On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

• Published in: Complex analysis and operator theory 2016-08, Vol.10 (6), p.1101-1131
• Main Authors: Karlovich, Alexei Yu, Karlovich, Yuri I., Lebre, Amarino B.
• Format: Article
• Language: English
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Operator Theory
• ISSN: 1661-8254; 1661-8262
• DOI: 10.1007/s11785-015-0452-0

Let α , β be orientation-preserving diffeomorphism (shifts) of R + = ( 0 , ∞ ) onto itself with the only fixed points 0 and ∞ and U α , U β be the isometric shift operators on L p ( R + ) given by U α f = ( α ′ ) 1 / p ( f ∘ α ) , U β f = ( β ′ ) 1 / p ( f ∘ β ) , and P 2 ± = ( I ± S 2 ) / 2 where ( S 2 f ) ( t ) : = 1 π i ∫ 0 ∞ t τ 1 / 2 - 1 / p f ( τ ) τ - t d τ , t ∈ R + , is the weighted Cauchy singular integral operator. We prove that if α ′ , β ′ and c , d are continuous on R + and slowly oscillating at 0 and ∞ , and lim sup t → s | c ( t ) | < 1 , lim sup t → s | d ( t ) | < 1 , s ∈ { 0 , ∞ } , then the operator ( I - c U α ) P 2 + + ( I - d U β ) P 2 - is Fredholm on L p ( R + ) and its index is equal to zero. Moreover, its regularizers are described.