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This note describes two results: (i) a sharp Hausdorff-Young inequality for the Fourier transform on Lp(Rn) which extends an earlier result of Babenko; and (ii) a sharp form of Young's inequality for the convolution of functions on Rn. That is, best possible constants are obtained for the following Lp(Rn) inequalities: [Formula: see text]
Keywords: Fourier transform, convolution, Hermite semigroup
