Very weak notions of differentiability.

*(English)*Zbl 1167.26001The authors study some weak notions of differentiability arising in connection with the spatial regularity of flows associated with non-smooth vector fields. The main difference from other similar concepts is that the convergence of difference quotients has to be understood as convergence in measure. So, the following definitions are introduced.

Let \(\Omega\) be an open subset of \(\mathbb{R}^{ N}\). A measurable function \(f: \Omega\rightarrow \mathbb{R}\) is called:

– (Fréchet-)differentiable in measure if there exists a measurable function \(g: \Omega \rightarrow \mathbb{R}^{ N}\) such that \[ \lim_{h\to 0}{{f(x+h)+f(x)-g(x)\cdot h}\over {|h|}} = 0 \] locally in measure with respect to \(x\in \Omega\),

– Gateaux differentiable in measure if there exists a measurable function \(g: \Omega \rightarrow \mathbb{R}^{ N}\) such that \[ \lim_{\delta\to 0}{{f(x+\delta y)+f(x)-\delta g(x)\cdot y}\over \delta} = 0 \] for each \(y\in \mathbb{R}^{ N}\), locally in measure with respect to \(x\in\Omega\), and

– directionally differentiable in measure if there exists a measurable function \(W: \Omega\times \mathbb{R}^{ N}\rightarrow \mathbb{R}\) such that \[ \lim_{\delta\to 0}{{f(x+\delta y)+f(x)-\delta W(x, y)}\over \delta} = 0 \] locally in measure with respect to \((x,y)\in \Omega \times \mathbb{R}^{N}\).

The main theorem states that all these notions of differentiability are equivalent. Moreover, the authors show that classical approximate differentiability is stronger than differentiability in measure.

Let \(\Omega\) be an open subset of \(\mathbb{R}^{ N}\). A measurable function \(f: \Omega\rightarrow \mathbb{R}\) is called:

– (Fréchet-)differentiable in measure if there exists a measurable function \(g: \Omega \rightarrow \mathbb{R}^{ N}\) such that \[ \lim_{h\to 0}{{f(x+h)+f(x)-g(x)\cdot h}\over {|h|}} = 0 \] locally in measure with respect to \(x\in \Omega\),

– Gateaux differentiable in measure if there exists a measurable function \(g: \Omega \rightarrow \mathbb{R}^{ N}\) such that \[ \lim_{\delta\to 0}{{f(x+\delta y)+f(x)-\delta g(x)\cdot y}\over \delta} = 0 \] for each \(y\in \mathbb{R}^{ N}\), locally in measure with respect to \(x\in\Omega\), and

– directionally differentiable in measure if there exists a measurable function \(W: \Omega\times \mathbb{R}^{ N}\rightarrow \mathbb{R}\) such that \[ \lim_{\delta\to 0}{{f(x+\delta y)+f(x)-\delta W(x, y)}\over \delta} = 0 \] locally in measure with respect to \((x,y)\in \Omega \times \mathbb{R}^{N}\).

The main theorem states that all these notions of differentiability are equivalent. Moreover, the authors show that classical approximate differentiability is stronger than differentiability in measure.

Reviewer: Uta Freiberg (Canberra)

##### MSC:

26A24 | Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems |

26B05 | Continuity and differentiation questions |

28A15 | Abstract differentiation theory, differentiation of set functions |