Gagliardo–Nirenberg Inequalities in Lorentz Type Spaces

In this paper, we derive some new Gagliardo–Nirenberg type inequalities in Lorentz type spaces without restrictions on the second index of Lorentz norms, which generalize almost all known corresponding results. Our proof mainly relies on the Bernstein inequalities in Lorentz spaces, the embedding relation among various Lorentz type spaces, and Littlewood–Paley decomposition techniques.