We discuss the Hausdorff–Young inequality in the context of maximal integral estimates, including the case of Hermite and Laguerre expansions. We establish a maximal inequality for integral operators with bounded kernel on \({\mathbb {R}}\) , which in particular allows for the pointwise evaluation of these operators, including the Fourier transform, for functions in appropriate Lorentz and Orlicz spaces. In the case of the Hermite expansions we prove a refined Hausdorff–Young inequality, further sharpened by considering the maximal Hermite coefficients in place of the Hermite coefficients when estimating the appropriate Lorentz and Orlicz norms. We also consider the refined companion Hausdorff–Young inequality and Hardy–Littlewood type inequalities for the Hermite expansions. Similar results are proved for the Laguerre expansions.
