TY - JOUR

T1 - Fredholmness and index of simplest weighted singular integral operators with two slowly oscillating shifts

AU - Karlovych, Oleksiy

N1 - This work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project PEst-OE/MAT/UIO297/2014 (Centro de Matemdtica e Aplicacoes). The authors would like to thank the anonymous referee for useful remarks and for informing about the work [5].

PY - 2015

Y1 - 2015

N2 - Let α and β be orientation-preserving diffeomorphisms (shifts) of R+ = (0, ∞) onto itself with the only fixed points 0 and ∞, where the derivatives α' and β' may have discontinuities of slowly oscillating type at 0 and ∞ For p ∈ (1 ∞), we consider the weighted shift operators Uα and Uβ given on the Lebesgue space Lp(R+) by Uαf = (α')1/p(f o α) and Uβf = (β')1/p(f o β). For i, j ∈ Z we study the simplest weighted singular integral operators with two shifts Aij = Uα iP+ γ + Uβ jP- γ on Lp(R+), where P± γ = (I ± Sγ)/2 are operators associated to theweighted Cauchy singular integral operator with γ ∈ C satisfying 0 < 1/p + Rγ < 1. We prove that the operator Aij is a Fredholm operator on Lp(R+) and has zero index if where wij (t) = log[αi (β-j (t))/t] and αi, β-j are iterations of α, β. This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for γ = 0.

AB - Let α and β be orientation-preserving diffeomorphisms (shifts) of R+ = (0, ∞) onto itself with the only fixed points 0 and ∞, where the derivatives α' and β' may have discontinuities of slowly oscillating type at 0 and ∞ For p ∈ (1 ∞), we consider the weighted shift operators Uα and Uβ given on the Lebesgue space Lp(R+) by Uαf = (α')1/p(f o α) and Uβf = (β')1/p(f o β). For i, j ∈ Z we study the simplest weighted singular integral operators with two shifts Aij = Uα iP+ γ + Uβ jP- γ on Lp(R+), where P± γ = (I ± Sγ)/2 are operators associated to theweighted Cauchy singular integral operator with γ ∈ C satisfying 0 < 1/p + Rγ < 1. We prove that the operator Aij is a Fredholm operator on Lp(R+) and has zero index if where wij (t) = log[αi (β-j (t))/t] and αi, β-j are iterations of α, β. This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for γ = 0.

KW - weighted singular integraloperator

KW - slowly oscillating shift

KW - index

KW - Mellin pseudodierential operator

KW - Fredholmness

KW - Fredholmness

KW - Index

KW - Mellin pseudodifferential operator

KW - Slowly oscillating shift

KW - Weighted singular integral operator

U2 - 10.15352/bjma/09-3-3

DO - 10.15352/bjma/09-3-3

M3 - Article

SN - 1735-8787

VL - 9

SP - 24

EP - 42

JO - Banach Journal Of Mathematical Analysis

JF - Banach Journal Of Mathematical Analysis

IS - 3

ER -