Abstract
We establish sharp affine weighted L p L^p Sobolev type inequalities by using the L p L_p BusemannâPetty centroid inequality proved by Lutwak, Yang, and Zhang. Our approach consists of combining the latter with a suitable family of sharp weighted L p L^p Sobolev type inequalities obtained by Nguyen and allows us to characterize all extremizers in some cases. The new inequalities do not rely on any euclidean geometric structure.
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