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 L(lq)(p) (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 L(lq)(pi,1) (r(n-1) dr) to L(lq)(pi,infinity) (r(n-1) dr) for i = 0, 1 when p(0) = 2n/n+1, p(1) = 2n/n-1, p(0) < q < p(1,) and we consider radial functions.