Gaussian Poincaré inequalities on the half-space with singular weights

2024/07/24 by Negro, Luigi, Spina, Chiara · 1 citation
#35B65 #35J70 #35J75 #35K08 #35K67 #35k08 #47D07 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.17096

Abstract

We prove Rellich-Kondrachov type theorems and weighted Poincaré inequalities on the half-space ℝ^N+1₊=\z=(x,y): x ∈ ℝ^N, y>0\ endowed with the weighted Gaussian measure μ:=y^ce^-a|z|2dz where c+1>0 and a>0. We prove that for some positive constant C>0 one has ‖u- u‖_{L²μ(ℝ^N+1+)}≤ C ‖∇ u‖_{L²μ(ℝ^N+1+)}, ∀ u∈ H¹_μ(ℝ^N+1₊) where u=\frac 1μ(ℝ^N+1₊)∫_ℝ^N+1+ u dμ(z). Besides this we also consider the local case of bounded domains of ℝ^N+1₊ where the measure μ is y^cdz.

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• Operator Decompositions and Weighted Frame Inequalities for Total Sequences in Hilbert Spaces

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