Sharp Adams type inequalities in Sobolev spaces for arbitrary integer m

The main purpose of our paper is to prove sharp Adams type inequalities in unbounded domains of Rn for the Sobolev space for any positive integer m less than n. Our results complement those of Ruf and Sani (in press) [35], where such inequalities have been established for even integer m. We extend the main techniques of Ruf and Sani (in press) [35], , which are the combinations of the comparison principle of Talenti (1976) [36], and Trombetti and Vázquez (1985) [38] for polyharmonic operators and a symmetrization argument together with constructions of radial auxiliary functions, to the case when m is odd. Moreover, we offer a completely different but much simpler approach to prove the comparison principle using the power of Bessel potentials and the Riesz rearrangement (see Remarks 3.2 and 3.3). This approach is of independent interest and works for any differential operators with appropriate radial kernels. As corollaries of our main theorems, we will derive the Adams type inequalities in the case when n=2m for all positive integer m by using different Sobolev norms.