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Abstract

This paper considers to the equation [\int_{S} \frac{U(Q)}{|P-Q|^{N-1}} dS(Q) = F(P), P \in S,] where the surface S is the graph of a Lipschitz function \phi on R^N, which has a small Lipschitz constant. The integral in the left-hand side is the single layer potential corresponding to the Laplacian in R^{N+1}. Let \Lambda(r) be a Lipschitz constant of \phi on the ball centered at the origin with radius 2r. Our analysis is carried out in local L^p-spaces and local Sobolev spaces, where 1 < p < \infty, and results are presented in terms of \Lambda(r). Estimates of solutions to the equation are provided, which can be used to obtain knowledge about the behaviour of the solutions near a point on the surface. The estimates are given in terms of seminorms. Solutions are also shown to be unique if they are subject to certain growth conditions. Local estimates are provided and some applications are supplied.

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... This case is particularly interesting because of the connection to boundary integral methods for solving Laplace's equation and the single layer potential; see Hsiao and Wendland [4]. Furthermore, in [5], we employed the type of seminorm structure presented below to study the single layer potential on a Lipschitz surface with small Lipschitz constant using inversion results for Riesz potentials of order one developed in [6]. ...
... which should be interpreted as a principal value. With this in mind, we prove the following asymptotic result for solutions to (5) in Sect. 4.2. ...
... By means of a weighted Hardy inequality, see, e.g., Maz'ya [7], one can use the seminorm estimate provided in Proposition 2 to obtain two weight estimates for solutions to (5). Specifically, we obtain Theorem 4. ...
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It is known that radial symmetry is preserved by the Riesz potential operators and also by the hypersingular Riesz fractional derivatives typically used for inversion. In this paper, we collect properties, asymptotics, and estimates for the radial and spherical parts of Riesz potentials and for solutions to the Riesz potential equation of order one. Sharp estimates for spherical functions are provided in terms of seminorms, and a careful analysis of the radial part of a Riesz potential is carried out in elementary terms. As an application, we provide a two weight estimate for the inverse of the Riesz potential operator of order one acting on spherical functions.
... Specifically, we consider two-weighted estimates for solutions to (1.2) in weighted L p -spaces, with right-hand side in weighted homogeneous Sobolev spaces, similar to those found in Section 7.5 in [3], or in Section 8 of [6] for the Riesz potential case with power exponential weights. We will rely on results from Kozlov, Thim, and Turesson [4], where we investigated the influence of perturbations of a surface like a cone by a small Lipschitz perturbation and results were expressed in terms of seminorms and the function Λ(r). This functions is defined as the Lipschitz constant of ϕ on a ball of radius 2r: ...
... We denote the global Lipschitz constant of ϕ by Λ 0 . Note that we only consider small perturbations in the sense that Λ 0 is assumed to be sufficiently small, which was the setting in [4] due to the application of a fixed point theorem in locally convex spaces [5]. Moreover, in this article, the function Λ will be assumed to satisfy a Dini-type condition: ...
... This simplifies the conditions significantly. The existence and uniqueness results follow from corresponding theorems for local spaces derived previously in Theorems 1.1 and 1.2 of [4]. Indeed, Theorem 1.1 of [4] states that there exist positive constants Λ * , c 1 , and c 2 , depending only on N and p, such that if Λ 0 ≤ Λ * , we obtain existence and uniqueness results for the equation in (1.2) under certain restrictions on p and the involved functions. ...
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This article considers two weight estimates for the single layer potential --- corresponding to the Laplace operator in RN+1\mathbf{R}^{N+1} --- on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.
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