We prove rearrangement inequalities for multiple integrals, using the polarization technique. Polarization refers to rearranging a function with respect to a hyperplane. Then we derive sharp inequalities for ratios of integrals of heat kernels of Schrödinger operators, using our polarization inequalities. These ratio inequalities imply inequalities for the partition functions and extend the results of R. Bañuelos, P.J. Méndez-Hernández and D. You. (2000): 26D15, 28A25, 35J10. Mathematics Subject Classifications

Abstract. We prove a general rearrangement inequality for multiple integrals, using polarization. We introduce a special class of kernels for which the product inequality holds, and then we prove that it also holds when the product is replaced by a so-called function ALm.

In this paper we maximize a class of functionals under certain constraints. We find necessary and sufficient conditions for these maximizers to exist and be unique. Moreover, we characterize them and discuss the optimality of our results by constructing counterexamples when one of the hypotheses does not hold.

We prove that the integral of n functions over a symmetric set L in R n , with additional properties, increases when the functions are replaced by their symmetric decreasing rearrangements. The result is known when L is a centrally symmetric convex set, and our result extends it to nonconvex sets. We deduce as consequences, inequalities for the average of a function whose level sets are of the same type as L, over measurable sets in R n . The average of such a function on E is maximized by the average over the symmetric set E * .

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