# 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

G. De Souza. Spaces formed by Special Atoms. Ph.D. dissertation, SUNY at Albany, (1980).

G. De Souza, R. O'Neil and G. Sampson. Several Characterizations for the Special Atom Spaces with Applications. Revista Matematica Iberoamericana, 2, No. 3, 333-355 (1986).

G. De Souza. The Dyadic Special Atom Space. Proceedings of the Conference on Harmonic Analysis. Minneapolis. Lecture Notes in Math, 908, Springer - Verlag, 297-899 (1982).

G. De Souza. A Proof of Carleson Theorem Based on a New Characterization of Lorentz Spaces L(p; 1) and Other Applications. To appear in Real Analysis Exchange.

G. De Souza and S. Bloom. Weighted Lipschitz Spaces and Their Analytic Characterizations. Journal of Constructive Approximations, 10, 339-376 (1994).

M. W. Hirsch, B. Mazur. Smoothing of piecewise-linear manifolds. Princeton, 3, 357-388 (1974).

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*Journal of the Nigerian Mathematical Society*,

*38*(3), 469–489. Retrieved from https://ojs.ictp.it/jnms/index.php/jnms/article/view/509