Polar factorization and pseudo-rearrangements: applications to Pólya–Szegö type inequalities Original Research Article

We are interested in the polar factorization of a function f defined in an open bounded set View the MathML source. It is well known that there exists a measure preserving map σ such that f=f∗∘σ where f∗ is the decreasing rearrangement of f. We prove that, under suitable assumptions, besides the classical polar factorization of f we have View the MathML source where View the MathML source is a pseudo-rearrangement of f with respect to the measurable function u and σ is the measure preserving map such that u=u∗∘σ. As an application, we characterize those functions that realize equality in the Pólya–Szegö inequality.