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The poisson problem for the fractional hardy operator: Distributional identities and singular solutions

Transactions of the American Mathematical Society, ISSN: 1088-6850, Vol: 374, Issue: 10, Page: 6881-6925
2021
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Article Description

The purpose of this paper is to study and classify singular solutions of the Poisson problem _Ls μu = f in Ω \ {0}, u = 0 in RN \ Ω for the fractional Hardy operator Ls μu = (-Δ)su+ μ |x|2s u in a bounded domain Ω RN (N ≥ 2) containing the origin. Here (-Δ)s, s ∈ (0, 1), is the fractional Laplacian of order 2s, and μ ≥ μ0, where μ0 = -22s Γ2(N+2s 4 ) Γ2(N-2s 4 ) < 0 is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case μ = μ0 which requires more subtle estimates than the case μ > μ0.

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