Choice problem of weight functions in Hardytype inequalities and applications to PDEs in full Euclidean space

We consider the following questions:
1. Constructive description of all possible weighting functions in Hardy-type inequalities;
2. Multidimensional weighted inequalities of the Friedrichs and Poincaré type in full Euclidean space;
3. Elliptic equations in the Sobolev scale of functions in the full Euclidean space;
4. Decomposition of the Sobolev spaces and gradient-divergence spaces in the sum of solenoidal and potential subspaces;
5. Divergence and rotor variants of the Stokes systems in the full Euclidean space (“explicit” solutions);
6. Stationary Fokker–Plank–Kolmogorov equations (continuum of nontrivial solutions).