For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a m...

We prove a sharp pointwise estimate of the nonincreasing rearrangement of the fractional maximal function of f, M f, by an expression involving the nonincreasing rearrangement of f. This estimate is used to obtain necessary and suucient conditions for the boundedness of M between classical Lorentz spaces.

We examine the maximal-element rationalizability of choice functions with arbitrary domains. While rationality formulated in terms of the choice of greatest elements according to a rationalizing relation has been analyzed relatively thoroughly in the earlier literature, this is not the case for maximal-element rationalizability, except when it coincides with greatest-element rationalizability b...