Remarks on the Gagliardo-Nirenberg Type Inequality in the Besov and the Triebel-Lizorkin Spaces in the Limiting Case

We consider the generalized Gagliardo-Nirenberg inequality in R-n including homogeneous Besov space (B) over dot(r,rho)(s)(R-n) with the critical order s = n/r, which describes the continuous embedding such as L-p(R-n) boolean AND (B) over dot(r,rho)(n/r) (R-n) subset of L-q(R-n) for all q with p <= q < infinity, where 1 <= p <= r < infinity and 1 < rho <= infinity. Indeed, the following inequality holds: parallel to u parallel to(Lq)(R-n) <= C q(1-1/rho)parallel to u parallel to(p/q)(Lp(Rn)) parallel to u parallel to((B) over dotr,rho n/r) (1-p/q)(,) where C is a constant depending only on r. In this inequality, we have the exact order 1 - 1/rho of divergence to the power q tending to the infinity. Furthermore, as a corollary of this inequality, we obtain the Gagliardo-Nirenberg inequality with the homogeneous Triebel-Lizorkin space (F) over dot(r,rho)(n/r) (R-n), which implies the usual Sobolev imbedding with the critical Sobolev space (H) over dot(r)(n/r) (R-n).Moreover, as another corollary, we shall prove the Trudinger-Moser type inequality in (B) over dot(r,rho)(n/r) (R-n).