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Abstract

We consider the following equation for the Riesz potential of order one: Uniqueness is proved in the class of solutions for which the integral is absolutely convergent for almost every x. We also prove an existence result and derive an asymptotic formula for solutions near the origin. Our analysis is carried out in local L p -spaces and Sobolev spaces, which allows us to obtain optimal results concerning the class of right-hand sides and solutions. We also apply our results to weighted L p -spaces and homogenous Sobolev spaces.
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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]. ...
... The expression in the left-hand side is the norm on X p α (R N ) and it should be noted that if 1 ≤ p ≤ q < ∞, then X q α (R N ) ⊂ X p α (R N ). Moreover, in [6], it was shown that the Riesz potential operator I α is defined on X p α (R N ) and also that X 1 α (R N ) is the largest possible domain if we require absolute convergence almost everywhere. This was shown for α = 1, but it is straightforward to generalize the argument. ...
... However, in the case when α = 1, it is possible to represent the inverse of I α as a derivative in a more classical sense. In [6], we proved that I α has an inverse R under certain restrictions, where ...
Article
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.
... The results presented in this paper generalize those found in Kozlov, Thim, and Turesson [6] for Riesz potentials on R N . We will now focus our attention on investigating the equation ...
... 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). ...
... We specifically note that with α = 0 (which satisfies the condition above), we obtain an isomorphism between L p (R N ) and the homogeneous Sobolev space BL 1,p Γ (R N ) (with Γ = 1). Moreover, we also note that these results reduce to the corresponding results for Riesz potentials in the case when Λ 0 = 0, i.e., the hyperplane case x N +1 = 0; we refer to Section 8 in [6]. ...
Article
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.
... We take this expression as the norm on X p (R N ). This space is the natural domain, in terms of the seminorms N p , for the operator S in the case that S is the hyperplane x N +1 = 0; this is discussed further by the authors in [10]. We also remark that, if 1 ≤ p < N , then L p (R N ) ⊂ X p (R N ). ...
... Maz'ya for ordinary differential equations and ordinary differential equations with operator coefficients; see Section 6.4 in [8] and Section 6.3 in [9]. It is possible to use (1.7) to obtain two-weighted estimates for solutions to (1.4) in weighted L p -spaces and weighted Sobolev spaces similar to those found in Section 7.5 in [9]; see Section 8 in the authors' article [10] for an example of this procedure when the surface S is the hyperplane x N +1 = 0. Furthermore, one can also compare with the boundedness results for Riesz potentials in local Morrey-type spaces found in Burenkov et al. [1,2] and references cited therein. If Λ 0 = 0, we recover the same estimate in (1.7) for the solution as in the case when S is the hyperplane x N +1 = 0; compare with (2.5) below. ...
... The choice of weight is not obvious since we need to estimate both the potential and its derivative. The fact that the solution to the fixed point problem solves (1.4) follows from results derived earlier by the authors for the hyperplane case in [10]. A summary of the hyperplane case can be found in Section 2.1. ...
Article
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.
... In the first paper [22], we consider equation (1.4) in the case when S is the hyperplane x N +1 = 0. In this case, the operator S can be reduced to the Riesz potential operator of order one: ...
... where the constant C T only depends on N , p, and K. In Kozlov, Thim, and Turesson [7], this result is proved for the Riesz transform, but it holds with the obvious modifications for all operators of the type given in (40). ...
Article
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For a locally convex space with the topology given by a family {p(┬; α)} α ∈ ω of seminorms, we study the existence and uniqueness of fixed points for a mapping defined on some set . We require that there exists a linear and positive operatorK, acting on functions defined on the index set Ω, such that for everyu, Under some additional assumptions, one of which is the existence of a fixed point for the operator , we prove that there exists a fixed point of . For a class of elements satisfyingK n (p)u;┬))(α) → 0 asn → ∞, we show that fixed points are unique. This class includes, in particular, the class for which we prove the existence of fixed points. We consider several applications by proving existence and uniqueness of solutions to first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms.
Article
This article considers two weight estimates for the single layer potential — corresponding to the Laplace operator in 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.
Book
"The book under review is the first systematic and self-contained presentation of a theory of arbitrary order ordinary differential equations with unbounded operator coefficients in a Hilbert or Banach space … . this is an excellent book, that contains recent results of the topic, deep theoretical results and various applications to PDE-s. It is warmly recommended to specialists in ODE-s, PDE-s, functional analysis." (Jeno Hegedus, Acta Scientiarum Mathematicarum, Vol. 72, 2006)
Complex Variables and Elliptic Equations
  • V G Mazya
  • Sobolev
V.G. Mazya, Sobolev Spaces, Springer-Verlag, Berlin, 1985. Complex Variables and Elliptic Equations