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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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