Name of Researcher: Dr Bolys Sabitbek

PhD thesis “Hardy-Sobolev type inequalities on homogeneous groups and applications” successfully defended in 2019.

Local scientific supervisor: Kalmenov Tynysbek Sharipovich – Academician of NAS, Doctor of Physical and Mathematical Sciences, Professor, Chief Researcher of the Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan.

Foreign scientific supervisor: Michael Ruzhansky– PhD, Senior Full Professor of Mathematics at the Ghent University, Belgium, and Honorary Professor at Imperial College London, London, UK.

Topics treated in his PhD thesis as follow:

– Geometric Hardy and Hardy-Sobolev inequalities on the stratified groups. In this direction, we obtained the geometric Hardy and Hardy-Sobolev inequalities on the half-spaces. We presented and versions of the (subelliptic) geometric Hardy inequalities in half-spaces and convex domains on general stratified groups. As a consequence, we have derived the Hardy-Sobolev inequality in the half-space on the Heisenberg group. Moreover, the geometric Hardy inequality is established on the starshaped sets.

– Horizontal inequalities on the stratified groups. In the second direction, we study the following horizontal version of Hardy type inequalities. The version of horizontal weighted Hardy-Rellich type inequalities was obtained on the stratified Lie groups, as a consequence of this inequality Sobolev type spaces are defined on stratified Lie groups and the embedding theorems are proved for these functional spaces. Also, we have obtained the subelliptic Picone type identities, as a result, we proved the Hardy and Rellich type inequalities for the anisotropic -sub-Laplacians. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on the stratified groups.

– Hardy and Rellich type inequalities and the sub-Laplacian fundamental solutions. In the third direction, we investigate the following type of Hardy inequalities. Generalised weighted Hardy, Rellich, and Caffarelli-Kohn-Nirenberg type inequalities with boundary terms are obtained on the stratified groups. As consequences, most of the Hardy type inequalities and the Heisenberg-Pauli-Weyl type uncertainty principles are recovered on the stratified groups. Moreover, a weighted Rellich type inequality with the boundary term is obtained. We also present Hardy and Rellich inequalities for the sub-Laplacians in terms of their fundamental solutions on the quaternion Heisenberg group.

– Weighted Hardy and Rellich type inequalities for general vector fields. In this direction, we study the weighted Hardy and Rellich inequalities for general vector fields without a group structure. We establish the weighted anisotropic Hardy and Rellich type inequalities with boundary terms for general (real-valued) vector fields. As consequences, we derive new as well as many of the fundamental Hardy and Rellich type inequalities which are known in different settings.