TY - JOUR

T1 - Necessary Fredholm conditions for 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 Portuguese Foundation for Science and Technology, project Nos. UID/MAT/00297/2013) (Centro de Matematica e Aplicagoes) and UID/MAT/04721/2013 (Centro de Analise Funcional, Estruturas Lineares e Aplicagoes). The second author was also supported by the SEP-CONACYT, project Nos. 168104 and 169496 (Mexico).

PY - 2017

Y1 - 2017

N2 - We extend the main result of [12] to the case of more general weighted singular integral operators with two shifts of the form (aI - bUα)Pγ + + (cI - dUβ)Pγ -, acting on the space Lp(ℝ+), 1 < p < ∞, where Pγ ± = (I ± Sγ)/2 are operators associated with the weighted Cauchy singular integral operator Sγ, given by with γ εℂ satisfying 0 < 1/p + γ < 1, and Uα, Uβ are the isometric shift operators given by Uαf = (α')1/p(f ο α), Uβf = (β')1/p(f ο β), generated by diffeomorphisms α β of ℝ+ onto itself having only two βxed points at the endpoints 0 and ∞, 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 - We extend the main result of [12] to the case of more general weighted singular integral operators with two shifts of the form (aI - bUα)Pγ + + (cI - dUβ)Pγ -, acting on the space Lp(ℝ+), 1 < p < ∞, where Pγ ± = (I ± Sγ)/2 are operators associated with the weighted Cauchy singular integral operator Sγ, given by with γ εℂ satisfying 0 < 1/p + γ < 1, and Uα, Uβ are the isometric shift operators given by Uαf = (α')1/p(f ο α), Uβf = (β')1/p(f ο β), generated by diffeomorphisms α β of ℝ+ onto itself having only two βxed points at the endpoints 0 and ∞, 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 - Orientation-preserving shift

KW - Slowly oscillating function

KW - Weighted Cauchy singular integral operator

UR - http://www.scopus.com/inward/record.url?scp=85011568264&partnerID=8YFLogxK

U2 - 10.1216/JIE-2017-29-3-365

DO - 10.1216/JIE-2017-29-3-365

M3 - Article

AN - SCOPUS:85011568264

SN - 0897-3962

VL - 29

SP - 365

EP - 399

JO - Journal of Integral Equations and Applications

JF - Journal of Integral Equations and Applications

IS - 3

ER -