Abstract
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 Lp-logarithmic Gagliardo–Nirenberg and Sobolev inequalities. Motivated by the L2-logarithmic Sobolev inequality, we obtain a fractional logarithmic Sobolev trace inequality in terms of the restriction τku of u from Rn to Rn−k. Finally, we prove the fractional Hardy inequality under Lorentz norms.
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