Hardy inequalities, Rellich inequalities and local Dirichlet forms

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 ⊆ R 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 R d \{0} and to a general class of operators of Grushin type.

On self-adjointness of symmetric diffusion operators

“…(see [Rob17]) which immediately leads to the estimate by the Hardy inequality (22). Thus inserting these estimates into (23) and taking the limit n → ∞ gives…”

“…Secondly, fix ϕ ∈ D(H F ) with supp ϕ ⊂ Γ r where r < 1. If ϕ p = ϕ(1 + ϕ/p) −1 with p ≥ 1 then ϕ p ∈ D(h) ∩ L ∞ (Ω), supp ϕ p = supp ϕ ⊂ Γ r and ϕ p − ϕ D(h) → 0 as p → ∞, by Lemma 2.6 of [Rob17]. Moreover, supp β n,m ϕ p ⊆ Γ r ∩ Ω 1/n and β n,m ϕ p ∈ D(h).…”

“…The β m form a sequence of bounded functions on Ω which converges monotonically upward to β as m → ∞. Further, set β n,m = ρ n β m with ρ n the approximate identity of Proposition 3.1 in [Rob17]. In particular it follows from the proposition that 0 ≤ ρ n ≤ 1, supp ρ n ⊂ Ω 1/n , ρ n → 1 and (ϕ, Γ(ρ n )ϕ) → 0 as n → ∞ for all ϕ ∈ D(h).…”

“…Secondly there is a correction term, which depends on |∇ρ n | 2 , given by τ r (ϕ p , ad δ Γ |∇ρ n | 2 β 2 m ϕ p ). But this is bounded by τ r β m 2 ∞ (ϕ p , ad δ Γ |∇ρ n | 2 ϕ p ) and the latter expression tends to zero as n → ∞ since (ψ, Γ(ρ n )ψ) → 0 for all ψ ∈ D(h) (see [Rob17], Proposition 3.1). Here it is essential that ϕ p ∈ D(h).…”

“…The leading term is given by the square of the contribution in which the derivatives are on the d Γ . This can be calculated exactly as in [Rob17] and one obtains an estimate…”