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This article considers two weight estimates for the single layer potential--- corresponding to the Laplace operator in $\mathbf{R}^{N+1}$ --- onLipschitz surfaces with small Lipschitz constant. We present conditions on theweights to obtain solvability and uniqueness results in weighted Lebesguespaces and weighted homogeneous Sobolev spaces, where the weights are assumedto be radial and doubling. In the case when the weights are additionallyassumed to be differentiable almost everywhere, simplified conditions in termsof the logarithmic derivative are presented, and as an application, we provethat the operator corresponding to the single layer potential in question is anisomorphism between certain weighted spaces of the type mentioned above.Furthermore, we consider several explicit weight functions. In particular, wepresent results for power exponential weights which generalize known resultsfor 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.

Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant