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International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
May 29, 2015 10:40–11:20, Ïëåíàðíûå äîêëàäû, Moscow, Steklov Mathematical Institute of RAS

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

V. S. Guliyevab

a Ahi Evran University, Turkey
b Institute of Mathematics and Mechanics of NAS of Azerbaijan
 Video records: MP4 1,055.5 Mb MP4 267.8 Mb Materials: Adobe PDF 132.6 Kb

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Abstract: 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 $f \in L_{p}^loc(\mathbb{G})$ with finite quasi-norms
\begin{align*} \| f \|_{LM_{p\theta,w}(\mathbb{G})} = ( \int_0^{\infty} w(r)^{\theta} ( \int_{\{x\in \mathbb{G} : \rho(x)<r\}} |f(x)|^p  dx )^{\theta/p}  dr )^{1/\theta},
\| f \|_{GM_{p\theta,w}(\mathbb{G})} = \sup_{x \in \mathbb{G}} ( \int_0^{\infty} w(r)^{\theta} ( \int_{\{y\in \mathbb{G} : \rho(y^{-1} \cdot x)<r\}} |f(y)|^p dy )^{\theta/p}  dr )^{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.

Materials: abstract.pdf (132.6 Kb)

Language: English

References
1. V. S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in $\mathbb{R}^n$, Doctor's degree dissertation, Mat. Inst. Steklov, Moscow, 1994, 329 pp. (in Russian)
2. V. S. Guliyev, Function spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups. Some Applications, Chashioglu, Baku, 1999, 332 pp. (in Russian)
3. V. I. Burenkov, H. V. Guliyev,, “Necessary and sufficient conditions for boundedness of the maximal operator in the local Morrey-type spaces”, Studia Math., 163:2 (2004), 157–176
4. by V. I. Burenkov, H. V. Guliyev, V. S. Guliyev “Necessary and sufficient conditions for boundedness of the fractional maximal operators in the local Morrey-type spaces”, J. Comput. Appl. Math., 208:1 (2007), 280–301
5. V. I. Burenkov, V. S. Guliyev, “Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces”, Potential Anal., 30:3 (2009), 211–249
6. V. Burenkov, V. S. Guliyev, A. Serbetci, T. V. Tararykova, “Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey type spaces”, Eurasian Math. J., 1 (2010), 32–53

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