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On summability of nonlinear mappings: a new approach. (English) Zbl 1245.46034
Given Banach spaces \(X_1,\dots, X_n\) and \(Y\) over the same field \({\mathbb K}\) and \(0<p_1,\dots,p_n<\infty\), a mapping \(f: X_1\times\cdots\times X_n\to Y\) is said to be \((p_1,\dots,p_n)\)-dominated at \((a_1,\dots,a_n)\in X_1\times \cdots\times X_n\) if there are \(C>0\) and Borel probabilities \(\mu_k\) on \(B_{X_k'}\) for \(k=1,\dots, n\) such that \[ \left\|f(a_1+x^{(1)},\dots,a_n+x^{(n)})-f(a_1,\dots,a_n)\right\|\leq C\prod_{k=1 }^n\left(\int_{B_{X_k'}}|\phi(x^{(k)})|^{p_k}\,d\mu_k\right)^{1/p_k} \] for all \(x^{(j)}\in X_j\), \(j=1,\dots, n\). Using Ky Fan’s lemma the authors show that \(f: X_1\times\cdots\times X_n\to Y\) is \((p_1,\dots,p_n)\)-dominated at \((a_1,\dots,a_n)\) if and only if there is \(C>0\) such that \[ \begin{aligned} \left(\sum_{j=1}^m\left(\left|b_j^{(1)}\cdots \;b_j^{(n)} \right| \cdot \left\| f\left(a_1+x^{(1)}_j, \dots,a_n+x^{(n)}_j\right)-f\left(a_1,\dots,a_n\right) \right\|\right)^p\right)^{1/p}\\ \leq C\prod_{k=1}^n\sup_{\phi\in B_{X_j'}}\left(\sum_{j=1}^m\left(\left|b_j^{(k)}\right|\cdot \left|\phi\left(x_j^{(k)}\right)\right|\right) ^{p_k}\right)^{1/p_k}\end{aligned} \] for all positive integers \(m\) and all \((x_j^{(k)},b_j^{(k)})\) in \(X_k\times {\mathbb K}\), \(1\leq j\leq m\), \(1\leq k\leq n\) where \(1/p=\sum_{j=1}^n1/p_j\).

MSC:
46G20 Infinite-dimensional holomorphy
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