NED sets on a hyperplane.
Let E₀, E₁ be two subsets of the closure D̅ of a domain D of the Euclidean n-space and Γ(E₀,E₁,D) the family of arcs joining E₀ to E₁ in D. We establish new cases of equality , where is the p-module of the arc family Γ(E₀,E₁,D), while is the p-capacity of E₀,E₁ relative to D and p > 1. One of these cases is when p = n, E̅₀ ∩ E̅₁ = ∅, , is inaccessible from D by rectifiable arcs, is open relative to D̅ or to the boundary ∂D of D, is at most countable, is closed (i = 0,1) and D...
The purpose of this paper is to derive norm inequalities for potentials of the formTf(x) = ∫(Rn) f(y)K(x,y)dy, x ∈ Rn,when K is a Kernel which satisfies estimates like those that hold for the Green function associated with the degenerate elliptic equations studied in [3] and [4].
The notion of the extremal length and the module of families of curves has been studied extensively and has given rise to a lot of applications to complex analysis and the potential theory. In particular, the coincidence of the p-module and the p-capacity plays an mportant role. We consider this problem on the Carnot group. The Carnot group G is a simply connected nilpotent Lie group equipped vith an appropriate family of dilations. Let omega be a bounded domain on G and Ko, K1 be disjoint non-empty...
Let , and let , be given. In this paper we study the dimension of -harmonic measures that arise from non-negative solutions to the -Laplace equation, vanishing on a portion of , in the setting of -Reifenberg flat domains. We prove, for , that there exists small such that if is a -Reifenberg flat domain with , then -harmonic measure is concentrated on a set of -finite -measure. We prove, for , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of -harmonic measure...
We study the existence of tangential boundary limits for harmonic functions in a Lipschitz domain, which belong to Orlicz-Sobolev classes. The exceptional sets appearing in this discussion are evaluated by use of Bessel-type capacities as well as Hausdorff measures.