Gagliardo-Nirenberg type inequalities on Lorentz, Marcinkiewicz and weak-𝐿^{∞} spaces

Abstract

We establish the Gagliardo-Nirenberg inequality, Trudinger-Moser inequality and John-Nirenberg inequality using the Lorentz spaces L p , α L^{p,\alpha } , the Marcinkiewicz space L q , ∞ L^{q,\infty } and the weak- L ∞ L^{\infty } space W W introduced by Bennett, DeVore and Sharpley [Ann. of Math. (2) 113 (1981), pp. 601–611]. As consequences, we obtain the Gagliardo-Nirenberg type inequality with weak- L ∞ L^{\infty } norm and BMO norm, Trudinger-Moser type inequality and John-Nirenberg type estimate with B M O BMO norm and weak- L 1 L^{1} norm.