Abstract
In this paper, we prove a Gagliardo-Nirenberg type inequality on the critical Sobolev-Lorentz space with an exact growth order for the embedding constant. It turns out that the second index for the Lorentz space plays an essential role to determine the optimal growth order for the embedding constant. On the other hand, we establish a similar type interpolation inequality on the critical Besov-Lorentz space. In this case, we see that the growth order for the embedding constant can be determined by the third index for the Besov space and it is independent of the second index for the Lorentz space. Our main objective of the paper is to describe the optimal growth orders of the embedding constants by giving exact quantitative descriptions for the Gagliardo-Nirenberg type inequalities.
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