A lower bound for the spectrum of a one-dimensional pseudo-differential operator

Abstract

In a 1928 paper [1], Hardy and Littlewood published the following inequality which is valid for all non-negative measurable functions ƒ and g on ( − ∞, ∞):where 1 < p, q < ∞, 0 < λ < 1, l/p + 1/q + λ = 2 and Cλp, q depends only on λ, p, q; ‖ƒ‖p and ‖g‖q are the Lp and Lq norms of ƒ and g respectively. See also p. 288 of the monograph [2] by Hardy, Littlewood and Pólya for a discussion of this inequality and related matters.