The boundedness of some singular integral operators on weighted Hardy spaces associated with Schr\"odinger operators

TL;DR

This paper studies the boundedness of certain singular integral operators related to Schrödinger operators on weighted Hardy spaces, introducing molecular characterizations and proving boundedness of imaginary powers and fractional integrals.

Contribution

It defines molecules for weighted Hardy spaces associated with Schrödinger operators and establishes their molecular characterizations, enabling new boundedness results for related operators.

Findings

Imaginary powers $L^{i heta}$ are bounded on $H^p_L(w)$ for specified $p$.

Fractional integrals $L^{-rac{eta}{2}}$ are bounded between weighted Hardy spaces.

Molecular and atomic decompositions are used to prove boundedness results.

Abstract

Let $L=-\Delta+V$ be a Schr\"odinger operator acting on $L^2(\mathbb R^n)$, $n\ge1$, where $V\not\equiv 0$ is a nonnegative locally integrable function on $\mathbb R^n$. In this paper, we first define molecules for weighted Hardy spaces $H^p_L(w)$($0<p\le1$) associated to $L$ and establish their molecular characterizations. Then by using the atomic decomposition and molecular characterization of $H^p_L(w)$, we will show that the imaginary power $L^{i\gamma}$ is bounded on $H^p_L(w)$ for $n/{(n+1)}<p\le1$, and the fractional integral operator $L^{-\alpha/2}$ is bounded from $H^p_L(w)$ to $H^q_L(w^{q/p})$, where $0<\alpha<\min\{n/2,1\}$, $n/{(n+1)}<p\le n/{(n+\alpha)}$ and $1/q=1/p-\alpha/n$.