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We investigate the connection between the validity of certain logarithmic Sobolev inequality and the validity of suitable generalizations of Gagliardo–Nirenberg inequalities. A similar connection holds between reverse logarithmic Sobolev inequalities and a new class of reverse Gagliardo–Nirenberg inequalities, valid for a suitable class of functions. 0. Introduction The main concern of this paper is to investigate the connections between logarithmic Sobolev inequalities (LSI in the sequel) and generalizations of Gagliardo–Nirenberg inequalities (GNI in the sequel). The typical LSI inequality we shall be concerned with in the first part of the paper will be of the form

Direct and Reverse Gagliardo–Nirenberg Inequalities from Logarithmic Sobolev Inequalities