Differentiation of Banach-space-valued additive processes (Q2759170)

Let \((X,\|\cdot\|)\) be a Banach space and \((\Omega,\Sigma,\mu)\) be a \(\sigma\)-finite measure space. Let \((L,\|\cdot\|_L)\) denote a Banach space of \(X\) valued strongly measurable functions on \((\Omega,\Sigma,\mu)\). \(\|\cdot\|_L\) satisfies NEWLINENEWLINENEWLINEIf \(f,g\in L\) and \(\|f(\omega)\|\leq\|g(\omega)\|\) for almost all \(\omega\in\Omega\), then \(\|f\|_L\leq\|g\|_L\). NEWLINENEWLINENEWLINEIf \(g\) is an \(X\) valued strongly measurable function on \(\Omega\) and if there exists an \(f\in L\) such that \(\|g(\omega)\|\leq\|f(\omega)\|\) for almost all \(\omega\), then \(g\in L\). NEWLINENEWLINENEWLINEIf \(E_n\in\Sigma\), \(E_n\supset E_{n+1}\) for each \(n\geq 1\), and \(\bigcap_{n=1}^\infty E_n=\emptyset\), then for any \(f\in L\) \(\lim_{n\to\infty}\|\chi_{E_n}f\|_L=0\), where \(\chi_E\) is the characteristic function of \(E_n\) \(f,g\in L\), \(\|f(\omega)\|\leq\|g(\omega)\|\) a.e. and \(\|f\|_L=\|g\|_L\), then \(\|f(\omega)=\|g(\omega)\|\) a.e. NEWLINE\[NEWLINE\{f_n\}\subset LNEWLINE\]NEWLINE such that \(\|f_n(\omega)\|\leq\|f_{n+1}(\omega)\|<\infty\), then there exists \(f\in L\) such that \(\|f_n(\omega)\|\leq\|f(\omega)\|\) a.e. Fix \(x_1\in X\) such that \(\|x_1\|=1\). \(L(R)\) is the set of real valued measurable functions \(\widetilde f\) on \((\Omega,\Sigma,\mu)\) such that \(f(\omega)=\widetilde f(\omega)x_1\in L\), and define \(\|\widetilde f\|_{L(R)}=\|f\|_L\). A positive operator \(P\) on \(L(R)\) is called a majorant of a linear operator \(U\) on \(L\) if \(\|Uf(\omega)\|\leq[P\|f(\cdot)\|](\omega)\) a.e.NEWLINENEWLINENEWLINELet \(P_d=\{u=(u_1,\ldots,u_d): u_i>0, 1\leq i\leq d\}\) and \({\mathcal I}_d\) denotes the set of all the bounded intervals in \(P_d\).NEWLINENEWLINENEWLINEConsider a strongly continuous \(d\)-dimensional semigroup \(T=\{T(u): u\in P_d\}\). A set function \(F:{\mathcal I}_d\to L\) is called a bounded process if NEWLINE\[NEWLINE\sup\left\{{\|F(I)_L\over\lambda_d(I)}: I\in{\mathcal I}_d, \lambda_d(I)>0\right\}<\infty,NEWLINE\]NEWLINE wheere \(\lambda_d\) denotes the \(d\)-dimensional Lebesgue measure. It is called additive if NEWLINE\[NEWLINET(u)F(I)=F(u+I)NEWLINE\]NEWLINE NEWLINE\[NEWLINEI_1,\ldots,I_k\in{\mathcal I}_dNEWLINE\]NEWLINE are pairwise disjoint, then \(F(\cup I_i)=\sum F(I_i)\). Let \(D\) be a countable dense subset of positive numbers, and denote by \(q-\lim_{\alpha\to 0}\) means the limit \(\alpha\to 0\) through \(\alpha\in D\). NEWLINENEWLINENEWLINEThe main result in this paper is NEWLINENEWLINENEWLINETheorem. Assume that each \(T(u)\) \(u\in P_d\) has a contraction majorant \(P(u)\) on \(L(R)\) and that the strong limit \(T(0)=\lim_{u\to 0}T(u)\) exists. Then for each \(f\in L\), NEWLINE\[NEWLINET(0)f(\omega) =q-\lim_{\alpha\to 0}\alpha^{-d} \left(\int_{(0,\alpha]^d}T(u)f du\right)(\omega)\quad a.e.NEWLINE\]NEWLINE If \(X\) is assumed to be a reflexive Banach space, then to each bounded additive process \(F\colon{\mathcal I}_d\to L\) there corresponds a function \(f\in L\) with \(T(0)f=f\) for which NEWLINE\[NEWLINEf(\omega)=q-\lim_{\alpha\to 0}\alpha^{-d}F((0,\alpha]^d)(\omega)\quad a.e.NEWLINE\]NEWLINE This paper is an extension for the case that \(L\) is \(X\) valued \(L^p\) space with \(1\leq p<\infty\) by the same author.