Necessary and sufficient conditions for the boundedness of the fractional integral operators in the local Morrey-type spaces on Carnot groups

Let $\mathbb{G}$ be a Carnot group (nilpotent stratified Lie group), $\rho$ its homogeneous norm and $Q$ its homogeneous dimension. The fractional integral $I_{\alpha}f$ on Carnot group $\mathbb{G}$ is defined by
$$ I_{\alpha} f(x)=\int_{\mathbb{G}} \rho(y^{-1} x)^{\alpha-Q} f(y) \, dy, \qquad 0<\alpha <Q. $$

Let $0 < p, \theta \le \infty $ and let $w$ be a non-negative measurable function on $(0,\infty)$. We denote by $LM_{p\theta,w}(\mathbb{G})$, $GM_{p\theta,w}(\mathbb{G})$, the local Morrey-type spaces, the global Morrey-type spaces respectively, which are the spaces of all functions $f \in L_{p}^{\rm loc}(\mathbb{G})$ with finite quasi-norms
\begin{align*} \| f \|_{LM_{p\theta,w}(\mathbb{G})} = \biggl( \int_0^{\infty} w(r)^{\theta} \biggl( \int_{\{x\in \mathbb{G} : \rho(x)<r\}} |f(x)|^p \,dx \biggr)^{\theta/p}\, dr \biggr)^{1/\theta}, \\ \| f \|_{GM_{p\theta,w}(\mathbb{G})} = \sup_{x \in \mathbb{G}} \biggl( \int_0^{\infty} w(r)^{\theta} \biggl( \int_{\{y\in \mathbb{G} : \rho(y^{-1} \cdot x)<r\}} |f(y)|^p\,dy \biggr)^{\theta/p} \,dr \biggr)^{1/\theta} \end{align*}
respectively. For $\theta=\infty$ and $w(r)=r^{-\frac{\lambda}{p}}$ with $0<\lambda<Q$ the space $M_{p,\lambda}(\mathbb{G})\equiv GM_{p\infty,r^{-{\lambda}/{p}}}(\mathbb{G})$ is the Morrey space, for $\theta=\infty$ the space $M_{p,w}(\mathbb{G})\equiv GM_{p\infty,w}(\mathbb{G})$ is the generalized Morrey space on Carnot group $\mathbb{G}$.
A survey will be given of recent results in which, for certain ranges of the numerical parameters $n$, $p_1$, $\theta_1$, $p_2$, $\theta_2$ necessary and sufficient conditions on the functions $w_1$ and $w_2$ are established ensuring the boundedness of the fractional integral operators from one local Morrey-type space $LM_{p_1\theta_1,w_1}(\mathbb{G})$ to another one $LM_{p_2\theta_2,w_2}(\mathbb{G})$.
It is shown that from the above result the Sobolev-Morrey embeddings for Carnot groups follow easily. A priori estimates for the sub-Laplacian in corresponding Besov-Morrey spaces are also proved.
Note that, the local Morrey-type spaces $LM_{p\theta,w}(\mathbb{G})$ defined on homogeneous Lie groups $\mathbb{G}$ were introduced in doctoral thesis [N224:GulDoc] by Guliyev (see also [N224:GulBook]) and the global Morrey-type spaces $GM_{p\theta,w}(\mathbb{R}^n)$ defined on $n$-dimensional Euclidian space $\mathbb{R}^n$ were introduced in [N224:BurHus1] by Burenkov and Guliyev (see also [N224:BurGulHus1], [N224:BurGul2]). The main purpose of [N224:GulDoc] (also of [N224:GulBook]) is to give some sufficient conditions for the boundedness of fractional integral operators and singular integral operators defined on homogeneous Lie groups in the local Morrey-type space $LM_{p\theta,w}(\mathbb{G})$. In a series of papers by Burenkov, H. Guliyev and V. Guliyev, etc. (see [N224:BurHus1], [N224:BurGulHus1], [N224:BurGul2], [N224:BurGulSerbTar]), some necessary and sufficient conditions for the boundedness of fractional maximal operators, fractional integral operators and singular integral operators in local Morrey-type spaces $LM_{p\theta,w}(\mathbb{R}^n)$ were given.
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This research was supported by the grant of Science Development Foundation under the President of the Republic of Azerbaijan Grant EIF-2013-9(15)-46/10/1 and by the grant of Presidium Azerbaijan National Academy of Science 2015.
Joint work with Dr. S.Q. Hasanov.