Morrey spaces related to certain nonnegative potentials and fractional integrals on the Heisenberg groups

TL;DR

This paper introduces new Morrey and weak Morrey spaces on the Heisenberg group related to a Schrödinger operator with nonnegative potential and proves the boundedness of fractional integrals on these spaces.

Contribution

It defines Morrey-type spaces associated with nonnegative potentials on the Heisenberg group and establishes the boundedness of fractional integrals on these spaces.

Findings

Boundedness of fractional integrals on Morrey and weak Morrey spaces.

Introduction of BMO and Hölder spaces adapted to the potential.

Extension of classical analysis to noncommutative setting with potentials.

Abstract

Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on the Heisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is the sub-Laplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_s$ with $s\geq Q/2$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. For given $\alpha\in(0,Q)$, the fractional integrals associated to the Schr\"odinger operator $\mathcal L$ is defined by $\mathcal I_{\alpha}={\mathcal L}^{-{\alpha}/2}$. In this article, we first introduce the Morrey space $L^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$ and weak Morrey space $WL^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$ related to the nonnegative potential $V$. Then we establish the boundedness of fractional integrals ${\mathcal L}^{-{\alpha}/2}$ on these new spaces. Furthermore, in order to deal with certain extreme cases, we also introduce the spaces $\mathrm{BMO}_{\rho,\infty}(\mathbb H^n)$ and $\mathcal{C}^{\beta}_{\rho,\infty}(\mathbb H^n)$ with exponent $\beta\in(0,1]$.