Remainder term analysis for Hardy type inequalities

Grant funding 2021-2023
Ministry of Education and Sciences of the Republic of Kazakhstan

The main goal of this research project is to analyze and construct new methods that answer the following three questions simultaneously: 1. Proofs of Hardy type inequalities with some constants; 2. Characterization of the best constant and its existence; 3. Characterization of nontrivial extremizers and their existence.

The basic idea of this project is to analyse remainder terms of the Hardy type inequalities. Project supervisor Corresponding Member of NAS RK, Assoc. Prof., Dr. Durvudkhan Suragan is a co-author of the international award-winning book:
Ruzhansky M., Suragan D., Hardy Inequalities on Homogeneous Groups, Progress in Mathematics // – V. 327. Birkhauser. – 2019. – P. xiii+588.
The winner of the Ferran Sunyer I Balaguer Prize 2018.
The present project plan is an essential continuation of the PI’s previous studies.

"We will obtain:
• Remainders for Steklov and Friedrichs inequalities
• Remainders for Hardy type inequalities (also, in generalized grand Lebesgue spaces)
• Remainders for weighted Hardy type inequalities
As applications we will establish:
• Proofs of new sharp Hardy inequalities with super weights
• Proofs of new sharp critical Hardy inequalities
• Proofs of new sharp critical Hardy inequalities of logarithmic type
• New remainder estimates: a new family of remainder estimates for the weighted L^{p}-Hardy inequalities.
• Stability analysis of Hardy type inequalities

Achieved results

The remainder term of the Steklov and Friedrichs inequalities are obtained; - Generalized nonlinear Picone identities on general stratified Lie groups are obtained; - Hardy-type inequalities are obtained for generalized quaternion-valued functions.
"