Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-trees

  • Nicola ArcozziEmail author
  • Pavel Mozolyako
  • Karl-Mikael Perfekt


In this note we investigate the multi-parameter Potential Theory on the weighted d-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish some basic properties of the respective potentials, capacities and equilibrium measures (in particular in the case of product polynomial weights). We explore multi-parameter Hardy inequality and its trace measures, and discuss some open problems of potential-theoretic and combinatorial nature.

Mathematics Subject Classification

31B15 31B30 32A37 42B25 


Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest.


  1. 1.
    Adams, D.R., Hedberg, L.I.: Function spaces and potential theory. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314. Springer, Berlin (1996). ISBN: 3-540-57060-8Google Scholar
  2. 2.
    Arcozzi, N., Holmes, I., Mozolyako, P., Volberg, A.: Bellman function sitting on a tree (2018). arXiv:1809.03397
  3. 3.
    Arcozzi, N., Holmes, I., Mozolyako, P., Volberg, A.: Bi-parameter embedding and measures with restriction energy condition (2018). arXiv:1811.00978
  4. 4.
    Arcozzi, N., Mozolyako, P., Perfekt, K.-M., Sarfatti, G.: Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc (2018). arXiv:1811.04990
  5. 5.
    Arcozzi, N., Rochberg, R., Sawyer, E.: Carleson measures for analytic Besov spaces (English summary). Rev. Mat. Iberoam. 18(2), 443–510 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Arcozzi, N., Rochberg, R., Sawyer, E.: Carleson measures for the Drury–Arveson Hardy space and other Besov–Sobolev spaces on complex balls (English summary). Adv. Math. 218(4), 1107–1180 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Arcozzi, N., Rochberg, R., Sawyer, E.T., Wick, B.D.: Potential theory on trees, graphs and Ahlfors-regular metric spaces. Potential Anal. 41(2), 317–366 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chang, S.-Y.A.: Carleson measure on the bi-disc. Ann. Math. 109(3), 613–620 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jessen, B., Marcinkiewicz, J., Zygmund, A.: Note on the differentiability of multiple integrals. Fundam. Math. 5, 217–234 (1935)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kerman, R., Sawyer, E.: The trace inequality and eigenvalue estimates for Schrödinger operators. Ann. Inst. Fourier (Grenoble) 36(4), 207–228 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kerman, R., Sawyer, E.: Carleson measures and multipliers of Dirichlet-type spaces. Trans. Am. Math. Soc. 309(1), 87–98 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sawyer, E.: Weighted inequalities for the two-dimensional Hardy operator. Stud. Math. 82(1), 1–16 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Stegenga, D.A.: Multipliers of the Dirichlet space. Ill. J. Math. 24(1), 113–139 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Talenti, G.: Osservazioni sopra una classe di disuguaglianze. (Italian) Rend. Sem. Mat. Fis. Milano 39, 171–185 (1969)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Università di BolognaBolognaItaly
  2. 2.Department of Mathematics and StatisticsUniversity of ReadingReadingUnited Kingdom

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