Hardy inequalities, Rellich inequalities and local Dirichlet forms

2017/01/19 by Derek W. Robinson, Robinson, Derek W.
Mathematics · #31C25 #39B62 #47D07 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1701.05629

openalex publication_date 2017/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

First the Hardy and Rellich inequalities are defined for the submarkovian operator associated with a local Dirichlet form. Secondly, two general conditions are derived which are sufficient to deduce the Rellich inequality from the Hardy inequality. In addition the Rellich constant is calculated from the Hardy constant. Thirdly, we establish that the criteria for the Rellich inequality are verified for a large class of weighted second-order operators on a domain Ω⊆ \Ri^d. The weighting near the boundary ∂ Ω can be different from the weighting at infinity. Finally these results are applied to weighted second-order operators on \Ri^d\backslash\0\ and to a general class of operators of Grushin type.

Citations

• Analysis of degenerate elliptic operators of Grušin type

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