Characterization of Carleson Measures by the Hausdorff–Young Property

It is shown that the Laplace transform of an $L^p$ ($1<p\le 2$) function defined on the positive semiaxis satisfies the Hausdorff–Young type inequality with a positive weight in the right complex half-plane if and only if the weight is a Carleson measure. In addition, Carleson's weighted $L^p$ inequality for the harmonic extension is given with a numeric constant.