Supersolution Construction and Optimal Hardy Inequality for Fractional Laplacians

2026/07/27 by Philipp Hake, Matthias Keller, Felix Pogorzelski
#math.AP #math-ph #math.MP #math.SP

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Abstract

We give a new criterion to show optimality of Hardy weights for general operators on graphs via the supersolution construction. For Laplacians on graphs without killing terms this always gives rise to an optimal Hardy weight via the Green's function without any further assumptions. Furthermore, in contrast to earlier results, our result is not restricted to locally finite graphs. This allows us in particular to obtain optimal Hardy weights for the fractional Laplacian on general graphs. For the fractional Laplacian on the Euclidean lattice, we then obtain an optimal Hardy weight with the constant and asymptotics as it is expected from the continuous setting.

Citations

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Related

• An optimal fractional Hardy inequality on the discrete half-line
• Optimal Hardy inequality for the fractional Laplacian on L^p
• On the Optimality and Decay of p-Hardy Weights on Graphs
• Optimal Hardy inequalities for Schrödinger operators on graphs
• The best constant in a fractional Hardy inequality