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

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
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