We establish maximal inequalities for integral operators on \({{\mathbb {R}}}^n\) with bounded kernel which allow for the pointwise evaluation of these operators, including the Fourier transform, for functions in Lorentz and Orlicz spaces. We introduce and estimate the maximal Fourier coefficients with respect to a uniformly bounded ONS on \(L^2_\mu (I)\) for functions in \(L^p_\mu (I)\) , with \(1< p\le 2\) , where \(\mu \) is a positive regular measure on a closed bounded interval I in the line. We obtain an improved Hausdorff–Young inequality with the maximal Fourier coefficients in place of the Fourier coefficients for \(1<p<3/2\) , and a complementary result for \(1<p\le 2\) when \(\mu (I)<\infty \) . We also discuss Paley’s theorem. These results are considered in the setting of Lorentz and Orlicz spaces as well. We illustrate the results with the Jacobi expansions.
