Polar factorization and pseudo-rearrangements:applications to Polya-Szego type inequalities

We are interested in the polar factorization of a function f defined in an open bounded subset of R^N. It is well known that there exists a measure preserving map s such that f = f*o s where f* is the decreasing rearrangement of f. We prove that, under suitable assumptions, besides the classical polar factorization of f we have f = f_u o s where f_u is a pseudo-rearrangement of f with respect to the measurable function u and s is the measure preserving map such that u = u* o s. As an application, we characterize those functions that realize equality in the Polya-Szego inequality