  We present a Gagliardo-Nirenberg inequality which bounds Lorentz norms of the
function by Sobolev norms and homogeneous Besov quasinorms with negative
smoothness. We prove also other versions involving Besov or Triebel-Lizorkin
quasinorms. These inequalities can be considered as refinements of Sobolev type
embeddings. They can also be applied to obtain Gagliardo-Nirenberg inequalities
in some limiting cases. Our methods are based on estimates of rearrangements in
terms of heat kernels. These methods enable us to cover also the case of
Sobolev norms with p = 1.
