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

Abstract Let L=−ΔHn+V $\mathcal{L}=-\Delta _{\mathbb{H}^{n}}+V$ be a Schrödinger operator on the Heisenberg group Hn $\mathbb{H}^{n}$, where ΔHn $\Delta _{\mathbb{H}^{n}}$ is the sub-Laplacian on Hn $\mathbb{H}^{n}$ and the nonnegative potential V belongs to the reverse Hölder class RHs $RH_{s}$ with s≥Q/2 $s\geq Q/2$. Here Q=2n+2 $Q=2n+2$ is the homogeneous dimension of Hn $\mathbb{H}^{n}$. For given α∈(0,Q) $\alpha \in (0,Q)$, the fractional integrals associated to the Schrödinger operator L $\mathcal{L}$ is defined by Iα=L−α/2 $\mathcal{I}_{\alpha }={\mathcal{L}}^{-{\alpha }/2}$. In this article, we first introduce the Morrey space Lρ,∞p,κ(Hn) $L^{p,\kappa }_{\rho, \infty }(\mathbb{H}^{n})$ and weak Morrey space WLρ,∞p,κ(Hn) $WL^{p,\kappa }_{ \rho,\infty }(\mathbb{H}^{n})$ related to the nonnegative potential V. Then we establish the boundedness of fractional integrals L−α/2 ${\mathcal{L}}^{-{\alpha }/2}$ on these new spaces. Furthermore, in order to deal with certain extreme cases, we also introduce the spaces BMOρ,∞(Hn) $\operatorname{BMO}_{\rho,\infty }(\mathbb{H}^{n})$ and Cρ,∞β(Hn) $\mathcal{C}^{ \beta }_{\rho,\infty }(\mathbb{H}^{n})$ with exponent β∈(0,1] $\beta \in (0,1]$.

Abstract licence: CC BY-NC-ND

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