Abstract. We derive a sharp uncertainty inequality of the formwith Λ 0 = 0.428368 . . . . As a consequence of this inequality we derive an upper bound for the so-called Laue constant, that is, the infimum λ 0 of the functional λ(p) = 4π 2 x 2 p 1 x 2p 1 /(p(0)p (0)), taken over all p ≥ 0 witĥ p ≥ 0 (p ≡ 0). Precisely, we obtain that λ 0 ≤ 2Λ 0 = 0.85673673 . . . , which improves a previous bound of T. Gneiting.
We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure |E| of such a set. We exploit a rational change of variable of the type used by George Boole in his paper "On the comparison of transcendents, with certain applications to the theory of definite integrals" together with the observation that if two functions f and g have the same L p norm in a range of exponents p 1 < p < p 2 then their distribution functions coincide.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.