Integral Operators in Potential Theory

Boundedness of the solution of the third problem for the Laplace equation

“…Then the following assertions are equivalent: [30], Corollary 2.1.8). According to Lemma 1 and Lemma 2 there is a constant c such that [12], Theorem 2.16, we haveμ ∈ C b (∂G) by Theorem 1. Since |u| K λ-a.e., u + λ, u − λ ∈ C b (∂G) by [25], Proposition 6 and µ =μ + u + λ − u − λ ∈ C b (∂G).…”

“…According to [12], Lemma 2.18 this is true if there are constants α > m − 2 and k > 0 such that λ(Ω r (x)) kr α for all x ∈ m and all r > 0.…”

“…(see [12]). There are more geometrical characterizations of v If G has a finite perimeter (which is fulfilled if…”

“…Since U (gH m ) ∈ C 1 ( m ) (see [6], Theorem A.6, Theorem A.11), ϕ ε * U (gH m ) → U (gH m ), ϕ ε * ∇U (gH m ) → ∇U (gH m ) locally uniformly as ε 0 (see [30], Theorem 1.6.1, [27], § 12). The Divergence Theorem (see [12], p. 49) and [6], Theorem A. 16 (G) of the third problem for the Laplace equation (14) with the boundary condition L ≡ Lμ.…”

Some Historical Remarks on the Positivity of Boundary Integral Operators

“…As a consequence, subsequent generalizations of his techniques were confined to hard harmonic analysis, too, see [24] and [25] for overviews.…”