SINGULAR INTEGRAL OPERATORS ON REARRANGEMENT-INVARIANT BANACH FUNCTION SPACES

• Autor(a) principal: Turcan, Oleg
• Data de Publicação: 2023
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10362/160215
• topic: Banach function space; rearrangement-invariant space; Fredholm operator; compact operator; Cauchy singular integral operator; Boyd indices; Domínio/Área Científica::Ciências Naturais::Matemáticas

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.