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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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