Fractional Hardy inequality with a remainder term. (English) Zbl 1228.26022

Dyda, Bartłomiej

Colloq. Math. 122, No. 1, 59-67 (2011).

Recently M. Loss and C. Sloane [J. Funct. Anal. 259, No. 6, 1369–1379 (2010; Zbl 1194.26011)] proved a fractional Hardy inequality on convex domains \(D \subset \mathbb R^n.\)
In this paper the author proves the fractional Hardy inequality for the fractional Laplacian on the interval \((a,b) \) being \(- \infty < a < b < + \infty\), with the optimal constant and additional lower order term.
As a consequence, it is also obtained a fractional Hardy inequality with the best constant and an extra lower order term for general domains.
The method developed yields an inequality with a “remainder” for general domains. The main new ingredient is this remainder with the smaller singularity at the boundary of \(D\) when it is bounded.
We point out that the calculation uses the Kelvin transform.

Reviewer: Maria Alessandra Ragusa (Catania)

MSC:

26D10	Inequalities involving derivatives and differential and integral operators
31C25	Dirichlet forms
46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

fractional Hardy inequality; best constant; interval; fractional Laplacian; Censore stable process; convex domain; ground state representation