an:00023286 Zbl 0746.26002 Mařík, Jan; Weil, Clifford E. Sums of powers of derivatives EN Proc. Am. Math. Soc. 112, No. 3, 807-817 (1991). 0002-9939 1088-6826 1991
j
26A24 26A15 nonnegative derivatives; approximate continuity; sums of powers of positive derivatives; approximately continuous functions 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. L.Mišík (Bratislava) 0517.26006