English

If sufficient demand

On completion of the course, participants shall be able to:
• Define Hardy spaces on the unit disc and describe their basic properties, especially the boundary behaviour of functions in these spaces, the use of non-tangential maximal functions and their relation to the Hardy-Littlewood maximal function.
• Describe the bounded Carleson-embeddings of these spaces into L^p-spaces corresponding to measures on the unit disc.
• Describe the inner-outer factorization of functions in Hardy spaces and its applications, for example the Phragmén-Lindelöf principle. Describe the inner-outer factorization of a concrete function.
• State and prove the M. Riesz theorem about conjugate harmonic functions, as well as Fefferman’s duality theorem in the limit case p=1.
• Reflect about the strength of these methods in the study of analytic functions on a disc, but also about their limitations concerning functions with fast growth near the boundary.

Poisson integrals and their boundary behaviour. Fatou’s theorem via the Hardy- Littlewood maximal function. Carleson measures. Outer functions. Hardy spaces and the inner-outer factorization in these spaces. The M. Riesz theorem, the Fefferman duality theorem, and analytic BMO. The John-Nirenberg inequality.

Define Hardy spaces on the unit disc and describe their basic properties, especially the boundary behaviour of functions in these spaces, the use of non-tangential maximal functions and their relation to the Hardy-Littlewood maximal function.

Describe the bounded Carleson-embeddings of these spaces into L^p-spaces corresponding to measures on the unit disc.

Describe the inner-outer factorization of functions in Hardy spaces and its applications, for example the Phragmén-Lindelöf principle. Describe the inner-outer factorization of a concrete function.

State and prove the M. Riesz theorem about conjugate harmonic functions, as well as Fefferman’s duality theorem in the limit case p=1.

Reflect about the strength of these methods in the study of analytic functions on a disc, but also about their limitations concerning functions with fast growth near the boundary.

Lectures

Seminars

Oral exam

Failed, pass

Courses in Analytic functions, Integration Theory, and a Specialized course in Integration Theory or corresponding pre-knowledge are obligatory.

A course in Linear Functional Analysis is recommended, but not compulsory.

Aleman, A.: Introduction to Hardy spaces.

Lecturer's own notes from previous years.

FMA330F

2024-08-27

Maria Sandsten