Potential Analysis

, Volume 50, Issue 2, pp 221–244 | Cite as

Littlewood-Paley Formulas and Carleson Measures for Weighted Fock Spaces Induced by \(A_{\infty }\)-Type Weights

  • Carme Cascante
  • Joan Fàbrega
  • José A. PeláezEmail author


We obtain Littlewood-Paley formulas for Fock spaces \({\mathcal {F}}_{\beta ,\omega }^{q}\) induced by weights \(\omega \in {A}_{\infty }^{restricted} = \cup _{1 \le p < \infty } {A}_{p}^{restricted}\), where \( {A}_{p}^{restricted} \) is the class of weights such that the Bergman projection Pα, on the classical Fock space \({\mathcal {F}}_{\alpha }^{2}\), is bounded on
$${\mathcal{L}}_{\alpha,\omega}^{p} := \left\{f:\, {\int}_{\mathbb{C}}|f(z)|^{p} e^{-p\frac{\alpha}{2}|z|^{2}}\,\omega(z)dA(z)<\infty \right\}. $$
Using these equivalent norms for \({\mathcal {F}}_{\beta ,\omega }^{q}\) we characterize the Carleson measures for weighted Fock-Sobolev spaces \({\mathcal {F}}_{\beta ,\omega }^{q,n}\).


Fock spaces Littlewood-Paley formula Carleson measures Pointwise multipliers 

Mathematics Subject Classification (2010)

30H20 42B25 46E35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We would like to thank the referee his/her suggestions and observations which improved the exposition of the paper.


  1. 1.
    Aleman, A., Constantin, O.: Spectra of integration operators on weighted Bergman spaces. J. Anal. Math. 109, 199–231 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bekollé, D., Bonami, A.: Inégalités á poids pour le noyau de Bergman, (French). C. R. Acad. Sci. Paris Sr. A-B 286(18), 775–778 (1978)zbMATHGoogle Scholar
  3. 3.
    Cho, H.R., Zhu, K.H.: Fock Sobolev spaces and their Carleson measures. J. Funct. Anal. 263, 2483–2506 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Constantin, O.: A Volterra-type integration operator on Fock spaces. Proc. Amer. Math. Soc. 140, 4247–4257 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Constantin, O., Peláez, J.A.: Integral operators, Embedding theorems and a Littlewood-Paley formula on Fock spaces. J. Geom. Anal. 26(2), 1109–1154 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Duoandikoetxea, J.: Fourier Analysis Graduate Studies in Mathematics, vol. 29. Amer. Math. Soc., Providence (2001)Google Scholar
  7. 7.
    Duoandikoetxea, J., Martin-Reyes, F., Ombrosi, S.: On the \(A_{\infty }\) conditions for general bases. Math. Z. 282(3), 955–972 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Duren, P.L.: Theory of H p Spaces. Academic Press, New York-London (1970). Reprint: Dover, Mineola, New York (2000)Google Scholar
  9. 9.
    Garcia-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)zbMATHGoogle Scholar
  10. 10.
    Isralowitz, J.: Invertible Toeplitz products, weighted norm inequalities, and A p weights. J. Oper. Theory 2, 381–410 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Isralowitz, J., Zhu, K.: Toeplitz operators on the Fock space. Int. Equ. Oper. Th. 66(4), 593–611 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kerman, R.A., Torchinsky, A.: Integral inequalities with weights for the maximal function. Studia Math. 71(3), 277–284 (1981/82)Google Scholar
  13. 13.
    Luecking, D.H.: Embedding theorems for spaces of analytic functions via Khinchine’s inequality. Michigan Math. J. 40, 333–358 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mengestie, T.: Carleson measures for Fock-Sobolev spaces. Complex Anal. Oper. Theory 8(6), 1225–1256 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ortega-Cerdá, J., Seip, K.: Beurling-type density theorems for weighted L p spaces of entire functions. J. Anal. Math. 75, 247–266 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Peláez, J.A., Rättyä, J.: Two weight inequality for Bergman projection. J. Math. Pures Appl. (9) 105(1), 102–130 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  18. 18.
    Tung, J.: Fock Spaces, PhD dissertation, University of Michigan (2005)Google Scholar
  19. 19.
    Zhu, K.: Operator Theory in Function Spaces, 2nd edn. Math Surveys and Monographs, vol. 138. American Mathematical Society, Providence (2007)CrossRefGoogle Scholar
  20. 20.
    Zhu, K.: Analysis on Fock Spaces. Springer, New York (2012)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Carme Cascante
    • 1
  • Joan Fàbrega
    • 1
  • José A. Peláez
    • 2
    Email author
  1. 1.Departament de Matemàtiques i InformàticaUniversitat de BarcelonaBarcelonaSpain
  2. 2.Departamento de Análisis MatemáticoFacultad de CienciasMálagaSpain

Personalised recommendations