Periodic Reporting for period 1 - LARGE BERGMAN (Toeplitz and related operators in large Bergman spaces)
Reporting period: 2023-05-01 to 2025-04-30
The topic and research objectives of this project belong to mathematical analysis, more precisely, to the theory of linear operators in Banach and Hilbert spaces of analytic functions. In general, the techniques of operator theory form a standard approach and toolkit for solving partial and ordinary differential equations arising in physics, chemistry, biology, economy and other branches of science and its applications. For example, spectra of linear operators are directly connected with central physical concepts: in particular they can be identified with the frequencies of propagating (electromagnetic, acoustic, elastic, quantum etc.) waves in various types of media or materials. Central mathematical questions for linear operators concern their boundedness, compactness, invertibility and Fredholm and spectral properties. For linear operators, the property of invertibility amounts to the question of the solvability of an equation containing the operator, and boundedness is equivalent to the continuity of the operator, which is relevant for the stability of the solution under small errors and perturbations for initial data contained in an equation.
The project focuses on the properties of special classes of linear operators in Bergman spaces of analytic functions. For example, Toeplitz operators are defined with the help of the so called Bergman kernel, which is the integral kernel of a natural orthogonal projection between the involved Hilbert spaces. The kernel depends on the domains and weights and can be explicitly presented in very simple cases only. Due to its applications and connections to other problems and areas in analysis, the Bergman kernel is in itself a much studied object. It is also closely connected or even an essential part of the definition of other operators studied in this project. Central, standard techniques for the study of these operators concern characterizations of the properties of the operators in terms of Carleson measures, averaging functions and Berezin transforms.
The predecessor of the study of operator theory in Bergman spaces is the corresponding research in Hardy spaces of analytic functions. The theory of Hardy spaces is a central subject in modern analysis with many deep problems and results. It has close connections to many phenomena in harmonic, complex and functional analysis and even in partial differential equations. Most of the central theory of Hardy spaces and operators on them were well understood already by the 1970's, but their natural counterparts and analogous problems for Bergman spaces were generally viewed as essentially more difficult, and only some isolated progress was achieved. The terminology goes back to the classical book by Stefan Bergman, which contained the first systematic treatment of the Hilbert space of square integrable analytic functions with respect to Lebesgue area measure on the unit disc, which is nowadays known as the standard or classical Bergman space; the reproducing kernel used by Bergman became known as the Bergman kernel function. Operator theory in Bergman spaces was initiated in 1980's with new contributions by several authors, but the main breakthroughs came only in the 1990's, when a number of difficult problems were solved in a several important contributions. Also, a direct application of Toeplitz operators in analytic function spaces was established via the so called deformation quantization, which is a special case of the canonical quantization in theoretical physics.
The Bergman spaces under consideration are equipped with weighted norms vanishing rapidly at the boundary of the underlying complex domain. Operator theory on such spaces have been studied actively mainly only during the last 10 years, although some techniques go back to Hörmander's studies on several complex variables in 1960's. In contrast, spaces with slower, so called doubling, weights, and operators on them, have been investigated actively already since 1980's. Such studies heavily use the classical hyperbolic geometry of the underlying domain, but in the case of exponentially decreasing weigths, the connection with the hyperbolic geometry is broken, and nontrivial changes are needed. These have been introduced by many authors in the literature during the last years. In the present project we have adapted and developed these techniques and obtained a number of new results on central questions in theory. In particular the project has focused on improving the existing techniques involving Carleson measures, averaging functions, Berezin transforms, Kahane-Khinchine inequalities, and others.
The project focuses on the properties of special classes of linear operators in Bergman spaces of analytic functions. For example, Toeplitz operators are defined with the help of the so called Bergman kernel, which is the integral kernel of a natural orthogonal projection between the involved Hilbert spaces. The kernel depends on the domains and weights and can be explicitly presented in very simple cases only. Due to its applications and connections to other problems and areas in analysis, the Bergman kernel is in itself a much studied object. It is also closely connected or even an essential part of the definition of other operators studied in this project. Central, standard techniques for the study of these operators concern characterizations of the properties of the operators in terms of Carleson measures, averaging functions and Berezin transforms.
The predecessor of the study of operator theory in Bergman spaces is the corresponding research in Hardy spaces of analytic functions. The theory of Hardy spaces is a central subject in modern analysis with many deep problems and results. It has close connections to many phenomena in harmonic, complex and functional analysis and even in partial differential equations. Most of the central theory of Hardy spaces and operators on them were well understood already by the 1970's, but their natural counterparts and analogous problems for Bergman spaces were generally viewed as essentially more difficult, and only some isolated progress was achieved. The terminology goes back to the classical book by Stefan Bergman, which contained the first systematic treatment of the Hilbert space of square integrable analytic functions with respect to Lebesgue area measure on the unit disc, which is nowadays known as the standard or classical Bergman space; the reproducing kernel used by Bergman became known as the Bergman kernel function. Operator theory in Bergman spaces was initiated in 1980's with new contributions by several authors, but the main breakthroughs came only in the 1990's, when a number of difficult problems were solved in a several important contributions. Also, a direct application of Toeplitz operators in analytic function spaces was established via the so called deformation quantization, which is a special case of the canonical quantization in theoretical physics.
The Bergman spaces under consideration are equipped with weighted norms vanishing rapidly at the boundary of the underlying complex domain. Operator theory on such spaces have been studied actively mainly only during the last 10 years, although some techniques go back to Hörmander's studies on several complex variables in 1960's. In contrast, spaces with slower, so called doubling, weights, and operators on them, have been investigated actively already since 1980's. Such studies heavily use the classical hyperbolic geometry of the underlying domain, but in the case of exponentially decreasing weigths, the connection with the hyperbolic geometry is broken, and nontrivial changes are needed. These have been introduced by many authors in the literature during the last years. In the present project we have adapted and developed these techniques and obtained a number of new results on central questions in theory. In particular the project has focused on improving the existing techniques involving Carleson measures, averaging functions, Berezin transforms, Kahane-Khinchine inequalities, and others.
The research work has been conducted by the researcher of the project in co-operation with the operator theory research groups of the supervisors at the University of Helsinki and also at the University of Reading during the secondmend period. The research has also benefitted from the contacts with a number of other institutions, for example at the Hebei University of Technology in PRC.
During the project it became clear that the Carleson measure-type characterizations of the properties of linear operators are most useful for the problems considered in the project. Such characterizations existed in the setting of large Bergman spaces for some classes of operators, but during the project, these have been extended to new areas. Indeed, we ave studied extensions of the concept of Carleson measures with applications to the questions of boundedness of integration operators in Fock spaces, which are closely related to large Bergman spaces. We have also characterized the boundedness of area operators in large Bergman spaces in terms of Carleson conditions of a measure related to the symbol, or in terms of the related conditions for the averaging operator. Other results include a study of the boundedness of the Bergman projection in Banach spaces with weighted sup-norms. Here, the focus was to consider a pair of weights, one connected with the space and another one with the operator. A complete solution was found in the setting of doubling, radial weights. A couple of articles have been devoted to study the compactness of commutators of bounded operators in analytic function spaces.
One of the last, yet unpublished, manuscripts, contains novel estimates for the derivatives of the Bergman kernel of large Bergman spaces. Completing the lengthy and technical proof for such estimates took considerable efforts and much more time than expected, but the result has been completed by now and is about to be submitted. See the next item for a remark concerning this study.
During the project it became clear that the Carleson measure-type characterizations of the properties of linear operators are most useful for the problems considered in the project. Such characterizations existed in the setting of large Bergman spaces for some classes of operators, but during the project, these have been extended to new areas. Indeed, we ave studied extensions of the concept of Carleson measures with applications to the questions of boundedness of integration operators in Fock spaces, which are closely related to large Bergman spaces. We have also characterized the boundedness of area operators in large Bergman spaces in terms of Carleson conditions of a measure related to the symbol, or in terms of the related conditions for the averaging operator. Other results include a study of the boundedness of the Bergman projection in Banach spaces with weighted sup-norms. Here, the focus was to consider a pair of weights, one connected with the space and another one with the operator. A complete solution was found in the setting of doubling, radial weights. A couple of articles have been devoted to study the compactness of commutators of bounded operators in analytic function spaces.
One of the last, yet unpublished, manuscripts, contains novel estimates for the derivatives of the Bergman kernel of large Bergman spaces. Completing the lengthy and technical proof for such estimates took considerable efforts and much more time than expected, but the result has been completed by now and is about to be submitted. See the next item for a remark concerning this study.
As mentioned, the theory of large Bergman spaces has been under active international research especially during the last 10 years. One of the central questions concerns pointwise estimates of the Bergman kernel in such spaces. Such satisfactory estimates for the kernel itself have been available in the literature recently, but it turned out that some standard literature references contained inaccurate statements about the derivatives of the kernel. Since a proper estimate was needed in our work mentioned above, we have produced a correction for the problem, which will be published in our paper that is at the moment at the final steps of preparation.