In this study, we extend Beckner’s seminal work on the Fourier transform to the domain of Cayley–Dickson algebras, establishing a precise form of the Hausdorff–Young inequality for functions that take values in these algebras. Our extension faces significant hurdles due to the unique characteristics of the Cayley–Dickson Fourier transform. This transformation diverges from the classical Fourier transform in several key aspects: it does not conform to the Plancherel theorem, alters the interplay between derivatives and multiplication, and the product of algebra elements does not necessarily maintain the magnitude relationships found in classical settings. To overcome these challenges, our approach involves constructing the Cayley–Dickson Fourier transform by sequentially applying classical Fourier transforms. A pivotal part of our strategy is the utilization of a theorem that facilitates the norm-preserving extension of linear operators between spaces \(L^p\) and \(L^q.\) Furthermore, our investigation brings new insights into the complexities surrounding the Beckner–Hirschman Entropic inequality in the context of non-associative algebras.

在这项研究中，我们将贝克纳关于傅立叶变换的开创性工作扩展到凯莱-迪克森代数领域，为取这些代数中的值的函数建立了精确形式的豪斯多夫-杨不等式。由于凯莱-迪克森傅里叶变换的独特特性，我们的扩展面临着重大障碍。这种变换在几个关键方面与经典傅里叶变换不同：它不符合 Plancherel 定理，改变了导数和乘法之间的相互作用，并且代数元素的乘积不一定保持经典设置中的量值关系。为了克服这些挑战，我们的方法包括通过顺序应用经典傅里叶变换来构建凯莱-迪克森傅里叶变换。我们策略的关键部分是利用一个定理，该定理有助于空间\(L^p\)和\(L^q\)之间线性算子的范数保持扩展。此外，我们的研究为复杂性带来了新的见解围绕非结合代数背景下的贝克纳-赫希曼熵不等式。