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Transactions of the American Mathematical Society
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
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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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: http://dx.doi.org/10.1090/S0002-9947-2010-05137-2
PII: S 0002-9947(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.