Carleson measures on Dirichlet-type spaces

Chacón, Gerardo

doi:10.1090/S0002-9939-2011-10823-2

Carleson measures on Dirichlet-type spaces
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by Gerardo R. Chacón PDF

Proc. Amer. Math. Soc. 139 (2011), 1605-1615 Request permission

Abstract:

We show that a maximal inequality holds for the non-tangential maximal operator on Dirichlet spaces with harmonic weights on the open unit disc. We then investigate two notions of Carleson measures on these spaces and use the maximal inequality to give characterizations of the Carleson measures in terms of an associated capacity.

References

Jim Agler, A disconjugacy theorem for Toeplitz operators, Amer. J. Math. 112 (1990), no. 1, 1–14. MR 1037599, DOI 10.2307/2374849
A. Aleman, The Multiplication Operators on Hilbert Spaces of Analytic Functions, Habilitationsschrift, Fernuniversität Hagen (1993).
Rick Chartrand, Toeplitz operators on Dirichlet-type spaces, J. Operator Theory 48 (2002), no. 1, 3–13. MR 1926041
Rick Chartrand, Multipliers and Carleson measures for $D(\mu )$, Integral Equations Operator Theory 45 (2003), no. 3, 309–318. MR 1965337, DOI 10.1007/s000200300007
P. Duren, Theory of $H^p$ Spaces, Dover Publications (2000).
Stefan Richter, A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328 (1991), no. 1, 325–349. MR 1013337, DOI 10.1090/S0002-9947-1991-1013337-1
Stefan Richter and Carl Sundberg, A formula for the local Dirichlet integral, Michigan Math. J. 38 (1991), no. 3, 355–379. MR 1116495, DOI 10.1307/mmj/1029004388
Stefan Richter and Carl Sundberg, Multipliers and invariant subspaces in the Dirichlet space, J. Operator Theory 28 (1992), no. 1, 167–186. MR 1259923
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Donald Sarason, Local Dirichlet spaces as de Branges-Rovnyak spaces, Proc. Amer. Math. Soc. 125 (1997), no. 7, 2133–2139. MR 1396993, DOI 10.1090/S0002-9939-97-03896-3
Donald Sarason, Harmonically weighted Dirichlet spaces associated with finitely atomic measures, Integral Equations Operator Theory 31 (1998), no. 2, 186–213. MR 1623461, DOI 10.1007/BF01214250
António Serra, Interpolating sequences in harmonically weighted Dirichlet spaces, Proc. Amer. Math. Soc. 131 (2003), no. 9, 2809–2817. MR 1974338, DOI 10.1090/S0002-9939-03-06846-1
Serguei Shimorin, Complete Nevanlinna-Pick property of Dirichlet-type spaces, J. Funct. Anal. 191 (2002), no. 2, 276–296. MR 1911187, DOI 10.1006/jfan.2001.3871
S. M. Shimorin, Reproducing kernels and extremal functions in Dirichlet-type spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 255 (1998), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 26, 198–220, 254 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 107 (2001), no. 4, 4108–4124. MR 1692900, DOI 10.1023/A:1012453003423
David A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), no. 1, 113–139. MR 550655
Zhijian Wu, Carleson measures and multipliers for Dirichlet spaces, J. Funct. Anal. 169 (1999), no. 1, 148–163. MR 1726750, DOI 10.1006/jfan.1999.3490
Kehe Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, Providence, RI, 2007. MR 2311536, DOI 10.1090/surv/138