European Mathematical Society - p. duren
https://euro-math-soc.eu/author/p-duren
enBergman Spaces
https://euro-math-soc.eu/review/bergman-spaces
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Bergman spaces Ap are subspaces of holomorphic Lp functions on the unit disc. In many respects, they resemble the Hardy spaces Hp, which were intensively studied from the 30s till the 50s. However, a majority of problems turned out to be much more difficult in the Bergman space setting and for decades they remained essentially intractable. This situation has changed radically in the 90s. Many significant advances have taken place, attracting in turn a lot of research activity in the area. The present book gives a systematic overview of the current state of this exciting subject.<br />
The first two chapters present a crash course on the classical theory of Hardy spaces, the Bergman kernel function, hyperbolic geometry, biharmonic Green functions and a lot of other prerequisites, thus making the book very self-contained and accessible to anyone with basic knowledge of complex function theory and functional analysis. Chapter 3 deals with properties of individual functions in Bergman spaces (growth and boundary behaviour, Taylor coefficients, etc.). Chapters 4 and 5 develop properties of zero-sets of Ap functions and of the Hedenmalm canonical zero-divisors (analogues of Blaschke products), respectively. Chapters 6 and 7 contain an exposition of Seip's beautiful theory of interpolation and sampling in Ap spaces. Finally, Chapters 8 and 9 are devoted to the structure of invariant subspaces of Ap spaces, including the study of cyclic elements and the proofs of what may be the most profound result in the area, the Aleman-Richter-Sundberg analogue of Beurling's theorem. The exposition is on a masterly level, neatly and tightly organized, and yet highly readable.<br />
So is, by the way, an earlier book on the subject, Theory of Bergman spaces by Hedenmalm, Korenblum and Zhu (Springer, 2000); there is, of course, a lot of overlap between the two books, but the current one contains more of the prerequisites (especially on Hp spaces), discusses some material not covered by the other book and treats some material in a different way. Similarly, the HKZ book contains several topics barely dealt with in the present volume, such as the invertible noncyclic functions or the log-subharmonic weights. It is extremely likely that both books are going to become standard references on the subject and should not be missing on the shelf of anyone seriously interested in this area.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">men</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/p-duren" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">p. duren</a></li><li class="vocabulary-links field-item odd"><a href="/author/schuster" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. schuster</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/american-mathematical-society-providence" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">american mathematical society, providence</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2004</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">0-8218-0810-9 </div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">79 USD</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/30-functions-complex-variable" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30 Functions of a complex variable</a></li></ul></span>Thu, 16 Jun 2011 19:46:00 +0000Anonymous39592 at https://euro-math-soc.euHarmonic Mappings in the Plane
https://euro-math-soc.eu/review/harmonic-mappings-plane
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>A complex valued function of a complex variable is called a harmonic function if it is a solution of the Laplace equation componentwise. Harmonic mappings are then univalent harmonic functions. Any harmonic function can be locally decomposed as a sum of an analytic function and a co-analytic function. If it is a harmonic mapping, then one of those parts is strictly majorized by the other one. If, say, the analytic part prevails, then the mapping is sense preserving. In many aspects, the theory resembles the theory of conformal mappings, but the class of harmonic mappings is much less stable. For example, an inverse of a harmonic mapping typically fails to be harmonic. The theory of harmonic mappings, besides of its own interest, has many applications (e.g., to the theory of minimal surfaces). After several introductory results, the Radó-Kneser-Choquet theorem is presented. This shows that any homeomorphism of the unit circle onto a boundary of a convex domain can be extended to a harmonic mapping of the full disc onto the closure of the domain. Of course, the extension is nothing more than the solution of the Dirichlet problem but the main point of the theorem is to show that the solution is univalent. The shear construction, which leads to interesting examples of explicit harmonic mappings, is described and applied. Another class of explicit harmonic mappings with dilatation of the type of a Blaschke product is used to map the disc onto a convex polygon. The harmonic Koebe function is a very interesting mapping, which is extremal for many problems.<br />
A part of the book is devoted to a study of analogues of the Riemann mapping theorem. The situation is much more complicated here than in the conformal case. It is also interesting to observe what is known for multiply connected domains. Estimates in Hardy spaces are represented by a few results. Many nice results relating coefficients of the Taylor expansion of the analytic and coanalytic part with the image of the mapping are formulated. The last part of the book shows how the theory can be applied to minimal surface problems. The Weierstrass-Enneper representation of minimal surfaces is explained and minimal graphs are studied. The theory of harmonic mappings is applied to curvature estimates of minimal surfaces. The roots of the theory of harmonic mappings can be considered as classical, its development is fluent, and interesting problems still wait for their solution. The book by Peter Duren is the first comprehensive treatment of the topic. Any friend of complex analysis will admire the beauty of this extension of the theory so nicely presented in the volume.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">jama</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/p-duren" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">p. duren</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">cambridge university press</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2004</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">0-521-64121-7</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">£40</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/32-several-complex-variables-and-analytic-spaces" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">32 Several complex variables and analytic spaces</a></li></ul></span>Sun, 29 May 2011 11:15:42 +0000Anonymous39222 at https://euro-math-soc.eu