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

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.