Relationship between limiting K-spaces and J-spaces in the real interpolation

TL;DR

This paper explores the relationships between limiting K-spaces and J-spaces in real interpolation, especially when certain conditions are not met, providing new representations and norm equivalences.

Contribution

It extends the understanding of K- and J-space relationships in real interpolation by expressing spaces as each other's limiting forms without the previous conditions.

Findings

Expressed limiting K-spaces as limiting J-spaces with convenient weights.

Expressed limiting J-spaces as limiting K-spaces with convenient weights.

Established equivalent norms and density theorems for these spaces.

Abstract

In the paper Description of the $K$-spaces by means of $J$-spaces and the reverse problem, Math. Nachr. 296 (2023), no. 9, 4002--4031, we have establish conditions under which the limiting $K$-space $(X_0,X_1)_{0,q,b;K}$, involving a slowly varying function $b$, can be described by means of the $J$-space $(X_0,X_1)_{0,q,a;J}$, with a convenient slowly varying function $a$, and we have also solved the reverse problem. It has been shown that if these conditions are not satisfied that the given problem may not have a solution. In this paper we assume that these conditions are not satisfied. Nevertheless, our aim is to express the limiting $K$-space $(X_0,X_1)_{0,q,b;K}$ as some limiting $J$-space $(Y_0,Y_1)_{0,q,A;J}$, and, similarly, to express the limiting $J$-space $(X_0,X_1)_{0,q,a;J}$ as a convenient limiting $K$-space $(Z_0,Z_1)_{0,q,B;K}$. To be more precise, we show that $$(X_0,X_1)_{0,q,b;K}=(X_0,X_0+X_1)_{0,q,A;J}=X_0+(X_0,X_1)_{0,q,A;J}$$ and $$(X_0,X_1)_{0,q,a;J}=(X_0,X_0\cap X_1)_{0,q,B;K}=X_0\cap (X_0,X_1)_{0,q,B;K},$$ where $A$ and $B$ are convenient weights. Moreover, we establish equivalent norms in the above mentioned spaces. The obtained results are applied to get density theorems for spaces in question.