Revista Matemática Iberoamericana

Carleson measures for analytic Besov spaces

Nicola Arcozzi, Richard Rochberg, and Eric Sawyer

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Abstract

We characterize Carleson measures for the analytic Besov spaces. The problem is first reduced to a discrete question involving measures on trees which is then solved. Applications are given to multipliers for the Besov spaces and to the determination of interpolating sequences. The discrete theorem is also applied to analysis of function space on trees.

Article information

Source
Rev. Mat. Iberoamericana, Volume 18, Number 2 (2002), 443-510.

Dates
First available in Project Euclid: 28 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1051544245

Mathematical Reviews number (MathSciNet)
MR1949836

Zentralblatt MATH identifier
1059.30051

Subjects
Primary: 30H05: Bounded analytic functions 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 46E39: Sobolev (and similar kinds of) spaces of functions of discrete variables

Keywords
Carleson measures discrete Sobolev inequalities

Citation

Arcozzi, Nicola; Rochberg, Richard; Sawyer, Eric. Carleson measures for analytic Besov spaces. Rev. Mat. Iberoamericana 18 (2002), no. 2, 443--510. https://projecteuclid.org/euclid.rmi/1051544245


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References

  • Adams, D. R.: A trace inequality for generalized potentials. Studia Math. 48 (1973), 99-105.
  • Arcozzi, N.: Carleson measures for analytic Besov spaces: the upper triangle case. Preprint, 2000.
  • Arcozzi, N., Rochberg, R.: Topics in dyadic Dirichlet spaces. Preprint, 2000.
  • Bekollé, D.: Inégalités à poids puor le projecteur de Bergman dans la boule unité de $\mathbbC^n$. Studia Math. 71(1982), 305-323.
  • Bekollé, D., Bonami, A,: Inégalités à poids pour le noyau de Bergman. C. R. Acad. Sci. Paris Sér. A-B 286 (1978), A775-A778.
  • Böe, B.: Interpolating Sequences for Besov Spaces. Preprint, 2000.
  • Carleson, L.: Interpolation by bounded analytic functions and the corona problem. Ann. of Math. 76 (1962), 547-559.
  • Cartier, P.: Fonctions harmoniques sur un arbre. Symposia Math. IX, Academic Press, (1972), 203-270.
  • Cascante, C., Ortega, J. M.: Carleson measures on spaces of Hardy-Sobolev type. Can. J. Math. 47 (1995), 1177-1200.
  • Cascante, C., Ortega, J. M.: On $q$-Carleson measures for spaces of $\mathcalM$-harmonic functions. Can. J. Math. 49 (1997), 653-674.
  • Coifman, R., Rochberg, R.: Representation theorems for holomorphic and harmonic functions in $L^p$. Astérisque 77 (1980), 11-65.
  • Di Biase, F.: Fatou type theorems: maximal functions and approach regions. Birkhäuser, 1997.
  • Durrett, R.: Brownian motion and martingales in analysis. Wadsworth Inc., 1984.
  • Evans, W. D., Harris, D. J., Pick, L.: Weighted Hardy and Poincaré inequalities on trees. J. London Math. Soc. 52 (1995), 121-136.
  • Kerman, R., Sawyer, E.: Carleson measures and multipliers of Dirichlet-type spaces. Trans. Amer. Math. Soc. 309 (1988), 87-98.
  • Korányi, A., Picardello, M., Taibleson, M.: Hardy spaces on nonhomogeneous trees. With an appendix by Picardello and Wolfgang Woess. Symposia Mathematica XXIX (Cortona, 1984), 205--265, Academic Press, 1987.
  • Luecking, D.: Representation and duality in weighted spaces of analytic functions. Indiana Univ. J. 34 (1985), 319-336.
  • Leucking, D.: Embedding theorems for spaces of analytic functions via Kinchine's inequality. Michigan Math. J. 40 (1993), 333-358.
  • Marshall, D. E., Sundberg, C.: Interpolating sequences for the multipliers of the Dirichlet space. Preprint.
  • Muckenhoupt, B.: Weighted norm inequalities for classical operators. Proc. Symp. in Pure Math. XXXV, Amer. Math. Soc. (1979), 69-84.
  • Rochberg, R.: Interpolating by functions in Bergman spaces. Mich. Math. J. 29 (1982), 229-236.
  • Rochberg, R., Wu, Z.: A new characterization of Dirichlet type spaces and applications. Illinois J. Math. 37 (1993), 101-122.
  • Sawyer, E.: Weighted inequalities for the two-dimensional Hardy operator. Studia Math. LXXXII (1985), 1-15.
  • Sawyer, E.: A two weight weak type inequality for fractional integrals. Trans. Amer. Math. Soc. 281 (1984), 339-345.
  • Sawyer, E., Wheeden, R. L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Amer. J. Math. 114 (1992), 813-874.
  • Soardi, P. M.: Potential theory on infinite networks. Lecture Notes in Math. 1590, Springer, 1994.
  • Stegenga, D. A.: Multipliers of the Dirichlet space. Illinois J. Math. 24 (1980), 113-139.
  • Stein, E.: Singular integrals and differentiability properties of functions. Princeton Univesity Press, 1970.
  • Taibleson, M.: Hardy spaces of harmonic functions on homogeneous isotropic trees. Math. Nachr. 133 (1987), 273-288.
  • Verbitsky, I. È.: Multipliers in spaces with ``fractional" norms, and inner functions. (Russian) Sibirsk. Mat. Zh. 26 (1985), 51-72, 221. Translated in Siber. J. Math. 26 (1985), 198-216.
  • Wang, J.: Ph. D. Thesis. Washington University in St. Louis, 1995.
  • Wu, Z.: Carleson measures and multipliers for Dirichlet spaces. J. Funct. Anal. 169 (1999), 148-163.
  • Zhu, K.: Operator theory on function spaces. Marcel Dekker Inc., 1990.