Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 51–85

# Multilinear Marcinkiewicz-Zygmund Inequalities

• Daniel Carando
• Martin Mazzitelli
• Sheldy Ombrosi
Article

## Abstract

We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on $$\ell ^r$$-valued extensions of linear operators. We show that for certain $$1 \le p, q_1, \dots , q_m, r \le \infty$$, there is a constant $$C\ge 0$$ such that for every bounded multilinear operator $$T:L^{q_1}(\mu _1) \times \cdots \times L^{q_m}(\mu _m) \rightarrow L^p(\nu )$$ and functions $$\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(\mu _1), \dots , \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(\mu _m)$$, the following inequality holds
\begin{aligned} \left\| \left( \sum _{k_1, \dots , k_m} |T(f_{k_1}^1, \dots , f_{k_m}^m)|^r\right) ^{1/r} \right\| _{L^p(\nu )} \le C \Vert T\Vert \prod _{i=1}^m \left\| \left( \sum _{k_i=1}^{n_i} |f_{k_i}^i|^r\right) ^{1/r} \right\| _{L^{q_i}(\mu _i)}. \end{aligned}
(1)
In some cases we also calculate the best constant $$C\ge 0$$ satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.

## Keywords

Vector-valued inequalities Multilinear operators Calderón-Zygmund operators

## Mathematics Subject Classification

Primary 47H60 47A63 Secondary 42B20

## Notes

### Acknowledgements

The authors wish to thank C. Muscalu for his valuable comments regarding this work. This project was supported in part by CONICET PIP 11220130100329CO, ANPCyT PICT 2015-2299 and UBACyT 20020130100474. The second author has a postdoctoral position from CONICET.

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## Authors and Affiliations

• Daniel Carando
• 1
• 2
• Martin Mazzitelli
• 3
• 4
• Sheldy Ombrosi
• 5
• 6
Email author
1. 1.Departamento de Matemática - Pab I, Facultad de Cs. Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
2. 2.IMAS-CONICETBuenos AiresArgentina
3. 3.Instituto BalseiroUniversidad Nacional de Cuyo - C.N.E.A.Buenos AiresArgentina
4. 4.Departamento de Matemática, Centro Regional Universitario BarilocheUniversidad Nacional del ComahueSan Carlos de BarilocheArgentina
5. 5.Departamento de MatemáticaUniversidad Nacional del SurBahía BlancaArgentina
6. 6.INMABB-CONICETBahía BlancaArgentina