In this note, we investigate orthogonal expansions adapted to exponential weights and function spaces endowed with Hermite-frequency decompositions. Our objective is to formulate the Hausdorff–Young inequality within the framework of Hermite based Triebel–Lizorkin type spaces. The results rest on two underlying principles. First, since the Hermite functions \(\{{\mathcal {H}}_m(x)\}\) , along with their generalized counterparts, are particular instances of generalized Freud functions associated with generalized Freud weights, we rely on the \(L^\infty \) and \(L^q\) , \(q<\infty \) , estimates of Kasuga and Sakai which extend Hille’s estimate \(|{\mathcal {H}}_m(x)|\lesssim m^{-1/12}\) , \(m=1,2,\ldots \) Second, we apply interpolation techniques involving the Orlicz–Lorentz spaces \(\Lambda (\varphi _X, C)\) , where C is a Young (or concave) function and \(\varphi _X\) denotes the fundamental function of a rearrangement-invariant space X. This approach enables us to derive Hausdorff–Young inequalities in \(L^p\) , Lorentz, Orlicz, and \(\Lambda (\varphi _X, C)\) based Hermite Triebel–Lizorkin spaces. Finally, in the Coda we present a general Hausdorff–Young inequality for the generalized Freud coefficients of functions \(f\in L^{p_0}({\mathbb {R}})+L^{p_1}({\mathbb {R}})\) , where \(1\le p_0<p_1\le 2\) , and a weighted version of the Hausdorff–Young inequality for generalized Freud expansions.
