Heisenberg群上非负势Schr dinger算子、分数次积分算子和Adams不等式的Morrey空间
Morrey spaces for Schrödinger operators with nonnegative potentials, fractional integral operators and the Adams inequality on the Heisenberg groups
摘要：
设$\mathcal L=-Δ_{\mathbb H^n}+V$是海森堡群$\mathbb H^n$上的薛定谔算子，其中$Δ_{\mathbb H^n}$是$\mathbb H^n$上的次拉普拉斯算子，非负势$V$属于逆Hölder类$RH_s$，其中$s\in[Q/2,\infty)$。这里$Q=2n+2$是$\mathbb H^n$的同质维度。对于给定的$α\in(0,Q)$，与薛定谔算子$\mathcal L$相关联的分数次积分算子定义为$\mathcal I_α={\mathcal L}^{-α/2}$。本文介绍了Morrey空间$L^{p,κ}_{ρ,\infty}(\mathbb H^n)$和与$\mathcal L$相联系的弱Morrey空间$WL^{p,κ}_{ρ,\infty}(\mathbb H^n)$，其中$(p,κ)\in[1,\infty)\times[0,1)$和$ρ(\cdot)$是与非负势$V$有关的辅助函数.建立了海森堡群上的分数次积分算子与极大算子之间的关系。在此基础上，进一步得到了这些新空间上的Adams（Morrey-Sobolev）不等式.证明了分数次积分算子$\mathcal I_α={\mathcal L}^{-α/2}$从$L^{p,κ}_{ρ,\infty}(\mathbb H^n)$到$L^{q,κ}_{ρ,\infty}(\mathbb H^n)$有界且有$0<α<Q$，$1<p<Q/α$，$0<κ<1-{(αp)}/Q$和$1/q=1/p-α/{Q(1-κ)}$，从$L^{1,κ}_{ρ,\infty}(\mathbb H^n)$到$WL^{q,κ}_{ρ,\infty}(\mathbb H^n)$有界且有$0<α<Q$，$0<κ<1-α/Q$和$1/q=1-α/{Q(1-κ)}$ .此外，为了处理$κ\geq 1-{(αp)}/Q$的极端情形，作者还引入了$\mathrm{BMO}_{ρ,\infty}(\mathbb H^n)$和$\mathcal{C}^β_{ρ,\infty}(\mathbb H^n)$空间，以及与$\mathcal L$相联系的$β\in(0,1]$.
英文摘要：
Let $\mathcal L=-Δ_{\mathbb H^n}+V$ be a Schrödinger operator on the Heisenberg group $\mathbb H^n$, where $Δ_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $RH_s$ with $s\in[Q/2,\infty)$. Here $Q=2n+2$ is the homogeneous dimension of $\mathbb H^n$. For given $α\in(0,Q)$, the fractional integral operator associated with the Schrödinger operator $\mathcal L$ is defined by $\mathcal I_α={\mathcal L}^{-α/2}$. In this article, the author introduces the Morrey space $L^{p,κ}_{ρ,\infty}(\mathbb H^n)$ and weak Morrey space $WL^{p,κ}_{ρ,\infty}(\mathbb H^n)$ associated with $\mathcal L$, where $(p,κ)\in[1,\infty)\times[0,1)$ and $ρ(\cdot)$ is an auxiliary function related to the nonnegative potential $V$. The relation between the fractional integral operator and the maximal operator on the Heisenberg group is established. From this, the author further obtains the Adams (Morrey-Sobolev) inequality on these new spaces. It is shown that the fractional integral operator $\mathcal I_α={\mathcal L}^{-α/2}$ is bounded from $L^{p,κ}_{ρ,\infty}(\mathbb H^n)$ to $L^{q,κ}_{ρ,\infty}(\mathbb H^n)$ with $0<α<Q$, $1<p<Q/α$, $0<κ<1-{(αp)}/Q$ and $1/q=1/p-α/{Q(1-κ)}$, and bounded from $L^{1,κ}_{ρ,\infty}(\mathbb H^n)$ to $WL^{q,κ}_{ρ,\infty}(\mathbb H^n)$ with $0<α<Q$, $0<κ<1-α/Q$ and $1/q=1-α/{Q(1-κ)}$. Moreover, in order to deal with the extreme cases $κ\geq 1-{(αp)}/Q$, the author also introduces the spaces $\mathrm{BMO}_{ρ,\infty}(\mathbb H^n)$ and $\mathcal{C}^β_{ρ,\infty}(\mathbb H^n)$, $β\in(0,1]$ associated with $\mathcal L$.
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