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

Abstract

Let $\mathcal{L}=-{\Delta }_{{\mathbb{H}}^n}+V$ be a Schrödinger 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 Hölder class $R{H}_s$ with $s\ge Q/2$. Here $Q=2n+2$ is the homogeneous dimension of ${\mathbb{H}}^n$. For given $\alpha \in (0,Q)$, the fractional integrals associated to the Schrödinger operator $\mathcal{L}$ is defined by ${\mathcal{I}}_{\alpha }={\mathcal{L}}^{-\alpha /2}$. In this article, we first introduce the Morrey space $L_{\rho ,\infty }^{p,\kappa }({\mathbb{H}}^n)$ and weak Morrey space $W{L}_{\rho ,\infty }^{p,\kappa }({\mathbb{H}}^n)$ 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 ${\mathcal{C}}_{\rho ,\infty }^{\beta }({\mathbb{H}}^n)$ with exponent $\beta \in (0,1]$.

22 pages. arXiv admin note: text overlap with arXiv:1802.02481