2. Theses and Dissertations

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    Functional inequalities on Lie groups and applications
    (Faculty of Engineering and Natural Sciences, 2023) Kalaman M.
    In this thesis we discuss sharp remainder formulae for the cylindrical extensions of the improved Hardy inequalities. For more general p we obtain cylindrical improved L?-Hardy identities for all real-valued functions f € Cg°(R"\{2x' = 0}), while in L? case we have them for any complex-valued function f € Cg°(R"\{z' = 0}). Moreover, we show cylindrical L?-Hardy inequalities for all complex-valued functions f € Cp°(IR"\{a’ = 0}). As applications, we establish Heisenberg-PaulWeyl type uncertainty principles and Caffarelli-Kohn-Nirenberg type inequalities. In particular cases, these inequalities imply new functional inequalities, which are not covered by the classical Caffarelli-Kohn-Nirenberg inequalities. Furthermore, the thesis contains L* and L? identities with logarithmic type functions on the quasi-ball B(0,R) with R > 0. In addition, we also discuss the results in the setting of homogeneous Lie groups.