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Sums of powers of derivatives. (English) Zbl 0746.26002
The authors prove analogous results for sums of powers of positive derivatives as they proved for products in the paper in Trans. Am. Math. Soc. 276, 361-373 (1983; Zbl 0517.26006). I mention only the last Theorem:
Let $$p\in(1,\infty)$$, let $$f_ 1,\dots,f_ m\in M$$, let $$\varphi=\|(f_ 1,\dots,f_ m)\|_ p$$, where $$\|.\|_ p$$ denotes the $$p$$-norm on $$R^ m$$ and $$\phi=\varphi^ p$$. Let us suppose that $$\liminf_{y\to t}\text{ap }\varphi(y)>0$$ for each $$t\in R$$. Then the following conditions are equivalent: (i) $$\phi\in Q_ p$$, where $$Q_ p$$ is the system of all functions $$\Omega$$ with the following property: there exist a natural number $$r$$, positive numbers $$q_ j$$ and nonnegative derivatives $$h_ j$$, $$j\in\{1,\dots,r\}$$, such that $$q_ 1+\cdots+q_ r\leq p$$ and $$\Omega=h^{q_ 1}_ 1\cdots h^{q_ r}_ r$$;
(ii) $$\varphi\in D$$, where $$D$$ is the system of all derivatives on $$R$$;
(iii) there exist approximately continuous functions $$\alpha_ 1,\dots,\alpha_ m$$ such that $$f_ 1=\alpha_ 1\varphi,\dots,f_ m=\alpha_ m\varphi$$;
(iv) there exist functions $$\psi$$, $$\alpha_ 1,\dots,\alpha_ m$$ such that $$\psi$$ is a non-negative derivative, $$\alpha_ 1,\dots,\alpha_ m$$ are approximately continuous and $$f_ 1=\alpha_ 1\psi,\dots,f_ m=\alpha_ m\psi$$;
(v) $$\varphi\in M$$, where $$M$$ is the system of all derivatives $$f$$ on $$R$$ such that $$fg$$ is a derivative for any bounded approximately continuous function $$g$$.
At the end of the paper there are three counterexamples.

##### MSC:
 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems 26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
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