Extremal Vector Valued Inequalities for Hankel Transforms

The disc multiplier may be seen as a vector valued operator when we consider its projections in terms of the spherical harmonics. In this form, it represents a vector valued Hankel transform. We know that, for radial functions, it is bounded on the spaces Lplq (r n−1 dr) when 2n n+1 < p, q < 2n n−1 . Here we prove that there exist weak-type estimates for this operator for the extremal exponents, that is, it is bounded from Li lq (r n−1 dr) to Lpi,∞ lq (r n−1 dr) for i = 0, 1 when p0 = 2n n+1 , p1 = 2n n−1 , p0 < q < p1, and we consider radial functions.