# Two types of Rubio de Francia operators on Triebel--Lizorkin and Besov spaces

## TL;DR

This paper explores two variants of Rubio de Francia operators on Triebel--Lizorkin and Besov spaces, analyzing their boundedness and the necessity of rotation factors across different smoothness spaces using interpolation methods.

## Contribution

It introduces a detailed analysis of two types of Rubio de Francia operators on these function spaces, highlighting when rotation factors are essential for boundedness.

## Key findings

- Rotation factor is necessary for boundedness in some smooth spaces.
- In other spaces, the rotation factor is not essential.
- Interpolation methods are used to study operators on end spaces of the scale.

## Abstract

We discuss generalizations of Rubio de Francia's inequality for Triebel--Lizorkin and Besov spaces, continuing the research from [5]. Two versions of Rubio de Francia's operator are discussed: it is shown that a rotation factor is needed for the boundedness of the operator in some smooth spaces while it is not essential in other spaces. We study the operators on some "end" spaces of the Triebel--Lizorkin scale and then use usual interpolation methods.