Relationship Between Limiting K‐Spaces and J‐Spaces in the Real Interpolation

In the paper, “Description of the K$K$‐Spaces by Means of J$J$‐Spaces and the Reverse Problem,” Mathematische Nachrichten 296, no. 9 (2023), 4002–4031, we have established conditions under which the limiting K$K$‐space (X0,X1)0,q,b;K$(X_0,X_1)_{0,q,b;K}$, involving a slowly varying function b$b$, can be described by means of the J$J$‐space (X0,X1)0,q,a;J$(X_0,X_1)_{0,q,a;J}$, with a convenient slowly varying function a$a$, and we have also solved the reverse problem. It has been shown that if these conditions are not satisfied, then 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$K$‐space (X0,X1)0,q,b;K$(X_0,X_1)_{0,q,b;K}$ as some limiting J$J$‐space (Y0,Y1)0,q,A;J$(Y_0,Y_1)_{0,q,A;J}$, and, similarly, to express the limiting J$J$‐space (X0,X1)0,q,a;J$(X_0,X_1)_{0,q,a;J}$ as a convenient limiting K$K$‐space (Z0,Z1)0,q,B;K$(Z_0,Z_1)_{0,q,B;K}$. To be more precise, we show that (X0,X1)0,q,b;K=(X0,X0+X1)0,q,A;J=X0+(X0,X1)0,q,A;J$$\begin{equation*} (X_0,X_1)_{0,q,b;K}= (X_0,X_0+X_1)_{0,q,A;J}=X_0+(X_0,X_1)_{0,q,A;J} \end{equation*}$$and (X0,X1)0,q,a;J=(X0,X0∩X1)0,q,B;K=X0∩(X0,X1)0,q,B;K,$$\begin{equation*} (X_0,X_1)_{0,q,a;J}= (X_0,X_0\cap X_1)_{0,q,B;K}=X_0\cap (X_0,X_1)_{0,q,B;K}, \end{equation*}$$where A$A$ and B$B$ 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. As an example, we prove the density of the set C0∞(IRn)$C_0^\infty ({\rm I\hspace{-1.69998pt}R}^n)$ in the Besov space Bp,q0,b(IRn)$B^{0,b}_{p,q}({\rm I\hspace{-1.69998pt}R}^n)$ involving the zero classical smoothness and a slowly varying smoothness b$b$. Note also that our results play important role in calculation of duals of limiting interpolation spaces (X0,X1)0,q,b;K$(X_0, X_1)_{0, q, b;K}$ and (X0,X1)0,q,b;J$(X_0, X_1)_{0, q, b;J}$.