@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},
}