Sobolev's imbedding theorem in the limiting case with Lorentz space and BMO

We consider the Gagliardo-Nirenberg type inequality in R. Let Ω be an arbitrary domain inR. It is well known that the Sobolev space H 0 (Ω), 1 < p < ∞, is continuously embedded into L(Ω) for all q with p 5 q < ∞. However, we cannot take q = ∞ in such an embedding. When Ω = R, Ogawa [11] and Ogawa-Ozawa [12] treated the Hilbert space H(R) and then Ozawa [15] gave the following general embedding theorem in the Sobolev space H(R) of the fractional derivatives which states that ‖Φp(α|u|p)‖L1(Rn) 5 C‖u‖pLp(Rn) (0.1)