| We prove rearrangement inequalities for multiple integrals in the Euclidean space, the unit sphere and the hyperbolic space, using the polarization technique.; Polarization refers to rearranging a function with respect to a hyperplane; in Rn , for example, hyperplanes are affine subspaces. Given a hyperplane H which does not pass through the origin, we denote by H + the half space that contains the origin and by H- the other half space. For a function f, we define its polarization fH as follows: For each pair of points x,y symmetric with respect to H, with x in H +, we define fH(x) to be the maximum of f at x and y, and fH(y) to be the minimum of f at x and y.; We show that certain multiple integrals increase when the functions are replaced by their polarization. Once we have these results for the polarization case, we can apply a variation of a result of Baernstein-Taylor to get them for general symmetric decreasing rearrangement. Given a function f, we associate to it its symmetric decreasing re-arrangement, f♯, which is equimeasurable with f (i.e. it preserves the measure of the level sets), is constant on each sphere centered at the origin and decreases as the radius of the sphere increases.; The results complement those of Brascamp-Lieb-Luttinger and have application to ratio inequalities of heat kernels for Schrodinger operators in bounded domains. |
