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On Nemyckii operators and integral representation of local functionals. (English) Zbl 0536.47027
In this paper, given a measure space \((\Omega,\mathfrak F,\mu)\) and a functional \(F(u,B)\) defined for \(\mu\)-measurable functions \(u\) and \(\mu\)-measurable sets \(B\), we study the problem of representing \(F\) in the integral form \[ F(u,B)=\int_{B}f(x,u(x))\,d\mu(x) \] for a suitable integrand \(f\). The main result of the paper is the following.
Suppose that \(F\) satisfies the following conditions \((1\leq p<+\infty)\):
1) \(F(\cdot,\Omega)\) is lower semicontinuous with respect to the strong \(L^ p(\Omega,\mathbb R^ n)\) topology;
2) \(F\) is local on \({\mathfrak F}\) (i.e. \(u|_ B=v|_ B \mu\)-a.e. \(\Rightarrow F(u,B)=F(v,B));\)
3) \(F(u,\cdot)\) is finitely additive on \(\mathfrak F\) for every \(u\in L^ p(\Omega;{\mathbb R}^ n);\)
4) \(F(0,B)=0\) for every \(B\in {\mathfrak F}\).
Then we have \(F(u,B)=\int_{B}f(x,u(x))\,d\mu(x),\) where \(f(x,s)\) is a measurable function, lower semicontinuous in \(s\), and such that \(f(x,s)\geq -[a(x)+b| s|^ p]\) \((a\in L^ 1(\Omega)\), \(b\geq 0).\)
In Section 3 we show by a counterexample that hypothesis 1) cannot be dropped.
An interesting consequence of our result is the following.
Let \(T:L^ p(\Omega,{\mathbb R}^ n)\to L^ q(\Omega,{\mathbb R}^ m)\) be a continuous mapping which is locally defined (i.e. \(u=v\) \(\mu\)-a.e. on \(B\Rightarrow Tu=Tv\) \(\mu\)-a.e. on \(B\)). Then \(T\) is a Nemyckii operator, that is \((Tu)(x)=f(x,u(x)),\) where \(f\) is a Carathéodory function such that \(| f(x,s)| \leq a(x)+b| s|^{p/q}\) \((a\in L^ q(\Omega), b\geq 0)\).

47B38 Linear operators on function spaces (general)
46G10 Vector-valued measures and integration
47Gxx Integral, integro-differential, and pseudodifferential operators