Weighted estimates for the Hankel transform transplantation operator

TLDR: In this paper, a suitably established local version of the Calderon-Zygmund operator theory was used to obtain weighted norm inequalities with weights more general than previously considered power weights.

Abstract: The Hankel transform transplantation operator is investigated by means of a suitably established local version of the Calderon-Zygmund operator theory. This approach produces weighted norm inequalities with weights more general than previously considered power weights. Moreover, it also allows to obtain weighted weak type $(1,1)$ inequalities, which seem to be new even in the unweighted setting. As a typical application of the transplantation, multiplier results in weighted $L^p$ spaces with general weights are obtained for the Hankel transform of any order $\alpha > -1$ greater than $-1$ by transplanting cosine transform multiplier results.