Abstract
We examine versions of the classical inequalities of Paley and Zygmund for functions of several variables. A sharp multiplier inclusion theorem and variants on the real line are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Note that since \(\Vert H_{{\mathbb {T}}} (f) \Vert _{L^1({\mathbb {T}}) } \lesssim 1 + \int _{{\mathbb {T}}} |f| \log (1+|f|)\), where \(H_{{\mathbb {T}}}\) is the periodic Hilbert transform, one deduces that \(L \log L ({\mathbb {T}}) \subset H^1 ({\mathbb {T}})\) and hence, one trivially has \(\mathcal {M}_{H^1 ({\mathbb {T}}) \rightarrow L^2 ({\mathbb {T}})} \subset \mathcal {M}_{L \log L ({\mathbb {T}}) \rightarrow L^2 ({\mathbb {T}})}\).
References
Blasco, O., Pelczynski, A.: Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces. Trans. Am. Math. Soc. 323(1), 335–367 (1991)
Blei, R.: Analysis in Integer and Fractional Dimensions. Cambridge Studies in Advanced Mathematics, 71. Cambridge University Press, Cambridge (2001)
Bonami, A.: Étude des coefficients de Fourier des fonctions de \(L^p (G)\). Ann. Inst. Fourie 20(2), 335–402 (1970)
Bourgain, J.: Sidon sets and Riesz products. Ann. Inst. Fourier 35(1), 137–148 (1985)
Bourgain, J., Lewko, M.: Sidonicity and variants of Kaczmarz’s problem. Ann. Inst. Fourier 67(3), 1321–1352 (2017)
Duren, P., Shields, A.: Coefficient multipliers of \(H^p\) and \(B^p\) spaces. Pac. J. Math. 32, 69–78 (1970)
Fournier, J.J.F.: On a theorem of Paley and the Littlewood conjecture. Ark. Mat. 17, 199–216 (1979)
Graham, C.C., Hare, K.E.: Interpolation and Sidon Sets for Compact Groups. Springer, New York (2013)
Hardy, G.H., Littlewood, J.E.: Notes on the theory of series (XX): generalizations of a theorem of Paley. Q. J. Math. Oxf. Ser. 8, 161–171 (1937)
Ingham, A.E.: Note on a certain power series. Ann. Math. 2(31), 241–250 (1930)
Kahane, J.P.: Sur les fonctions moyenne-périodiques bornées. Ann. Inst. Fourier 7, 293–314 (1957)
Lust-Piquard, F., Pisier, G.: Non commutative Khintchine and Paley inequalities. Ark. Mat. 29(2), 241–260 (1991)
Marcus, M.B., Pisier, G.: Random Fourier Series with Applications to Harmonic Analysis. Annals of Mathematics Studies, 101, p. 150. Princeton University Press and University of Tokyo Press, Princeton (1981)
McCall, J.D.: A multiplier theorem for Fourier transforms. Trans. Am. Math. Soc. 189, 359–369 (1974)
Oberlin, D.M.: Two multiplier theorems for \(H^1(U^2)\). Proc. Edinb. Math. Soc. II. Ser. 22, 43–47 (1979)
Paley, R.E.A.C.: On some problems connected with Weierstrass’s non-differentiable function. Proc. Lond. Math. Soc. 2(31), 301–328 (1930)
Paley, R.E.A.C.: A note on power series. J. Lond. Math. Soc. 7, 122–130 (1932)
Paley, R.E.A.C.: On the lacunary coefficients of power series. Ann. Math. 2(34), 615–616 (1933)
Pisier, G.: Ensembles de Sidon et processus gaussiens. C. R. Acad. Sci., Paris, Sér. A 286, 671–674 (1978)
Pisier, G.: Sur l’espace de Banach des séries de Fourier aléatoires presque sûrement continues. Sem. Geom. des Espaces de Banach, Ec. Polytech. Cent. Math., 1977–1978, Exposes No.12, 13 (1978)
Rider, D.: Randomly continuous functions and Sidon sets. Duke Math. J. 42, 759–764 (1975)
Rudin, W.: Remarks on a theorem of Paley. J. Lond. Math. Soc. 32, 307–311 (1957)
Rudin, W.: Trigonometric series with gaps. J. Math. Mech. 9, 203–227 (1960)
Rudin, W.: Fourier Analysis on Groups. Wiley, New York (1990)
Sidon, S.: Verallgemeinerung eines Satzes über die absolute Konvergenz von Fourierreihen mit Lücken. Math. Ann. 97, 675–676 (1927)
Stein, E.M.: \(H^p\)-classes, multiplicateurs et fonctions de Littlewood-Paley. Applications de résultats anterieurs. C. R. Acad. Sci. Paris Sér. A 263, 780–781 (1966)
Tao, T., Wright, J.: Endpoint multiplier theorems of Marcinkiewicz type. Rev. Mat. Iberoam. 17(3), 521–558 (2001)
Yudin, V.A.: Multidimensional versions of Paley’s inequality. Math. Notes 70(6), 860–865 (2001); translation from Mat. Zametki 70(6), 941–947 (2001)
Zygmund, A.: On the convergence of lacunary trigonometric series. Fundam. Math. 16, 90–107 (1930)
Zygmund, A.: Trigonometric series. Vol. I and II. 2nd reprint of the 2nd ed, 3rd edn. Cambridge University Press, Cambridge (2002)
Acknowledgements
This work was conducted during the author’s Ph.D. studies at the University of Edinburgh under the supervision of Professor Jim Wright. The author would like to thank and acknowledge his Ph.D. supervisor for his continuous help, support and guidance on this work and for all his useful comments and suggestions that improved the presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yura Lyubarskii.
Rights and permissions
About this article
Cite this article
Bakas, O. Variants of the Inequalities of Paley and Zygmund. J Fourier Anal Appl 25, 1113–1133 (2019). https://doi.org/10.1007/s00041-018-9605-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-018-9605-7