Carleson measures and reproducing kernel thesis in Dirichlet-type spaces

Chacón, G.; Fricain, E.; Shabankhah, M.

doi:10.1090/S1061-0022-2013-01269-6

Carleson measures and reproducing kernel thesis in Dirichlet-type spaces
HTML articles powered by AMS MathViewer

by G. R. Chacón, E. Fricain and M. Shabankhah

St. Petersburg Math. J. 24 (2013), 847-861

DOI: https://doi.org/10.1090/S1061-0022-2013-01269-6

Published electronically: September 23, 2013

PDF | Request permission

Abstract:

In the paper, a generalization of a Richter and Sundberg representation theorem is employed to obtain a new characterization of Carleson measures for the Dirichlet-type space $\mathcal {D}(\mu )$ when $\mu$ is a finite sum of point masses. A reproducing kernel thesis result is also established in this case.

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, Fernuniv. Hagen, 1993.
Nicola Arcozzi, Daniel Blasi, and Jordi Pau, Interpolating sequences on analytic Besov type spaces, Indiana Univ. Math. J. 58 (2009), no. 3, 1281–1318. MR 2541368, DOI 10.1512/iumj.2009.58.3589
Anton Baranov, Emmanuel Fricain, and Javad Mashreghi, Weighted norm inequalities for de Branges-Rovnyak spaces and their applications, Amer. J. Math. 132 (2010), no. 1, 125–155. MR 2597508, DOI 10.1353/ajm.0.0094
Oscar Blasco, Operators on weighted Bergman spaces $(0<p\leq 1)$ and applications, Duke Math. J. 66 (1992), no. 3, 443–467. MR 1167102, DOI 10.1215/S0012-7094-92-06614-2
Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
Gerardo R. Chacón, Carleson measures on Dirichlet-type spaces, Proc. Amer. Math. Soc. 139 (2011), no. 5, 1605–1615. MR 2763750, DOI 10.1090/S0002-9939-2011-10823-2
Gerardo R. Chacón, Interpolating sequences in harmonically weighted Dirichlet spaces, Integral Equations Operator Theory 69 (2011), no. 1, 73–85. MR 2749449, DOI 10.1007/s00020-010-1820-9
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
Nicolas Chevrot, Dominique Guillot, and Thomas Ransford, De Branges-Rovnyak spaces and Dirichlet spaces, J. Funct. Anal. 259 (2010), no. 9, 2366–2383. MR 2674117, DOI 10.1016/j.jfa.2010.07.004
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
John B. Garnett, Bounded analytic functions, 1st ed., Graduate Texts in Mathematics, vol. 236, Springer, New York, 2007. MR 2261424
V. P. Havin and N. K. Nikolski, Stanislav Aleksandrovich Vinogradov, his life and mathematics, Complex analysis, operators, and related topics, Oper. Theory Adv. Appl., vol. 113, Birkhäuser, Basel, 2000, pp. 1–18. MR 1771747
Peter W. Jones, Carleson measures and the Fefferman-Stein decomposition of $\textrm {BMO}(\textbf {R})$, Ann. of Math. (2) 111 (1980), no. 1, 197–208. MR 558401, DOI 10.2307/1971197
D. E. Marshall and C. Sundberg, Interpolating sequences for the multipliers of the Dirichlet space, Manuscript, 1994; Avalaible at http://www.math.washington.edu/marshall/preprints/preprints.html.
Scott McCullough and Tavan T. Trent, Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal. 178 (2000), no. 1, 226–249. MR 1800795, DOI 10.1006/jfan.2000.3664
Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. MR 1892647
Stefan Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988), 205–220. MR 936999, DOI 10.1515/crll.1988.386.205
Stefan Richter and Carl Sundberg, Multipliers and invariant subspaces in the Dirichlet space, J. Operator Theory 28 (1992), no. 1, 167–186. MR 1259923
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
Donald Sarason, Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR 1289670
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
Kristian Seip, Interpolation and sampling in spaces of analytic functions, University Lecture Series, vol. 33, American Mathematical Society, Providence, RI, 2004. MR 2040080, DOI 10.1090/ulect/033
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
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
David A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), no. 1, 113–139. MR 550655