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We investigate a novel connection between the weighted isoperimetric problemsand the weighted Poisson integrals of the extension problems for nonlocalelliptic operators. We first derive sharp inequalities for the weighted Poissonintegrals associated with degenerated elliptic equations on the half-space andthe unit ball, and classify their extremizers. The equations arise from theCaffarelli-Silvestre extension for the fractional Laplacian on the Euclideanspace and its conformal transformation via the M\"obius transformation. We nextinterpret the above sharp inequalities in a conformal geometric viewpoint. Forthis aim, we formulate a variational problem involving a weighted isoperimetricratio on a smooth metric measure space induced by a conformally compactEinstein (CCE) manifold. Then, we prove that the variational problem is closelylinked to the Chang-Gonz\'alez extension for a conformal fractional Laplacianon the conformal infinity of the CCE manifold, and is reduced to the sharpinequality if the CCE manifold is either the Poincar\'e half-space or ballmodel. We also find a criterion that ensures the existence of a smoothextremizer of the variational problem, and present a relevant conjecture.

Weighted isoperimetric ratios and extension problems for conformal fractional Laplacians