Sharp convolution estimates for measures on flat surfaces (Q1900387)

The author proves the following estimate \[ |f*\mu|_{L^q(\mathbb{R}^n)}\leq C|f|_{L^\Phi(\mathbb{R}^n)}, \] with the optimal Young's function \(\Phi= \Phi_q\) for characteristic functions \(f\) of Borel sets \(E\subset \mathbb{R}^n\). This result is extended to the class of all (integrable) simple functions \(f\). A similar theorem for the Fourier restriction operator is proved, too.