Best Constants for Certain Multilinear Integral Operators

Hilbert’s proof, apart from the determination of the best possible constant π csc(π/p), was published by Weyl [7]. The calculation of the constant, and the integral analogue of Hilbert’s double series theorem (for p = 2) are due to Schur [6]. The generalizations to other p′s of both the discrete and integral versions of this result were discovered later on by Hardy and Riesz and published by Hardy in [4]. The statement of the integral analogue of the theorem above is the following. If p > 1, p′ = p/(p− 1), ∫∞ 0 f p(x)dx ≤ F, ∫∞ 0 g p(y)dy ≤G, then ∫∫∞