A Hardy-type inequality and its applications

We prove a Hardy-type inequality that provides a lower bound for the integral $\int_0^\infty|f(r)|^pr^{p-1}\,dr$, $p>1$. In the scale of classical Hardy inequalities, this integral corresponds to the value of the exponential parameter for which neither direct nor inverse Hardy inequalities hold. However, the problem of estimating this integral and its multidimensional generalization from below arises in some practical questions. These are, for example, the question of solvability of elliptic equations in the scale of Sobolev spaces in the whole Euclidean space $\mathbb R^n$, some questions in the theory of Sobolev spaces, hydrodynamic problems, etc. These questions are studied in the present paper.