  In a recent work of the author, a parabolic extension of the elliptic Ogawa
type inequality has been established. This inequality is originated from the
Brezis-Gallouet-Wainger logarithmic type inequalities revealing Sobolev
embeddings in the critical case. In this paper, we improve the parabolic
version of Ogawa inequality by allowing it to cover not only the class of
functions from Sobolev spaces, but the wider class of Holder continuous
functions.
