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Harmonic Analysis

Harmonic Analysis has its root dating back to the 1800's, a time when Fourier made his contributions. That is to say, classical Harmonic Analysis treats problems concerning the Fourier transform and Fourier series. Overtime, it has broadened its investigation and influenced the development of mathematics in a substantial way.

The Harmonic Analysis group in Taiwan, although small, focuses on a wide variety of topics which include the classical two weight problems, theory of function spaces, operator theory and numerical range. Our current research problems focus on

  • Hardy spaces, Besov spaces and BMO spaces associated to differential operators/equations
  • Two weight problem for singular integral operators
  • Geometric and combinatorial analysis
  • Bergman-type singular integral operators and applications to operator theory

The study of Hardy spaces Hp, p≦1, has over time been transformed into a rich and multi-faceted theory, providing basic insights into topics such as maximal functions, singular integrals, and Lp spaces. For p = 1, H1 is a natural substitute for L1, and its dual is BMO space which plays a similar role with respect to L. We will study the Hardy spaces and Besov spaces associated with differential operators or differential equations, and then characterizetheir dual spaces. To achieve such a goal, one has to develop a corresponding covering lemma, and then to set up a corresponding Calderon-Zygmund decomposition. For the real variable theory related to the above Hardy/Besov spaces, one has to modify the classical Calderon-Zygmund operators to fit the frame, and show the boundedness of these singular integral operators acting on the new Hardy/Besov spaces and their duals.

One of the major open problems in the nonhomogeneous harmonic analysis was to find the necessary and sufficient conditions for the boundedness of two weight Calderon-Zygmund operators. In part, the solution will have extremely important applications to a number of subjects, including embedding inequalities for model space, interpolating sequences for Paley-Wiener space, spectral theory for perturbed operators and PDE. The characterization has been formulated to be a T1 type condition which is a profound extension of the famous David-Journe T1 theorem. The current approach involves a delicate combination of ideas: random grids, weighted Haar functions and Carleson embeddings, stopping intervals, and culminates in the use of these techniques with several corona decompositions.

Singular integral operators on the whole Euclidean space have evolved into a rather large and sophisticated machinery. Yet in applications one often lacks tools on domains with boundary, for which the theory is from being well-established, and the most important case is probably Bergman-type operators, which connects harmonic analysis with complex analysis and complex geometry. The new approach we take toward these important operators is to build on dyadic models, a very recent technique showing up in this area. Then we are planning to apply our findings to the study of concrete Toeplitz operators and composition operators on Hilbert spaces of analytic functions. This will probably form a new direction in operator theory. We also investigate the connection with quasiconformal mappings, and this appears to be the first time in the literature for someone to investigate these two largely independent areas in analysis together.

 

 
 
 

 

 
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