Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 51–85 | Cite as

Multilinear Marcinkiewicz-Zygmund Inequalities

  • Daniel Carando
  • Martin Mazzitelli
  • Sheldy OmbrosiEmail author


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
$$\begin{aligned} \left\| \left( \sum _{k_1, \dots , k_m} |T(f_{k_1}^1, \dots , f_{k_m}^m)|^r\right) ^{1/r} \right\| _{L^p(\nu )} \le C \Vert T\Vert \prod _{i=1}^m \left\| \left( \sum _{k_i=1}^{n_i} |f_{k_i}^i|^r\right) ^{1/r} \right\| _{L^{q_i}(\mu _i)}. \end{aligned}$$
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.


Vector-valued inequalities Multilinear operators Calderón-Zygmund operators 

Mathematics Subject Classification

Primary 47H60 47A63 Secondary 42B20 



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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Copyright information

© Springer Science+Business Media, LLC 2017

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

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