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 LpLp-logarithmic Gagliardo–Nirenberg and Sobolev inequalities. Motivated by the L2L2-logarithmic Sobolev inequality, we obtain a fractional logarithmic Sobolev trace inequality in terms of the restriction τkuτku of uu from RnRn to Rn−kRn−k. Finally, we prove the fractional Hardy inequality under Lorentz norms.