Optimal discrete Hardy-Rellich-Birman inequalities

2024/05/13 by Štampach, František, Waclawek, Jakub · 1 citation
#26D15 #39A12 #47B39 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2405.07742

Abstract

We prove sufficient conditions on a parameter sequence to determine optimal weights in inequalities for an integer power ℓ of the discrete Laplacian on the half-line. By a concrete choice of the parameter sequence, we obtain explicit optimal discrete Rellich (ℓ=2) and Birman (ℓ≥3) weights. For ℓ=1, we rediscover the optimal Hardy weight of Keller-Pinchover-Pogorzelski. For ℓ=2, we improve upon the best known Rellich weights due to Gerhat-Krejčiřík-Štampach and Huang-Ye. For ℓ≥3, our main result proves a conjecture by Gerhat-Krejčiřík-Štampach and improves the discrete analogue of the classical Birman weight due to Huang-Ye to the optimal.

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