On a commutative hypergroup the Fourier transform is norm-decreasing from L^p into the conjugate Lebesgue space for exponents between one and two, and the question is which functions turn that inequality into an equality. Any such function, for an exponent strictly between one and two, must have compact open support, as must its transform, and it is forced to be a constant multiple of a character carried on a translate of a compact open subhypergroup; conversely the indicator of such a subhypergroup always achieves equality. When the centre of the hypergroup carries positive Haar measure, or the hypergroup is compact, this yields a clean dichotomy: the hypergroup admits a non-trivial function and transform pair both of compact support exactly when one is the best constant in the inequality. The classical locally compact abelian case is recovered as a special case, but characters need not be unimodular and translation need not be isometric, so the classical arguments require genuine repair.
Bandyopadhyay and Mohanty (Banach J. Math. Anal. 16, 62 (2022)) ask when the Hausdorff-Young inequality is sharp on a commutative hypergroup. A hypergroup is a locally compact space whose measure alg…