Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Morrey spaces for Schr\"odinger operators with nonnegative potentials,
  fractional integral operators and the Adams inequality on the Heisenberg
  groups   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
sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the
reverse H\"older class $RH_s$ with $s\in[Q/2,\infty)$. Here $Q=2n+2$ is the
homogeneous dimension of $\mathbb H^n$. For given $\alpha\in(0,Q)$, the
fractional integral operator associated with the Schr\"odinger operator
$\mathcal L$ is defined by $\mathcal I_{\alpha}={\mathcal L}^{-{\alpha}/2}$. In
this article, the author introduces the Morrey space
$L^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$ and weak Morrey space
$WL^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$ associated with $\mathcal L$, where
$(p,\kappa)\in[1,\infty)\times[0,1)$ and $\rho(\cdot)$ 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 $\mathcal I_{\alpha}={\mathcal L}^{-{\alpha}/2}$ is bounded from
$L^{p,\kappa}_{\rho,\infty}(\mathbb H^n)$ to
$L^{q,\kappa}_{\rho,\infty}(\mathbb H^n)$ with $0<\alpha<Q$, $1<p<Q/{\alpha}$,
$0<\kappa<1-{(\alpha p)}/Q$ and $1/q=1/p-{\alpha}/{Q(1-\kappa)}$, and bounded
from $L^{1,\kappa}_{\rho,\infty}(\mathbb H^n)$ to
$WL^{q,\kappa}_{\rho,\infty}(\mathbb H^n)$ with $0<\alpha<Q$,
$0<\kappa<1-\alpha/Q$ and $1/q=1-{\alpha}/{Q(1-\kappa)}$. Moreover, in order to
deal with the extreme cases $\kappa\geq 1-{(\alpha p)}/Q$, the author also
introduces the spaces $\mathrm{BMO}_{\rho,\infty}(\mathbb H^n)$ and
$\mathcal{C}^{\beta}_{\rho,\infty}(\mathbb H^n)$, $\beta\in(0,1]$ associated
with $\mathcal L$.

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The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom