Hardy inequalities with a piecewise power weight and their applications (Q1048526)

From the introduction: This paper concerns inequalities with piecewise power weight functions that follow from the classical Hardy inequalities. A feature of the weight functions is that they define a finite measure on the real semiaxis. As a result, when passing to the multidimensional case, we can use the factorization approach and obtain weighted inequalities of the Poincaré type for functions with a zero mean on a sphere in \({\mathbb R}^n\). The applications considered include the solvability of Poisson's equation in \({\mathbb R^n}\), the decomposition of Sobolev spaces into the sum of solenoidal and potential subspaces, and the existence of nontrivial solutions to the stationary Kolmogorov-Fokker-Planck equation in \({\mathbb R}^n\).