NOTE ON A GENERALIZATION OF THE SPACE OF DERIVATIVES OF LIPSCHITZ FUNCTIONS
Abstract
In this note, we denote by $(Lip^1)'$ the space of derivatives of Lipschitz functions of order 1. We propose a generalization of the space $(Lip^1)'$ on the interval $[0,2\pi]$ for general measures on subsets of $[0,2\pi]$ with respect to the representation of the norm. As a byproduct, we obtain H\"{o}lder's type inequalities and duality results between the space $(Lip^1)'$ as well as its generalization, and the special atoms spaces $B$ and $B(\mu,1)$, spaces first introduced by De Souza in his PhD thesis. Another byproduct is a relation between the space $(Lip^1)'$ as well as its generalization, and the space $L_\infty$. As a result we prove that the special atom space is a simple characterization of $L_1$.References
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