Paley inequality for the Weyl transform and its applications

The Hausdorff-Young inequality is a foundational result in Fourier analysis which admits several generalizations. The main aim of the paper under consideration is to study Paley’s extension of the Hausdorff-Young inequality and to establish variants for the Weyl transform associated to locally compact abelian groups. Following a presentation of the historical and conceptual context (Section 1), and a detailed presentation of the results on which the proofs of the main theorems are based (Section 2), two generalizations of the Hausdorff-Young theorem are obtained. The first extends the Hausdorff-Young theorem to Lorentz spaces while the second (a version of the Paley inequality) is a more generalized result that extends the Hausdorff-Young theorem to non-commutative Lorentz spaces on the Banach algebra of all bounded operators on L2(G) where G is a locally compact abelian group. It was in this context that the French mathematician Louis Gérard defended, in 1892 at the Faculty of Sciences in Paris, a thesis titled Sur la géométrie non euclidienne [On Non-Euclidean Geometry]. Gérard’s thesis – the most significant work specifically on this subject published in French-speaking countries since the Memoirs of Joseph-Marie De Tilly [1870] and Camille Flye Sainte-Marie [1871] – has not yet been the subject of an in-depth study, and this gap is what the article under review seeks to address. First, a word about Louis Gérard. Born in 1859 in Grand, in the historical region of Lorraine, in eastern France, Gérard spent his entire career in secondary education. His thesis on non-Euclidean geometry thus represents his main contribution to mathematical research. The date of his death is unknown but is after 1939. As this article demonstrates, Gérard’s thesis serves as a transitional work between the contributions of the inventors of non-Euclidean geometry and the purely axiomatic research of the late 19th and early 20th centuries. For this reason, it deserves our attention.