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In this paper we establish sharp invertibility results for the elastostatics and hydrostatics single and double layer potential type operators acting on , , whenever is an infinite sector in . This analysis is relevant to the layer potential treatment of a variety of boundary value problems for the Lam, system of elastostatics and the Stokes system of hydrostatics in the class of curvilinear polygons in two dimensions, such as the Dirichlet, the Neumann, and the Regularity problems. Mellin transform techniques are used to identify the critical integrability indices for which invertibility of these layer potentials fails. Computer-aided proofs are produced to further study the monotonicity properties of these indices relative to parameters determined by the aperture of the sector and the differential operator in question.