# Article

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Keywords:
spaces of derivatives; Peano derivatives; Lipschitz function; multiplication operator
Summary:
For subspaces, $X$ and $Y$, of the space, $D$, of all derivatives $M(X,Y)$ denotes the set of all $g\in D$ such that $fg \in Y$ for all $f \in X$. Subspaces of $D$ are defined depending on a parameter $p \in [0,\infty ]$. In Section 6, $M(X,D)$ is determined for each of these subspaces and in Section 7, $M(X,Y)$ is found for $X$ and $Y$ any of these subspaces. In Section 3, $M(X,D)$ is determined for other spaces of functions on $[0,1]$ related to continuity and higher order differentiation.
References:
[1] A. M. Bruckner, J. Mařík, C. E. Weil: Some aspects of products of derivatives. Amer. Math. Monthly 99 (1992), 134–145. DOI 10.2307/2324182 | MR 1144354
[2] R. J. Fleissner: Distant bounded variation and products of derivatives. Fundam. Math. 94 (1977), 1–11. MR 0425041 | Zbl 0347.26009
[3] R. J. Fleissner: Multiplication and the fundamental theorem of calculus: A survey. Real Anal. Exchange 2 (1976), 7–34. MR 0507383
[4] J. Mařík: Multipliers of summable derivatives. Real Anal. Exchange 8 (1982–83), 486–493. DOI 10.2307/44153420 | MR 0700199
[5] J. Mařík: Transformation and multiplication of derivatives. Classical Real Analysis, Proc. Spec. Sess. AMS, 1982, AMS, Contemporary Mathematics 42, 1985, pp. 119–134. MR 0807985
[6] J. Mařík, C. E. Weil: Sums of powers of derivatives. Proc. Amer. Math. Soc. 112, 807–817. MR 1042268
[7] W. Wilkocz: Some properties of derivative functions. Fund. Math. 2 (1921), 145–154. DOI 10.4064/fm-2-1-145-154
[8] S. Saks: Theory of the integral. Dover Publications, 1964. MR 0167578

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