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On local derivatives. (English) Zbl 0537.46041

Local derivatives of functions of several variables having values in a fixed Banach space are considered. Local derivatives of functions of one real variable with values in a Banach space were considered by J. Mikusiński [in ”The Bochner Integral”, (1978; Zbl 0369.28010)], and earlier, in a Hilbert space by K. Skórnik [in ”The form of locally integrable function whose m-th local derivative vanishes almost everywhere” (in Polish), Zesz. Nauk. WSP Katowice (1966)] for functions of any number of real variables.
By the m-th local derivative of a function f in \(R^ q\) we mean the local limit of the expression \(1/h^ m\Delta^{(m,h)}f(x)\) as \(h\to 0 (x,h\in R^ q\), m is a non-negative integer point of \(R^ q\) and \(\Delta^{(m,h)}f\) difference operator of order m). The m-th local derivative of f is denoted by \(D^ m_{loc}f.\)
Let f and g be locally integrable functions and let one of them have a bounded support. If the m-th local derivative of f and the k-th local derivative of g exist, then there also exist the \((m+k)\)-th local derivative of the convolution f*g and we have \(D^{m+k}_{loc}(f*g)=(D^ m_{loc}f)*(D^ k_{loc}g).\)
A locally integrable function f has its m-th local derivative equal to 0, iff, for each fixed \(h\in R^ q\), the equation \(\Delta^{(m,h)}f(x)=0\) holds for almost all \(x\in R^ q.\)
The local derivative (of order \(e=(1,...,1))\) of an integral equals to the integrand.
If a locally integrable function f has the local derivative \(D^ e_{loc}f\), then \(\Delta^{(e,h)}f(x)=\int^{x+h}_{x}D^ e_{loc}f(t)dt\) for each \(h\in R^ q\) and almost all \(x\in R^ q\).

MSC:

46G05 Derivatives of functions in infinite-dimensional spaces
46G10 Vector-valued measures and integration

Citations:

Zbl 0369.28010
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