Norm Of The Generalized Stieltjes Transform

The generalized Stieltjes transform of a function f(t),

g(x) = \int_0^\infty \frac{f(t)\,dt}{(x+t)^\mu}\ ,

occurs for example in superstring theory. For f(t) in the Lebesgue class L_p, Hardy et al. showed that for under certain conditions on p, q and \mu,

\| g\|_q \;<\; C \|f\|_p

where C(p,\mu) is some constant. The best possible value of C is the norm of the transform and has only been found for special cases of the parameters.

The current approach used a parameterised family of functions for which the integral above can be found explicitly. The norms were computed numerically and Nimrod/O used to maximize the value of C. Results of the research have been published here.

Reference

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Book
Hardy G. H. , Littlewood J. E. and Polya G. , Inequalities, Second edition, CUP, 1952. Abstract
Journal Article
Peachey, T. C. and Enticott, C. M., "Determination of the Best Constant in an Inequality of Hardy, Littlewood and Polya", Experimental Mathematics, v 15(1), pp 43-50, 2006. Abstract
He Y.-H., Schwarz J. H., Spradlin M., and Volovich A., Explicit formulas for Neumann coefficients in the plane-wave geometry, Physical Review D, 67, 2003, article 086005. Abstract http://prola.aps.org/abstract/PRD/v67/i8/e086005

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