Linear functions and duality on the infinite polytorus

  • Ole Fredrik BrevigEmail author


We consider the following question: are there exponents \(2<p<q\) such that the Riesz projection is bounded from \(L^q\) to \(L^p\) on the infinite polytorus? We are unable to answer the question, but our counter-example improves a result of Marzo and Seip by demonstrating that the Riesz projection is unbounded from \(L^\infty \) to \(L^p\) if \(p\ge 3.31138\). A similar result can be extracted for any \(q>2\). Our approach is based on duality arguments and a detailed study of linear functions. Some related results are also presented.

Mathematics Subject Classification

Primary 42B05 Secondary 42B30 46E30 



The author would like to extend his gratitude to A. Bondarenko, H. Hedenmalm, E. Saksman and K. Seip for an interesting discussion which culminated in the material presented in Sect. 3 and to the referee for a helpful suggestion.


  1. 1.
    Borwein, J.M., Nuyens, D., Straub, A., Wan, J.: Some arithmetic properties of short random walk integrals. Ramanujan J. 26(1), 109–132 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brevig, O.F., Ortega-Cerdà, J., Seip, K., Zhao, J.: Contractive inequalities for Hardy spaces. Funct. Approx. Comment. Math. 59(1), 41–56 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brevig, O.F., Perfekt, K.-M.: Failure of Nehari’s theorem for multiplicative Hankel forms in Schatten classes. Stud. Math. 228(2), 101–108 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cole, B.J., Gamelin, T.W.: Representing measures and Hardy spaces for the infinite polydisk algebra. Proc. Lond. Math. Soc. (3) 53(1), 112–142 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duren, P.L.: Theory of \(H^{p}\) Spaces, Pure and Applied Mathematics, vol. 38. Academic, New York (1970)Google Scholar
  6. 6.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 8th edn. Elsevier, Amsterdam (2015)zbMATHGoogle Scholar
  7. 7.
    König, H., Kwapień, S.: Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors. Positivity 5(2), 115–152 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Marzo, J., Seip, K.: \(L^\infty \) to \(L^p\) constants for Riesz projections. Bull. Sci. Math. 135(3), 324–331 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ortega-Cerdà, J., Seip, K.: A lower bound in Nehari’s theorem on the polydisc. J. Anal. Math. 118(1), 339–342 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway

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