On exact inequalities of Hardy-Littlewood-Polya type

The following problem is investigated for certain Hilbert function spaces: for a given A greater than or equal to 0 find the infimum of the set of B g reater than or equal to 0 such that the inequality parallel to x((k))parallel to(2)(2) less than or equal to A parallel to x p arallel to(2)(2) + B parallel to x((r))parallel to(2)(2), for k, r is an element of N boolean OR {0}, 0 less than or equal to k < r, holds for all sufficiently smooth functions. An analogous problem is invest igated under some restrictions on the spectrum of functions. (C) 2000 Acade mic Press.