Full paper in PDF format:
$%E. Romera, Extremal vector vInequalities for Hankel transforms, Rev. Mat. Complut. 22 (2009), no. 1, 153–163.%$

Extremal Vector Valued Inequalities for Hankel Transforms

Elena ROMERA

Departamento de Matemáticas
Universidad Carlos III de Madrid
Avenida de la Universidad, 30
28911 Leganés (Madrid) — Spain
eromera@math.uc3m.es

Received: February 15, 2008
Accepted: June 5, 2008

ABSTRACT

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 $Lpq(rn-1dr) l$ when $-2n-< p,q <-2n n+1 n-1$ . Here we prove that there exist weak-type estimates for this operator for the extremal exponents, that is, it is bounded from $Lpilq,1(rn-1dr)$ to $Lpliq,∞(rn-1dr)$ for $i= 0,1$ when $p0 = 2nn+1$ , $p1 = 2nn-1$ , $p0 < q < p1$ , and we consider radial functions.

Key words: disc multiplier, Fourier-Hankel transforms.
2000 Mathematics Subject Classification: 42B10.