Fractional Gagliardo–Nirenberg and Hardy inequalities under Lorentz norms

In this paper, we establish the Gagliardo–Nirenberg inequality under Lorentz norms for fractional Laplacian. Based on special cases of this inequality under Lebesgue norms, we prove the L p -logarithmic Gagliardo–Nirenberg and Sobolev inequalities. Motivated by the L 2 -logarithmic Sobolev inequality, we obtain a fractional logarithmic Sobolev trace inequality in terms of the restriction τ k u of u from R n to R n − k . Finally, we prove the fractional Hardy inequality under Lorentz norms.