Fractional Gagliardo-Nirenberg and Hardy inequalities under Lorentz norms

[摘要] 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 tau(k)u of u from R-n to Rn-k. Finally, we prove the fractional Hardy inequality under Lorentz norms. (C) 2012 Elsevier Inc. All rights reserved.

[发布日期] 2012-12-15

[关键词] Fractional Laplacian;Gagliardo-Nirenberg inequality;Sobolev inequality;Logarithmic Sobolev inequality;Hardy inequality