TY - JOUR

T1 - The index of weighted singular integral operators with shifts and slowly oscillating data

AU - Karlovych, Oleksiy

AU - Karlovich, Yuri I.

AU - Lebre, Amarino B.

N1 - This work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UM/MAT/00297/2013 (Centro de Matematica e Aplicacoes) and UID/MAT/04721/2013 (Centro de Analise Funcional, Estruturas Linares e Aplicacoes). The second author was also supported by the SEP-CONACYT Projects No. 168104 and No. 169496 (Mexico).

PY - 2017/6/1

Y1 - 2017/6/1

N2 - Let α and β be orientation-preserving diffeomorphism (shifts) of R+=(0,∞) onto itself with the only fixed points 0 and ∞. We establish a Fredholm criterion and calculate the index of the weighted singular integral operator with shifts (aI−bUα)Pγ ++(cI−dUβ)Pγ −, acting on the space Lp(R+), where Pγ ±=(I±Sγ)/2 are the operators associated to the weighted Cauchy singular integral operator Sγ given by (Sγf)(t)=1πi∫R+(tτ)γf(τ)τ−tdτ with γ∈C satisfying 0<1/p+ℜγ<1, and Uα,Uβ are the isometric shift operators given by Uαf=(α′)1/p(f∘α),Uβf=(β′)1/p(f∘β), under the assumptions that the coefficients a,b,c,d and the derivatives α′,β′ of the shifts are bounded and continuous on R+ and admit discontinuities of slowly oscillating type at 0 and ∞.

AB - Let α and β be orientation-preserving diffeomorphism (shifts) of R+=(0,∞) onto itself with the only fixed points 0 and ∞. We establish a Fredholm criterion and calculate the index of the weighted singular integral operator with shifts (aI−bUα)Pγ ++(cI−dUβ)Pγ −, acting on the space Lp(R+), where Pγ ±=(I±Sγ)/2 are the operators associated to the weighted Cauchy singular integral operator Sγ given by (Sγf)(t)=1πi∫R+(tτ)γf(τ)τ−tdτ with γ∈C satisfying 0<1/p+ℜγ<1, and Uα,Uβ are the isometric shift operators given by Uαf=(α′)1/p(f∘α),Uβf=(β′)1/p(f∘β), under the assumptions that the coefficients a,b,c,d and the derivatives α′,β′ of the shifts are bounded and continuous on R+ and admit discontinuities of slowly oscillating type at 0 and ∞.

KW - Fredholmness

KW - Index

KW - Orientation-preserving shift

KW - Semi-almost periodic function

KW - Slowly oscillating function

KW - Weighted Cauchy singular integral operator

UR - http://www.scopus.com/inward/record.url?scp=85010850270&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2017.01.052

DO - 10.1016/j.jmaa.2017.01.052

M3 - Article

AN - SCOPUS:85010850270

VL - 450

SP - 606

EP - 630

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -