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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
##### Keywords:
dominated mapping; Ky Fan’s lemma
Full Text:
##### References:
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