# Joint weak type interpolation on Lorentz-Karamata spaces

## TL;DR

This paper establishes sharp interpolation theorems for a broad class of quasilinear operators acting between Lorentz-Karamata spaces, including all limiting cases, with implications for integral operators and Hardy inequalities.

## Contribution

It provides the first comprehensive characterization of joint weak type interpolation on Lorentz-Karamata spaces, including new limiting cases of Hardy-type inequalities.

## Key findings

- Sharp interpolation theorems for quasilinear operators
- Optimality results in Lorentz-Karamata spaces
- New limiting cases of Hardy inequalities

## Abstract

We present sharp interpolation theorems, including all limiting cases, for a class of quasilinear operators of joint weak type acting between Lorentz-Karamata spaces over $\sigma$-finite measure. This class contains many of the important integral operators. The optimality in the scale of Lorentz-Karamata spaces is also discussed. The proofs of our results rely on a characterization of Hardy-type inequalities restricted to monotone functions and with power-slowly varying weights. Some of the limiting cases of these inequalities have not been considered in the literature so far.