# Weighted norm inequalities for generalized Fourier-type transforms and applications

## TL;DR

This paper establishes precise weight conditions for the boundedness of generalized Fourier-type transforms between weighted L^p and L^q spaces, with applications to sine, Hankel, and other power-type transforms.

## Contribution

It provides necessary and sufficient weight conditions for boundedness, avoiding reliance on decreasing rearrangements, which advances understanding of Fourier-type transform inequalities.

## Key findings

- Derived explicit weight conditions for boundedness
- Applied results to sine, Hankel, and power-type transforms
- Enhanced previous criteria by removing the need for decreasing rearrangements

## Abstract

We obtain necessary and sufficient conditions on weights for the generalized Fourier-type transforms to be bounded between weighted $L^p-L^q$ spaces. As an important example, we investigate transforms with kernel of power type, as for instance the sine, Hankel or $\mathscr{H}_\alpha$ transforms. The obtained necessary and sufficient conditions are given in terms of weights, but not in terms of their decreasing rearrangements, as in several previous investigations.