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

TL;DR

The paper advances the theory of limiting real interpolation by linking the limiting $K$- and $J$-spaces through slowly varying weights, including cases where previous conditions fail. It demonstrates that $ (X_0,X_1)_{0,q,b;K} = (X_0,X_0+X_1)_{0,q,B;K} = (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,A;J} = X_0\cap (X_0,X_1)_{0,q,B;K}$ with appropriate slowly varying weights, establishing equivalent norms and density theorems. It also extends these identifications to endpoint cases ($q=\infty$ and $q=1$) and provides complementary representations when the OG conditions are not satisfied. The results enhance the understanding of the relationship between limiting K- and J-spaces and supply practical density conclusions for the associated interpolation spaces. Overall, the work clarifies how to translate limiting K-spaces into limiting J-spaces (and vice versa) via auxiliary weights, enriching the toolkit of real interpolation theory.

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.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2: Density theorem
• Theorem 1.3
• Theorem 1.4: Density theorem
• Remark 1.5
• Theorem 1.6: OG
• Remark 1.7
• Theorem 1.8: OG
• Remark 1.9
• Theorem 1.10: Density theorem, OG