Interpolation spaces of Besov hierarchical spaces and non-linearities defined by vertex K functional and grid topology

TL;DR

This paper uses wavelets to analyze the nonlinear and topological structures of interpolation spaces of Besov hierarchical spaces, solving a longstanding open problem in the field.

Contribution

It introduces a wavelet-based approach to fully characterize the real interpolation spaces of Besov hierarchical spaces, addressing Peetre's open problem.

Findings

Complete solution to Peetre's open problem

Wavelet transformation of nonlinear structures

New insights into interpolation space topology

Abstract

In 1967, Peetre proposed to give a precise description of the real interpolation space for Besov hierarchical spaces $l^{s,q}(A)$. In 1974, Cwikel proved that the Lions-Peetre formula for $(l^{q_0}(A_0), l^{q_1}(A_1))_{\theta,r}$ have no reasonable generalization for any $r\neq q$. In this paper, we apply wavelets to transform the study of real interpolation space into {\bf the study of nonlinear functional structure and nonlinear topological structure}. We solve completely Peetre's longstanding open problem.