Publication
SINGULAR INTEGRAL OPERATORS ON REARRANGEMENT-INVARIANT BANACH FUNCTION SPACES
| Summary: | Given the Cauchy singular integral operator S acting on a reflexive rearrangementinvariant Banach function space, our goal is to study the Fredholmness of the operator aP + Q where P = 1 2 (I + S), Q = 1 2 (I − S) and a is a semi-almost periodic function. We start by equipping the space of the measurable functions finite μ-a.e., defined on a σ-finite measure space, with a metric. Then, we define the Banach function spaces and we proceed to a detailed study of their properties. Next up, we define the rearrangement invariant spaces and study the boundedness of the Hilbert transform H acting on these spaces. The operators S and H share the same behaviour, since S = iH, where i represents the imaginary unit. Further, we develop the necessary theory of compact and Fredholm operators. Finally, we prove our main result saying that, if a is a semi-almost periodic function and the operator aP + Q is Fredholm, then alP + Q and arP + Q are invertible on the reflexive rearrangement-invariant space, where al and ar are the left and right almost periodic representatives of a, respectively, provided by the Sarason theorem. If a is a purely almost periodic function, then a = al = ar and the above result implies that the invertibility and the Fredholmness of this operator are equivalent. |
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| Main Authors: | Turcan, Oleg |
| Subject: | Banach function space rearrangement-invariant space Fredholm operator compact operator Cauchy singular integral operator Boyd indices |
| Year: | 2023 |
| Country: | Portugal |
| Document type: | master thesis |
| Access type: | open access |
| Associated institution: | Universidade Nova de Lisboa |
| Language: | English |
| Origin: | Repositório Institucional da UNL |
| Summary: | Given the Cauchy singular integral operator S acting on a reflexive rearrangementinvariant Banach function space, our goal is to study the Fredholmness of the operator aP + Q where P = 1 2 (I + S), Q = 1 2 (I − S) and a is a semi-almost periodic function. We start by equipping the space of the measurable functions finite μ-a.e., defined on a σ-finite measure space, with a metric. Then, we define the Banach function spaces and we proceed to a detailed study of their properties. Next up, we define the rearrangement invariant spaces and study the boundedness of the Hilbert transform H acting on these spaces. The operators S and H share the same behaviour, since S = iH, where i represents the imaginary unit. Further, we develop the necessary theory of compact and Fredholm operators. Finally, we prove our main result saying that, if a is a semi-almost periodic function and the operator aP + Q is Fredholm, then alP + Q and arP + Q are invertible on the reflexive rearrangement-invariant space, where al and ar are the left and right almost periodic representatives of a, respectively, provided by the Sarason theorem. If a is a purely almost periodic function, then a = al = ar and the above result implies that the invertibility and the Fredholmness of this operator are equivalent. |
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