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Summation of coefficients of polynomials on \(\ell_{p}\) spaces. (English) Zbl 1378.46032
The main result of this paper is:
Theorem 2.1. Let \(Y\) be a cotype \(q\) space and \(v:X\rightarrow Y\) an \(\left( r,1\right) \)-summing operator (with \(1\leq r\leq q\)). For \(1\leq p_{1},\dots,p_{m}\leq \infty \) with \(\frac{1}{p_{1}}+\cdot \cdot \cdot +\frac{1 }{p_{m}}<\frac{1}{r}\), we define \(\frac{1}{\lambda }=\frac{1}{r}-\left( \frac{ 1}{p_{1}}+\cdot \cdot \cdot +\frac{1}{p_{m}}\right) \) and \(\frac{1}{\mu }= \frac{1}{m\lambda }+\frac{m-1}{mq}\). Then there exists \(C>0\) such that, for every \(m\)-linear \(T:l_{p_{1}}\times \cdot \cdot \cdot \times l_{p_{m}}\rightarrow X\) with coefficients \(\left( a_{i_{1},\dots,i_{m}}\right) \), we have:
(i) If \(\lambda \geq q\), then \(\left( \sum\limits_{i_{1},\dots,i_{m}=1}^{\infty }\left\| va_{i_{1},\dots,i_{m}}\right\| ^{\lambda }\right) ^{\frac{1}{\lambda }}\leq C\left\| T\right\| \).
(ii) If \(\lambda <q\), then \(\left( \sum\limits_{i_{1},\dots,i_{m}=1}^{\infty }\left\| va_{i_{1},\dots,i_{m}}\right\| ^{\mu }\right) ^{\frac{1}{\mu } }\leq C\left\| T\right\| \).
The proof is very ingenious and uses an inequality for mixed sums and complex interpolation. As a consequence we mention:
Proposition 4.1. Let \(1\leq p_{1},\dots,p_{m}\leq \infty \) such that \(\frac{1}{ p_{1}}+\cdot \cdot \cdot +\frac{1}{p_{m}}<1\). Consider the exponents \(\frac{1 }{\lambda }=1-\left( \frac{1}{p_{1}}+\cdot \cdot \cdot +\frac{1}{p_{m}} \right) \) and \(\frac{1}{\mu }=\frac{1}{m\lambda }+\frac{m-1}{2m}\). Then there exists \(C>0\) such that, for every \(m\)-linear \(T:l_{p_{1}}\times \cdot \cdot \cdot \times l_{p_{m}}\rightarrow \mathbb{C}\) with coefficients \( \left( a_{i_{1},\dots,i_{m}}\right) \), we have:
(i) If \(\frac{1}{2}\leq \frac{1}{p_{1}}+\cdot \cdot \cdot +\frac{1}{p_{m}}<1\), then \(\left( \sum\limits_{i_{1},\dots,i_{m}=1}^{\infty }\left| a_{i_{1},\dots,i_{m}}\right| ^{\lambda }\right) ^{\frac{1}{\lambda }}\leq C\left\| T\right\| \).
(ii) If \(0\leq \frac{1}{p_{1}}+\cdot \cdot \cdot +\frac{1}{p_{m}}<\frac{1}{2} \), then \(\left( \sum\limits_{i_{1},\dots,i_{m}=1}^{\infty }\left| a_{i_{1},\dots,i_{m}}\right| ^{\mu }\right) ^{\frac{1}{\mu }}\leq C\left\| T\right\| \).
Moreover, the exponents are optimal.
This result extends an older result of G. H. Hardy and J. E. Littlewood [Q. J. Math., Oxf. Ser. 5, 241–254 (1934; Zbl 0010.36101)].

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
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