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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Multipliers and integration operators on Dirichlet spaces


Authors: Petros Galanopoulos, Daniel Girela and José Ángel Peláez
Journal: Trans. Amer. Math. Soc. 363 (2011), 1855-1886
MSC (2010): Primary 30H20, 30H99, 47B38
DOI: https://doi.org/10.1090/S0002-9947-2010-05137-2
Published electronically: November 16, 2010
MathSciNet review: 2746668
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Abstract: For $ 0<p<\infty $ and $ \alpha>-1, $ we let $ \mathcal D^p_{\alpha}$ denote the space of those functions $ f$ which are analytic in the unit disc $ \mathbb{D}$ in $ \mathbb{C}$ and satisfy $ \int_{\mathbb{D}}(1-\vert z\vert\sp 2)\sp {\alpha}\vert f'(z)\vert \sp p dx dy <\infty $. Of special interest are the spaces $ \mathcal D^p_{p-1} $ ( $ 0<p<\infty $) which are closely related with Hardy spaces and the analytic Besov spaces $ B^p=\mathcal D^p_{p-2}$ ( $ 1<p<\infty $). A good number of results on the boundedness of integration operators and multipliers from $ \mathcal D^p_{\alpha}$ to $ \mathcal D^q_{\beta}$ are known in the case $ p<q$. Here we are mainly concerned with the upper triangle case $ 0<q\le p$. We describe the boundedness of these operators from $ \mathcal D^p_{\alpha}$ to $ \mathcal D^q_{\beta}$ in the case $ 0<q<p$. Among other results we prove that if $ 0<q<p$ and $ \frac{p\beta-q\alpha}{p-q}\le -1$, then the only pointwise multiplier from $ \mathcal D^p_{\alpha}$ to $ \mathcal D^q_{\beta}$ is the trivial one. In particular, we have that 0 is the only multiplier from $ \mathcal D^p_{p-1}$ to $ \mathcal D^q_{q-1} $ if $ p\neq q$, and from $ B^p$ to $ B^q$ if $ 1<q<p$. Also, we give a number of explicit examples of multipliers from $ \mathcal D^p_{\alpha}$ to $ \mathcal D^q_{\beta} $ in the remaining case $ \frac{p\beta-q\alpha}{p-q}> -1$. Furthermore, we present a number of results on the self-multipliers of $ \mathcal D^p_{\alpha}$ ( $ 0<p<\infty $, $ \alpha >-1$). We prove that 0 is the only compact multiplier from $ \mathcal D^p_{p-1}$ to itself ( $ 0<p<\infty $) and we give a number of explicit examples of functions which are self-multipliers of $ \mathcal D^p_{\alpha}$.

We also consider the closely related question of characterizing the Carleson measures for the spaces $ \mathcal D^p_{\alpha}$. In particular, we prove constructively that a result of Arcozzi, Rochberg and Sawyer characterizing the Carleson measures for $ \mathcal D^p_{\alpha}$ in the range $ -1<\alpha <p-1$ cannot be extended to cover the case $ \alpha =p-1$ and we find a certain condition on a measure $ \mu $ which is necessary for $ \mu $ to be a $ q$-Carleson measure for $ \mathcal D^p_{\alpha} $ ( $ 0<q<p, \alpha >-1$). This result plays a basic role in our work concerning integration operators.


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  • 1. A. Aleman, A class of integral operators on spaces of analytic functions, in: “ Topics in Complex Analysis and Operator Theory” (D. Girela and C. González, eds.), University of Málaga, Spain, 2007, pp. 3-30. MR 2394654
  • 2. A. Aleman and J. A. Cima, An integral operator on $ H\sp p$ and Hardy's inequality, J. Anal. Math. 85 (2001), 157-176. MR 1869606 (2002k:30068)
  • 3. A. Aleman and A. G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337-356. MR 1481594 (99b:47039)
  • 4. J. Arazy, S. D. Fisher and J. Peetre, Möbius invariant function spaces, J. Reine Angew. Math. 363 (1985), 110-145. MR 814017 (87f:30104)
  • 5. N. Arcozzi, R. Rochberg and E. Sawyer, Carleson measures for analytic Besov spaces, Rev. Mat. Iberoamericana 18 (2002), 443-510. MR 1949836 (2003j:30080)
  • 6. N. Arcozzi, Carleson measures for analytic Besov spaces: the upper triangle case, J. Inequal. Pure and Appl. Math. 6 (2005), no. 1, 1-15. MR 2122940 (2005k:30104)
  • 7. S. M. Buckley, P. Koskela and D. Vukotić, Fractional integration, differentiation, and weighted Bergman spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999), 369-385. MR 1670257 (2000m:46050)
  • 8. L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921-930. MR 0117349 (22:8129)
  • 9. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in $ L^p$, Astérisque 77 (1980), 11-66. MR 604369 (82j:32015)
  • 10. J. J. Donaire, D. Girela and D. Vukotić, On univalent functions in some Möbius invariant spaces, J. Reine Angew. Math. 553 (2002), 43-72. MR 1944807 (2003k:30076)
  • 11. P. L. Duren, Extension of a Theorem of Carleson, Bull. Amer. Math. Soc. 75 (1969), 143-146. MR 0241650 (39:2989)
  • 12. P. L. Duren, Theory of $ H\sp{p} $ Spaces, Academic Press, New York-London, 1970. Reprint: Dover, Mineola, New York, 2000. MR 0268655 (42:3552)
  • 13. P. L. Duren - A. P. Schuster, Bergman Spaces, Math. Surveys and Monographs, Vol. 100, American Mathematical Society, Providence, Rhode Island, 2004. MR 2033762 (2005c:30053)
  • 14. P. L. Duren, A. P. Schuster and D. Vukotić, On uniformly discrete sequences in the disk, in: “ Quadrature domains and their applications” , The Harold S. Shapiro Anniversary Volume (P. Ebenfelt, B. Gustafsson, D. Khavinson, and M. Putinar, editors) 131-150, Oper. Theory Adv. Appl., 156, Birkhäuser, Basel, 2005. MR 2129739 (2005k:30105)
  • 15. T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 756-765. MR 0304667 (46:3799)
  • 16. D. Girela and J. A. Peláez, Growth properties and sequences of zeros of analytic functions in spaces of Dirichlet type, J. Australian Math. Soc. 80 (2006), 397-418. MR 2236047 (2007e:30037)
  • 17. D. Girela, M. Pavlović and J. A. Peláez, Spaces of analytic functions of Hardy-Bloch type, J. Anal. Math. 100 (2006), 53-81. MR 2303304 (2008a:30048)
  • 18. D. Girela and J. A. Peláez, Carleson measures for spaces of Dirichlet type, Integral Equations and Operator Theory 55 (2006), no. 3, 415-427. MR 2244197 (2007e:30079)
  • 19. D. Girela and J. A. Peláez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Analysis 241 (2006), no. 1, 334-358. MR 2264253 (2007m:46034)
  • 20. D. Gnuschke, Relations between certain sums and integrals concerning power series with Hadamard gaps, Complex Variables Theory Appl. 4 (1984), 89-100. MR 770989 (86e:30004)
  • 21. C. Horowitz, Zeros of functions in the Bergman spaces, Duke Math. J. 41 (1974), 693-710. MR 0357747 (50:10215)
  • 22. H. Hedenmalm, B. Korenblum - K. Zhu, Theory of Bergman Spaces, Graduate Texts in Mathematics, Vol. 199, Springer, New York, Berlin, 2000. MR 1758653 (2001c:46043)
  • 23. J. E. Littlewood and R. E. A. C. Paley, Theorems on Fourier series and power series. II, Proc. London Math. Soc. 42 (1936), 52-89.
  • 24. D. H. Luecking, Forward and reverse inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985), 85-111. MR 778090 (86g:30002)
  • 25. D. H. Luecking, Multipliers of Bergman spaces into Lebesgue spaces, Proc. Edinburgh Math. Soc. (2) 29 (1986), 125-131. MR 829188 (87e:46034)
  • 26. D. H. Luecking, Embedding derivatives of Hardy spaces into Lebesgue spaces, Proc. London Math. Soc. 63 (1991), no. 3, 595-619. MR 1127151 (92k:42030)
  • 27. D. H. Luecking, Embedding theorems for spaces of analytic functions via Khinchine's inequality, Michigan Math. J. 40 (1993), no. 2, 333-358. MR 1226835 (94e:46046)
  • 28. M. Mateljevic and M. Pavlovic, $ L^p$- behaviour of power series with positive coefficients and Hardy spaces, Proc. Amer. Math. Soc., 87 (1983), 309-316. MR 681840 (84g:30034)
  • 29. M. M. Peloso, Möbius invariant spaces on the unit ball, Michigan Math. J. 39 (1992), 509-536. MR 1182505 (93k:46018)
  • 30. R. Rochberg, Decomposition theorems for Bergman spaces and their applications, in: “ Operator and function theory” (S. C.  Power, ed.), Reidel, Dordrecht, The Netherlands, 1985, pp. 225-278. MR 810448 (87c:46032)
  • 31. S. A. Vinogradov, Multiplication and division in the space of analytic functions with area integrable derivative, and in some related spaces (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), Issled. po Linein. Oper. i Teor. Funktsii 23, 45-77, 308; translation in J. Math. Sci. (New York) 87, no. 5 (1997), 3806-3827. MR 1359994 (96m:30053)
  • 32. R. Wallstén, Hankel operators between weighted Bergman spaces in the ball, Ark. Mat. 28 (1990), 182-192. MR 1049650 (91i:47041)
  • 33. Z. Wu and L. Yang, Multipliers between Dirichlet spaces, Integral Equations and Operator Theory 32 (1998), 482-492. MR 1656473 (99j:47048)
  • 34. Z. Wu, Carleson measures and multipliers for Dirichlet spaces, J. Funct. Anal. 169 (1999), 148-163. MR 1726750 (2000k:30081)
  • 35. R. Zhao, Pointwise multipliers from weighted Bergman spaces and Hardy spaces to weighted Bergman spaces, Ann. Acad. Sci. Fenn. 29 (2004), 139-150. MR 2041703 (2004m:30059)
  • 36. K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990. Reprint: Math. Surveys and Monographs, Vol. 138, American Mathematical Society, Providence, Rhode Island, 2007. MR 1074007 (92c:47031)
  • 37. K. Zhu, Analytic Besov spaces, J. Math. Anal. Appl. 157 (1991), 318-336. MR 1112319 (93e:30092)
  • 38. A. Zygmund, Trigonometric Series, Vol. I and Vol. II, Second edition, Camb. Univ. Press, Cambridge, 1959. MR 0107776 (21:6498)

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Additional Information

Petros Galanopoulos
Affiliation: Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
Address at time of publication: Departamento de Análisis Matemático, Universidad de Málaga, Facultad de Ciencias, Campus de Teatinos, 29071 Málaga, Spain
Email: galanopoulos_petros@yahoo.gr

Daniel Girela
Affiliation: Departamento de Análisis Matemático, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
Email: girela@uma.es

José Ángel Peláez
Affiliation: Departamento de Matemáticas, Universidad de Córdoba, Edificio Einstein, Campus de Rabanales, 14014 Córdoba, Spain
Address at time of publication: Departamento de Análisis Matemático, Universidad de Málaga, Facultad de Ciencias, Campus de Teatinos, 29071 Málaga, Spain
Email: ma1pemaj@uco.es, japelaez@uma.es

DOI: https://doi.org/10.1090/S0002-9947-2010-05137-2
Keywords: Carleson measures, multiplication operators, integration operators, Dirichlet spaces, Bergman spaces
Received by editor(s): September 23, 2008
Published electronically: November 16, 2010
Additional Notes: This research was partially supported by grants from \lq\lq the Ministerio de Educación y Ciencia, Spain\rq\rq(MTM2007-60854, MTM2007-30904-E and Ingenio Mathematica (i-MATH) No. CSD2006-00032); from \lq\lq La Junta de Andalucía\rq\rq(FQM210 and P06-FQM01504) and from the European Networking Programme \lq\lq HCAA\rq\rq of the European Science Foundation.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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