ИСТИНА

It is observed that due to Carleson's harmonic extension theorem some classical inequalities, such as those of Hilbert and Hausdorff-Young can be stated in a stronger "maximal" form. For instance, if $f\in L^p(R)$ ($p>1$) and $g=Hf$ is its conjugate function, then the function $g^*(r)=\sup_{0<\theta<\pi} g(r e^{i\theta})$ belongs to $L^p(R_+)$. Abstracting the essence of a proof, we are led to a measure-theoretic construction involving the notions of conditional essential supremum and conditional Carleson measure.