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Factorization of p-dominated polynomials through \({L}^p\)-spaces. (English) Zbl 1305.46042
Let \(E\) and \(F\) be Banach spaces over \({\mathbb K}\) (\({\mathbb K}={\mathbb R}\) or \({\mathbb C}\)) and \(n\) be a positive integer. A (continuous) mapping \(P: E\to F\) is said to be an \(n\)-homogeneous polynomial if there is a (continuous) symmetric \(n\)-linear mapping \(\check{P}: E\times E\times \dots\times E\to F\) such that \(P(x)=\check{P}(x,\dots,x)\) for all \(x\) in \(E\). For \(p\geq 1\), the authors introduce the class of factorable \(p\)-dominated polynomials, as those polynomials, \(P\), for which there is \(C>0\) such that for every set of vectors, \((x_j^i)\), \(1\leq i\leq k\), \(1\leq j\leq m\), and every set of scalars, \((\lambda_j^i)\), \(1\leq i\leq k\), \(1\leq j\leq m\), \[ \left(\sum_{j=1}^m\left(\sum_{i=1}^k\lambda_j^i\|P(x_j^i)\|\right)^p\right)^ {1/p}\leq C\sup_{\|\phi\|\leq 1}\left(\sum_{j=1}^m\left(\sum_{i=1}^k\lambda_j^i \phi(P(x_j^i))\right)^p\right)^{1/p}. \] It is shown that an \(n\)-homogeneous polynomial \(P: E\to F\) is factorable \(p\)-dominated if and only if there are a regular Borel measure \(\mu\) in \(B_{E'}\) and \(C>0\) such that \[ \left\|\sum_{i=1}^k\lambda_jP(x_j)\right\|\leq C\left(\int_{B_{E'}}\left| \sum_{i=1}^k\lambda_j\phi(x_j)^n\right|^p\, d\mu(\phi)\right)^{1/p} \] for each set of vectors, \((x_j)\), \(1\leq j\leq k\), and each set of scalars, \(( \lambda_j)\), \(1\leq j\leq k\). Using this result, it is shown that a polynomial, \(P\), is factorable \(p\)-dominated if and only if there is a regular Borel measure \(\mu\) on \(B_{E'}\) such that \(P\) has a canonical factorisation through a certain closed subspace of \(L^p(\mu)\).

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
46M05 Tensor products in functional analysis
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