A fractional order Hardy inequality

On comparability of integral forms

“…The application was an important motivation for the study of Hardy type inequalities in [7] and in the present paper.…”

“…We obtain the inequality as a consequence of Theorem 1 and the isotropic Hardy inequality of [7] (see (19) below). When α ∈ (0, 2) and p = 2, the integral forms in (1) give by polarization certain symmetric Dirichlet forms with core C ∞ c (D) [2,9,15].…”

“…In particular the methods we use do not depend on the general context of the theory of Dirichlet forms, Markov processes, or their generators. We apply a real-analytic technique initiated in [7], which is reminiscent of the technique of Balayage in potential theory.…”

“…Except for the case α = 1, this result may be obtained independently from the difficult considerations of [2] because for 0 < α < 1 [7] gives an explicit example of C ∞ c (D) functions approximating 1 D in F .…”

Banach Envelopes of Monogenic Hardy Spaces

“…Having disposed off (3.3), the next step is to use a smooth partition of unity to reduce matters when Ω is the domain lying above the graph of a Lipschitz function ϕ : R m−1 → R and when F ∈ Lip comp (Ω). In this scenario, the implication (3.1) is then proved by relying on a version of Hardy's inequality (see[9, Theorem 1.1(3)] and subsequent remarks) to the effect thatΩ |u(x)| dist (x, ∂Ω) −s dx ≤ C Ω Ω |u(x) − u(y)||x − y| m+s dxdy,(3.4) …”