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.