Sums of powers of derivatives.

*(English)*Zbl 0746.26002The 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.

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.

Reviewer: L.Mišík (Bratislava)

##### 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 |