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

A remark on a generalization of a logarithmic Sobolev inequality to the Holder class