Advertisement

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
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.

References

  1. 1.
    Benea, C., Muscalu, C.: Multiple vector valued inequalities via the helicoidal method. Anal. PDE 9, 1931–1988 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benea, C., Muscalu, C.: Quasi-Banach valued inequalities via the helicoidal method. Preprint, arXiv:1609.01090
  3. 3.
    Bergh, J., L\(\ddot{\text{o}}\)fstr\(\ddot{\text{ o }}\)m, J.: Interpolation Spaces. An Introduction. Grund- lehren der mathematischen Wissenschaften, vol. 223. Springer, Berlin (1976)Google Scholar
  4. 4.
    Boas, H.: Majorant series. J. Korean Math. Soc. 37(2), 321–337 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bombal, F., Pérez-García, D., Villanueva, I.: Multilinear extensions of Grothendieck’s theorem. Quart. J. Math. 55(4), 441–450 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. 32(3), 600–622 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Culiuc, A., Di Plinio, F., Ou, Y.: Domination of multilinear singular integrals by positive sparse forms. Preprint, arXiv:1603.05317
  8. 8.
    Cruz-Uribe, D., Martell, J.M., Pérez, C.: Extrapolation from \(A_{\infty }\) weights and applications. J. Funct. Anal. 213(2), 412–439 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Curbera, G., García-Cuerva, J., Martell, J.M., Pérez, C.: Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals. Adv. Math. 203, 256–318 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Davie, A.M.: Quotient algebras of uniform algebras. J. London Math. Soc. 7(2), 31–40 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland Publishing Co., Amsterdam (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    Defant, A., Junge, M.: Best constants and asymptotics of Marcinkiewicz-Zygmund inequalities. Studia Math. 125(3), 271–287 (1997)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Defant, A., Mastyło, M.: Interpolation of Fremlin tensor products and Schur factorization of matrices. J. Funct. Anal. 262, 3981–3999 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Defant, A., Sevilla-Peris, P.: A new multilinear insight on littlewood’s 4/3-inequality. J. Funct. Anal. 256(5), 1642–1664 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Di Plinio, F., Ou, Y.: Banach-valued multilinear singular integrals. Preprint, arXiv:1506.05827. To appear in Indiana Univ. Math. J
  16. 16.
    García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, vol. 116. North-Holland Publishing Co., Amsterdam (1985)CrossRefzbMATHGoogle Scholar
  17. 17.
    Gasch, J., Maligranda, L.: On vector-valued inequalities of Marcinkiewicz-Zygmund, Herz and Krivine type. Math. Nachr. 167, 95–129 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grafakos, L., Martell, J.M.: Extrapolation of weighted norm inequalities for multivariable operators and applications. J. Geom. Anal. 14(1), 19–46 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grafakos, L., Torres, R.H.: Multilinear Calderón-Zygmund theory. Adv. Math. 165(1), 124–164 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Grafakos, L., Torres, R.H.: Maximal operator and weighted norm inequalities for multilinear singular integrals. Indiana Univ. Math. J. 51(5), 1261–1276 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Herz, C.: Theory of \(p\)-spaces with an application to convolution operators. Trans. Am. Math. Soc. 154, 69–82 (1971)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kaijser, S.: Some results in the metric theory of tensor products. Studia Math. 63(2), 157–170 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lerner, A., Ombrosi, S., Pérez, C., Torres, R., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Marcinkiewicz, J., Zygmund, A.: Quelques inégalités pour les opérations linéaires. Fund. Math. 32, 113–121 (1939)zbMATHGoogle Scholar
  25. 25.
    Pérez, C., Torres, R.H.: Sharp maximal function estimates for multilinear singular integrals. Contemp. Math. 320, 323–333 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pietsch, A.: Operator Ideals. North-Holland Publishing Co., Amsterdam (1980)zbMATHGoogle Scholar
  27. 27.
    Quefflec, H.H.: Bohr’s vision of ordinary Dirichlet series; old and new results. J. Anal. 3, 43–60 (1995)MathSciNetGoogle Scholar

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

Personalised recommendations