An optimal fractional Hardy inequality on the discrete half-line

Abstract In the context of Hardy inequalities for the fractional Laplacian $$(-\Delta _{\mathbb {N}})^{\sigma }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>σ</mml:mi> </mml:msup> </mml:math> on the discrete half-line $$\mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> , we provide an optimal Hardy-weight $$W^{\textrm{op}}_{\sigma }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mi>σ</mml:mi> <mml:mtext>op</mml:mtext> </mml:msubsup> </mml:math> for exponents $$\sigma \in \left( 0,1\right] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>∈</mml:mo> <mml:mfenced> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> </mml:mrow> </mml:math> . As a consequence, we provide an estimate of the sharp constant in the fractional Hardy inequality with the classical Hardy-weight $$n^{-2\sigma }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> <mml:mi>σ</mml:mi> </mml:mrow> </mml:msup> </mml:math> on $$\mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> . It turns out that for $$\sigma =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> the Hardy-weight $$W^{\textrm{op}}_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mn>1</mml:mn> <mml:mtext>op</mml:mtext> </mml:msubsup> </mml:math> is pointwise larger than the optimal Hardy-weight obtained by Keller–Pinchover–Pogorzelski near infinity. As an application of our main result, we obtain unique continuation results at infinity for the solutions of some fractional Schrödinger equation.