The main goal of this research project is to construct a theory for subelliptic (geometric) Hardy type functional inequalities and to carry out qualitative research of their extensions and applications. To achieve this aim, we propose to develop existent and create new methods of homogeneous (Lie) groups, then to study general subelliptic differential operators. This study will lead to a deeper understanding of the basic Lie group structure of functional inequalities and subelliptic differential operators. For instance, existing methods for proving subelliptic functional inequalities on nilpotent Lie groups are based on the study of the properties of a fixed homogeneous norm, for example, the L-gauge. Proposed in this research project, methods allow us to work with arbitrary quasi-norms, thus we believe the combination of our methods with the lifting theory will give new subelliptic functional inequalities on manifolds which will follow new results in both analysis on manifolds and theory of subelliptic differential equations.
The subject of Hardy inequalities has now been a fascinating subject of continuous research by numerous mathematicians for about more than one century, 1918-2019. The original inequality was published by G. H. Hardy in “Notes on some points in the integral calculus (51)”, Messenger of Mathematics, 48 (1918), P. 107-112.
The Hardy inequalities have numerous applications in different fields, for example in the spectral theory, leading to the lower bounds for the quadratic form associated with the Laplacian operator. They are also related to many other areas and fields, notably to the uncertainty principles. The uncertainty principle in physics is a fundamental concept going back to Heisenberg's work on quantum mechanics, as well as to its mathematical justification by Hermann Weyl. Over the last 100 years, the subject of Hardy inequalities and related analysis has been a topic of intensive research: currently, MathSciNet lists more than 800 papers containing words ‘Hardy inequality’ in the title, and almost 3500 papers containing the words ‘Hardy inequality’ in the abstract or in the review. The Hardy inequalities have been already presented in many monographs and reviews; here we can mention excellent books by Opic and Kufner in 1990, Davies in 1999, Edmunds and Evans in 2004, parts of Mazya's books in 1985 and 2011, Ghoussoub and Moradifam in 2013, and Balinsky, Evans, and Lewis in 2015, as well as many other books on different areas related to Hardy spaces.
In view of this wealth of information (and the page limit), we apologize for the inevitability of missing to mention many important contributions to the subject.
However, all of these presentations are largely confined to the Euclidean part of the available wealth of information on this subject.
At the same time, there is another layer of intensive research over the years related to Hardy type functional inequalities in subelliptic settings motivated by their applications to problems involving subelliptic differential equations. This is complemented by the more general anisotropic versions of the theory. In this direction, the subelliptic ideas of the analysis on the Heisenberg group, significantly advanced by Folland and Stein in [1] (see, also [3]), were subsequently consistently developed by Folland [2] leading to the foundations for analysis on stratified groups (or homogeneous Carnot groups). Thus, the intensive study of the subelliptic functional estimates started due to their importance for many questions involving subelliptic partial differential equations, unique continuation, sub-Riemannian geometry, subelliptic spectral theory, etc. As expected, the subelliptic Hardy inequality was obtained to the most important example of the Heisenberg group by Garofalo and Lanconelli [4] (see, Thangavelu’s book in 2004, also Roncal and Thangavelu’s works, e.g. [5] on recent advances on the Heisenberg group). The place where Hardy type inequalities and general homogeneous (Lie) groups meet is a beautiful area of mathematics which was not consistently treated in the project form. We took it as an incentive to write this project to extend and deepen the understanding of Hardy type inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups. While we will construct the general theory of Hardy type inequalities in the setting of general homogeneous groups, particular attention is paid to the special class of stratified groups and graded groups as well as extensions to manifolds (without group structures). In this setting, the theory of subelliptic functional inequalities becomes intricately intertwined with the properties of sub-Laplacians and more general subelliptic partial differential equations.
These topics constitute the core of this project with the results complemented with additional closely related topics such as uncertainty principles, the theory of linear and nonlinear subelliptic differential equations, subelliptic spectral theory as well as the theory of (anisotropic) function spaces.
Thus, the present project is devoted to the exposition of the research developments at the intersection of two active fields of mathematics: Hardy inequalities and related analysis, and the noncommutative analysis in the setting of nilpotent Lie groups. Both subjects are very broad and deserve separate studies on their own. However, a combination of the so-called lifting theory and our recent research techniques in the area does allow one to make a consistent treatment of `anisotropic' Hardy inequalities, their numerous features, and a number of related topics. This brings many new insights to the subject, also allowing to underline the interesting character of its subelliptic features.
This study has mainly fundamental character, makes a valuable contribution to the development of the theory of functional analysis on nilpotent Lie groups and the theory of subelliptic differential equations. Note that we will solve previously unsolved conjecture regarding the natural weight in the geometric Hardy inequality (please see Section 5 for specific tasks). Obtained results will be applied to solving various problems in mathematics and theoretical physics as well as may serve as fundaments for many new university courses. Particularly, we apply obtained inequalities to subelliptic partial differential equations and subelliptic spectral theory which may have interpretations in theoretical physics. For example, we will prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) and uniqueness of a positive solution to Dirichlet p-versions of sub-Laplacians (with nonlinear right-hand sight functions).
Expected social impacts: Most important contribution of this project to the Kazakhstani society and scientific community will be the training of four graduate students, who will get their Ph.D. and MS degrees working on this project. By conducting advanced research and publishing in top venues, they will become the next generation of academicians in Kazakhstan. Publication of fundamental results in prestigious mathematical journals promotes raising the image of the Republic of Kazakhstan in the scientific world.