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Basic functional and geometric inequalities for the fractional order operators on homogenous Lie groups

Aidyn Kassymov (UGent)
(2020)
Author
Promoter
(UGent) and Durvudkhan Suragan
Organization
Abstract
In this PhD dissertation, we study functional and geometric inequalities on homo- geneous Lie groups. As for direct inequalities we obtain fractional Hardy, Sobolev, Hardy-Sobolev, Gagliardo-Nirenberg, Caffarelli-Kohn-Nirenberg, logarithmic inequal- ities, Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups. Also, we obtain the integer order Sobolev-Folland-Stein inequality on strati- fied groups. For reverse inequalities, we prove reverse integral Hardy inequalities with pa- rametersq<0,p∈(0,1)and−∞<q≤p<0. Also,weshowreverseinte- gral Hardy inequalities on homogeneous Lie groups, hyperbolic spaces and Cartan- Hadamard manifolds with q < 0, p ∈ (0,1). As consequences, we show reverse Hardy-Littlewood-Sobolev, Stein-Weiss and improved version Stein-Weiss inequali- tiesforcasesq<0,p∈(0,1)and−∞<q≤p<0. Inaddition,weobtain reverse Hardy, Lp-Sobolev and Lp- Caffarelli-Kohn-Nirenberg inequalities with radial derivative on homogeneous Lie groups. Then we show some applications of these inequalities for linear and nonlinear PDEs on homogeneous groups. Also, we consider one-dimensional functional inequalities in Euclidean settings. We establish fractional Hardy, Poincar ́e type, Gagliardo-Nirenberg type and Caffarelli- Kohn-Nirenberg inequalities for fractional order differential operators as Caputo, Riemann-Liouville and Hadamard fractional derivatives. Also, we discuss applica- tions of these inequalities. In addition, we show Lyapunov and Hartman-Wintner- type inequalities for a fractional partial differential equation with Dirichlet condition, we give an application of these inequalities for the first eigenvalue and we show a de La Vall ́ee Poussin-type inequality for fractional elliptic boundary value problem.

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MLA
Kassymov, Aidyn. Basic Functional and Geometric Inequalities for the Fractional Order Operators on Homogenous Lie Groups. Universiteit Gent. Faculteit Wetenschappen ; Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics, 2020.
APA
Kassymov, A. (2020). Basic functional and geometric inequalities for the fractional order operators on homogenous Lie groups. Universiteit Gent. Faculteit Wetenschappen ; Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics.
Chicago author-date
Kassymov, Aidyn. 2020. “Basic Functional and Geometric Inequalities for the Fractional Order Operators on Homogenous Lie Groups.” Universiteit Gent. Faculteit Wetenschappen ; Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics.
Chicago author-date (all authors)
Kassymov, Aidyn. 2020. “Basic Functional and Geometric Inequalities for the Fractional Order Operators on Homogenous Lie Groups.” Universiteit Gent. Faculteit Wetenschappen ; Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics.
Vancouver
1.
Kassymov A. Basic functional and geometric inequalities for the fractional order operators on homogenous Lie groups. Universiteit Gent. Faculteit Wetenschappen ; Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics; 2020.
IEEE
[1]
A. Kassymov, “Basic functional and geometric inequalities for the fractional order operators on homogenous Lie groups,” Universiteit Gent. Faculteit Wetenschappen ; Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics, 2020.
@phdthesis{8686773,
  abstract     = {{In this PhD dissertation, we study functional and geometric inequalities on homo- geneous Lie groups. As for direct inequalities we obtain fractional Hardy, Sobolev, Hardy-Sobolev, Gagliardo-Nirenberg, Caffarelli-Kohn-Nirenberg, logarithmic inequal- ities, Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups. Also, we obtain the integer order Sobolev-Folland-Stein inequality on strati- fied groups.
For reverse inequalities, we prove reverse integral Hardy inequalities with pa- rametersq<0,p∈(0,1)and−∞<q≤p<0. Also,weshowreverseinte- gral Hardy inequalities on homogeneous Lie groups, hyperbolic spaces and Cartan- Hadamard manifolds with q < 0, p ∈ (0,1). As consequences, we show reverse Hardy-Littlewood-Sobolev, Stein-Weiss and improved version Stein-Weiss inequali- tiesforcasesq<0,p∈(0,1)and−∞<q≤p<0. Inaddition,weobtain reverse Hardy, Lp-Sobolev and Lp- Caffarelli-Kohn-Nirenberg inequalities with radial derivative on homogeneous Lie groups.
Then we show some applications of these inequalities for linear and nonlinear PDEs on homogeneous groups.
Also, we consider one-dimensional functional inequalities in Euclidean settings. We establish fractional Hardy, Poincar ́e type, Gagliardo-Nirenberg type and Caffarelli- Kohn-Nirenberg inequalities for fractional order differential operators as Caputo, Riemann-Liouville and Hadamard fractional derivatives. Also, we discuss applica- tions of these inequalities. In addition, we show Lyapunov and Hartman-Wintner- type inequalities for a fractional partial differential equation with Dirichlet condition, we give an application of these inequalities for the first eigenvalue and we show a de La Vall ́ee Poussin-type inequality for fractional elliptic boundary value problem.}},
  author       = {{Kassymov, Aidyn}},
  language     = {{eng}},
  pages        = {{205}},
  publisher    = {{Universiteit Gent. Faculteit Wetenschappen ; Al-Farabi Kazakh National University. Faculty of Mechanics and Mathematics}},
  school       = {{Ghent University}},
  title        = {{Basic functional and geometric inequalities for the fractional order operators on homogenous Lie groups}},
  year         = {{2020}},
}