $\mathcal{L}$ via $T1$ theorem, respectively.