From Hardy to Rellich inequalities on graphs. (English) Zbl 1464.35389
Summary: We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrödinger operators afterwards.
MSC:
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 39A12 | Discrete version of topics in analysis |
| 26D15 | Inequalities for sums, series and integrals |
| 31C20 | Discrete potential theory |
| 35B09 | Positive solutions to PDEs |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 58E35 | Variational inequalities (global problems) in infinite-dimensional spaces |
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