We prove a Hardy inequality for the fractional Laplacian on the interval with the optimal constant and additional lower order term. As a consequence, we also obtain a fractional Hardy inequality with the best constant and an extra lower order term for general domains, following the method of M. Loss and C. Sloane [J. Funct. Anal. 259 (2010)].
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We prove a fractional version of the Hardy-Sobolev-Maz'ya inequality for arbitrary domains and $L^{p}$ norms with p ≥ 2. This inequality combines the fractional Sobolev and the fractional Hardy inequality into a single inequality, while keeping the sharp constant in the Hardy inequality.
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For $C^{1,1}$ domains we give exact asymptotics near the domain's boundary for the Green function and Martin kernel of the rotation invariant α-stable Lévy process. We also obtain a relative Fatou theorem for harmonic functions of the stable process.
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